research article bibo stabilization of discrete-time stochastic control systems...
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 916357 11 pageshttpdxdoiorg1011552013916357
Research ArticleBIBO Stabilization of Discrete-Time Stochastic Control Systemswith Mixed Delays and Nonlinear Perturbations
Xia Zhou1 Yong Ren2 and Shouming Zhong3
1 School of Mathematics and Computational Science Fuyang Teachers College Fuyang 236037 China2Department of Mathematics Anhui Normal University Wuhu 241000 China3 College of Applied Mathematics University of Electronic Science and Technology of China Chengdu Sichuan 611731 China
Correspondence should be addressed to Xia Zhou zhouxia44185163com
Received 19 March 2013 Revised 23 May 2013 Accepted 6 June 2013
Academic Editor Sakthivel Rathinasamy
Copyright copy 2013 Xia Zhou et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The problem of bounded-input bounded-output (BIBO) stabilization in mean square for a class of discrete-time stochasticcontrol systems with mixed time-varying delays and nonlinear perturbations is investigated Some novel delay-dependent stabilityconditions for the previously mentioned system are established by constructing a novel Lyapunov-Krasovskii function Theseconditions are expressed in the forms of linearmatrix inequalities (LMIs) whose feasibility can be easily checked by usingMATLABLMI Toolbox Finally a numerical example is given to illustrate the validity of the obtained results
1 Introduction
Many dynamical systems not only depend on the presentstates but also involve the past ones generally called the time-delay systems Generally as a source of poor or significantlydeteriorated performance and instability for the concernedclosed-loop system the time delays are unavoidable intechnology and nature Many works have been done on thestability of time-delay systems one can see [1ndash23] and thereferences therein The dynamics analysis of continuous-time systems with distributed delay has been well studiedin [9ndash12 20] The aspect of simulation and application incontrol systems whereas discrete-time control systems play amore important role than their continuous-time counterpartsin the practical digital world If one wants to simulateor compute the continuous-time systems it is essential toformulate the discrete-time analogue so as to investigate thedynamical characteristics It is necessary to take continuousdistributed delays into account formodeling realistic systemsfor example neural networks due to the presence of anamount of parallel pathways of a variety of axon sizes andlengths a neural network usually has a spatial nature Veryrecently Liu et al introduced the infinite distributed delayand distributed delay in the form of constant delay into thedelay neural networks See [17ndash19]
In order to track out the reference input signal in realworld the bounded-input bounded-output stabilization hasbeen investigated by many researchers one can see [20ndash32] and the references therein In [22 23] the sufficientconditions for BIBO stabilization of control systems with nodelays were proposed by the Bihari type inequality In [9 10]by employing the parameters technique and the Gronwallinequality the authors investigated the BIBO stability ofthe systems without distributed time delays In [20 27 29]based on Riccati equations and by constructing appropriateLyapunov functions someBIBO stabilization conditions for aclass of delayed control systems with nonlinear perturbationswere established In [30] the BIBO stabilization problemof a class of piecewise switched linear systems was furtherinvestigated It should be pointed out that almost all resultsconcerning the BIBO stability for control systems mainlyconcentrate on continuous-time models Seldom works havebeen done for discrete-time control systems one can see[21 28] In addition the previously mentioned works justconsidered the deterministic systems (see eg [31 32]) Thedeterministic systems often fluctuate due to noise which israndom or at least appears to be soTherefore we must movefrom deterministic problems to stochastic ones So the BIBOstabilization for stochastic control systems case is necessaryand interesting To the best of our knowledge there is no
2 Abstract and Applied Analysis
work reported on the mean square BIBO stabilization forthe discrete-time stochastic control systemswithmixed time-varying delays
It is well known that the classical technique applied in thestudy of stability is based on the Lyapunov direct methodHowever the Lyapunov direct method has some difficultieswith the theory and application to specific problems whilediscussing the stability of solutions in stochastic systems withtime delay In [33] the midpoint in the time delayrsquos variationinterval is introduced and the variation interval is dividedinto two subintervals with equal length by constructing theLyapunov functional which involved midpoint to reduce theconservatism of stability conditions This method was firstproposed to study the stability and stabilization problems forlinear continuous-time systems and then many successfulapplications were found in [13ndash15] In this paper we willreconsider this method by introducing a new piecewise-like delay method given that the point of the time delayrsquosvariation interval is arbitrary point rather than midpoint
Motivated by the aforementioned works in this paperwe investigate BIBO stabilization in mean square for a classof discrete-time stochastic control systems with mixed time-varying delays and nonlinear perturbations Some noveldelay-dependent stability conditions for the previously men-tioned system are derived by constructing a novel Lyapunov-Krasovskii function These conditions are expressed in theforms of linear matrix inequalities whose feasibility can beeasily checked by using MATLAB LMI Toolbox Finally anumerical example is given to illustrate the validity of theobtained results
The paper is organized as follows In Section 2 somenotations and the problem formulation are proposed Themain results are given in Section 3 In Section 4 a numericalexample is given to illustrate the validity of the obtainedtheory results The conclusion is proposed in Section 5
2 Notations and Problem Formulation
Firstly we propose some notations which will be needed inthe sequel The notations are quite standard Let 119877119899 and 119877119899times119898denote respectively the 119899-dimensioned Euclidean space andthe set of all 119899 times119898 real matrices The superscript ldquo119879rdquo denotesthe transpose and the notation 119883 ge 119884 (respective 119883 gt 119884)means that119883 and 119884 are symmetric matrices and that119883minus119884 ispositive semidefinitive (respective positive definite) Let sdot denote the Euclidean norm in 119877119899 let119873+ denote the positiveinteger set and let 119868 be the identity matrix with compatibledimension If119860 is a matrix denote by 119860 its operator normthat is 119860 = sup119860119909 119909 = 1 = radic120582max(119860
119879119860) where120582max(119860) (resp 120582min(119860)) means the largest (resp smallest)value of 119860 Moreover let (ΩF F
119905119905ge0 119875) be a complete
probability space with a filtration F119905119905ge0
satisfying theusual conditions (ie the filtration contains all 119875-null setsand is right continuous) 119864sdot stands for the mathematicalexpectation operator with respect to the given probabilitymeasure 119875 The asterisk lowast in a matrix is used to denote termthat is induced by its symmetry Matrices if not explicitlystate are assumed to have compatible dimensions Denote
119873[119886 119887] = 119886 119886 + 1 119887 Sometimes the arguments of thefunctions will be omitted in the analysis without confusions
In this paper we consider the discrete-time stochasticcontrol systemwithmixed time-varying delays and nonlinearperturbations with the following form
119909 (119896 + 1) = 119860119909 (119896) + 119861119909 (119896 minus 120591 (119896)) + 119862119906 (119896)
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))
+ (119866119909 (119896) + 119867119909 (119896 minus 119889 (119896))) 120596 (119896)
119910 (119896) = 119872119909 (119896)
(1)
where 119909(119896) = [1199091(119896) 1199092(119896) 119909
119899(119896)]119879
isin 119877119899 denotes the
state vector 119906(119896) = [1199061(119896) 1199062(119896) 119906
119898(119896)]119879
isin 119877119898 is the
control input vector 119910(119896) = [1199101(119896) 1199102(119896) 119910
119899(119896)]119879
isin 119877119899 is
the control output vector ℎ(119909(119896)) = [ℎ1(119909(119896)) ℎ
2(119909(119896))
ℎ119899(119909(119896))]
119879
119860 119861 119863 119866 119862 119867 and 119872 represent the weight-ing matrices with appropriate dimension and the positiveintegers 120591(119896) and 119889(119896) are the discrete-time-varying delaydistributed time-varying delay and respectively satisfyingthat
1205911le 120591 (119896) le 120591
2 1198891le 119889 (119896) le 119889
2 119896 isin 119873
+
(2)
with 1205911 1205912 1198891 and 119889
2being four known positive integers For
any given 120591lowast isin (1205911 1205912) 119889lowast isin (119889
1 1198892) The initial conditions of
the system (1) are given by
119909 (119896) = 120601 (119896) 119896 isin [minusmax 1205912 1198892 0] (3)
Thenonlinear vector-valued perturbation119891(119896 119909(119896)119909(119896minus120591(119896))) satisfies that
1003817100381710038171003817119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))1003817100381710038171003817
2
le 1205721119909(119896)
2
+ 1205722119909(119896 minus 120591(119896))
2
(4)
where 1205721and 120572
2are two positive constants 120596(119896) is a scalar
Wiener process defined on (ΩF F119905119905ge0 119875) with
119864 (120596 (119896)) = 0 119864 (120596(119896)2
) = 1
119864 (120596 (119894) 120596 (119895)) = 0 119894 = 119895
(5)
Remark 1 The 120591lowast divides the discrete-time delayrsquos variationinterval into two subintervals that is [120591
1 120591lowast
] and (120591lowast 1205912] and
119889lowast divides the distributed time delayrsquos variation interval into
two subintervals that is [1198891 119889lowast
] and (119889lowast 1198892] We will dis-
cuss the variation of differences of the Lyapunov-Krasovskiifunctional119881(119905 119909(119905)) for each subinterval Compared with theprevious results in the works of [27ndash32] the BIBO stabilityconditions are derived in this paper by checking the variationof119881(119905 119909(119905)) in subintervals rather than in the whole variationinterval of the delays
In what follows we describe the controller with the form
119906 (119896) = 119870119909 (119896) + 119903 (119896) (6)
Abstract and Applied Analysis 3
where119870 is the feedback gain matrix and 119903(119896) is the referenceinput
Assumption 2 For any 1205851 1205852isin 119877 120585
1= 1205852 let
120574minus
119894leℎ119894(1205851) minus ℎ119894(1205852)
1205851minus 1205852
le 120574+
119894 (7)
where 120574minus119894and 120574+119894are known constant scalars
Remark 3 The constants 120574minus119894 120574+119894in Assumption 2 are allowed
to be positive negative or zero Hence the function ℎ(119909(119896))could be nonmonotonic and is more general than the usualsigmoid functions and the recently commonly used Lipschitzconditions
At the end of this section let us introduce some importantdefinitions and lemmas as which will be used in the sequel
Definition 4 (see [28 31]) A vector function 119903(119896) = (1199031(119896)
1199032(119896) 119903
119899(119896))119879 is said to be an element of 119871119899
infin if 119903
infin=
sup119896isin119873[0infin)
119903(119896) lt +infin where sdot denotes the Euclid normin 119877119899 or the norm of a matrix
Definition 5 (see [28 31]) The nonlinear stochastic controlsystem (1) is said to beBIBO stability inmean square if we canconstruct a controller (6) such that the output 119910(119896) satisfiesthat
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198731+ 11987321199032
infin (8)
where1198731and119873
2are two positive constants
Lemma 6 (see [28]) For any given vectors V119894isin 119877119899 119894 = 1
2 119899 the following inequality holds
[
119899
sum
119894=1
V119894]
119879
[
119899
sum
119894=1
V119894] le 119899
119899
sum
119894=1
V119879119894V119894 (9)
Lemma 7 (see [28]) Let 119909 119910 isin 119877119899 and any 119899 times 119899 positive-
definite matrix 119876 gt 0 Then one has
2119909119879
119910 le 119909119879
119876minus1
119909 + 119910119879
119876119910 (10)
Lemma 8 (see [28]) Given the constant matricesΩ1Ω2 and
Ω3with appropriate dimensions where Ω
1= Ω119879
1and Ω
2=
Ω119879
2gt 0 then Ω
1+ Ω119879
3Ωminus1
2Ω3lt 0 if and only if
(Ω1Ω119879
3
lowast minusΩ2
) lt 0 119900119903 (minusΩ2Ω3
lowast Ω1
) lt 0 (11)
3 BIBO Stabilization for the System (1)In this section we aim to establish our main results based onthe LMI approach For the conveniences we denote
Γ1= diag 120574minus
1120574+
1 120574minus
2120574+
2 120574
minus
119899120574+
119899
Γ3= diag 120574+
1 120574+
2 120574
+
119899
Γ2= diag
120574minus
1+ 120574+
1
2120574minus
2+ 120574+
2
2
120574minus
119899+ 120574+
119899
2
119886 = 1205912minus 1205911+ 1
119887 =
(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2 1198891le 119889 (119896) le 119889
lowast
(119889lowast
+ 1198892minus 1) (119889
2minus 119889lowast
) + 21198892
2 119889lowast
lt 119889 (119896) le 1198892
119888 = 119889lowast
1198891le 119889 (119896) le 119889
lowast
1198892 119889lowast
lt 119889 (119896) le 1198892
120579 (119896) = 119909 (119896 minus 120591
1) 1205911le 120591 (119896) le 120591
lowast
119909 (119896 minus 1205912) 120591
lowast
lt 120591 (119896) le 1205912
120591 = 120591lowast
minus 1205911 1205911le 120591 (119896) le 120591
lowast
1205912minus 120591lowast
120591lowast
lt 120591 (119896) le 1205912
120573 =
120591 (119896) minus 1205911
120591lowast minus 1205911
1205911le 120591 (119896) le 120591
lowast
1205912minus 120591 (119896)
1205912minus 120591lowast
120591lowast
lt 120591 (119896) le 1205912
(12)
Theorem9 For given positive integers 1205911 12059121198891 and119889
2 under
Assumption 2 the nonlinear discrete-time stochastic controlsystem (1) with the controller (6) is BIBO stabilization inmean square if there exist symmetric positive-definite matrix119875 119877 119876
119894 119894 = 1 2 5 119885
1 1198852 and 119883 with appropriate
dimensional positive-definite diagonal matrices Λ and somepositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868 (13)
Ξ
=
(((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))
)
le 0
(14)
4 Abstract and Applied Analysis
whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 119866
119879
120582lowast
119866 + 2120591lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860 minus 2119860119879
119883 minus 2119883119879
119860
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861 + 119866119879
120582lowast
119867
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ22= 119867119879
120582lowast
119867 + 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)
119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881
2(119896) + 119881
3(119896)
+ 1198814(119896) + 119881
5(119896) + 119881
6(119896)
(16)
where
1198811(119896 119909 (119896)) = 119909
119879
(119896) 119875119909 (119896)
1198812(119896) =
119896minus1
sum
119894=119896minus120591lowast
119909119879
(119894) 1198761119909 (119894)
+
119896minus1
sum
119894=119896minus119889lowast
119909119879
(119894) 1198762119909 (119894)
1198813(119896) =
119896minus1
sum
119894=119896minus120591(119896)
119909119879
(119894) 1198763119909 (119894) +
1205912minus1
sum
119894=1205911
119896minus1
sum
119895=119896minus119894
119909119879
(119895) 1198763119909 (119895)
+
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
119896minus1
sum
119895=119896minus1+119894
119909119879
(119895)1198764119909 (119895)
1198814(119896) = 120591
lowast
minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198851120578 (119895)
120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)
1198815(119896) =
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198765119909 (119894) + (120591
lowast
minus 1205911)
times
minus1205911minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
1205911le 120591 (119896) le 120591
lowast
119896minus1
sum
119894=119896minus1205912
119909119879
(119894) 1198765119909 (119894) + (120591
2minus 120591lowast
)
times
minus120591lowast
minus1
sum
119894=minus1205912
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
120591lowast
lt 120591 (119896) le 1205912
1198816(119896) =
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
1198891le 119889 (119896) le 119889
lowast
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus119889lowast
minus1
sum
119894=minus1198892
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
119889lowast
lt 119889 (119896) le 1198892
(17)
Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have
119864Δ1198811(119896 119909 (119896))
= 119864 [119909119879
(119896 + 1) 119875119909 (119896 + 1) minus 119909119879
(119896) 119875119909 (119896)]
= 119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896)]
119864Δ1198812(119896)
= 119864 [ 119909119879
(119896) 1198761119909 (119896) minus 119909
119879
(119896 minus 120591lowast
) 1198761119909 (119896 minus 120591
lowast
)
+119909119879
(119896) 1198762119909 (119896) minus 119909
119879
(119896 minus 119889lowast
) 1198762119909 (119896 minus 119889
lowast
)]
119864Δ1198813(119896)
= 119864[(
119896
sum
119894=119896+1minus120591(119896+1)
minus
119896minus1
sum
119894=119896minus120591(119896)
)119909119879
(119894) 1198763119909 (119894)
+
1205912minus1
sum
119894=1205911
(
119896
sum
119895=119896minus119894+1
minus
119896minus1
sum
119895=119896minus119894
)119909119879
(119895)1198763119909 (119895)
+ (
119896
sum
119894=119896minus1205911+1
minus
119896minus1
sum
119894=119896minus1205911
)119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
(
119896
sum
119895=119896+119894
minus
119896minus1
sum
119895=119896+119895minus1
)
times 119909119879
(119895)1198764119909 (119895) ]
]
Abstract and Applied Analysis 5
le 119864[
[
(
119896
sum
119894=119896+1minus1205912
119909119879
(119894) 1198763119909 (119894) minus
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894))
+
1205912minus1
sum
119894=1205911
(119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 119894) 1198763119909 (119896 minus 119894))
+ (119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1))
+
minus1205911+1
sum
119894=minus1205912+1
(119909119879
(119896) 1198764119909 (119896) minus 119909
119879
times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]
]
= 119864[
[
119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus 2119909119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1)
minus
119896minus1205911
sum
119894=119896minus1205912
119909119879
(119894) 1198764119909 (119894)]
]
le 119864 [119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus119909119879
(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]
119864Δ1198814(119896)
= 119864[
[
120591lowast
minus1
sum
119894=minus120591lowast
(
119896
sum
119895=119896+119894+1
minus
119896minus1
sum
119895=119896+119894
)120578119879
(119895) 1198851120578 (119895)]
]
= 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896) minus 120591
lowast
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851120578 (119894)]
le 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896)
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)]
(18)
Note that
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)
= (119909 (119896)
119909 (119896 minus 120591lowast
))
119879
(minus11988511198851
lowast minus1198851
)(119909 (119896)
119909 (119896 minus 120591lowast
))
(19)
119864Δ1198815(119896) = 119864 119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) minus 120591120595 (119896)
(20)
where
120595 (119896) =
119896minus1205911minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852120578 (119894) 120591
1le 120591 (119896) le 120591
lowast
119896minus120591lowast
minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894) 120591
lowast
lt 120591 (119896) le 1205912
(21)
When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that
minus 120591120595 (119896)
= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
)]
times
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852120578 (119894)
minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
))]
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894)
le minus120573
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894) minus (1 minus 120573)
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894)
(22)
When 1205911le 120591(119896) le 120591
lowast similarly we can have
minus120591120595 (119896) le minus (1 minus 120573)
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
minus 120573
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
(23)
6 Abstract and Applied Analysis
From (20) (22) and (23) we have
119864Δ1198815(119896) = 119864 [119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) + (
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
119879
times (
minus211988521198852
1198852
lowast minus1198852
0
lowast lowast minus1198852
)(
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
+ 120573(119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
))
119879
(minus11988521198852
lowast minus1198852
)
times (119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)) + (1 minus 120573)
times(119909 (119896 minus 120591 (119896))
120579 (119896))
119879
(minus11988521198852
lowast minus1198852
)
times(119909 (119896 minus 120591 (119896))
120579 (119896))]
(24)
When 1198891le 119889(119896) le 119889
lowast by Lemma 6 it is easy to get
119864Δ1198816(119896) = 119864[
[
(
minus1
sum
119894=minus119889(119896+1)
119896
sum
119895=119896+119894+1
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
)
times ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
(
119896
sum
119897=119896+119895+1
minus
119896minus1
sum
119897=119896+119895
)
times ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897)) ]
]
le 119864[
[
minus1
sum
119894=minus119889lowast
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1
sum
119894=minus119889lowast
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1
sum
119894=minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]
]
le 119864[(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119889lowast(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(25)
When 119889lowast lt 119889(119896) le 1198892 similarly we can have
119864Δ1198816(119896) le 119864[
(119889lowast
+ 1198892minus 1) (119889
1minus 119889lowast
) + 21198892
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
1198892
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(26)
From (25) and (26) we have
119864Δ1198816(119896) le 119864[119887ℎ
119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119888(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(27)
From (7) it follows that
(ℎ119894(119909 (119896)) minus 120574
+
119894119909119894(119896))
times (ℎ119894(119909 (119896)) minus 120574
minus
119894119909119894(119896)) le 0 119894 = 1 2 119899
(28)
which are equivalent to
(119909 (119896)
ℎ (119909 (119896)))
119879
(120574minus
119894120574+
119894119890119894119890119879
119894minus120574minus
119894+ 120574+
119894
2119890119894119890119879
119894
lowast 119890119894119890119879
119894
)
times(119909 (119896)
ℎ (119909 (119896))) le 0
(29)
where 119890119894denotes the unit column vector having one element
on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582
1 1205822
120582119899 gt 0 it follows that
(119909 (119896)
ℎ (119909 (119896)))
119879
(minusΓ1Λ Γ2Λ
lowast minusΛ)(
119909 (119896)
ℎ (119909 (119896))) ge 0 (30)
Abstract and Applied Analysis 7
Note that by Lemma 7 we get
119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896) + 120591lowast2
120578119879
(119896) 1198851120578 (119896)
+120591120578119879
(119896) 1198852120578 (119896)]
= 119864 [120578119879
(119896) (119875 + 120591lowast2
1198851+ 1205911198852) 120578 (119896) + 2120578
119879
(119896) 119875119909 (119896)]
= 119864 [119909119879
(119896 + 1) (119875 + 120591lowast2
1198851+ 1205911198852) 119909 (119896 + 1)
minus 2119909119879
(119896 + 1) (120591lowast2
1198851+ 1205911198852) 119909 (119896)
+ 119909119879
(119896) (120591lowast2
1198851+ 1205911198852minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) (119875 + 2120591lowast2
1198851+ 2120591119885
2) 119909 (119896 + 1)
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) 120582lowast
119868119909 (119896 + 1) + 119909119879
(119896)
times (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879
times 120582lowast
119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
times 120582lowast
119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]
119879
120582lowast
119868
times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
120582lowast
119868
times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
+ 120582lowast
119909119879
(119860 + 119862119870)119879
(119860 + 119862119870) 119909 (119896)
+ 120582lowast
119909119879
(119896 minus 120591 (119896)) 119861119879
119861119909 (119896 minus 120591 (119896))
+ 120582lowast
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
119863119879
119863
times (
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
+5120582lowast
1198622
1199032
infin
(31)
Then from (18) to (31) we have
119864Δ119881 (119896) le 119864120585119879
(119896) [Ξ1015840
+ (radic2119883 0 0 0 0 0 0 0)119879 1
120582lowast
times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)
+ 1205881199032
infin
(32)
where
Ξ1015840
119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582
lowast
1198632
120585119879
(119896) = [119909119879
(119896) 119909119879
(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast
) 119909 (119896 minus 119889lowast
)
120579119879
(119896) ℎ119879
(119909 (119896))
minus1
sum
minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119891119879
]
(33)
If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that
119864Δ119881 (119896) le minus120576119864119909 (119896)2
+ 1205881199032
infin (34)
It is easy to derive that
119864119881 (119896) le 1205831119864119909 (119896)
2
+ 1205832
119896minus1
sum
119894=119896minus1205912
119864119909 (119894)2
+ 1205833
119896minus1
sum
119894=119896minus1198892
119864119909 (119894)2
(35)
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
work reported on the mean square BIBO stabilization forthe discrete-time stochastic control systemswithmixed time-varying delays
It is well known that the classical technique applied in thestudy of stability is based on the Lyapunov direct methodHowever the Lyapunov direct method has some difficultieswith the theory and application to specific problems whilediscussing the stability of solutions in stochastic systems withtime delay In [33] the midpoint in the time delayrsquos variationinterval is introduced and the variation interval is dividedinto two subintervals with equal length by constructing theLyapunov functional which involved midpoint to reduce theconservatism of stability conditions This method was firstproposed to study the stability and stabilization problems forlinear continuous-time systems and then many successfulapplications were found in [13ndash15] In this paper we willreconsider this method by introducing a new piecewise-like delay method given that the point of the time delayrsquosvariation interval is arbitrary point rather than midpoint
Motivated by the aforementioned works in this paperwe investigate BIBO stabilization in mean square for a classof discrete-time stochastic control systems with mixed time-varying delays and nonlinear perturbations Some noveldelay-dependent stability conditions for the previously men-tioned system are derived by constructing a novel Lyapunov-Krasovskii function These conditions are expressed in theforms of linear matrix inequalities whose feasibility can beeasily checked by using MATLAB LMI Toolbox Finally anumerical example is given to illustrate the validity of theobtained results
The paper is organized as follows In Section 2 somenotations and the problem formulation are proposed Themain results are given in Section 3 In Section 4 a numericalexample is given to illustrate the validity of the obtainedtheory results The conclusion is proposed in Section 5
2 Notations and Problem Formulation
Firstly we propose some notations which will be needed inthe sequel The notations are quite standard Let 119877119899 and 119877119899times119898denote respectively the 119899-dimensioned Euclidean space andthe set of all 119899 times119898 real matrices The superscript ldquo119879rdquo denotesthe transpose and the notation 119883 ge 119884 (respective 119883 gt 119884)means that119883 and 119884 are symmetric matrices and that119883minus119884 ispositive semidefinitive (respective positive definite) Let sdot denote the Euclidean norm in 119877119899 let119873+ denote the positiveinteger set and let 119868 be the identity matrix with compatibledimension If119860 is a matrix denote by 119860 its operator normthat is 119860 = sup119860119909 119909 = 1 = radic120582max(119860
119879119860) where120582max(119860) (resp 120582min(119860)) means the largest (resp smallest)value of 119860 Moreover let (ΩF F
119905119905ge0 119875) be a complete
probability space with a filtration F119905119905ge0
satisfying theusual conditions (ie the filtration contains all 119875-null setsand is right continuous) 119864sdot stands for the mathematicalexpectation operator with respect to the given probabilitymeasure 119875 The asterisk lowast in a matrix is used to denote termthat is induced by its symmetry Matrices if not explicitlystate are assumed to have compatible dimensions Denote
119873[119886 119887] = 119886 119886 + 1 119887 Sometimes the arguments of thefunctions will be omitted in the analysis without confusions
In this paper we consider the discrete-time stochasticcontrol systemwithmixed time-varying delays and nonlinearperturbations with the following form
119909 (119896 + 1) = 119860119909 (119896) + 119861119909 (119896 minus 120591 (119896)) + 119862119906 (119896)
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))
+ (119866119909 (119896) + 119867119909 (119896 minus 119889 (119896))) 120596 (119896)
119910 (119896) = 119872119909 (119896)
(1)
where 119909(119896) = [1199091(119896) 1199092(119896) 119909
119899(119896)]119879
isin 119877119899 denotes the
state vector 119906(119896) = [1199061(119896) 1199062(119896) 119906
119898(119896)]119879
isin 119877119898 is the
control input vector 119910(119896) = [1199101(119896) 1199102(119896) 119910
119899(119896)]119879
isin 119877119899 is
the control output vector ℎ(119909(119896)) = [ℎ1(119909(119896)) ℎ
2(119909(119896))
ℎ119899(119909(119896))]
119879
119860 119861 119863 119866 119862 119867 and 119872 represent the weight-ing matrices with appropriate dimension and the positiveintegers 120591(119896) and 119889(119896) are the discrete-time-varying delaydistributed time-varying delay and respectively satisfyingthat
1205911le 120591 (119896) le 120591
2 1198891le 119889 (119896) le 119889
2 119896 isin 119873
+
(2)
with 1205911 1205912 1198891 and 119889
2being four known positive integers For
any given 120591lowast isin (1205911 1205912) 119889lowast isin (119889
1 1198892) The initial conditions of
the system (1) are given by
119909 (119896) = 120601 (119896) 119896 isin [minusmax 1205912 1198892 0] (3)
Thenonlinear vector-valued perturbation119891(119896 119909(119896)119909(119896minus120591(119896))) satisfies that
1003817100381710038171003817119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))1003817100381710038171003817
2
le 1205721119909(119896)
2
+ 1205722119909(119896 minus 120591(119896))
2
(4)
where 1205721and 120572
2are two positive constants 120596(119896) is a scalar
Wiener process defined on (ΩF F119905119905ge0 119875) with
119864 (120596 (119896)) = 0 119864 (120596(119896)2
) = 1
119864 (120596 (119894) 120596 (119895)) = 0 119894 = 119895
(5)
Remark 1 The 120591lowast divides the discrete-time delayrsquos variationinterval into two subintervals that is [120591
1 120591lowast
] and (120591lowast 1205912] and
119889lowast divides the distributed time delayrsquos variation interval into
two subintervals that is [1198891 119889lowast
] and (119889lowast 1198892] We will dis-
cuss the variation of differences of the Lyapunov-Krasovskiifunctional119881(119905 119909(119905)) for each subinterval Compared with theprevious results in the works of [27ndash32] the BIBO stabilityconditions are derived in this paper by checking the variationof119881(119905 119909(119905)) in subintervals rather than in the whole variationinterval of the delays
In what follows we describe the controller with the form
119906 (119896) = 119870119909 (119896) + 119903 (119896) (6)
Abstract and Applied Analysis 3
where119870 is the feedback gain matrix and 119903(119896) is the referenceinput
Assumption 2 For any 1205851 1205852isin 119877 120585
1= 1205852 let
120574minus
119894leℎ119894(1205851) minus ℎ119894(1205852)
1205851minus 1205852
le 120574+
119894 (7)
where 120574minus119894and 120574+119894are known constant scalars
Remark 3 The constants 120574minus119894 120574+119894in Assumption 2 are allowed
to be positive negative or zero Hence the function ℎ(119909(119896))could be nonmonotonic and is more general than the usualsigmoid functions and the recently commonly used Lipschitzconditions
At the end of this section let us introduce some importantdefinitions and lemmas as which will be used in the sequel
Definition 4 (see [28 31]) A vector function 119903(119896) = (1199031(119896)
1199032(119896) 119903
119899(119896))119879 is said to be an element of 119871119899
infin if 119903
infin=
sup119896isin119873[0infin)
119903(119896) lt +infin where sdot denotes the Euclid normin 119877119899 or the norm of a matrix
Definition 5 (see [28 31]) The nonlinear stochastic controlsystem (1) is said to beBIBO stability inmean square if we canconstruct a controller (6) such that the output 119910(119896) satisfiesthat
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198731+ 11987321199032
infin (8)
where1198731and119873
2are two positive constants
Lemma 6 (see [28]) For any given vectors V119894isin 119877119899 119894 = 1
2 119899 the following inequality holds
[
119899
sum
119894=1
V119894]
119879
[
119899
sum
119894=1
V119894] le 119899
119899
sum
119894=1
V119879119894V119894 (9)
Lemma 7 (see [28]) Let 119909 119910 isin 119877119899 and any 119899 times 119899 positive-
definite matrix 119876 gt 0 Then one has
2119909119879
119910 le 119909119879
119876minus1
119909 + 119910119879
119876119910 (10)
Lemma 8 (see [28]) Given the constant matricesΩ1Ω2 and
Ω3with appropriate dimensions where Ω
1= Ω119879
1and Ω
2=
Ω119879
2gt 0 then Ω
1+ Ω119879
3Ωminus1
2Ω3lt 0 if and only if
(Ω1Ω119879
3
lowast minusΩ2
) lt 0 119900119903 (minusΩ2Ω3
lowast Ω1
) lt 0 (11)
3 BIBO Stabilization for the System (1)In this section we aim to establish our main results based onthe LMI approach For the conveniences we denote
Γ1= diag 120574minus
1120574+
1 120574minus
2120574+
2 120574
minus
119899120574+
119899
Γ3= diag 120574+
1 120574+
2 120574
+
119899
Γ2= diag
120574minus
1+ 120574+
1
2120574minus
2+ 120574+
2
2
120574minus
119899+ 120574+
119899
2
119886 = 1205912minus 1205911+ 1
119887 =
(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2 1198891le 119889 (119896) le 119889
lowast
(119889lowast
+ 1198892minus 1) (119889
2minus 119889lowast
) + 21198892
2 119889lowast
lt 119889 (119896) le 1198892
119888 = 119889lowast
1198891le 119889 (119896) le 119889
lowast
1198892 119889lowast
lt 119889 (119896) le 1198892
120579 (119896) = 119909 (119896 minus 120591
1) 1205911le 120591 (119896) le 120591
lowast
119909 (119896 minus 1205912) 120591
lowast
lt 120591 (119896) le 1205912
120591 = 120591lowast
minus 1205911 1205911le 120591 (119896) le 120591
lowast
1205912minus 120591lowast
120591lowast
lt 120591 (119896) le 1205912
120573 =
120591 (119896) minus 1205911
120591lowast minus 1205911
1205911le 120591 (119896) le 120591
lowast
1205912minus 120591 (119896)
1205912minus 120591lowast
120591lowast
lt 120591 (119896) le 1205912
(12)
Theorem9 For given positive integers 1205911 12059121198891 and119889
2 under
Assumption 2 the nonlinear discrete-time stochastic controlsystem (1) with the controller (6) is BIBO stabilization inmean square if there exist symmetric positive-definite matrix119875 119877 119876
119894 119894 = 1 2 5 119885
1 1198852 and 119883 with appropriate
dimensional positive-definite diagonal matrices Λ and somepositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868 (13)
Ξ
=
(((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))
)
le 0
(14)
4 Abstract and Applied Analysis
whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 119866
119879
120582lowast
119866 + 2120591lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860 minus 2119860119879
119883 minus 2119883119879
119860
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861 + 119866119879
120582lowast
119867
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ22= 119867119879
120582lowast
119867 + 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)
119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881
2(119896) + 119881
3(119896)
+ 1198814(119896) + 119881
5(119896) + 119881
6(119896)
(16)
where
1198811(119896 119909 (119896)) = 119909
119879
(119896) 119875119909 (119896)
1198812(119896) =
119896minus1
sum
119894=119896minus120591lowast
119909119879
(119894) 1198761119909 (119894)
+
119896minus1
sum
119894=119896minus119889lowast
119909119879
(119894) 1198762119909 (119894)
1198813(119896) =
119896minus1
sum
119894=119896minus120591(119896)
119909119879
(119894) 1198763119909 (119894) +
1205912minus1
sum
119894=1205911
119896minus1
sum
119895=119896minus119894
119909119879
(119895) 1198763119909 (119895)
+
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
119896minus1
sum
119895=119896minus1+119894
119909119879
(119895)1198764119909 (119895)
1198814(119896) = 120591
lowast
minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198851120578 (119895)
120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)
1198815(119896) =
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198765119909 (119894) + (120591
lowast
minus 1205911)
times
minus1205911minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
1205911le 120591 (119896) le 120591
lowast
119896minus1
sum
119894=119896minus1205912
119909119879
(119894) 1198765119909 (119894) + (120591
2minus 120591lowast
)
times
minus120591lowast
minus1
sum
119894=minus1205912
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
120591lowast
lt 120591 (119896) le 1205912
1198816(119896) =
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
1198891le 119889 (119896) le 119889
lowast
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus119889lowast
minus1
sum
119894=minus1198892
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
119889lowast
lt 119889 (119896) le 1198892
(17)
Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have
119864Δ1198811(119896 119909 (119896))
= 119864 [119909119879
(119896 + 1) 119875119909 (119896 + 1) minus 119909119879
(119896) 119875119909 (119896)]
= 119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896)]
119864Δ1198812(119896)
= 119864 [ 119909119879
(119896) 1198761119909 (119896) minus 119909
119879
(119896 minus 120591lowast
) 1198761119909 (119896 minus 120591
lowast
)
+119909119879
(119896) 1198762119909 (119896) minus 119909
119879
(119896 minus 119889lowast
) 1198762119909 (119896 minus 119889
lowast
)]
119864Δ1198813(119896)
= 119864[(
119896
sum
119894=119896+1minus120591(119896+1)
minus
119896minus1
sum
119894=119896minus120591(119896)
)119909119879
(119894) 1198763119909 (119894)
+
1205912minus1
sum
119894=1205911
(
119896
sum
119895=119896minus119894+1
minus
119896minus1
sum
119895=119896minus119894
)119909119879
(119895)1198763119909 (119895)
+ (
119896
sum
119894=119896minus1205911+1
minus
119896minus1
sum
119894=119896minus1205911
)119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
(
119896
sum
119895=119896+119894
minus
119896minus1
sum
119895=119896+119895minus1
)
times 119909119879
(119895)1198764119909 (119895) ]
]
Abstract and Applied Analysis 5
le 119864[
[
(
119896
sum
119894=119896+1minus1205912
119909119879
(119894) 1198763119909 (119894) minus
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894))
+
1205912minus1
sum
119894=1205911
(119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 119894) 1198763119909 (119896 minus 119894))
+ (119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1))
+
minus1205911+1
sum
119894=minus1205912+1
(119909119879
(119896) 1198764119909 (119896) minus 119909
119879
times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]
]
= 119864[
[
119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus 2119909119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1)
minus
119896minus1205911
sum
119894=119896minus1205912
119909119879
(119894) 1198764119909 (119894)]
]
le 119864 [119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus119909119879
(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]
119864Δ1198814(119896)
= 119864[
[
120591lowast
minus1
sum
119894=minus120591lowast
(
119896
sum
119895=119896+119894+1
minus
119896minus1
sum
119895=119896+119894
)120578119879
(119895) 1198851120578 (119895)]
]
= 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896) minus 120591
lowast
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851120578 (119894)]
le 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896)
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)]
(18)
Note that
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)
= (119909 (119896)
119909 (119896 minus 120591lowast
))
119879
(minus11988511198851
lowast minus1198851
)(119909 (119896)
119909 (119896 minus 120591lowast
))
(19)
119864Δ1198815(119896) = 119864 119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) minus 120591120595 (119896)
(20)
where
120595 (119896) =
119896minus1205911minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852120578 (119894) 120591
1le 120591 (119896) le 120591
lowast
119896minus120591lowast
minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894) 120591
lowast
lt 120591 (119896) le 1205912
(21)
When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that
minus 120591120595 (119896)
= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
)]
times
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852120578 (119894)
minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
))]
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894)
le minus120573
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894) minus (1 minus 120573)
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894)
(22)
When 1205911le 120591(119896) le 120591
lowast similarly we can have
minus120591120595 (119896) le minus (1 minus 120573)
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
minus 120573
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
(23)
6 Abstract and Applied Analysis
From (20) (22) and (23) we have
119864Δ1198815(119896) = 119864 [119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) + (
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
119879
times (
minus211988521198852
1198852
lowast minus1198852
0
lowast lowast minus1198852
)(
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
+ 120573(119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
))
119879
(minus11988521198852
lowast minus1198852
)
times (119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)) + (1 minus 120573)
times(119909 (119896 minus 120591 (119896))
120579 (119896))
119879
(minus11988521198852
lowast minus1198852
)
times(119909 (119896 minus 120591 (119896))
120579 (119896))]
(24)
When 1198891le 119889(119896) le 119889
lowast by Lemma 6 it is easy to get
119864Δ1198816(119896) = 119864[
[
(
minus1
sum
119894=minus119889(119896+1)
119896
sum
119895=119896+119894+1
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
)
times ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
(
119896
sum
119897=119896+119895+1
minus
119896minus1
sum
119897=119896+119895
)
times ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897)) ]
]
le 119864[
[
minus1
sum
119894=minus119889lowast
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1
sum
119894=minus119889lowast
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1
sum
119894=minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]
]
le 119864[(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119889lowast(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(25)
When 119889lowast lt 119889(119896) le 1198892 similarly we can have
119864Δ1198816(119896) le 119864[
(119889lowast
+ 1198892minus 1) (119889
1minus 119889lowast
) + 21198892
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
1198892
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(26)
From (25) and (26) we have
119864Δ1198816(119896) le 119864[119887ℎ
119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119888(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(27)
From (7) it follows that
(ℎ119894(119909 (119896)) minus 120574
+
119894119909119894(119896))
times (ℎ119894(119909 (119896)) minus 120574
minus
119894119909119894(119896)) le 0 119894 = 1 2 119899
(28)
which are equivalent to
(119909 (119896)
ℎ (119909 (119896)))
119879
(120574minus
119894120574+
119894119890119894119890119879
119894minus120574minus
119894+ 120574+
119894
2119890119894119890119879
119894
lowast 119890119894119890119879
119894
)
times(119909 (119896)
ℎ (119909 (119896))) le 0
(29)
where 119890119894denotes the unit column vector having one element
on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582
1 1205822
120582119899 gt 0 it follows that
(119909 (119896)
ℎ (119909 (119896)))
119879
(minusΓ1Λ Γ2Λ
lowast minusΛ)(
119909 (119896)
ℎ (119909 (119896))) ge 0 (30)
Abstract and Applied Analysis 7
Note that by Lemma 7 we get
119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896) + 120591lowast2
120578119879
(119896) 1198851120578 (119896)
+120591120578119879
(119896) 1198852120578 (119896)]
= 119864 [120578119879
(119896) (119875 + 120591lowast2
1198851+ 1205911198852) 120578 (119896) + 2120578
119879
(119896) 119875119909 (119896)]
= 119864 [119909119879
(119896 + 1) (119875 + 120591lowast2
1198851+ 1205911198852) 119909 (119896 + 1)
minus 2119909119879
(119896 + 1) (120591lowast2
1198851+ 1205911198852) 119909 (119896)
+ 119909119879
(119896) (120591lowast2
1198851+ 1205911198852minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) (119875 + 2120591lowast2
1198851+ 2120591119885
2) 119909 (119896 + 1)
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) 120582lowast
119868119909 (119896 + 1) + 119909119879
(119896)
times (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879
times 120582lowast
119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
times 120582lowast
119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]
119879
120582lowast
119868
times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
120582lowast
119868
times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
+ 120582lowast
119909119879
(119860 + 119862119870)119879
(119860 + 119862119870) 119909 (119896)
+ 120582lowast
119909119879
(119896 minus 120591 (119896)) 119861119879
119861119909 (119896 minus 120591 (119896))
+ 120582lowast
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
119863119879
119863
times (
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
+5120582lowast
1198622
1199032
infin
(31)
Then from (18) to (31) we have
119864Δ119881 (119896) le 119864120585119879
(119896) [Ξ1015840
+ (radic2119883 0 0 0 0 0 0 0)119879 1
120582lowast
times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)
+ 1205881199032
infin
(32)
where
Ξ1015840
119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582
lowast
1198632
120585119879
(119896) = [119909119879
(119896) 119909119879
(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast
) 119909 (119896 minus 119889lowast
)
120579119879
(119896) ℎ119879
(119909 (119896))
minus1
sum
minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119891119879
]
(33)
If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that
119864Δ119881 (119896) le minus120576119864119909 (119896)2
+ 1205881199032
infin (34)
It is easy to derive that
119864119881 (119896) le 1205831119864119909 (119896)
2
+ 1205832
119896minus1
sum
119894=119896minus1205912
119864119909 (119894)2
+ 1205833
119896minus1
sum
119894=119896minus1198892
119864119909 (119894)2
(35)
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
where119870 is the feedback gain matrix and 119903(119896) is the referenceinput
Assumption 2 For any 1205851 1205852isin 119877 120585
1= 1205852 let
120574minus
119894leℎ119894(1205851) minus ℎ119894(1205852)
1205851minus 1205852
le 120574+
119894 (7)
where 120574minus119894and 120574+119894are known constant scalars
Remark 3 The constants 120574minus119894 120574+119894in Assumption 2 are allowed
to be positive negative or zero Hence the function ℎ(119909(119896))could be nonmonotonic and is more general than the usualsigmoid functions and the recently commonly used Lipschitzconditions
At the end of this section let us introduce some importantdefinitions and lemmas as which will be used in the sequel
Definition 4 (see [28 31]) A vector function 119903(119896) = (1199031(119896)
1199032(119896) 119903
119899(119896))119879 is said to be an element of 119871119899
infin if 119903
infin=
sup119896isin119873[0infin)
119903(119896) lt +infin where sdot denotes the Euclid normin 119877119899 or the norm of a matrix
Definition 5 (see [28 31]) The nonlinear stochastic controlsystem (1) is said to beBIBO stability inmean square if we canconstruct a controller (6) such that the output 119910(119896) satisfiesthat
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198731+ 11987321199032
infin (8)
where1198731and119873
2are two positive constants
Lemma 6 (see [28]) For any given vectors V119894isin 119877119899 119894 = 1
2 119899 the following inequality holds
[
119899
sum
119894=1
V119894]
119879
[
119899
sum
119894=1
V119894] le 119899
119899
sum
119894=1
V119879119894V119894 (9)
Lemma 7 (see [28]) Let 119909 119910 isin 119877119899 and any 119899 times 119899 positive-
definite matrix 119876 gt 0 Then one has
2119909119879
119910 le 119909119879
119876minus1
119909 + 119910119879
119876119910 (10)
Lemma 8 (see [28]) Given the constant matricesΩ1Ω2 and
Ω3with appropriate dimensions where Ω
1= Ω119879
1and Ω
2=
Ω119879
2gt 0 then Ω
1+ Ω119879
3Ωminus1
2Ω3lt 0 if and only if
(Ω1Ω119879
3
lowast minusΩ2
) lt 0 119900119903 (minusΩ2Ω3
lowast Ω1
) lt 0 (11)
3 BIBO Stabilization for the System (1)In this section we aim to establish our main results based onthe LMI approach For the conveniences we denote
Γ1= diag 120574minus
1120574+
1 120574minus
2120574+
2 120574
minus
119899120574+
119899
Γ3= diag 120574+
1 120574+
2 120574
+
119899
Γ2= diag
120574minus
1+ 120574+
1
2120574minus
2+ 120574+
2
2
120574minus
119899+ 120574+
119899
2
119886 = 1205912minus 1205911+ 1
119887 =
(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2 1198891le 119889 (119896) le 119889
lowast
(119889lowast
+ 1198892minus 1) (119889
2minus 119889lowast
) + 21198892
2 119889lowast
lt 119889 (119896) le 1198892
119888 = 119889lowast
1198891le 119889 (119896) le 119889
lowast
1198892 119889lowast
lt 119889 (119896) le 1198892
120579 (119896) = 119909 (119896 minus 120591
1) 1205911le 120591 (119896) le 120591
lowast
119909 (119896 minus 1205912) 120591
lowast
lt 120591 (119896) le 1205912
120591 = 120591lowast
minus 1205911 1205911le 120591 (119896) le 120591
lowast
1205912minus 120591lowast
120591lowast
lt 120591 (119896) le 1205912
120573 =
120591 (119896) minus 1205911
120591lowast minus 1205911
1205911le 120591 (119896) le 120591
lowast
1205912minus 120591 (119896)
1205912minus 120591lowast
120591lowast
lt 120591 (119896) le 1205912
(12)
Theorem9 For given positive integers 1205911 12059121198891 and119889
2 under
Assumption 2 the nonlinear discrete-time stochastic controlsystem (1) with the controller (6) is BIBO stabilization inmean square if there exist symmetric positive-definite matrix119875 119877 119876
119894 119894 = 1 2 5 119885
1 1198852 and 119883 with appropriate
dimensional positive-definite diagonal matrices Λ and somepositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868 (13)
Ξ
=
(((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))
)
le 0
(14)
4 Abstract and Applied Analysis
whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 119866
119879
120582lowast
119866 + 2120591lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860 minus 2119860119879
119883 minus 2119883119879
119860
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861 + 119866119879
120582lowast
119867
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ22= 119867119879
120582lowast
119867 + 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)
119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881
2(119896) + 119881
3(119896)
+ 1198814(119896) + 119881
5(119896) + 119881
6(119896)
(16)
where
1198811(119896 119909 (119896)) = 119909
119879
(119896) 119875119909 (119896)
1198812(119896) =
119896minus1
sum
119894=119896minus120591lowast
119909119879
(119894) 1198761119909 (119894)
+
119896minus1
sum
119894=119896minus119889lowast
119909119879
(119894) 1198762119909 (119894)
1198813(119896) =
119896minus1
sum
119894=119896minus120591(119896)
119909119879
(119894) 1198763119909 (119894) +
1205912minus1
sum
119894=1205911
119896minus1
sum
119895=119896minus119894
119909119879
(119895) 1198763119909 (119895)
+
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
119896minus1
sum
119895=119896minus1+119894
119909119879
(119895)1198764119909 (119895)
1198814(119896) = 120591
lowast
minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198851120578 (119895)
120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)
1198815(119896) =
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198765119909 (119894) + (120591
lowast
minus 1205911)
times
minus1205911minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
1205911le 120591 (119896) le 120591
lowast
119896minus1
sum
119894=119896minus1205912
119909119879
(119894) 1198765119909 (119894) + (120591
2minus 120591lowast
)
times
minus120591lowast
minus1
sum
119894=minus1205912
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
120591lowast
lt 120591 (119896) le 1205912
1198816(119896) =
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
1198891le 119889 (119896) le 119889
lowast
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus119889lowast
minus1
sum
119894=minus1198892
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
119889lowast
lt 119889 (119896) le 1198892
(17)
Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have
119864Δ1198811(119896 119909 (119896))
= 119864 [119909119879
(119896 + 1) 119875119909 (119896 + 1) minus 119909119879
(119896) 119875119909 (119896)]
= 119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896)]
119864Δ1198812(119896)
= 119864 [ 119909119879
(119896) 1198761119909 (119896) minus 119909
119879
(119896 minus 120591lowast
) 1198761119909 (119896 minus 120591
lowast
)
+119909119879
(119896) 1198762119909 (119896) minus 119909
119879
(119896 minus 119889lowast
) 1198762119909 (119896 minus 119889
lowast
)]
119864Δ1198813(119896)
= 119864[(
119896
sum
119894=119896+1minus120591(119896+1)
minus
119896minus1
sum
119894=119896minus120591(119896)
)119909119879
(119894) 1198763119909 (119894)
+
1205912minus1
sum
119894=1205911
(
119896
sum
119895=119896minus119894+1
minus
119896minus1
sum
119895=119896minus119894
)119909119879
(119895)1198763119909 (119895)
+ (
119896
sum
119894=119896minus1205911+1
minus
119896minus1
sum
119894=119896minus1205911
)119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
(
119896
sum
119895=119896+119894
minus
119896minus1
sum
119895=119896+119895minus1
)
times 119909119879
(119895)1198764119909 (119895) ]
]
Abstract and Applied Analysis 5
le 119864[
[
(
119896
sum
119894=119896+1minus1205912
119909119879
(119894) 1198763119909 (119894) minus
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894))
+
1205912minus1
sum
119894=1205911
(119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 119894) 1198763119909 (119896 minus 119894))
+ (119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1))
+
minus1205911+1
sum
119894=minus1205912+1
(119909119879
(119896) 1198764119909 (119896) minus 119909
119879
times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]
]
= 119864[
[
119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus 2119909119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1)
minus
119896minus1205911
sum
119894=119896minus1205912
119909119879
(119894) 1198764119909 (119894)]
]
le 119864 [119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus119909119879
(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]
119864Δ1198814(119896)
= 119864[
[
120591lowast
minus1
sum
119894=minus120591lowast
(
119896
sum
119895=119896+119894+1
minus
119896minus1
sum
119895=119896+119894
)120578119879
(119895) 1198851120578 (119895)]
]
= 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896) minus 120591
lowast
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851120578 (119894)]
le 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896)
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)]
(18)
Note that
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)
= (119909 (119896)
119909 (119896 minus 120591lowast
))
119879
(minus11988511198851
lowast minus1198851
)(119909 (119896)
119909 (119896 minus 120591lowast
))
(19)
119864Δ1198815(119896) = 119864 119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) minus 120591120595 (119896)
(20)
where
120595 (119896) =
119896minus1205911minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852120578 (119894) 120591
1le 120591 (119896) le 120591
lowast
119896minus120591lowast
minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894) 120591
lowast
lt 120591 (119896) le 1205912
(21)
When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that
minus 120591120595 (119896)
= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
)]
times
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852120578 (119894)
minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
))]
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894)
le minus120573
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894) minus (1 minus 120573)
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894)
(22)
When 1205911le 120591(119896) le 120591
lowast similarly we can have
minus120591120595 (119896) le minus (1 minus 120573)
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
minus 120573
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
(23)
6 Abstract and Applied Analysis
From (20) (22) and (23) we have
119864Δ1198815(119896) = 119864 [119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) + (
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
119879
times (
minus211988521198852
1198852
lowast minus1198852
0
lowast lowast minus1198852
)(
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
+ 120573(119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
))
119879
(minus11988521198852
lowast minus1198852
)
times (119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)) + (1 minus 120573)
times(119909 (119896 minus 120591 (119896))
120579 (119896))
119879
(minus11988521198852
lowast minus1198852
)
times(119909 (119896 minus 120591 (119896))
120579 (119896))]
(24)
When 1198891le 119889(119896) le 119889
lowast by Lemma 6 it is easy to get
119864Δ1198816(119896) = 119864[
[
(
minus1
sum
119894=minus119889(119896+1)
119896
sum
119895=119896+119894+1
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
)
times ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
(
119896
sum
119897=119896+119895+1
minus
119896minus1
sum
119897=119896+119895
)
times ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897)) ]
]
le 119864[
[
minus1
sum
119894=minus119889lowast
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1
sum
119894=minus119889lowast
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1
sum
119894=minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]
]
le 119864[(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119889lowast(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(25)
When 119889lowast lt 119889(119896) le 1198892 similarly we can have
119864Δ1198816(119896) le 119864[
(119889lowast
+ 1198892minus 1) (119889
1minus 119889lowast
) + 21198892
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
1198892
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(26)
From (25) and (26) we have
119864Δ1198816(119896) le 119864[119887ℎ
119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119888(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(27)
From (7) it follows that
(ℎ119894(119909 (119896)) minus 120574
+
119894119909119894(119896))
times (ℎ119894(119909 (119896)) minus 120574
minus
119894119909119894(119896)) le 0 119894 = 1 2 119899
(28)
which are equivalent to
(119909 (119896)
ℎ (119909 (119896)))
119879
(120574minus
119894120574+
119894119890119894119890119879
119894minus120574minus
119894+ 120574+
119894
2119890119894119890119879
119894
lowast 119890119894119890119879
119894
)
times(119909 (119896)
ℎ (119909 (119896))) le 0
(29)
where 119890119894denotes the unit column vector having one element
on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582
1 1205822
120582119899 gt 0 it follows that
(119909 (119896)
ℎ (119909 (119896)))
119879
(minusΓ1Λ Γ2Λ
lowast minusΛ)(
119909 (119896)
ℎ (119909 (119896))) ge 0 (30)
Abstract and Applied Analysis 7
Note that by Lemma 7 we get
119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896) + 120591lowast2
120578119879
(119896) 1198851120578 (119896)
+120591120578119879
(119896) 1198852120578 (119896)]
= 119864 [120578119879
(119896) (119875 + 120591lowast2
1198851+ 1205911198852) 120578 (119896) + 2120578
119879
(119896) 119875119909 (119896)]
= 119864 [119909119879
(119896 + 1) (119875 + 120591lowast2
1198851+ 1205911198852) 119909 (119896 + 1)
minus 2119909119879
(119896 + 1) (120591lowast2
1198851+ 1205911198852) 119909 (119896)
+ 119909119879
(119896) (120591lowast2
1198851+ 1205911198852minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) (119875 + 2120591lowast2
1198851+ 2120591119885
2) 119909 (119896 + 1)
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) 120582lowast
119868119909 (119896 + 1) + 119909119879
(119896)
times (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879
times 120582lowast
119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
times 120582lowast
119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]
119879
120582lowast
119868
times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
120582lowast
119868
times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
+ 120582lowast
119909119879
(119860 + 119862119870)119879
(119860 + 119862119870) 119909 (119896)
+ 120582lowast
119909119879
(119896 minus 120591 (119896)) 119861119879
119861119909 (119896 minus 120591 (119896))
+ 120582lowast
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
119863119879
119863
times (
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
+5120582lowast
1198622
1199032
infin
(31)
Then from (18) to (31) we have
119864Δ119881 (119896) le 119864120585119879
(119896) [Ξ1015840
+ (radic2119883 0 0 0 0 0 0 0)119879 1
120582lowast
times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)
+ 1205881199032
infin
(32)
where
Ξ1015840
119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582
lowast
1198632
120585119879
(119896) = [119909119879
(119896) 119909119879
(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast
) 119909 (119896 minus 119889lowast
)
120579119879
(119896) ℎ119879
(119909 (119896))
minus1
sum
minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119891119879
]
(33)
If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that
119864Δ119881 (119896) le minus120576119864119909 (119896)2
+ 1205881199032
infin (34)
It is easy to derive that
119864119881 (119896) le 1205831119864119909 (119896)
2
+ 1205832
119896minus1
sum
119894=119896minus1205912
119864119909 (119894)2
+ 1205833
119896minus1
sum
119894=119896minus1198892
119864119909 (119894)2
(35)
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
whereΞ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 119866
119879
120582lowast
119866 + 2120591lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860 minus 2119860119879
119883 minus 2119883119879
119860
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861 + 119866119879
120582lowast
119867
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ22= 119867119879
120582lowast
119867 + 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(15)Proof We construct the following Lyapunov-Krasovskiifunction for the system (1)
119881 (119896 119909 (119896)) = 1198811(119896 119909 (119896)) + 119881
2(119896) + 119881
3(119896)
+ 1198814(119896) + 119881
5(119896) + 119881
6(119896)
(16)
where
1198811(119896 119909 (119896)) = 119909
119879
(119896) 119875119909 (119896)
1198812(119896) =
119896minus1
sum
119894=119896minus120591lowast
119909119879
(119894) 1198761119909 (119894)
+
119896minus1
sum
119894=119896minus119889lowast
119909119879
(119894) 1198762119909 (119894)
1198813(119896) =
119896minus1
sum
119894=119896minus120591(119896)
119909119879
(119894) 1198763119909 (119894) +
1205912minus1
sum
119894=1205911
119896minus1
sum
119895=119896minus119894
119909119879
(119895) 1198763119909 (119895)
+
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
119896minus1
sum
119895=119896minus1+119894
119909119879
(119895)1198764119909 (119895)
1198814(119896) = 120591
lowast
minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198851120578 (119895)
120578 (119896) = 119909 (119896 + 1) minus 119909 (119896)
1198815(119896) =
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198765119909 (119894) + (120591
lowast
minus 1205911)
times
minus1205911minus1
sum
119894=minus120591lowast
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
1205911le 120591 (119896) le 120591
lowast
119896minus1
sum
119894=119896minus1205912
119909119879
(119894) 1198765119909 (119894) + (120591
2minus 120591lowast
)
times
minus120591lowast
minus1
sum
119894=minus1205912
119896minus1
sum
119895=119896+119894
120578119879
(119895) 1198852120578 (119895)
120591lowast
lt 120591 (119896) le 1205912
1198816(119896) =
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
1198891le 119889 (119896) le 119889
lowast
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus119889lowast
minus1
sum
119894=minus1198892
minus1
sum
119895=119894+1
119896minus1
sum
119897=119896+119895
ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897))
119889lowast
lt 119889 (119896) le 1198892
(17)
Calculating the difference of 119881(119896 119909(119896)) and taking the math-ematical expectation by Lemma 6 we have
119864Δ1198811(119896 119909 (119896))
= 119864 [119909119879
(119896 + 1) 119875119909 (119896 + 1) minus 119909119879
(119896) 119875119909 (119896)]
= 119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896)]
119864Δ1198812(119896)
= 119864 [ 119909119879
(119896) 1198761119909 (119896) minus 119909
119879
(119896 minus 120591lowast
) 1198761119909 (119896 minus 120591
lowast
)
+119909119879
(119896) 1198762119909 (119896) minus 119909
119879
(119896 minus 119889lowast
) 1198762119909 (119896 minus 119889
lowast
)]
119864Δ1198813(119896)
= 119864[(
119896
sum
119894=119896+1minus120591(119896+1)
minus
119896minus1
sum
119894=119896minus120591(119896)
)119909119879
(119894) 1198763119909 (119894)
+
1205912minus1
sum
119894=1205911
(
119896
sum
119895=119896minus119894+1
minus
119896minus1
sum
119895=119896minus119894
)119909119879
(119895)1198763119909 (119895)
+ (
119896
sum
119894=119896minus1205911+1
minus
119896minus1
sum
119894=119896minus1205911
)119909119879
(119894) 1198763119909 (119894)
+
minus1205911+1
sum
119894=minus1205912+1
(
119896
sum
119895=119896+119894
minus
119896minus1
sum
119895=119896+119895minus1
)
times 119909119879
(119895)1198764119909 (119895) ]
]
Abstract and Applied Analysis 5
le 119864[
[
(
119896
sum
119894=119896+1minus1205912
119909119879
(119894) 1198763119909 (119894) minus
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894))
+
1205912minus1
sum
119894=1205911
(119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 119894) 1198763119909 (119896 minus 119894))
+ (119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1))
+
minus1205911+1
sum
119894=minus1205912+1
(119909119879
(119896) 1198764119909 (119896) minus 119909
119879
times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]
]
= 119864[
[
119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus 2119909119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1)
minus
119896minus1205911
sum
119894=119896minus1205912
119909119879
(119894) 1198764119909 (119894)]
]
le 119864 [119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus119909119879
(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]
119864Δ1198814(119896)
= 119864[
[
120591lowast
minus1
sum
119894=minus120591lowast
(
119896
sum
119895=119896+119894+1
minus
119896minus1
sum
119895=119896+119894
)120578119879
(119895) 1198851120578 (119895)]
]
= 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896) minus 120591
lowast
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851120578 (119894)]
le 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896)
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)]
(18)
Note that
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)
= (119909 (119896)
119909 (119896 minus 120591lowast
))
119879
(minus11988511198851
lowast minus1198851
)(119909 (119896)
119909 (119896 minus 120591lowast
))
(19)
119864Δ1198815(119896) = 119864 119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) minus 120591120595 (119896)
(20)
where
120595 (119896) =
119896minus1205911minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852120578 (119894) 120591
1le 120591 (119896) le 120591
lowast
119896minus120591lowast
minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894) 120591
lowast
lt 120591 (119896) le 1205912
(21)
When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that
minus 120591120595 (119896)
= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
)]
times
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852120578 (119894)
minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
))]
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894)
le minus120573
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894) minus (1 minus 120573)
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894)
(22)
When 1205911le 120591(119896) le 120591
lowast similarly we can have
minus120591120595 (119896) le minus (1 minus 120573)
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
minus 120573
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
(23)
6 Abstract and Applied Analysis
From (20) (22) and (23) we have
119864Δ1198815(119896) = 119864 [119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) + (
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
119879
times (
minus211988521198852
1198852
lowast minus1198852
0
lowast lowast minus1198852
)(
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
+ 120573(119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
))
119879
(minus11988521198852
lowast minus1198852
)
times (119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)) + (1 minus 120573)
times(119909 (119896 minus 120591 (119896))
120579 (119896))
119879
(minus11988521198852
lowast minus1198852
)
times(119909 (119896 minus 120591 (119896))
120579 (119896))]
(24)
When 1198891le 119889(119896) le 119889
lowast by Lemma 6 it is easy to get
119864Δ1198816(119896) = 119864[
[
(
minus1
sum
119894=minus119889(119896+1)
119896
sum
119895=119896+119894+1
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
)
times ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
(
119896
sum
119897=119896+119895+1
minus
119896minus1
sum
119897=119896+119895
)
times ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897)) ]
]
le 119864[
[
minus1
sum
119894=minus119889lowast
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1
sum
119894=minus119889lowast
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1
sum
119894=minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]
]
le 119864[(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119889lowast(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(25)
When 119889lowast lt 119889(119896) le 1198892 similarly we can have
119864Δ1198816(119896) le 119864[
(119889lowast
+ 1198892minus 1) (119889
1minus 119889lowast
) + 21198892
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
1198892
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(26)
From (25) and (26) we have
119864Δ1198816(119896) le 119864[119887ℎ
119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119888(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(27)
From (7) it follows that
(ℎ119894(119909 (119896)) minus 120574
+
119894119909119894(119896))
times (ℎ119894(119909 (119896)) minus 120574
minus
119894119909119894(119896)) le 0 119894 = 1 2 119899
(28)
which are equivalent to
(119909 (119896)
ℎ (119909 (119896)))
119879
(120574minus
119894120574+
119894119890119894119890119879
119894minus120574minus
119894+ 120574+
119894
2119890119894119890119879
119894
lowast 119890119894119890119879
119894
)
times(119909 (119896)
ℎ (119909 (119896))) le 0
(29)
where 119890119894denotes the unit column vector having one element
on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582
1 1205822
120582119899 gt 0 it follows that
(119909 (119896)
ℎ (119909 (119896)))
119879
(minusΓ1Λ Γ2Λ
lowast minusΛ)(
119909 (119896)
ℎ (119909 (119896))) ge 0 (30)
Abstract and Applied Analysis 7
Note that by Lemma 7 we get
119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896) + 120591lowast2
120578119879
(119896) 1198851120578 (119896)
+120591120578119879
(119896) 1198852120578 (119896)]
= 119864 [120578119879
(119896) (119875 + 120591lowast2
1198851+ 1205911198852) 120578 (119896) + 2120578
119879
(119896) 119875119909 (119896)]
= 119864 [119909119879
(119896 + 1) (119875 + 120591lowast2
1198851+ 1205911198852) 119909 (119896 + 1)
minus 2119909119879
(119896 + 1) (120591lowast2
1198851+ 1205911198852) 119909 (119896)
+ 119909119879
(119896) (120591lowast2
1198851+ 1205911198852minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) (119875 + 2120591lowast2
1198851+ 2120591119885
2) 119909 (119896 + 1)
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) 120582lowast
119868119909 (119896 + 1) + 119909119879
(119896)
times (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879
times 120582lowast
119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
times 120582lowast
119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]
119879
120582lowast
119868
times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
120582lowast
119868
times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
+ 120582lowast
119909119879
(119860 + 119862119870)119879
(119860 + 119862119870) 119909 (119896)
+ 120582lowast
119909119879
(119896 minus 120591 (119896)) 119861119879
119861119909 (119896 minus 120591 (119896))
+ 120582lowast
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
119863119879
119863
times (
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
+5120582lowast
1198622
1199032
infin
(31)
Then from (18) to (31) we have
119864Δ119881 (119896) le 119864120585119879
(119896) [Ξ1015840
+ (radic2119883 0 0 0 0 0 0 0)119879 1
120582lowast
times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)
+ 1205881199032
infin
(32)
where
Ξ1015840
119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582
lowast
1198632
120585119879
(119896) = [119909119879
(119896) 119909119879
(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast
) 119909 (119896 minus 119889lowast
)
120579119879
(119896) ℎ119879
(119909 (119896))
minus1
sum
minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119891119879
]
(33)
If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that
119864Δ119881 (119896) le minus120576119864119909 (119896)2
+ 1205881199032
infin (34)
It is easy to derive that
119864119881 (119896) le 1205831119864119909 (119896)
2
+ 1205832
119896minus1
sum
119894=119896minus1205912
119864119909 (119894)2
+ 1205833
119896minus1
sum
119894=119896minus1198892
119864119909 (119894)2
(35)
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
le 119864[
[
(
119896
sum
119894=119896+1minus1205912
119909119879
(119894) 1198763119909 (119894) minus
119896minus1
sum
119894=119896minus1205911
119909119879
(119894) 1198763119909 (119894))
+
1205912minus1
sum
119894=1205911
(119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 119894) 1198763119909 (119896 minus 119894))
+ (119909119879
(119896) 1198763119909 (119896) minus 119909
119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1))
+
minus1205911+1
sum
119894=minus1205912+1
(119909119879
(119896) 1198764119909 (119896) minus 119909
119879
times (119896 + 119894 minus 1)1198764119909 (119896 + 119894 minus 1) )]
]
= 119864[
[
119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus 2119909119879
(119896 minus 1205911) 1198763119909 (119896 minus 120591
1)
minus
119896minus1205911
sum
119894=119896minus1205912
119909119879
(119894) 1198764119909 (119894)]
]
le 119864 [119909119879
(119896) [119886 (1198763+ 1198764) + 1198763] 119909 (119896)
minus119909119879
(119896 minus 120591 (119896)) 1198764119909 (119896 minus 120591 (119896))]
119864Δ1198814(119896)
= 119864[
[
120591lowast
minus1
sum
119894=minus120591lowast
(
119896
sum
119895=119896+119894+1
minus
119896minus1
sum
119895=119896+119894
)120578119879
(119895) 1198851120578 (119895)]
]
= 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896) minus 120591
lowast
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851120578 (119894)]
le 119864[120591lowast2
120578119879
(119896) 1198851120578 (119896)
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)]
(18)
Note that
minus
119896minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198851
119896minus1
sum
119894=119896minus120591lowast
120578 (119894)
= (119909 (119896)
119909 (119896 minus 120591lowast
))
119879
(minus11988511198851
lowast minus1198851
)(119909 (119896)
119909 (119896 minus 120591lowast
))
(19)
119864Δ1198815(119896) = 119864 119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) minus 120591120595 (119896)
(20)
where
120595 (119896) =
119896minus1205911minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852120578 (119894) 120591
1le 120591 (119896) le 120591
lowast
119896minus120591lowast
minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894) 120591
lowast
lt 120591 (119896) le 1205912
(21)
When 120591lowast lt 120591(119896) le 1205912 it is easy to compute that
minus 120591120595 (119896)
= minus [(1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
)]
times
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852120578 (119894)
minus [((1205912minus 120591 (119896)) + (120591 (119896) minus 120591
lowast
))]
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852120578 (119894)
le minus120573
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591lowast
minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894) minus (1 minus 120573)
times
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus1205912
120578 (119894)
(22)
When 1205911le 120591(119896) le 120591
lowast similarly we can have
minus120591120595 (119896) le minus (1 minus 120573)
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus1205911minus1
sum
119894=119896minus120591(119896)
120578119879
(119894) 1198852
119896minus120591minus1
sum
119894=119896minus120591(119896)
120578 (119894)
minus
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
minus 120573
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578119879
(119894) 1198852
119896minus120591(119896)minus1
sum
119894=119896minus120591lowast
120578 (119894)
(23)
6 Abstract and Applied Analysis
From (20) (22) and (23) we have
119864Δ1198815(119896) = 119864 [119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) + (
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
119879
times (
minus211988521198852
1198852
lowast minus1198852
0
lowast lowast minus1198852
)(
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
+ 120573(119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
))
119879
(minus11988521198852
lowast minus1198852
)
times (119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)) + (1 minus 120573)
times(119909 (119896 minus 120591 (119896))
120579 (119896))
119879
(minus11988521198852
lowast minus1198852
)
times(119909 (119896 minus 120591 (119896))
120579 (119896))]
(24)
When 1198891le 119889(119896) le 119889
lowast by Lemma 6 it is easy to get
119864Δ1198816(119896) = 119864[
[
(
minus1
sum
119894=minus119889(119896+1)
119896
sum
119895=119896+119894+1
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
)
times ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
(
119896
sum
119897=119896+119895+1
minus
119896minus1
sum
119897=119896+119895
)
times ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897)) ]
]
le 119864[
[
minus1
sum
119894=minus119889lowast
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1
sum
119894=minus119889lowast
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1
sum
119894=minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]
]
le 119864[(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119889lowast(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(25)
When 119889lowast lt 119889(119896) le 1198892 similarly we can have
119864Δ1198816(119896) le 119864[
(119889lowast
+ 1198892minus 1) (119889
1minus 119889lowast
) + 21198892
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
1198892
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(26)
From (25) and (26) we have
119864Δ1198816(119896) le 119864[119887ℎ
119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119888(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(27)
From (7) it follows that
(ℎ119894(119909 (119896)) minus 120574
+
119894119909119894(119896))
times (ℎ119894(119909 (119896)) minus 120574
minus
119894119909119894(119896)) le 0 119894 = 1 2 119899
(28)
which are equivalent to
(119909 (119896)
ℎ (119909 (119896)))
119879
(120574minus
119894120574+
119894119890119894119890119879
119894minus120574minus
119894+ 120574+
119894
2119890119894119890119879
119894
lowast 119890119894119890119879
119894
)
times(119909 (119896)
ℎ (119909 (119896))) le 0
(29)
where 119890119894denotes the unit column vector having one element
on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582
1 1205822
120582119899 gt 0 it follows that
(119909 (119896)
ℎ (119909 (119896)))
119879
(minusΓ1Λ Γ2Λ
lowast minusΛ)(
119909 (119896)
ℎ (119909 (119896))) ge 0 (30)
Abstract and Applied Analysis 7
Note that by Lemma 7 we get
119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896) + 120591lowast2
120578119879
(119896) 1198851120578 (119896)
+120591120578119879
(119896) 1198852120578 (119896)]
= 119864 [120578119879
(119896) (119875 + 120591lowast2
1198851+ 1205911198852) 120578 (119896) + 2120578
119879
(119896) 119875119909 (119896)]
= 119864 [119909119879
(119896 + 1) (119875 + 120591lowast2
1198851+ 1205911198852) 119909 (119896 + 1)
minus 2119909119879
(119896 + 1) (120591lowast2
1198851+ 1205911198852) 119909 (119896)
+ 119909119879
(119896) (120591lowast2
1198851+ 1205911198852minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) (119875 + 2120591lowast2
1198851+ 2120591119885
2) 119909 (119896 + 1)
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) 120582lowast
119868119909 (119896 + 1) + 119909119879
(119896)
times (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879
times 120582lowast
119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
times 120582lowast
119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]
119879
120582lowast
119868
times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
120582lowast
119868
times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
+ 120582lowast
119909119879
(119860 + 119862119870)119879
(119860 + 119862119870) 119909 (119896)
+ 120582lowast
119909119879
(119896 minus 120591 (119896)) 119861119879
119861119909 (119896 minus 120591 (119896))
+ 120582lowast
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
119863119879
119863
times (
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
+5120582lowast
1198622
1199032
infin
(31)
Then from (18) to (31) we have
119864Δ119881 (119896) le 119864120585119879
(119896) [Ξ1015840
+ (radic2119883 0 0 0 0 0 0 0)119879 1
120582lowast
times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)
+ 1205881199032
infin
(32)
where
Ξ1015840
119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582
lowast
1198632
120585119879
(119896) = [119909119879
(119896) 119909119879
(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast
) 119909 (119896 minus 119889lowast
)
120579119879
(119896) ℎ119879
(119909 (119896))
minus1
sum
minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119891119879
]
(33)
If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that
119864Δ119881 (119896) le minus120576119864119909 (119896)2
+ 1205881199032
infin (34)
It is easy to derive that
119864119881 (119896) le 1205831119864119909 (119896)
2
+ 1205832
119896minus1
sum
119894=119896minus1205912
119864119909 (119894)2
+ 1205833
119896minus1
sum
119894=119896minus1198892
119864119909 (119894)2
(35)
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
From (20) (22) and (23) we have
119864Δ1198815(119896) = 119864 [119909
119879
(119896) 1198765119909 (119896) minus 120579
119879
(119896) 1198765120579 (119896)
+ 1205912
120578119879
(119896) 1198852120578 (119896) + (
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
119879
times (
minus211988521198852
1198852
lowast minus1198852
0
lowast lowast minus1198852
)(
119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)
120579 (119896)
)
+ 120573(119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
))
119879
(minus11988521198852
lowast minus1198852
)
times (119909 (119896 minus 120591 (119896))
119909 (119896 minus 120591lowast
)) + (1 minus 120573)
times(119909 (119896 minus 120591 (119896))
120579 (119896))
119879
(minus11988521198852
lowast minus1198852
)
times(119909 (119896 minus 120591 (119896))
120579 (119896))]
(24)
When 1198891le 119889(119896) le 119889
lowast by Lemma 6 it is easy to get
119864Δ1198816(119896) = 119864[
[
(
minus1
sum
119894=minus119889(119896+1)
119896
sum
119895=119896+119894+1
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894
)
times ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
(
119896
sum
119897=119896+119895+1
minus
119896minus1
sum
119897=119896+119895
)
times ℎ119879
(119909 (119897)) 119877ℎ (119909 (119897)) ]
]
le 119864[
[
minus1
sum
119894=minus119889lowast
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
minus
minus1
sum
119894=minus119889(119896)
119896minus1
sum
119895=119896+119894+1
ℎ119879
(119909 (119895)) 119877ℎ (119909 (119895))
+
minus1
sum
119894=minus119889lowast
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1
sum
119894=minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119877ℎ (119909 (119896 + 119894))
+
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus
minus1198891minus1
sum
119894=minus119889lowast
minus1
sum
119895=119894+1
ℎ119879
(119909 (119896 + 119895)) 119877ℎ (119909 (119896 + 119895))]
]
le 119864[(119889lowast
+ 1198891minus 1) (119889
lowast
minus 1198891) + 2119889
lowast
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119889lowast(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(25)
When 119889lowast lt 119889(119896) le 1198892 similarly we can have
119864Δ1198816(119896) le 119864[
(119889lowast
+ 1198892minus 1) (119889
1minus 119889lowast
) + 21198892
2
times ℎ119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
1198892
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(26)
From (25) and (26) we have
119864Δ1198816(119896) le 119864[119887ℎ
119879
(119909 (119896)) 119877ℎ (119909 (119896))
minus1
119888(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
times 119877(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))]
(27)
From (7) it follows that
(ℎ119894(119909 (119896)) minus 120574
+
119894119909119894(119896))
times (ℎ119894(119909 (119896)) minus 120574
minus
119894119909119894(119896)) le 0 119894 = 1 2 119899
(28)
which are equivalent to
(119909 (119896)
ℎ (119909 (119896)))
119879
(120574minus
119894120574+
119894119890119894119890119879
119894minus120574minus
119894+ 120574+
119894
2119890119894119890119879
119894
lowast 119890119894119890119879
119894
)
times(119909 (119896)
ℎ (119909 (119896))) le 0
(29)
where 119890119894denotes the unit column vector having one element
on its 119894th row and zeros elsewhereThen from (29) for any matrices Λ = diag120582
1 1205822
120582119899 gt 0 it follows that
(119909 (119896)
ℎ (119909 (119896)))
119879
(minusΓ1Λ Γ2Λ
lowast minusΛ)(
119909 (119896)
ℎ (119909 (119896))) ge 0 (30)
Abstract and Applied Analysis 7
Note that by Lemma 7 we get
119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896) + 120591lowast2
120578119879
(119896) 1198851120578 (119896)
+120591120578119879
(119896) 1198852120578 (119896)]
= 119864 [120578119879
(119896) (119875 + 120591lowast2
1198851+ 1205911198852) 120578 (119896) + 2120578
119879
(119896) 119875119909 (119896)]
= 119864 [119909119879
(119896 + 1) (119875 + 120591lowast2
1198851+ 1205911198852) 119909 (119896 + 1)
minus 2119909119879
(119896 + 1) (120591lowast2
1198851+ 1205911198852) 119909 (119896)
+ 119909119879
(119896) (120591lowast2
1198851+ 1205911198852minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) (119875 + 2120591lowast2
1198851+ 2120591119885
2) 119909 (119896 + 1)
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) 120582lowast
119868119909 (119896 + 1) + 119909119879
(119896)
times (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879
times 120582lowast
119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
times 120582lowast
119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]
119879
120582lowast
119868
times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
120582lowast
119868
times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
+ 120582lowast
119909119879
(119860 + 119862119870)119879
(119860 + 119862119870) 119909 (119896)
+ 120582lowast
119909119879
(119896 minus 120591 (119896)) 119861119879
119861119909 (119896 minus 120591 (119896))
+ 120582lowast
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
119863119879
119863
times (
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
+5120582lowast
1198622
1199032
infin
(31)
Then from (18) to (31) we have
119864Δ119881 (119896) le 119864120585119879
(119896) [Ξ1015840
+ (radic2119883 0 0 0 0 0 0 0)119879 1
120582lowast
times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)
+ 1205881199032
infin
(32)
where
Ξ1015840
119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582
lowast
1198632
120585119879
(119896) = [119909119879
(119896) 119909119879
(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast
) 119909 (119896 minus 119889lowast
)
120579119879
(119896) ℎ119879
(119909 (119896))
minus1
sum
minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119891119879
]
(33)
If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that
119864Δ119881 (119896) le minus120576119864119909 (119896)2
+ 1205881199032
infin (34)
It is easy to derive that
119864119881 (119896) le 1205831119864119909 (119896)
2
+ 1205832
119896minus1
sum
119894=119896minus1205912
119864119909 (119894)2
+ 1205833
119896minus1
sum
119894=119896minus1198892
119864119909 (119894)2
(35)
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
Note that by Lemma 7 we get
119864 [120578119879
(119896) 119875120578 (119896) + 2120578119879
(119896) 119875119909 (119896) + 120591lowast2
120578119879
(119896) 1198851120578 (119896)
+120591120578119879
(119896) 1198852120578 (119896)]
= 119864 [120578119879
(119896) (119875 + 120591lowast2
1198851+ 1205911198852) 120578 (119896) + 2120578
119879
(119896) 119875119909 (119896)]
= 119864 [119909119879
(119896 + 1) (119875 + 120591lowast2
1198851+ 1205911198852) 119909 (119896 + 1)
minus 2119909119879
(119896 + 1) (120591lowast2
1198851+ 1205911198852) 119909 (119896)
+ 119909119879
(119896) (120591lowast2
1198851+ 1205911198852minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) (119875 + 2120591lowast2
1198851+ 2120591119885
2) 119909 (119896 + 1)
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
le 119864 [119909119879
(119896 + 1) 120582lowast
119868119909 (119896 + 1) + 119909119879
(119896)
times (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)]
= 119864 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]119879
times 120582lowast
119868 [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) + 119862119903 (119896) ]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
times 120582lowast
119868 [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
le 119864[ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+ 119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894))
+119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896))) ]
119879
120582lowast
119868
times [ (119860 + 119862119870) 119909 (119896) + 119861119909 (119896 minus 120591 (119896))
+119863
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)) + 119891 (119896 119909 (119896) 119909 (119896 minus 120591 (119896)))]
+ [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]119879
120582lowast
119868
times [119866119909 (119896) + 119867119909 (119896 minus 120591 (119896))]
+ 119909119879
(119896) (2120591lowast2
1198851+ 2120591119885
2minus 119875) 119909 (119896)
+ 120582lowast
119909119879
(119860 + 119862119870)119879
(119860 + 119862119870) 119909 (119896)
+ 120582lowast
119909119879
(119896 minus 120591 (119896)) 119861119879
119861119909 (119896 minus 120591 (119896))
+ 120582lowast
(
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
119879
119863119879
119863
times (
minus1
sum
119894=minus119889(119896)
ℎ (119909 (119896 + 119894)))
+5120582lowast
1198622
1199032
infin
(31)
Then from (18) to (31) we have
119864Δ119881 (119896) le 119864120585119879
(119896) [Ξ1015840
+ (radic2119883 0 0 0 0 0 0 0)119879 1
120582lowast
times (radic2119883 0 0 0 0 0 0 0) ] 120585 (119896)
+ 1205881199032
infin
(32)
where
Ξ1015840
119894119895= Ξ119894119895 119894 119895 = 1 2 8 120588 = 5120582
lowast
1198632
120585119879
(119896) = [119909119879
(119896) 119909119879
(119896 minus 120591 (119896)) 119909 (119896 minus 120591lowast
) 119909 (119896 minus 119889lowast
)
120579119879
(119896) ℎ119879
(119909 (119896))
minus1
sum
minus119889(119896)
ℎ119879
(119909 (119896 + 119894)) 119891119879
]
(33)
If the LMI (14) holds by using Lemma 8 it follows that thereexists a sufficient small positive 120576 gt 0 such that
119864Δ119881 (119896) le minus120576119864119909 (119896)2
+ 1205881199032
infin (34)
It is easy to derive that
119864119881 (119896) le 1205831119864119909 (119896)
2
+ 1205832
119896minus1
sum
119894=119896minus1205912
119864119909 (119894)2
+ 1205833
119896minus1
sum
119894=119896minus1198892
119864119909 (119894)2
(35)
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
with
1205831= 120582max (119875)
1205832= 120582max (1198761) + (119886 + 1) 120582max (1198763)
+ 2120582max (1198764) + 4120591lowast2
120582max (1198851)
+ 4(1205912minus 1205911)2
120582max (1198852)
1205833= 120582max (1198762) + [1198892 + (1198892 minus 1198891)
times (1198892minus 1198891)]
times Γ2
120582max (119877)
(36)
For any 120579 gt 1 it follows from (34) and (35) that
119864 [120579119895+1
119881 (119895 + 1) minus 120579119895
119881 (119895)]
= 120579119895+1
119864Δ119881 (119895) + 120579119895
(120579 minus 1) 119864119881 (119895)
le 120579119895 [
[
(minus120576120579 + (120579 minus 1) 1205831) 1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ (120579 minus 1) 1205832
119895minus1
sum
119894=119895minus1205912
119864119909 (119894)2
+ 1205881205791199032
infin
+ (120579 minus 1) 1205833
119895minus1
sum
119894=119895minus1198892
119864119909 (119894)2]
]
(37)
Summing up both sides of (37) from 0 to 119896minus1 we can obtain
120579119896
119864119881 (119896) minus 119864119881 (0)
le (1205831(120579 minus 1) minus 120576120579)
119896minus1
sum
119895=0
120579119895
1198641003817100381710038171003817119909 (119895)
1003817100381710038171003817
2
+ 1205832(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
+ 1205833(120579 minus 1)
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
(38)
Also it is easy to compute that
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1205912
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1205912
119894+1205912
sum
119895=0
+
119896minus1minus1205912
sum
119894=0
119894+1205912
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1205912
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 12059121205791205912 sup119904isin[minus120591
20]
119864119909 (119904)2
+ 12059121205791205912
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
119896minus1
sum
119895=0
119895minus1
sum
119894=119895minus1198892
120579119895
119864119909 (119894)2
le (
minus1
sum
119894=minus1198892
119894+1198892
sum
119895=0
+
119896minus1minus1198892
sum
119894=0
119894+1198892
sum
119895=119894+1
+
119896minus1
sum
119894=119896minus1198892
119896minus1
sum
119895=119894+1
)
times 120579119895
119864119909 (119894)2
le 11988921205791198892 sup119904isin[minus1198892 0]
119864119909 (119904)2
+ 11988921205791198892
times
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(39)Substituting (39) into (38) leads to
120579119896
119864119881 (119896) minus 119864119881 (0)
le 1205781(120579) sup119904isin[minus1205912 0]
119864119909 (119904)2
+ 120588
119896minus1
sum
119895=0
120579119895+1
1199032
infin
+ 1205782(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
+ 1205783(120579)
119896minus1
sum
119894=0
120579119894
119864119909 (119894)2
(40)
where 1205781(120579) = 120583
2(120579 minus 1)120591
21205791205912 + 1205833(120579 minus 1)119889
21205791198892 1205782(120579) = 120583
2(120579 minus
1)12059121205791205912 +1205831(120579minus1)minus120576120579 120578
3(120579) = 120583
3(120579minus1)119889
21205791198892 +1205831(120579minus1)minus120576120579
Since 1205782(1) lt 0 120578
3(1) lt 0 there must exist a positive
1205790gt 1 such that 120578
2(1205790) lt 0 120578
3(1205790) lt 0 Then we have
119864119881 (119896)
le 1205781(1205790) (
1
1205790
)
119896
sup119904isin[minus12059120]
119864119909 (119904)2
+ (1
1205790
)
119896
119864119881 (0) + 120588
119896minus1
sum
119895=0
1
120579119896minus119895minus1
0
1199032
infin
+ 1205782(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
+ 1205783(1205790)
119896minus1
sum
119894=0
1
120579119896minus119894
0
119864119909 (119894)2
le (1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+120588
1205790minus 1
1199032
infin
(41)On the other hand by (16) we can get
119864119881 (119896) ge 120582min (119875) 119864119909 (119896)2
(42)Combining (41) with (42) we have
119864119909 (119896)2
le1205781(1205790) + 1205831+ 12058321205912+ 12058331198892
120582min (119875)
times sup119904isin[minusmax120591211988920]
119864119909 (119904)2
+1
120582min (119875)
120588
1205790minus 1
1199032
infin
(43)
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
Thus
E1003817100381710038171003817119910 (119896)
1003817100381710038171003817
2
le 1198722
119864119909 (119896)2
le 1198731+ 11987321199032
infin (44)
where 1198731= 119872
2
((1205781(1205790) + 1205831+ 12058321205912+ 12058331198892)120582min(119875))
sup119904isin[minusmax120591
211988920]119864119909(119904)
2 1198732
= (1120582min(119875))(120588(1205790 minus
1))1198722 By Definition 5 the nonlinear discrete-time
stochastic control system (1) is BIBO stability in meansquare This completes the proof
If the stochastic term 120596(119870) is removed in (1) then thefollowing results can be obtained
Theorem 10 For given positive integers 1205911 1205912 1198891 and 119889
2
under Assumption 2 the nonlinear discrete-time stochasticcontrol system (1) with the controller (6) is BIBO stabilizationin mean square if there exist symmetric positive-definitematrix 119875 119877119876
119894 119894 = 1 2 5119885
11198852 and119883with appropriate
dimensional positive-definite diagonal matrices Λ and twopositive constants 120577 and 120582lowast such that the following two LMIshold
119875 + 2 (120591lowast2
1198851+ 1205912
1198852) le 120582lowast
119868
Ξ
=
(((((((((((
(
Ξ11Ξ12
1198851
0 0 Γ2Λ Ξ17
Ξ18
radic2119883
lowast Ξ22Ξ23
0 Ξ25
0 119861119879
119863 119861119879
0
lowast lowast Ξ33
0 0 0 0 0 0
lowast lowast lowast minus1198762
0 0 0 0 0
lowast lowast lowast lowast Ξ55
0 0 0 0
lowast lowast lowast lowast lowast Ξ66
0 0 0
lowast lowast lowast lowast lowast lowast Ξ77
120582lowast
119863119879
0
lowast lowast lowast lowast lowast lowast lowast minus120577119868 0
lowast lowast lowast lowast lowast lowast lowast lowast minus120582lowast
119868
)))))))))))
)
le 0
(45)
where
Ξ11= 1198761+ 1198762+ 1198763+ 119886 (119876
3+ 1198764) minus 1198851
+ 1198765minus Γ1Λ + 120577119868 + 2120591
lowast2
1198851+ 21205912
1198852
+ 2120582lowast
1205721119868 + 1205721120577119868 minus 119875 + 2120582
lowast
119860119879
119860
minus 2119860119879
119883 minus 2119883119879
119860
Ξ22= 2120582lowast
119861119879
119861 minus 1198764minus 21198852minus 1205731198852
+ 2120582lowast
1205722119868 + 1205722120577119868 minus (1 minus 120573)119885
2
Ξ12= 120582lowast
119860119879
119861 minus 119883119879
119861
Ξ17= 120582lowast
119860119879
119863 minus 119883119879
119863
Ξ18= 120582lowast
119860119879
minus 119883119879
Ξ25= 1198852+ (1 minus 120573)119885
2
Ξ23= 1198852+ 1205731198852
Ξ33= minus1198761minus 1198851minus 1198852minus 1205731198852
Ξ55= minus1198765minus 1198852minus (1 minus 120573)119885
2
Ξ66= 119887119877 minus Λ
Ξ77= 2120582lowast
119863119879
119863 minus1
119888119877
119883 = minus120582lowast
119862119870
(46)
Proof The proof is straightforward and hence omitted
Corollary 11 System (1) is also stabilization in mean squarewhen all the conditions in Theorems 9 and 10 are satisfied ifthe bounded input 119903(119905) = 0 in (6)
Remark 12 In this paper a novel BIBO stability criterion forsystem (1) is derived by checking the variation of derivativesof the Lyapunov-Krasovskii functionals for each subintervalIt is different from [27ndash32] which checked the variation ofthe Lyapunov functional in the whole variation interval of thedelay
Remark 13 The BIBO stabilization criteria for discrete-timesystems have been investigated in the recently reported paper[28] However the stochastic disturbances and nonlinearperturbations have not been taken into account in the controlsystems In [28] the time delay is constant time which is aspecial case of this paper when 120591
1= 1205912
Remark 14 The mean square stabilization conditions inTheorem 9 in this paper depend on the time-delays upperbounds and the lower bounds time-delays interval and time-delay interval segmentation point and relate to the delaysthemselves
Remark 15 In [33] the time-delay interval is divided intotwo equal subintervals the interval segmentation point ismidpoint In this paper the time-delay interval is divided intotwo any subintervals the interval segmentation point is anypoint in the time-delay interval
4 An Example
In this section a numerical example will be presented to showthe validity of the main results derived in Section 3
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
Table 1 For 120591(119896) = 1 2 3 4 5 120573
120591(119896) 1 2 3 4 5
120573 0 1 12 12 0
Example 1 As a simple application of Theorem 9 considerthe stochastic control system (1) with the control law (6) theparameters are given by
119860 = (minus01 0
01 minus02) 119861 = (
minus01 01
minus01 01)
119862 = (01 01
05 03) 119863 = (
01 01
0 02)
(47)
119866 = 0001119868 119867 = 002119868 119891 = [01119909(119896) radic02119909(119896 minus 120591(119896))]119879
ℎ1(119904) = sin(02119904) minus 06 cos(119904) ℎ
2(119904) = tanh(minus04119904) 120591
1= 1
1205912= 5 119889
1= 2 119889
2= 7
It is easy to verify that 119886 = 5 120591lowast = 3 119889lowast = 4 120591 = 2 and
Γ1= (
minus064 0
0 0) Γ
2= (
0 0
0 minus02)
120591lowast
=1205911+ 1205912
2minusmin (minus1)1205911+1205912 0
2
119889lowast
=1205911+ 1205912
2+min (minus1)1205911+1205912 0
2
119887 = 9 119889
1le 119889 (119896) le 119889
lowast
20 119889lowast
lt 119889 (119896) le 1198892
119888 = 4 119889
1le 119889 (119896) le 119889
lowast
7 119889lowast
lt 119889 (119896) le 1198892
(48)
Meanwhile the corresponding values of120573 for various 120591(119896)are listed in Table 1
By using the MATLAB LMI Toolbox we solve LMIs (13)(14) and obtain six groups of feasible solutions we list onecase as follows
When 120573 = 1 119887 = 9 119888 = 4
119875 = (1450684 minus32651
minus32651 1477799) 119885
1= (
00478 00298
00298 00230)
1198852= (
03443 02142
02142 01663) 119876
1= (
28401 17808
17808 13624)
1198762= (
37464 23462
23462 17985) 119876
3= (
02731 01713
01713 01309)
1198764= (
153027 minus68645
minus68645 155746) 119876
5= (
31500 19740
19740 15118)
119877 = (1282646 minus921100
minus921100 2390910) 119873 = (
04512 0
0 343341)
119883 = (133495 31121
31121 07692) 119870 = (
minus14212 minus03303
05268 01218)
(49)
and 120577 = 820259 120582lowast = 1502996The system (1) exhibits stabilization in mean square
behavior as shown in Figure 1
0 20 40 60 80 100 1200
20
40
60
k
0 20 40 60 80 100 120minus05
0
05
1
Varia
tion
of120591k
Figure 1
5 Conclusions
In this paper we have derived some conditions for theBIBO stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varyingdelaysThe results have been obtained by constructing a novelLyapunov-Krasovskii function The conditions are expressedin the forms of linear matrix inequalities which can beeasily checked by usingMATLAB LMI Toolbox A numericalexample is given to illustrate the validity of the obtainedresults
Acknowledgments
The work of Xia Zhou is supported by the National NaturalScience Foundation of China (no 11226140) and the AnhuiProvincial Colleges and Universities Natural Science Foun-dation (no KJ2013Z267) The work of Yong Ren is supportedby the National Natural Science Foundation of China (nos10901003 and 11126238) the Distinguished Young Scholars ofAnhui Province (no 1108085J08) the Key Project of ChineseMinistry of Education (no 211077) and the Anhui ProvincialNatural Science Foundation (no 10040606Q30)
References
[1] R Sakthivel K Mathiyalagan and S M Anthoni ldquoRobuststability and control for uncertain neutral time delay systemsrdquoInternational Journal of Control vol 85 no 4 pp 373ndash383 2012
[2] S Lakshmanan JH ParkDH JiH Y Jung andGNagamanildquoState estimation of neural networks with time-varying delaysand Markovian jumping parameter based on passivity theoryrdquoNonlinear Dynamics vol 70 no 2 pp 1421ndash1434 2012
[3] R Sakthivel KMathiyalagan and SMarshal Anthoni ldquoRobust119867infin
control for uncertain discrete-time stochastic neural net-works with time-varying delaysrdquo IET Control Theory amp Appli-cations vol 6 no 9 pp 1220ndash1228 2012
[4] P Balasubramaniam S Lakshmanan and A ManivannanldquoRobust stability analysis for Markovian jumping interval neu-ral networks with discrete and distributed time-varying delaysrdquoChaos Solitons and Fractals vol 45 no 4 pp 483ndash495 2012
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
[5] K Mathiyalagan R Sakthivel and S Marshal Anthoni ldquoExpo-nential stability result for discrete-time stochastic fuzzy uncer-tain neural networksrdquo Physics Letters A vol 376 no 8-9 pp901ndash912 2012
[6] A Arunkumar R Sakthivel K Mathiyalagan and S MarshalAnthoni ldquoRobust stability criteria for discrete-time switchedneural networks with various activation functionsrdquo AppliedMathematics andComputation vol 218 no 22 pp 10803ndash108162012
[7] S Lakshmanan and P Balasubramaniam ldquoNew results of robuststability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parametersrdquo CanadianJournal of Physics vol 89 no 8 pp 827ndash840 2011
[8] P Vadivel R Sakthivel K Mathiyalagan and P ThangarajldquoRobust stabilization of nonlinear uncertain Takagi-Sugenofuzzy systems by H controlrdquo IET Control Theory and Applica-tions vol 6 pp 2556ndash2566 2012
[9] Z Liu S Lu S Zhong andMYe ldquoImproved robust stability cri-teria of uncertain neutral systems with mixed delaysrdquo Abstractand Applied Analysis Article ID 294845 18 pages 2009
[10] F Qiu B Cui and Y Ji ldquoDelay-dividing approach for absolutestability of Lurie control system with mixed delaysrdquo NonlinearAnalysis Real World Applications vol 11 no 4 pp 3110ndash31202010
[11] X Wang Q Guo and D Xu ldquoExponential 119901-stability ofimpulsive stochastic Cohen-Grossberg neural networks withmixed delaysrdquo Mathematics and Computers in Simulation vol79 no 5 pp 1698ndash1710 2009
[12] W Zhou H Lu and C Duan ldquoExponential stability of hybridstochastic neural networks with mixed time delays and nonlin-earityrdquo Neurocomputing vol 72 no 13ndash15 pp 3357ndash3365 2009
[13] Y Zhang S Xu and Z Zeng ldquoNovel robust stability criteriaof discrete-time stochastic recurrent neural networks with timedelayrdquo Neurocomputing vol 72 no 13ndash15 pp 3343ndash3351 2009
[14] C Peng and Y-C Tian ldquoDelay-dependent robust stabilitycriteria for uncertain systems with interval time-varying delayrdquoJournal of Computational and AppliedMathematics vol 214 no2 pp 480ndash494 2008
[15] ZWu H Su J Chu andW Zhou ldquoImproved result on stabilityanalysis of discrete stochastic neural networks with time delayrdquoPhysics Letters A vol 373 no 17 pp 1546ndash1552 2009
[16] R Sakthivel S Santra and K Mathiyalagan ldquoAdmissibilityanalysis and control synthesis for descriptor systems with ran-dom abrupt changesrdquo Applied Mathematics and Computationvol 219 no 18 pp 9717ndash9730 2013
[17] Y Liu Z Wang and X Liu ldquoState estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delaysrdquo Physics Letters A vol 372 no 48 pp 7147ndash7155 2008
[18] Y Tang J-A Fang M Xia and D Yu ldquoDelay-distribution-dependent stability of stochastic discrete-time neural networkswith randomly mixed time-varying delaysrdquo Neurocomputingvol 72 no 16ndash18 pp 3830ndash3838 2009
[19] Y Liu ZWang and X Liu ldquoAsymptotic stability for neural net-works with mixed time-delays the discrete-time caserdquo NeuralNetworks vol 22 no 1 pp 67ndash74 2009
[20] P Li and S-m Zhong ldquoBIBO stabilization for system withmultiple mixed delays and nonlinear perturbationsrdquo AppliedMathematics andComputation vol 196 no 1 pp 207ndash213 2008
[21] T Bose and M Q Chen ldquoBIBO stability of the discrete bilinearsystemrdquoDigital Signal Processing vol 5 no 3 pp 160ndash166 1995
[22] SM Zhong andYQHUang ldquoBIBO stabilization of nonlinearsystem with time-delayrdquo Journal of University of ElectronicScience and Technology of China vol 32 no 4 pp 655ndash6572000
[23] K C Cao S M Zhong and B S Liu ldquoBIBO and robuststabilization for system with time-delay and nonlinear per-turbationsrdquo Journal of University of Electronic Science andTechnology of China vol 32 no 6 pp 787ndash789 2003
[24] J R Partington and C Bonnet ldquo119867infinand BIBO stabilization of
delay systems of neutral typerdquo SystemsampControl Letters vol 52no 3-4 pp 283ndash288 2004
[25] A T Tomerlin and W W Edmonson ldquoBIBO stability of119889-dimensional filtersrdquo Multidimensional Systems and SignalProcessing vol 13 no 3 pp 333ndash340 2002
[26] Y Q Huang W Zeng and S M Zhong ldquoBIBO stabukuty ofcontinuous time systemsrdquo Journal of University of ElectronicScience and Technology of China vol 3 no 2 pp 178ndash181 2005
[27] P Li and S-m Zhong ldquoBIBO stabilization of time-delayedsystem with nonlinear perturbationrdquo Applied Mathematics andComputation vol 195 no 1 pp 264ndash269 2008
[28] Z X Liu S Lv and S M Zhong ldquoAugmented Lyapunovmethod for BIBO stabilization of discrete systemrdquo Journal ofMathematics Research vol 2 pp 116ndash122 2011
[29] P Li and S-M Zhong ldquoBIBO stabilization of piecewiseswitched linear systems with delays and nonlinear perturba-tionsrdquo Applied Mathematics and Computation vol 213 no 2pp 405ndash410 2009
[30] P Li S-m Zhong and J-z Cui ldquoDelay-dependent robust BIBOstabilization of uncertain system via LMI approachrdquo ChaosSolitons and Fractals vol 40 no 2 pp 1021ndash1028 2009
[31] Y Fu and X Liao ldquoBIBO stabilization of stochastic delaysystems with uncertaintyrdquo Institute of Electrical and ElectronicsEngineers vol 48 no 1 pp 133ndash138 2003
[32] X Zhou and S Zhong ldquoRiccati equations and delay-dependentBIBO stabilization of stochastic systems with mixed delaysand nonlinear perturbationsrdquoAdvances in Difference EquationsArticle ID 494607 14 pages 2010
[33] D Yue ldquoRobust stabilization of uncertain systems withunknown input delayrdquo Automatica vol 40 no 2 pp 331ndash3362004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of