research article synchronization of discontinuous neural...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 147164 9 pageshttpdxdoiorg1011552013147164

Research ArticleSynchronization of Discontinuous Neural Networks with Delaysvia Adaptive Control

Xinsong Yang1 and Jinde Cao23

1 Department of Mathematics Chongqing Normal University Chongqing 401331 China2Department of Mathematics Southeast University Nanjing 210096 China3Department of Mathematics King Abdulaziz University Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Jinde Cao jdcaoseueducn

Received 29 September 2012 Accepted 23 January 2013

Academic Editor Sridhar Seshagiri

Copyright copy 2013 X Yang and J Cao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The drive-response synchronization of delayed neural networks with discontinuous activation functions is investigated via adaptivecontrolThe synchronization of this paper means that the synchronization error approaches to zero for almost all time as time goesto infinity The discontinuous activation functions are assumed to be monotone increasing which can be unbounded Due to themild condition on the discontinuous activations adaptive control technique is utilized to control the response system Under theframework of Filippov solution by using Lyapunov function and chain rule of differential inclusion rigorous proofs are given toshow that adaptive control can realize complete synchronization of the consideredmodelThe results of this paper are also applicableto continuous neural networks since continuous function is a special case of discontinuous function Numerical simulations verifythe effectiveness of the theoretical results Moreover when there are parameter mismatches between drive and response neuralnetworks with discontinuous activations numerical example is also presented to demonstrate the complete synchronization byusing discontinuous adaptive control

1 Introduction

In the last decades synchronization of coupled chaoticsystems (including fractional-order chaotic systems andinteger-order chaotic systems [1ndash3]) has received increasingresearch attention from different branches of science andapplication fields due to its potential applications such assecure communication biological systems and informationscience [4 5] Along with the presentation of different kindsof synchronization such as complete synchronization [6]lag synchronization [7 8] quasi-synchronization [9 10]projective synchronization [11ndash13] and generalized synchro-nization [14 15] many controlmethods have been developedfor instance state feedback control [9 16] and adaptivecontrol [7 15 17] The adaptive control technique derivesspecial attention since its control gains need not to be knownin advance and can self-adjust according to the designedadaptive law

Delayed neural networks as a class of important func-tional differential equations have witnessed many applica-tions in different areas such as signal processing associa-tive memories classification of patterns and optimizationTherefore investigating dynamical behaviors of neural net-works with various parameters has long been an intensiveresearch topic such as stability of equilibrium point [18]and chaos synchronization [7] However the activationfunctions in most of known models including those in [718] are accompanied by the assumption of continuity oreven Lipschitz continuity Actually neural networks withdiscontinuous neuron activations are ideal models for thecase where the gain of the neuron amplifiers is very high andis frequently arising in the applications [19 20] Thereforein the literature there were some results on dynamicalbehaviors of neural networks with discontinuous activationfunctions For instance Forti et al investigated the stabilityand global convergence of delayed and nondelayed neural

2 Discrete Dynamics in Nature and Society

networks with discontinuous activations in [19 20] authorsin [21ndash23] studied the global robust stability of delayedneural networks with discontinuous neuron activations in[24 25] authors considered existence and global convergenceof periodic and almost periodic solutions of neural networkswith discontinuous activations

On the other hand although there were many resultsconcerning synchronization of chaotic neural networks fewpublished papers considered the same issue for neural net-works with discontinuous activations and we only found[10 26] The difficulty comes from the discontinuity ofactivations The methods utilized to analyze the stabilityof neural networks with discontinuous activations cannotbe extended to chaos synchronization case directly In [10]authors investigated quasi-synchronization of discontinuousneural networks with and without parameter mismatchesthat is synchronization error can only be controlled to a smallregion around zero but cannot approach to zero Resultsof [10] revealed that complete synchronization is difficult tobe realized due to the discontinuity of activation functionIt is known that one of the most important applications ofchaos synchronization is in secure communication Whenchaos synchronization is applied to secure communicationonly when the drive and response systems achieve completesynchronization can the transmitted signal be fingered outTherefore it is necessary to study complete synchroniza-tion of neural networks with discontinuous activations In[26] complete synchronization of discontinuous neural net-works was investigated via approximation and linear matrixinequality (LMI) approach But we find that through theapproximation approach used in [26] the control gain isuncertain and may be very large which leads to inappli-cablility in practice On the other hand some results onsynchronization and control of discontinuous dynamicalsystems are complex which are difficult to be verified Forinstance synchronization criteria obtained in [27] were interms of integral inequality and the restrict condition onthe discontinuity of discontinuous function was weakenedthat is as time goes to infinity the discontinuous functionapproaches to a continuous function From the above analy-sis investigating the synchronization of neural networks withdiscontinuous activations is really a tough task

Being motivated by the above analysis this paper inves-tigates asymptotic complete synchronization of neural net-works with discontinuous activation functions via adaptivecontrol technique Because of the discontinuity of the acti-vation functions the solution is in the sense of differentialinclusion by the Filippov theory [28] and the complete syn-chronization of this paper means that the state error betweenthe derive (or master) and response systems approaches tozero for almost all (aa) time as time goes to infinity We donot impose the restriction conditions of growth condition onactivation function The discontinuous activations are onlyassumed to be monotone increasing and can be unboundedDue to the mild condition on the discontinuous activationsthe precise control gain is difficult to be determined andthe state feedback control is not so good as the adaptivecontrol technique Under the framework of Filippov solutionby using Lyapunov function and chain rule of differential

inclusion rigorous proofs are given for the asymptotic sta-bility of the error system of the coupled systems Numericalsimulations show the effectiveness of the theoretical resultsMoreover when there are parameter mismatches betweendriver and response neural networks with discontinuous acti-vations numerical example is presented to demonstrate thecomplete synchronization by using discontinuous adaptivecontrol

Notations In the sequel if not explicitly stated matricesare assumed to have compatible dimensions 119868

119898stands for

the identity matrix of 119898-dimension R is the space of realnumber The Euclidean norm in R119898 is denoted as sdot accordingly for vector 119909 isin R119898 119909 = radic119909119879119909 where 119879denotes transposition 119909 = 0 represents each component of119909 is zero 119860 = (119886

119894119895)119898times119898

denotes a matrix of 119898-dimension

119860 = radic120582max(119860119879119860)

The rest of this paper is organized as follows In Section 2a model of delayed neural networks with discontinuousactivation functions is described Some necessary assump-tions definitions and lemmas are also given in this sectionOur main results and their rigorous proofs are described inSection 3 In Section 4 two examples with their numericalsimulations are offered to show the effectiveness of ourresults In Section 5 conclusions are given and at lastacknowledgments

2 Preliminaries

In this paper we consider the delayed neural network whichis described as follows

(119905) = minus119862119909 (119905) + 119860119891 (119909 (119905)) + 119861119891 (119909 (119905 minus 120579)) + 119869 (1)

where 119909(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905)) isin R119899 is the state

vector 119862 = diag(1198881 1198882 119888

119899) in which c

119894gt 0 119894 =

1 2 119899 are the neuron self-inhibitions 120579 gt 0 is thetransmission delay 119860 = (119886

119894119895)119899times119899

and 119861 = (119887119894119895)119899times119899

arethe connection weight matrix and the delayed connectionweight matrix respectively the activation function119891(119909(119905)) =(1198911(1199091(119905)) 119891

2(1199092(119905)) 119891

119899(119909119899(119905)))119879 represents the output of

the network 119869 = (1198691 1198692 119869

119899) is the external input vector

As for neural networks (1) we give the following assump-tion condition

Assumption 1 For every 119894 = 1 2 119899 119891119894 R rarr R is

monotone nondecreasing and has at most a finite number ofjump discontinuities in every compact interval

Remark 2 Assumption 1 was used in [20] Any functionsatisfying this assumption condition does not need to becontinuously differentiable in compact interval However thecontinuous differentiability is necessary in [10 26] On theother hand the ldquomonotone nondecreasingrdquo can be replacedby ldquomonotonic functionrdquo For instance if we replace the

Discrete Dynamics in Nature and Society 3

matrices 119860 and 119861 in Example 13 (see Section 4 of this paper)with

119860 = (2 01

minus5 minus45) 119861 = (

minus15 01

minus02 4) (2)

1198911(1199091) is the same function as that in example and

1198912(1199092) =

minus tanh (1199092) minus 002radic1199092 minus 003 1199092 gt 0

minus tanh (1199092) minus 0018119909

2+ 003 119909

2lt 0

(3)

Obviously the 1198911is monotone increasing and the 119891

2is

monotone decreasing but the obtained system and theoriginal system are identical Furthermore the results of thispaper are also applicable to neural networks with continuousmonotone activation functions since continuous function isa special case of discontinuous function

Since119891(119909) is discontinuous at isolate jumping points onecannot define a solution in the conventional senseThereforewe resort to the notion of Filippov solution and stabilityresults on differential inclusion [28] Filippov solution is oneof notions to deal with the discontinuity that determinethe solution on the discontinuous surface with a set-valuedmapping

The Filippov set-valued map of 119891(119909) at 119909 isin R119899 is definedas follows [28]

119865 (119909) = ⋂

120575gt0

⋂

120583(119873)=0

119870[119891 (119861 (119909 120575) 119873)] (4)

where 119870[119864] is the closure of the convex hull of the set 119864119861(119909 120575) = 119910 119910minus119909 le 120575 and 120583(119873) is the Lebesguemeasureof set119873

When 119891(119909) satisfies Assumption 1 it is not difficult to getfrom (4) that

119865 (119909) = 119870 [119891 (119909)]

= (119870 [1198911(1199091)] 119870 [119891

2(1199092)] 119870 [119891

119899(1199091)])

(5)

where119870[119891119894(119909119894)] = [119891

minus

119894(119909119894) 119891+

119894(119909119894)] 119894 = 1 2 119899

Definition 3 (see [19]) A function 119909 [minus120579 119879) rarr R119899119879 isin (0 +infin] is a solution (in the sense of Filippov) of thediscontinuous system (1) on [minus120579 119879) if

(i) 119909 is continuous on [minus120579 119879) and absolutely continuouson [0 119879)

(ii) there exists a measurable function 120574 =

(1205741 1205742 120574

119899)119879

[minus120579 119879) rarr R119899 such that120574(119909(119905)) isin 119870[119891(119909(119905))] for almost all (aa) 119905 isin [minus120579 119879)and (119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869

for aa 119905 isin [0 119879) (6)

Note that the measurable function 120574(119909(119905)) is a single-value function which is called the measurable selection of119870[119891(119909(119905))] Any function 120574(119909(119905)) satisfying (6) is called anoutput associated to 119909(119905) In this paper we assume that thetrajectory of the solution 119909(119905) of neural network (1) is chaotic

The next definition is the initial value problem (IVP)associated to system (1)

Definition 4 ((IVP) see [20]) For any continuous function120601 [minus120579 0] rarr R119899 and measurable selection 120595 [minus120579 0] rarrR119899 such that120595(119904) isin 119870[119891(120601(119904))] for aa 119904 isin [minus120579 0] by an initialvalue problem associated to (1) with initial condition (120601 120595)one means the following problem find a couple of functions[119909 120574] [minus120579 119879) rarr R119899 timesR119899 such that 119909 is a solution of (1) on[minus120579 119879) for some 119879 gt 0 120574 is an output associated to 119909 and

(119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869

for aa 119905 isin [0 119879)

120574 (119909 (119905)) isin 119870 [119891 (119909 (119905))] for aa 119905 isin [0 119879)

119909 (119904) = 120601 (119904) forall119904 isin [minus120579 0]

120574 (119909 (119904)) = 120595 (119904) for aa 119904 isin [minus120579 0]

(7)

Lemma 5 (see [20]) Suppose that Assumption 1 is satisfiedThen any IVP for (1) has at least a local solution [119909 120574] definedon [0 119879) for some 119879 isin (0 +infin]

Since chaotic system has strange attractors there existsa bounded region containing all attractors of it such thatevery orbit of the system never leaves them Hence in view ofLemma 5 the solution of (1) is defined on [0 +infin)

Lemma 6 ((Chain rule) see [29]) If 119881(119909) R119899 rarr R isC-regular and 119909(119905) is absolutely continuous on any compactsubinterval of [0 +infin) Then 119909(119905) and 119881(119909(119905)) [0 +infin) rarrR are differentiable for aa 119905 isin [0 +infin) and

119889

119889119905119881 (119909 (119905)) = 120574 (119905) (119905) forall120574 isin 120597119881 (119909 (119905)) (8)

where 120597119881(119909(119905)) is the Clark generalized gradient of 119881 at 119909(119905)

Lemma 7 (see [30 page 174]) Let 119886 le 119887 119886 119887 isin R Assume[119886 119887] sub R 119909(119905) is a measurable function on [119886 119887] If 119909(119905) ismonotone on [119886 119887] then 119909(119905) is differentiable for aa 119905 isin [119886 119887]and

(i) minusinfin lt 119863minus119909(119905) = 119863

minus

119909(119905) = 119863+119909(119905) = 119863

+

119909(119905) lt +infinfor aa 119905 isin [119886 119887]

(ii) 119898119909 119886 lt 119909 lt 119887119863+119891(119909) = plusmninfin = 0where119863

minus119909(119905) = lim

ℎrarr0minus

(119891(119909+ℎ)minus119909(119905))ℎ 119863minus

119909(119905) =

limℎrarr0

minus(119891(119909+ℎ)minus119909(119905))ℎ119863+119909(119905) = lim

ℎrarr0+

(119891(119909+

ℎ) minus 119909(119905))ℎ 119863+

119909(119905) = limℎrarr0

+(119891(119909 + ℎ) minus 119909(119905))ℎ

Consider the neural network model (1) as the driversystem the controlled response system is

(119905) = minus119862119910 (119905) + 119860119891 (119910 (119905)) + 119861119891 (119910 (119905 minus 120579)) + 119869 + 119906 (119905)

(9)

where 119910(119905) = (1199101(119905) 1199102(119905) 119910

119899(119905))119879 is the state of the

response system 119906(119905) = (1199061(119905) 1199062(119905) 119906

119899(119905))119879 is the

controller to be designed and the other parameters are thesame as those defined in system (1)


Definition 8 The neural network (9) with discontinuousactivations is said to be asymptotically synchronized withsystem (1) if for any initial values there holds

lim119905rarr+infin

1003817100381710038171003817119910 (119905) minus 119909 (119905)1003817100381710038171003817 = 0 for aa 119905 isin R (10)

3 Main Results

In this section rigorous mathematical proofs about completesynchronization between systems (9) and (1) under adaptivecontrol are given Remarks are given to specify that underAssumption 1 state feedback control is not applicable

By virtue of the above preparations in order to study thesynchronization issue between (1) and (9) we only need toconsider the same problem of the following systems

(119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869 (11)

(119905) = minus119862119910 (119905) + 119860120574 (119910 (119905)) + 119861120574 (119910 (119905 minus 120579)) + 119869 + 119906 (119905)

(12)

Let 119890(119905) = (1198901(119905) 1198902(119905) 119890

119899(119905))119879

= 119910(119905) minus 119909(119905) Subtract-ing (11) from (12) yields the following error system

119890 (119905) = minus119862119890 (119905) + 119860120573 (119890 (119905)) + 119861120573 (119890 (119905 minus 120579)) + 119906 (119905) (13)

where 120573(119890(119905)) = 120574(119890(119905) + 119909(119905)) minus 120574(119909(119905))Obviously 119890(119905) = 0 is the equilibrium point of the error

system (13) when 119906(119905) = 0 If system (13) realizes globalasymptotical stability at the origin for any given initial con-dition then the global asymptotical synchronization between(11) and (12) (or (1) and (9)) is achieved

Theorem 9 Suppose that Assumption 1 is satisfied Then theneural networks (1) and (9) can achieve global asymptoticalsynchronization under the following adaptive controller

119906119894(119905) = minus119897

119894(119905) 119890119894(119905)

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

(14)

where 120576119894gt 0 is an arbitrary positive constant

Proof Since 119891(119909) satisfies Assumption 1 120574(119909) is a single-valued measurable function satisfying 120574(119909) isin 119870[119891(119909)]Therefore 120574

119894(120585) (119894 = 1 2 119899) are monotone increasing and

measurable functions on R In view of Lemma 7 120574119894(120585) is

differentiable for aa 120585 isin R and there exist positive constants119898119894such that 0 le 1205741015840

119894(120585) le 119898

119894 119894 = 1 2 119899 Consequently for

aa 119909 119910 isin R119899 there holds1003817100381710038171003817120574 (119909) minus 120574 (119910)

1003817100381710038171003817 le 1198981003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (15)

where119898 = max1198981 1198982 119898

119899

Define the following Lyapunov functional candidate

119881 (119905) =1

2119890119879

(119905) 119890 (119905) + 120583int

119905

119905minus120579

119890119879

(119904) 119890 (119904) 119889119904

+

119898

sum

119894=1

1

2120576119894

(119897119894(119905) minus 119896

119894)2

(16)

where 120583 and 119896119894119894 = 1 2 119899 are positive constants to be

determinedThen for aa 119905 isin [0 +infin) computing the derivative

of 119881(119905) along trajectories of error system (13) we get fromLemma 6 and the calculus for differential inclusion in [31]that

(119905)

= 119890119879

(119905) [minus119862119890 (119905) + 119860120573 (119890 (119905)) + 119861120573 (119890 (119905 minus 120579)) minus 119897 (119905) 119890 (119905)]

+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879

(119905 minus 120579) 119890 (119905 minus 120579) +

119898

sum

119894=1

(119897119894(119905) minus 119896

119894) 1198902

119894(119905)

= minus119890119879

(119905) 119862119890 (119905) + 119890119879

(119905) 119860120573 (119890 (119905)) + 119890119879

(119905) 119861120573 (119890 (119905 minus 120579))

+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879

(119905 minus 120579) 119890 (119905 minus 120579) minus 119890119879

(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 1198601003817100381710038171003817120573 (119890 (119905))

1003817100381710038171003817

+ 119890 (119905) 1198611003817100381710038171003817120573 (119890 (119905 minus 120579))

1003817100381710038171003817 + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(17)

where 119897(119905) = diag(1198971(119905) 1198972(119905) 119897

119899(119905))119870 = diag(119896

1 1198962

119896119899)It follows from (15) and (17) that

(119905) le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 119860119898 119890 (119905)

+ 119890 (119905) 119861119898 119890 (119905 minus 120579) + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119898 119860 119890 (119905)2

+1

2119898 119861 119890 (119905)

2

+1

2119898 119861 119890(119905 minus 120579)

2

+ 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(18)

Take 120583 = (12)119898119861 Then one derives from (18) that

(119905) le 119890119879

(119905) (minus119862 + 119898 119860 119868119899+ 119898 119861 119868

119899minus 119870) 119890 (119905) (19)

Take 119896119894= minus119888119894+ 119898119860 + 119898119861 + 1 Then

(119905) le minus119890119879

(119905) 119890 (119905) le 0 (20)

Therefore for aa 119905 isin [0 +infin) we have

lim119905rarr+infin

119890 (119905) = 0 (21)

According to Definition 8 the neural networks (1) and (9)achieve global asymptotical synchronizationMoreover from(16) 119897

119894(119905) 119894 = 1 2 119899 approach to some constants as

119890(119905) rarr 0 This completes the proof


Remark 10 Although for aa 120585 isin R there exist positiveconstants 119898

119894such that |1205741015840

119894(120585)| le 119898

119894 119894 = 1 2 119899 these 119898

119894

are usually unknown because the function 120574119894(120585) is uncertain

Hence in this paper for neural networks with discontinuousactivations using state feedback control to synchronize (1)and (8) is not good since the maximum value of control gaincannot be ascertained However adaptive control techniquecan synchronize this class of neural networks as the controlgains increase according to the adaptive lawsThis is themainreason why we choose adaptive control method to study thesynchronization issue of the considered model

Remark 11 Under Assumption 1 complete synchronizationof neural networks with discontinuous activation functionscan be achieved in this paper However based on the growthcondition used in [32 33] authors in [10] only got thequasi-synchronization criteria of systems (1) and (9) by statefeedback control Therefore results of this paper improvecorresponding parts of those in [10]

Remark 12 The synchronization criteria in this paper aresimple and can be easily verified in practice In [27] newconditions on synchronization of linearly coupled dynamicalnetworks with non-Lipschitz right-hand sides were derivedbut the discontinuous functions were weakened to be weak-QUAD and semi-QUAD which means that the discontin-uous function approaches to a continuous function andthe criteria were expressed in integral inequalities Suchsynchronization criteriamay be not easily verified in practiceespecially in the case that there are countable discontinuitiesfor the discontinuous functions Hence results of this paperimprove those in [27]

4 Numerical Examples

In this section we provide two examples to show that ourtheoretical results obtained above are effective Examplealso show that when the discontinuous neural networkshave parameters mismatches synchronization is still realizedunder the discontinuous adaptive control developed in ourprevious works

Example 13 Consider the delayed neural network model (1)with the following parameters 119909(119905) = (119909

1(119905) 1199092(119905))119879 119869 =

(0 0)119879 120579 = 1 119862 is identity matrix of 2-dimension and

119860 = (2 minus01

minus5 45) 119861 = (

minus15 minus01

minus02 minus4) (22)

the activation function is 119891(119909) = (1198911(1199091) 1198912(1199092)) with

119891119894(119909119894) =

tanh (119909119894) + 002radic119909119894 + 003 119909119894 gt 0 119894 = 1 2

tanh (119909119894) + 0018119909

119894minus 003 119909

119894lt 0 119894 = 1 2

(23)

Figure 1 shows chaotic-like trajectory of (1) with initialcondition 119909(119905) = (04 minus06)119879 119905 isin [minus1 0]

Obviously 119891119894(V) in this example is monotone increasing

and is discontinuous at V = 0 so the activation function

8

6

4

2

0

minus2

minus4

minus6

minus8minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

1199092(119905)

1199091(119905)

Figure 1 Trajectory of system (1) with initial value 119909(119905) =(04 minus06)

119879 119905 isin [minus1 0]

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 5 10 15 20 25 30 35 40

1198902(119905)

1198901(119905)

119890 1(119905)119890 2(119905)

119905

Figure 2 Time response of synchronization error 119890(119905) = 119910(119905)minus119909(119905)

satisfies Assumption 1 It follows fromTheorem 9 that system(9) can synchronize the driver system (1) under the adaptivecontroller (14)

In the numerical simulations we use the forward Eulermethod which was used in [34] to obtain numerical solutionof differential inclusions The parameters in the simulationsare taken as step-length is 001 119910(119905) = (minus02 11)119879 119897

1(119905) =

1198972(119905) = 1 for all 119905 isin [minus1 0] 120576

119894= 005 we get the simulation

results shown in Figures 2 and 3 Figure 2 describes thetrajectories of the error states Figure 3 represents the timeresponse of 119897(119905) = (119897

1(119905) 1198972(119905))119879 Figures 2 and 3 show that

synchronization error approaches to zero quickly as timegoes and the control gain 119897(119905) turns out to be some constantswhen the synchronization has been realized Numericalsimulations verify the theoretical results perfectly


14

135

13

125

12

115

11

105

1

095

090 5 10 15 20 25 30 35 40

1198972(119905)

1198971(119905)

119897 1(119905)119897 2(119905)

119905

Figure 3 Trajectories of control parameters 1198971(119905) and 119897

2(119905)

Remark 14 In Example 13 the activation function119891(119909) is notgeneralized differentiable at 119909 = 0 nor is global LipschitzHowever these two conditions are necessary in [26] Hencethis example demonstrates that results of this paper improvecorresponding results in [26]

When there are parameter mismatches between driveand response systems only quasi-synchronization is realizedif state feedback control technique or continuous adaptivecontrol technique is utilized In our previous research asimple but all-powerful discontinuous adaptive control wasdesigned to synchronize chaotic systems with uncertainperturbationsThe following example is given to demonstratethat the discontinuous adaptive control is also applicable tosynchronize neural networks with discontinuous activationsFor more details of the discontinuous adaptive controller see[6 7 15]Themodels used in the following example are takenfrom example in [10]

Example 15 Consider the delayed neural network model (1)with the activation function 119891(119909) = (119891

1(1199091) 1198912(1199092)) as

119891119894(119909119894) =

tanh (119909119894) + 0025119909

119894+ 0028 119909

119894gt 0 119894 = 1 2

tanh (119909119894) + 0025119909

119894minus 0028 119909

119894lt 0 119894 = 1 2

(24)

the other parameters are the same as those in Example 13Welabel system (1) with such activation as (lowast) The chaotic-liketrajectory of system (lowast) can be seen in Figure 4

The response system with parameter mismatches isassumed to be the same as that in [10] which is describedas follows

(119905) = minus (119905) 119909 (119905) + (119905) 119891 (119909 (119905)) + (119905) 119891 (119909 (119905 minus 120579))

+ (119905) + 119906 (t) (25)

8

6

4

2

0

minus2

minus4

minus6

minus8minus15 minus05 0 05 1 15

1199092(119905)

1199091(119905)

minus1

Figure 4 Trajectory of system (lowast) with initial value 119909(119905) =(04 minus06)

119879 119905 isin [minus1 0]

where

(119905) = (2 minus01

minus5 + 01 cos (119905) 45 )

(119905) = (minus15 minus01

minus02 minus4 + 01 sin (119905))

(119905) = diag (1 1 + 01 cos (119905)) (119905) = 0

(26)

The chaotic-like trajectory of system (25) is shown inFigure 5 which is different from that in Figure 4

According to the analysis in [7 15] the systems (lowast)and (25) can realize synchronization under the followingdiscontinuous adaptive controller

119906119894(119905) = minus119897

119894(119905) 119890119894(119905) minus 120572120573

119894(119905) sign (119890

119894(119905))

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

119894(119905) = 120585

119894

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816 119894 = 1 2 119899

(27)

where 120576119894gt 0 120585119894gt 0 and120572 gt 1 are arbitrary positive constants

In the numerical simulations we still use the EulermethodThe parameters in the simulations are taken as step-length is 0001 119910(119905) = (minus05 15)119879 119897

1(119905) = 119897

2(119905) = 01

1205731(119905) = 120573


119894= 120585119894= 005

119894 = 1 2 120572 = 2 We get the simulation results shown inFigures 6ndash8 Figure 6 describes the trajectories of the errorstates as time involves Figures 7 and 8 represent the timeresponse of 119897(119905) = (119897

1(119905) 1198972(119905))119879 and 120573(119905) = (120573

1(119905) 1205732(119905))119879

Simulations demonstrate that the neural networks with dis-continuous activations and parameters mismatches achievesynchronization by utilizing the discontinuous adaptive con-trol technique

Remark 16 It can be seen from Figures 7 and 8 that thefinal control gains are 119897

119894(119905) lt 09 and 120573

119894(119905) le 051

which are much smaller than 119866 = diag(246 246) In [10]


8

6

4

2

0

minus2

minus4

minus6

minus8minus15 minus1 minus05 0 05 1 15

1199092(119905)

1199091(119905)

Figure 5 Trajectory of system (25) with initial value 119909(119905) =(minus05 15)

119879 and 119906(119905) = 0 119905 isin [minus1 0]

4

3

2

1

0

minus1

minus2

minus3

minus40 5 10 15

1198902(119905)1198901(119905)

119890 1(119905)119890 2(119905)

119905


119906(119905) = minus119866(119910(119905) minus 119909(119905)) was utilized to control system (25)and only quasi-synchronization was achieved This exam-ple demonstrates that the designed discontinuous adaptivecontroller is really useful Since in our previous works suchadaptive controller has been discussed in details we use ithere without any proof

5 Conclusions

In this paper newdefinition of synchronization for discontin-uous dynamical systems is proposed Under this definitionsynchronization of delayed neural networks with discontin-uous activation functions via adaptive control is studied Thediscontinuous activations in the neural networks are assumedto be monotone increasing and can be unbounded By utiliz-ing the framework of Filippov solution Lyapunov function

1198972(119905)

1

09

08

07

06

05

04

03

02

01

00 5 10 15

119905

1198971(119905)

119897 1(119905)119897 2(119905)


2(119905)

0 5 10 15

06

055

05

045

04

035

03

025

02

015

01

1205731(119905)

1205731(119905)1205732(119905)

119905

1205732(119905)


2(119905)

and chain rule of differential inclusion sufficient conditionsguaranteeing the realization of asymptotic complete synchro-nization of the considered model are derived Numericalsimulations verify the effectiveness of the theoretical resultsWhen there are parametermismatches between the drive andresponse neural networks with discontinuous activationsa useful discontinuous adaptive controller can achieve thesame goal Results of this paper are also applicable to neuralnetworks with continuous monotone activation functions

Acknowledgments

This work was jointly supported by the National NaturalScience Foundation of China (NSFC) under Grants nos


61263020 61272530 11072059 the Natural Science Foun-dation of Jiangsu Province of China under Grants noBK2012741 and the Scientific Research Fund of YunnanProvince under Grant no 2010ZC150 the Scientific ResearchFund of Chongqing Normal University under Grants no12XLB031 and no 940115

References

[1] P Zhou and Y X Cao ldquoFunction projective synchronizationbetween fractional-order chaotic systems and integer-orderchaotic systemsrdquo Chinese Physics B vol 19 no 10 Article ID100507 2010

[2] L Pan and J Cao ldquoExponential synchronization for impulsivedynamical networksrdquo Discrete Dynamics in Nature and Societyvol 2012 Article ID 232794 20 pages 2012

[3] P Zhou R Ding and Y Cao ldquoMulti drive-one response syn-chronization for fractional-order chaotic systemsrdquo NonlinearDynamics vol 70 no 2 pp 1263ndash1271 2012

[4] S Sundar and A A Minai ldquoSynchronization of randomly mul-tiplexed chaotic systems with application to communicationrdquoPhysical Review Letters vol 85 no 25 pp 5456ndash5459 2000

[5] S Bowong F M Moukam Kakmeni and H Fotsin ldquoA newadaptive observer-based synchronization scheme for privatecommunicationrdquo Physics Letters Section A vol 355 no 3 pp193ndash201 2006

[6] X Yang J Cao and J Lu ldquoStochastic synchronization ofcomplex networks with nonidentical nodes via hybrid adaptiveand impulsive controlrdquo IEEE Transactions on Circuits andSystems I vol 59 no 2 pp 371ndash384 2012

[7] X Yang J Cao Y Long and W Rui ldquoAdaptive lag synchro-nization for competitive neural networks withmixed delays anduncertain hybrid perturbationsrdquo IEEE Transactions on NeuralNetworks vol 21 no 10 pp 1656ndash1667 2010

[8] E M Shahverdiev S Sivaprakasam and K A Shore ldquoLag syn-chronization in time-delayed systemsrdquo Physics Letters SectionA vol 292 no 6 pp 320ndash324 2002

[9] T Huang C Li W Yu and G Chen ldquoSynchronization ofdelayed chaotic systems with parameter mismatches by usingintermittent linear state feedbackrdquo Nonlinearity vol 22 no 3pp 569ndash584 2009

[10] X Liu T Chen J Cao and W Lu ldquoDissipativity and quasi-synchronization for neural networks with discontinuous activa-tions and parametersrdquo Neural Networks vol 24 pp 1013ndash10212011

[11] P Zhou and W Zhu ldquoFunction projective synchronizationfor fractional-order chaotic systemsrdquo Nonlinear Analysis RealWorld Applications vol 12 no 2 pp 811ndash816 2011

[12] X Wu C Xu J Feng Y Zhao and X Zhou ldquoGeneralizedprojective synchronization between two different neural net-workswithmixed time delaysrdquoDiscreteDynamics inNature andSociety vol 2012 Article ID 153542 19 pages 2012

[13] P Zhou R Ding and Y Cao ldquoHybrid projective synchroniza-tion for two identical fractional-order chaotic systemsrdquoDiscreteDynamics in Nature and Society vol 2012 Article ID 768587 11pages 2012

[14] N F Rulkov M M Sushchik L S Tsimring and H D I Abar-banel ldquoGeneralized synchronization of chaos in directionallycoupled chaotic systemsrdquo Physical Review E vol 51 no 2 pp980ndash994 1995

[15] X Yang Q Zhu and C Huang ldquoGeneralized lag-synchronization of chaotic mix-delayed systems with uncertainparameters and unknown perturbationsrdquo Nonlinear AnalysisReal World Applications vol 12 no 1 pp 93ndash105 2011

[16] B Xin and T Chen ldquoProjective synchronization of 119873-dimensional chaotic fractional-order systems via linear stateerror feedback controlrdquo Discrete Dynamics in Nature andSociety Article ID 191063 10 pages 2012

[17] X Yang and J Cao ldquoAdaptive pinning synchronization of com-plex networkswith stochastic perturbationsrdquoDiscreteDynamicsinNature and Society vol 2010Article ID416182 21 pages 2010

[18] H Hu ldquoOn stability of nonlinear continuous-time neuralnetworks with delayrdquo IEEE Transactions on Neural Networksvol 13 pp 1135ndash1143 2000

[19] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I vol 50 no 11 pp 1421ndash1435 2003

[20] M Forti P Nistri and D Papini ldquoGlobal exponential stabil-ity and global convergence in finite time of delayed neuralnetworks with infinite gainrdquo IEEE Transactions on NeuralNetworks vol 16 no 6 pp 1449ndash1463 2005

[21] Z Guo and L Huang ldquoLMI conditions for global robuststability of delayed neural networks with discontinuous neuronactivationsrdquoAppliedMathematics andComputation vol 215 no3 pp 889ndash900 2009

[22] Y Wang Y Zuo L Huang and C Li ldquoGlobal robust stabilityof delayed neural networks with discontinuous activation func-tionsrdquo IET Control Theory amp Applications vol 2 no 7 pp 543ndash553 2008

[23] Y Zuo Y Wang L Huang Z Wang X Liu and X WuldquoRobust stability criterion for delayed neural networks withdiscontinuous activation functionsrdquo Neural Processing Lettersvol 29 no 1 pp 29ndash44 2009

[24] X Liu and J Cao ldquoOn periodic solutions of neural networksvia differential inclusionsrdquo Neural Networks vol 22 no 4 pp329ndash334 2009

[25] W Lu and T Chen ldquoAlmost periodic dynamics of a classof delayed neural networks with discontinuous activationsrdquoNeural Computation vol 20 no 4 pp 1065ndash1090 2008

[26] X Liu and J Cao ldquoSynchronization control of discontinuousneural networks via approximationrdquo in Proceedings of theChinese Control and Decision Conference (CDC rsquo10) pp 782ndash787 May 2010

[27] B LiuW Lu and T Chen ldquoNew conditions on synchronizationof networks of linearly coupled dynamical systems with non-Lipschitz right-hand sidesrdquo Neural Networks vol 25 pp 5ndash132012

[28] A F Filippov ldquoDifferential equations with discontinuous right-hand sidesrdquo in Mathematics and Its Applications Soviet SeriesKluwer Academic Publishers Boston Mass USA

[29] F H ClarkeOptimization and Nonsmooth Analysis JohnWileyamp Sons New York NY USA 1983

[30] Z Jiang and Z Wu Real Analysis Higher Education PublisherBeijing China 2nd edition 2005

[31] B E Paden and S S Sastry ldquoA calculus for computing Filippovrsquosdifferential inclusion with application to the variable structurecontrol of robot manipulatorsrdquo IEEE Transactions on Circuitsand Systems vol 34 no 1 pp 73ndash82 1987

[32] M-F Danca ldquoSynchronization of switch dynamical systemsrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 12 no 8 pp 1813ndash1826 2002


[33] L Huang J Wang and X Zhou ldquoExistence and global asymp-totic stability of periodic solutions forHopfield neural networkswith discontinuous activationsrdquoNonlinear Analysis Real WorldApplications vol 10 no 3 pp 1651ndash1661 2009

[34] M F Danca ldquoControlling chaos in discontinuous dynamicalsystemsrdquo Chaos Solitons and Fractals vol 22 no 3 pp 605ndash612 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


networks with discontinuous activations in [19 20] authorsin [21ndash23] studied the global robust stability of delayedneural networks with discontinuous neuron activations in[24 25] authors considered existence and global convergenceof periodic and almost periodic solutions of neural networkswith discontinuous activations

On the other hand although there were many resultsconcerning synchronization of chaotic neural networks fewpublished papers considered the same issue for neural net-works with discontinuous activations and we only found[10 26] The difficulty comes from the discontinuity ofactivations The methods utilized to analyze the stabilityof neural networks with discontinuous activations cannotbe extended to chaos synchronization case directly In [10]authors investigated quasi-synchronization of discontinuousneural networks with and without parameter mismatchesthat is synchronization error can only be controlled to a smallregion around zero but cannot approach to zero Resultsof [10] revealed that complete synchronization is difficult tobe realized due to the discontinuity of activation functionIt is known that one of the most important applications ofchaos synchronization is in secure communication Whenchaos synchronization is applied to secure communicationonly when the drive and response systems achieve completesynchronization can the transmitted signal be fingered outTherefore it is necessary to study complete synchroniza-tion of neural networks with discontinuous activations In[26] complete synchronization of discontinuous neural net-works was investigated via approximation and linear matrixinequality (LMI) approach But we find that through theapproximation approach used in [26] the control gain isuncertain and may be very large which leads to inappli-cablility in practice On the other hand some results onsynchronization and control of discontinuous dynamicalsystems are complex which are difficult to be verified Forinstance synchronization criteria obtained in [27] were interms of integral inequality and the restrict condition onthe discontinuity of discontinuous function was weakenedthat is as time goes to infinity the discontinuous functionapproaches to a continuous function From the above analy-sis investigating the synchronization of neural networks withdiscontinuous activations is really a tough task

Being motivated by the above analysis this paper inves-tigates asymptotic complete synchronization of neural net-works with discontinuous activation functions via adaptivecontrol technique Because of the discontinuity of the acti-vation functions the solution is in the sense of differentialinclusion by the Filippov theory [28] and the complete syn-chronization of this paper means that the state error betweenthe derive (or master) and response systems approaches tozero for almost all (aa) time as time goes to infinity We donot impose the restriction conditions of growth condition onactivation function The discontinuous activations are onlyassumed to be monotone increasing and can be unboundedDue to the mild condition on the discontinuous activationsthe precise control gain is difficult to be determined andthe state feedback control is not so good as the adaptivecontrol technique Under the framework of Filippov solutionby using Lyapunov function and chain rule of differential

inclusion rigorous proofs are given for the asymptotic sta-bility of the error system of the coupled systems Numericalsimulations show the effectiveness of the theoretical resultsMoreover when there are parameter mismatches betweendriver and response neural networks with discontinuous acti-vations numerical example is presented to demonstrate thecomplete synchronization by using discontinuous adaptivecontrol

Notations In the sequel if not explicitly stated matricesare assumed to have compatible dimensions 119868

119898stands for

the identity matrix of 119898-dimension R is the space of realnumber The Euclidean norm in R119898 is denoted as sdot accordingly for vector 119909 isin R119898 119909 = radic119909119879119909 where 119879denotes transposition 119909 = 0 represents each component of119909 is zero 119860 = (119886

119894119895)119898times119898

denotes a matrix of 119898-dimension

119860 = radic120582max(119860119879119860)

The rest of this paper is organized as follows In Section 2a model of delayed neural networks with discontinuousactivation functions is described Some necessary assump-tions definitions and lemmas are also given in this sectionOur main results and their rigorous proofs are described inSection 3 In Section 4 two examples with their numericalsimulations are offered to show the effectiveness of ourresults In Section 5 conclusions are given and at lastacknowledgments

2 Preliminaries

In this paper we consider the delayed neural network whichis described as follows

(119905) = minus119862119909 (119905) + 119860119891 (119909 (119905)) + 119861119891 (119909 (119905 minus 120579)) + 119869 (1)

where 119909(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905)) isin R119899 is the state

vector 119862 = diag(1198881 1198882 119888

119899) in which c

119894gt 0 119894 =

1 2 119899 are the neuron self-inhibitions 120579 gt 0 is thetransmission delay 119860 = (119886

119894119895)119899times119899

and 119861 = (119887119894119895)119899times119899

arethe connection weight matrix and the delayed connectionweight matrix respectively the activation function119891(119909(119905)) =(1198911(1199091(119905)) 119891

2(1199092(119905)) 119891

119899(119909119899(119905)))119879 represents the output of

the network 119869 = (1198691 1198692 119869

119899) is the external input vector

As for neural networks (1) we give the following assump-tion condition

Assumption 1 For every 119894 = 1 2 119899 119891119894 R rarr R is

monotone nondecreasing and has at most a finite number ofjump discontinuities in every compact interval

Remark 2 Assumption 1 was used in [20] Any functionsatisfying this assumption condition does not need to becontinuously differentiable in compact interval However thecontinuous differentiability is necessary in [10 26] On theother hand the ldquomonotone nondecreasingrdquo can be replacedby ldquomonotonic functionrdquo For instance if we replace the



119860 = (2 01

minus5 minus45) 119861 = (

minus15 01

minus02 4) (2)


1198912(1199092) =


minus tanh (1199092) minus 0018119909

2+ 003 119909

2lt 0

(3)


2is




119865 (119909) = ⋂

120575gt0

⋂

120583(119873)=0

119870[119891 (119861 (119909 120575) 119873)] (4)



119865 (119909) = 119870 [119891 (119909)]

= (119870 [1198911(1199091)] 119870 [119891

2(1199092)] 119870 [119891

119899(1199091)])

(5)

where119870[119891119894(119909119894)] = [119891

minus

119894(119909119894) 119891+

119894(119909119894)] 119894 = 1 2 119899




(1205741 1205742 120574

119899)119879


for aa 119905 isin [0 119879) (6)




(119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869

for aa 119905 isin [0 119879)

120574 (119909 (119905)) isin 119870 [119891 (119909 (119905))] for aa 119905 isin [0 119879)


120574 (119909 (119904)) = 120595 (119904) for aa 119904 isin [minus120579 0]

(7)




119889

119889119905119881 (119909 (119905)) = 120574 (119905) (119905) forall120574 isin 120597119881 (119909 (119905)) (8)




minus

119909(119905) = 119863+119909(119905) = 119863

+



minus119909(119905) = lim

ℎrarr0minus

(119891(119909+ℎ)minus119909(119905))ℎ 119863minus

119909(119905) =

limℎrarr0


ℎrarr0+

(119891(119909+

ℎ) minus 119909(119905))ℎ 119863+

119909(119905) = limℎrarr0

+(119891(119909 + ℎ) minus 119909(119905))ℎ


(119905) = minus119862119910 (119905) + 119860119891 (119910 (119905)) + 119861119891 (119910 (119905 minus 120579)) + 119869 + 119906 (119905)

(9)

where 119910(119905) = (1199101(119905) 1199102(119905) 119910


response system 119906(119905) = (1199061(119905) 1199062(119905) 119906

119899(119905))119879 is the




lim119905rarr+infin


3 Main Results



(119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869 (11)

(119905) = minus119862119910 (119905) + 119860120574 (119910 (119905)) + 119861120574 (119910 (119905 minus 120579)) + 119869 + 119906 (119905)

(12)

Let 119890(119905) = (1198901(119905) 1198902(119905) 119890

119899(119905))119879


119890 (119905) = minus119862119890 (119905) + 119860120573 (119890 (119905)) + 119861120573 (119890 (119905 minus 120579)) + 119906 (119905) (13)




119906119894(119905) = minus119897

119894(119905) 119890119894(119905)

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

(14)






119894(120585) le 119898



1003817100381710038171003817 le 1198981003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (15)

where119898 = max1198981 1198982 119898

119899


119881 (119905) =1

2119890119879

(119905) 119890 (119905) + 120583int

119905

119905minus120579

119890119879

(119904) 119890 (119904) 119889119904

+

119898

sum

119894=1

1

2120576119894

(119897119894(119905) minus 119896

119894)2

(16)




(119905)

= 119890119879


+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879

(119905 minus 120579) 119890 (119905 minus 120579) +

119898

sum

119894=1

(119897119894(119905) minus 119896

119894) 1198902

119894(119905)

= minus119890119879

(119905) 119862119890 (119905) + 119890119879

(119905) 119860120573 (119890 (119905)) + 119890119879

(119905) 119861120573 (119890 (119905 minus 120579))

+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879


(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 1198601003817100381710038171003817120573 (119890 (119905))

1003817100381710038171003817

+ 119890 (119905) 1198611003817100381710038171003817120573 (119890 (119905 minus 120579))

1003817100381710038171003817 + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(17)

where 119897(119905) = diag(1198971(119905) 1198972(119905) 119897

119899(119905))119870 = diag(119896

1 1198962


(119905) le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 119860119898 119890 (119905)

+ 119890 (119905) 119861119898 119890 (119905 minus 120579) + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119898 119860 119890 (119905)2

+1

2119898 119861 119890 (119905)

2

+1

2119898 119861 119890(119905 minus 120579)

2

+ 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(18)


(119905) le 119890119879

(119905) (minus119862 + 119898 119860 119868119899+ 119898 119861 119868

119899minus 119870) 119890 (119905) (19)

Take 119896119894= minus119888119894+ 119898119860 + 119898119861 + 1 Then

(119905) le minus119890119879

(119905) 119890 (119905) le 0 (20)


lim119905rarr+infin

119890 (119905) = 0 (21)






119894such that |1205741015840

119894(120585)| le 119898

119894 119894 = 1 2 119899 these 119898

119894








1(119905) 1199092(119905))119879 119869 =


119860 = (2 minus01

minus5 45) 119861 = (

minus15 minus01



119891119894(119909119894) =

tanh (119909119894) + 002radic119909119894 + 003 119909119894 gt 0 119894 = 1 2

tanh (119909119894) + 0018119909

119894minus 003 119909

119894lt 0 119894 = 1 2

(23)




8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 119905 isin [minus1 0]

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 5 10 15 20 25 30 35 40

1198902(119905)

1198901(119905)

119890 1(119905)119890 2(119905)

119905




1(119905) =







14

135

13

125

12

115

11

105

1

095

090 5 10 15 20 25 30 35 40

1198972(119905)

1198971(119905)

119897 1(119905)119897 2(119905)

119905


2(119905)




1(1199091) 1198912(1199092)) as

119891119894(119909119894) =

tanh (119909119894) + 0025119909

119894+ 0028 119909

119894gt 0 119894 = 1 2

tanh (119909119894) + 0025119909

119894minus 0028 119909

119894lt 0 119894 = 1 2

(24)



(119905) = minus (119905) 119909 (119905) + (119905) 119891 (119909 (119905)) + (119905) 119891 (119909 (119905 minus 120579))

+ (119905) + 119906 (t) (25)

8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)

minus1


119879 119905 isin [minus1 0]

where

(119905) = (2 minus01

minus5 + 01 cos (119905) 45 )

(119905) = (minus15 minus01

minus02 minus4 + 01 sin (119905))

(119905) = diag (1 1 + 01 cos (119905)) (119905) = 0

(26)



119906119894(119905) = minus119897

119894(119905) 119890119894(119905) minus 120572120573

119894(119905) sign (119890

119894(119905))

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

119894(119905) = 120585

119894

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816 119894 = 1 2 119899

(27)



1(119905) = 119897

2(119905) = 01

1205731(119905) = 120573


119894= 120585119894= 005


1(119905) 1198972(119905))119879 and 120573(119905) = (120573

1(119905) 1205732(119905))119879



119894(119905) lt 09 and 120573

119894(119905) le 051



8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 and 119906(119905) = 0 119905 isin [minus1 0]

4

3

2

1

0

minus1

minus2

minus3

minus40 5 10 15

1198902(119905)1198901(119905)

119890 1(119905)119890 2(119905)

119905



5 Conclusions


1198972(119905)

1

09

08

07

06

05

04

03

02

01

00 5 10 15

119905

1198971(119905)

119897 1(119905)119897 2(119905)


2(119905)

0 5 10 15

06

055

05

045

04

035

03

025

02

015

01

1205731(119905)

1205731(119905)1205732(119905)

119905

1205732(119905)


2(119905)


Acknowledgments




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860 = (2 01

minus5 minus45) 119861 = (

minus15 01

minus02 4) (2)


1198912(1199092) =


minus tanh (1199092) minus 0018119909

2+ 003 119909

2lt 0

(3)


2is




119865 (119909) = ⋂

120575gt0

⋂

120583(119873)=0

119870[119891 (119861 (119909 120575) 119873)] (4)



119865 (119909) = 119870 [119891 (119909)]

= (119870 [1198911(1199091)] 119870 [119891

2(1199092)] 119870 [119891

119899(1199091)])

(5)

where119870[119891119894(119909119894)] = [119891

minus

119894(119909119894) 119891+

119894(119909119894)] 119894 = 1 2 119899




(1205741 1205742 120574

119899)119879


for aa 119905 isin [0 119879) (6)




(119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869

for aa 119905 isin [0 119879)

120574 (119909 (119905)) isin 119870 [119891 (119909 (119905))] for aa 119905 isin [0 119879)


120574 (119909 (119904)) = 120595 (119904) for aa 119904 isin [minus120579 0]

(7)




119889

119889119905119881 (119909 (119905)) = 120574 (119905) (119905) forall120574 isin 120597119881 (119909 (119905)) (8)




minus

119909(119905) = 119863+119909(119905) = 119863

+



minus119909(119905) = lim

ℎrarr0minus

(119891(119909+ℎ)minus119909(119905))ℎ 119863minus

119909(119905) =

limℎrarr0


ℎrarr0+

(119891(119909+

ℎ) minus 119909(119905))ℎ 119863+

119909(119905) = limℎrarr0

+(119891(119909 + ℎ) minus 119909(119905))ℎ


(119905) = minus119862119910 (119905) + 119860119891 (119910 (119905)) + 119861119891 (119910 (119905 minus 120579)) + 119869 + 119906 (119905)

(9)

where 119910(119905) = (1199101(119905) 1199102(119905) 119910


response system 119906(119905) = (1199061(119905) 1199062(119905) 119906

119899(119905))119879 is the




lim119905rarr+infin


3 Main Results



(119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869 (11)

(119905) = minus119862119910 (119905) + 119860120574 (119910 (119905)) + 119861120574 (119910 (119905 minus 120579)) + 119869 + 119906 (119905)

(12)

Let 119890(119905) = (1198901(119905) 1198902(119905) 119890

119899(119905))119879


119890 (119905) = minus119862119890 (119905) + 119860120573 (119890 (119905)) + 119861120573 (119890 (119905 minus 120579)) + 119906 (119905) (13)




119906119894(119905) = minus119897

119894(119905) 119890119894(119905)

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

(14)






119894(120585) le 119898



1003817100381710038171003817 le 1198981003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (15)

where119898 = max1198981 1198982 119898

119899


119881 (119905) =1

2119890119879

(119905) 119890 (119905) + 120583int

119905

119905minus120579

119890119879

(119904) 119890 (119904) 119889119904

+

119898

sum

119894=1

1

2120576119894

(119897119894(119905) minus 119896

119894)2

(16)




(119905)

= 119890119879


+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879

(119905 minus 120579) 119890 (119905 minus 120579) +

119898

sum

119894=1

(119897119894(119905) minus 119896

119894) 1198902

119894(119905)

= minus119890119879

(119905) 119862119890 (119905) + 119890119879

(119905) 119860120573 (119890 (119905)) + 119890119879

(119905) 119861120573 (119890 (119905 minus 120579))

+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879


(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 1198601003817100381710038171003817120573 (119890 (119905))

1003817100381710038171003817

+ 119890 (119905) 1198611003817100381710038171003817120573 (119890 (119905 minus 120579))

1003817100381710038171003817 + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(17)

where 119897(119905) = diag(1198971(119905) 1198972(119905) 119897

119899(119905))119870 = diag(119896

1 1198962


(119905) le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 119860119898 119890 (119905)

+ 119890 (119905) 119861119898 119890 (119905 minus 120579) + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119898 119860 119890 (119905)2

+1

2119898 119861 119890 (119905)

2

+1

2119898 119861 119890(119905 minus 120579)

2

+ 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(18)


(119905) le 119890119879

(119905) (minus119862 + 119898 119860 119868119899+ 119898 119861 119868

119899minus 119870) 119890 (119905) (19)

Take 119896119894= minus119888119894+ 119898119860 + 119898119861 + 1 Then

(119905) le minus119890119879

(119905) 119890 (119905) le 0 (20)


lim119905rarr+infin

119890 (119905) = 0 (21)






119894such that |1205741015840

119894(120585)| le 119898

119894 119894 = 1 2 119899 these 119898

119894








1(119905) 1199092(119905))119879 119869 =


119860 = (2 minus01

minus5 45) 119861 = (

minus15 minus01



119891119894(119909119894) =

tanh (119909119894) + 002radic119909119894 + 003 119909119894 gt 0 119894 = 1 2

tanh (119909119894) + 0018119909

119894minus 003 119909

119894lt 0 119894 = 1 2

(23)




8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 119905 isin [minus1 0]

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 5 10 15 20 25 30 35 40

1198902(119905)

1198901(119905)

119890 1(119905)119890 2(119905)

119905




1(119905) =







14

135

13

125

12

115

11

105

1

095

090 5 10 15 20 25 30 35 40

1198972(119905)

1198971(119905)

119897 1(119905)119897 2(119905)

119905


2(119905)




1(1199091) 1198912(1199092)) as

119891119894(119909119894) =

tanh (119909119894) + 0025119909

119894+ 0028 119909

119894gt 0 119894 = 1 2

tanh (119909119894) + 0025119909

119894minus 0028 119909

119894lt 0 119894 = 1 2

(24)



(119905) = minus (119905) 119909 (119905) + (119905) 119891 (119909 (119905)) + (119905) 119891 (119909 (119905 minus 120579))

+ (119905) + 119906 (t) (25)

8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)

minus1


119879 119905 isin [minus1 0]

where

(119905) = (2 minus01

minus5 + 01 cos (119905) 45 )

(119905) = (minus15 minus01

minus02 minus4 + 01 sin (119905))

(119905) = diag (1 1 + 01 cos (119905)) (119905) = 0

(26)



119906119894(119905) = minus119897

119894(119905) 119890119894(119905) minus 120572120573

119894(119905) sign (119890

119894(119905))

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

119894(119905) = 120585

119894

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816 119894 = 1 2 119899

(27)



1(119905) = 119897

2(119905) = 01

1205731(119905) = 120573


119894= 120585119894= 005


1(119905) 1198972(119905))119879 and 120573(119905) = (120573

1(119905) 1205732(119905))119879



119894(119905) lt 09 and 120573

119894(119905) le 051



8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 and 119906(119905) = 0 119905 isin [minus1 0]

4

3

2

1

0

minus1

minus2

minus3

minus40 5 10 15

1198902(119905)1198901(119905)

119890 1(119905)119890 2(119905)

119905



5 Conclusions


1198972(119905)

1

09

08

07

06

05

04

03

02

01

00 5 10 15

119905

1198971(119905)

119897 1(119905)119897 2(119905)


2(119905)

0 5 10 15

06

055

05

045

04

035

03

025

02

015

01

1205731(119905)

1205731(119905)1205732(119905)

119905

1205732(119905)


2(119905)


Acknowledgments




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











lim119905rarr+infin


3 Main Results



(119905) = minus119862119909 (119905) + 119860120574 (119909 (119905)) + 119861120574 (119909 (119905 minus 120579)) + 119869 (11)

(119905) = minus119862119910 (119905) + 119860120574 (119910 (119905)) + 119861120574 (119910 (119905 minus 120579)) + 119869 + 119906 (119905)

(12)

Let 119890(119905) = (1198901(119905) 1198902(119905) 119890

119899(119905))119879


119890 (119905) = minus119862119890 (119905) + 119860120573 (119890 (119905)) + 119861120573 (119890 (119905 minus 120579)) + 119906 (119905) (13)




119906119894(119905) = minus119897

119894(119905) 119890119894(119905)

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

(14)






119894(120585) le 119898



1003817100381710038171003817 le 1198981003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (15)

where119898 = max1198981 1198982 119898

119899


119881 (119905) =1

2119890119879

(119905) 119890 (119905) + 120583int

119905

119905minus120579

119890119879

(119904) 119890 (119904) 119889119904

+

119898

sum

119894=1

1

2120576119894

(119897119894(119905) minus 119896

119894)2

(16)




(119905)

= 119890119879


+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879

(119905 minus 120579) 119890 (119905 minus 120579) +

119898

sum

119894=1

(119897119894(119905) minus 119896

119894) 1198902

119894(119905)

= minus119890119879

(119905) 119862119890 (119905) + 119890119879

(119905) 119860120573 (119890 (119905)) + 119890119879

(119905) 119861120573 (119890 (119905 minus 120579))

+ 120583119890119879

(119905) 119890 (119905) minus 120583119890119879


(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 1198601003817100381710038171003817120573 (119890 (119905))

1003817100381710038171003817

+ 119890 (119905) 1198611003817100381710038171003817120573 (119890 (119905 minus 120579))

1003817100381710038171003817 + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(17)

where 119897(119905) = diag(1198971(119905) 1198972(119905) 119897

119899(119905))119870 = diag(119896

1 1198962


(119905) le minus119890119879

(119905) 119862119890 (119905) + 119890 (119905) 119860119898 119890 (119905)

+ 119890 (119905) 119861119898 119890 (119905 minus 120579) + 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

le minus119890119879

(119905) 119862119890 (119905) + 119898 119860 119890 (119905)2

+1

2119898 119861 119890 (119905)

2

+1

2119898 119861 119890(119905 minus 120579)

2

+ 120583119890119879

(119905) 119890 (119905)

minus 120583119890119879


(119905) 119870119890 (119905)

(18)


(119905) le 119890119879

(119905) (minus119862 + 119898 119860 119868119899+ 119898 119861 119868

119899minus 119870) 119890 (119905) (19)

Take 119896119894= minus119888119894+ 119898119860 + 119898119861 + 1 Then

(119905) le minus119890119879

(119905) 119890 (119905) le 0 (20)


lim119905rarr+infin

119890 (119905) = 0 (21)






119894such that |1205741015840

119894(120585)| le 119898

119894 119894 = 1 2 119899 these 119898

119894








1(119905) 1199092(119905))119879 119869 =


119860 = (2 minus01

minus5 45) 119861 = (

minus15 minus01



119891119894(119909119894) =

tanh (119909119894) + 002radic119909119894 + 003 119909119894 gt 0 119894 = 1 2

tanh (119909119894) + 0018119909

119894minus 003 119909

119894lt 0 119894 = 1 2

(23)




8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 119905 isin [minus1 0]

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 5 10 15 20 25 30 35 40

1198902(119905)

1198901(119905)

119890 1(119905)119890 2(119905)

119905




1(119905) =







14

135

13

125

12

115

11

105

1

095

090 5 10 15 20 25 30 35 40

1198972(119905)

1198971(119905)

119897 1(119905)119897 2(119905)

119905


2(119905)




1(1199091) 1198912(1199092)) as

119891119894(119909119894) =

tanh (119909119894) + 0025119909

119894+ 0028 119909

119894gt 0 119894 = 1 2

tanh (119909119894) + 0025119909

119894minus 0028 119909

119894lt 0 119894 = 1 2

(24)



(119905) = minus (119905) 119909 (119905) + (119905) 119891 (119909 (119905)) + (119905) 119891 (119909 (119905 minus 120579))

+ (119905) + 119906 (t) (25)

8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)

minus1


119879 119905 isin [minus1 0]

where

(119905) = (2 minus01

minus5 + 01 cos (119905) 45 )

(119905) = (minus15 minus01

minus02 minus4 + 01 sin (119905))

(119905) = diag (1 1 + 01 cos (119905)) (119905) = 0

(26)



119906119894(119905) = minus119897

119894(119905) 119890119894(119905) minus 120572120573

119894(119905) sign (119890

119894(119905))

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

119894(119905) = 120585

119894

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816 119894 = 1 2 119899

(27)



1(119905) = 119897

2(119905) = 01

1205731(119905) = 120573


119894= 120585119894= 005


1(119905) 1198972(119905))119879 and 120573(119905) = (120573

1(119905) 1205732(119905))119879



119894(119905) lt 09 and 120573

119894(119905) le 051



8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 and 119906(119905) = 0 119905 isin [minus1 0]

4

3

2

1

0

minus1

minus2

minus3

minus40 5 10 15

1198902(119905)1198901(119905)

119890 1(119905)119890 2(119905)

119905



5 Conclusions


1198972(119905)

1

09

08

07

06

05

04

03

02

01

00 5 10 15

119905

1198971(119905)

119897 1(119905)119897 2(119905)


2(119905)

0 5 10 15

06

055

05

045

04

035

03

025

02

015

01

1205731(119905)

1205731(119905)1205732(119905)

119905

1205732(119905)


2(119905)


Acknowledgments




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894such that |1205741015840

119894(120585)| le 119898

119894 119894 = 1 2 119899 these 119898

119894








1(119905) 1199092(119905))119879 119869 =


119860 = (2 minus01

minus5 45) 119861 = (

minus15 minus01



119891119894(119909119894) =

tanh (119909119894) + 002radic119909119894 + 003 119909119894 gt 0 119894 = 1 2

tanh (119909119894) + 0018119909

119894minus 003 119909

119894lt 0 119894 = 1 2

(23)




8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 119905 isin [minus1 0]

2

15

1

05

0

minus05

minus1

minus15

minus2

minus250 5 10 15 20 25 30 35 40

1198902(119905)

1198901(119905)

119890 1(119905)119890 2(119905)

119905




1(119905) =







14

135

13

125

12

115

11

105

1

095

090 5 10 15 20 25 30 35 40

1198972(119905)

1198971(119905)

119897 1(119905)119897 2(119905)

119905


2(119905)




1(1199091) 1198912(1199092)) as

119891119894(119909119894) =

tanh (119909119894) + 0025119909

119894+ 0028 119909

119894gt 0 119894 = 1 2

tanh (119909119894) + 0025119909

119894minus 0028 119909

119894lt 0 119894 = 1 2

(24)



(119905) = minus (119905) 119909 (119905) + (119905) 119891 (119909 (119905)) + (119905) 119891 (119909 (119905 minus 120579))

+ (119905) + 119906 (t) (25)

8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)

minus1


119879 119905 isin [minus1 0]

where

(119905) = (2 minus01

minus5 + 01 cos (119905) 45 )

(119905) = (minus15 minus01

minus02 minus4 + 01 sin (119905))

(119905) = diag (1 1 + 01 cos (119905)) (119905) = 0

(26)



119906119894(119905) = minus119897

119894(119905) 119890119894(119905) minus 120572120573

119894(119905) sign (119890

119894(119905))

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

119894(119905) = 120585

119894

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816 119894 = 1 2 119899

(27)



1(119905) = 119897

2(119905) = 01

1205731(119905) = 120573


119894= 120585119894= 005


1(119905) 1198972(119905))119879 and 120573(119905) = (120573

1(119905) 1205732(119905))119879



119894(119905) lt 09 and 120573

119894(119905) le 051



8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 and 119906(119905) = 0 119905 isin [minus1 0]

4

3

2

1

0

minus1

minus2

minus3

minus40 5 10 15

1198902(119905)1198901(119905)

119890 1(119905)119890 2(119905)

119905



5 Conclusions


1198972(119905)

1

09

08

07

06

05

04

03

02

01

00 5 10 15

119905

1198971(119905)

119897 1(119905)119897 2(119905)


2(119905)

0 5 10 15

06

055

05

045

04

035

03

025

02

015

01

1205731(119905)

1205731(119905)1205732(119905)

119905

1205732(119905)


2(119905)


Acknowledgments




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










14

135

13

125

12

115

11

105

1

095

090 5 10 15 20 25 30 35 40

1198972(119905)

1198971(119905)

119897 1(119905)119897 2(119905)

119905


2(119905)




1(1199091) 1198912(1199092)) as

119891119894(119909119894) =

tanh (119909119894) + 0025119909

119894+ 0028 119909

119894gt 0 119894 = 1 2

tanh (119909119894) + 0025119909

119894minus 0028 119909

119894lt 0 119894 = 1 2

(24)



(119905) = minus (119905) 119909 (119905) + (119905) 119891 (119909 (119905)) + (119905) 119891 (119909 (119905 minus 120579))

+ (119905) + 119906 (t) (25)

8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)

minus1


119879 119905 isin [minus1 0]

where

(119905) = (2 minus01

minus5 + 01 cos (119905) 45 )

(119905) = (minus15 minus01

minus02 minus4 + 01 sin (119905))

(119905) = diag (1 1 + 01 cos (119905)) (119905) = 0

(26)



119906119894(119905) = minus119897

119894(119905) 119890119894(119905) minus 120572120573

119894(119905) sign (119890

119894(119905))

119897119894(119905) = 120576

1198941198902

119894(119905) 119894 = 1 2 119899

119894(119905) = 120585

119894

1003816100381610038161003816119890119894 (119905)1003816100381610038161003816 119894 = 1 2 119899

(27)



1(119905) = 119897

2(119905) = 01

1205731(119905) = 120573


119894= 120585119894= 005


1(119905) 1198972(119905))119879 and 120573(119905) = (120573

1(119905) 1205732(119905))119879



119894(119905) lt 09 and 120573

119894(119905) le 051



8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 and 119906(119905) = 0 119905 isin [minus1 0]

4

3

2

1

0

minus1

minus2

minus3

minus40 5 10 15

1198902(119905)1198901(119905)

119890 1(119905)119890 2(119905)

119905



5 Conclusions


1198972(119905)

1

09

08

07

06

05

04

03

02

01

00 5 10 15

119905

1198971(119905)

119897 1(119905)119897 2(119905)


2(119905)

0 5 10 15

06

055

05

045

04

035

03

025

02

015

01

1205731(119905)

1205731(119905)1205732(119905)

119905

1205732(119905)


2(119905)


Acknowledgments




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










8

6

4

2

0

minus2

minus4

minus6


1199092(119905)

1199091(119905)


119879 and 119906(119905) = 0 119905 isin [minus1 0]

4

3

2

1

0

minus1

minus2

minus3

minus40 5 10 15

1198902(119905)1198901(119905)

119890 1(119905)119890 2(119905)

119905



5 Conclusions


1198972(119905)

1

09

08

07

06

05

04

03

02

01

00 5 10 15

119905

1198971(119905)

119897 1(119905)119897 2(119905)


2(119905)

0 5 10 15

06

055

05

045

04

035

03

025

02

015

01

1205731(119905)

1205731(119905)1205732(119905)

119905

1205732(119905)


2(119905)


Acknowledgments




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article synchronization of discontinuous neural...

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