robust stochastic stability analysis for uncertain neutral ...robust stochastic stability analysis...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 829594, 12 pagesdoi:10.1155/2012/829594

Research ArticleRobust Stochastic Stability Analysis for UncertainNeutral-Type Delayed Neural Networks Driven byWiener Process

Weiwei Zhang1 and Linshan Wang2

1 College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China2 Department of Mathematics, Ocean University of China, Qingdao 266100, China

Correspondence should be addressed to Linshan Wang, [email protected]

Received 9 July 2011; Revised 20 September 2011; Accepted 27 September 2011

Academic Editor: Shiping Lu

Copyright q 2012 W. Zhang and L. Wang. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The robust stochastic stability for a class of uncertain neutral-type delayed neural networks drivenby Wiener process is investigated. By utilizing the Lyapunov-Krasovskii functional and inequalitytechnique, some sufficient criteria are presented in terms of linear matrix inequality (LMI) toensure the stability of the system. A numerical example is given to illustrate the applicability ofthe result.

1. Introduction

In the past few years, neural networks and their various generalizations have drawn muchresearch attention owing to their promising potential applications in a variety of areas, such asrobotics, aerospace, telecommunications, pattern recognition, image processing, associativememory, signal processing, and combinatorial optimization [1–3]. In such applications, itis of prime importance to ensure the asymptotic stability of the designed neural networks.Because of this, the stability of neural networks has been deeply investigated in the literature[4–14].

It is known that time delays and stochastic perturbations are commonly encounteredin the implementation of neural networks, and may result in instability or oscillation. Soit is essential to investigate the stability of delayed stochastic neural networks [15, 16].Moreover, uncertainties are unavoidable in practical implementation of neural networksdue to modeling errors and parameter fluctuation, which also cause instability and poorperformance [15, 17, 18]. Therefore, it is significant to introduce such uncertainties intodelayed stochastic neural networks.

2 Journal of Applied Mathematics

On the other hand, because of the complicated dynamic properties of the neural cellsin the real world, it is natural and important that systems will contain some informationabout the derivative of the past state. Practically, such phenomenon always appears in thestudy of automatic control, circuit analysis, chemical process simulation, and populationdynamics, and so forth. Recently, there has been increasing interest in the study of delayedneural networks of neutral type, see [6–15, 18–24]. In [6, 8], the authors developed the globalasymptotic stability of neutral-type neural networks with delays by utilizing the Lyapunovstability theory and LMI technique. In [9, 10], the global exponential stability of neutral-typeneural networks with distributed delays is studied. However, the stochastic perturbationswere not taken into account in those delayed neural networks [6–10].

In [23, 24], the authors discussed the robust stability for uncertain stochastic neuralnetworks of neutral-type with time-varying delays. However, the distributed delays werenot taken into account in the models. So far, there are only a few papers that not only dealwith the stochastic stability analysis for delayed neural networks of neutral-type, but alsoconsider the parameter uncertainties.

To the best of our knowledge, there are very few results on the stochastic stabilityanalysis for uncertain neutral-type neural networks with both discrete and distributed delaysdriven by Wiener process. This motivates the research in this paper.

In this paper, a class of uncertain neutral-type delayed neural networks driven byWiener process is considered. By constructing a suitable Lyapunov functional, some newstability criteria to guarantee the system to be stochastically asymptotically stable in the meansquare are given, which are less conservative than some existing reports. The structure of theaddressed system is more general than in the other papers. The criteria can be checked easilyby the LMI control toolbox in MATLAB. Moreover, a numerical example is given to illustratethe effectiveness and improvement over some existing results.

2. Preliminaries

Notations. A < 0 denotes that A is a negative definite matrix. The superscript “T” stands forthe transpose of a matrix. (Ω, F, P) denotes a complete probability space, E(·) stands for themathematical expectation operator. ‖·‖ stands for the Euclidean norm. I is the identity matrixof appropriate dimension, and the symmetric terms in a symmetric matrix are denoted by ∗.

Consider the following class of uncertain neutral-type delayed neural networks drivenby Wiener process:

d[x(t) − Cx(t − h(t))] =

[−A(t)x(t) + B(t)f(x(t − τ(t))) +D(t)

∫ t

t−r(t)f(x(s))ds

]dt

+[H0(t)x(t) +H1(t)x(t − τ(t))]dw(t),

x(t0 + s) = ϕ(s), s ∈ [t0 − ρ, t0],

(2.1)

where x = (x1, x2, . . . , xn)T is the neuron state vector, A(t) = A + ΔA(t), B(t) = B + ΔB(t),

D(t) = D + ΔD(t), H0(t) = H0 + ΔH0(t), H1(t) = H1 + ΔH1(t), A = diag(ai)n×n is apositive diagonal matrix, B,C,D ∈ Rn×n are the connection weight matrices, H0,H1 ∈Rn×n are known real constant matrices, ΔA(t), ΔB(t), ΔD(t), ΔH0(t), ΔH1(t) represent thetime-varying parameter uncertain terms. f(x) = (f1(x1), f2(x2), . . . , fn(xn))

T is the neuron

Journal of Applied Mathematics 3

activation function with f(0) = 0.w(t) = (w1(t), w2(t), . . . , wn(t))T is an n-dimensionalWiener

process defined on a complete probability space (Ω, F, P). r(t), τ(t), h(t) are nonnegative,bounded, and differentiable time varying delays satisfying

0 < r(t) ≤ r < ∞,

0 < τ(t) ≤ τ < ∞, τ(t) ≤ η1 < ∞,

0 < h(t) ≤ h < ∞, h(t) ≤ η2 < ∞.

(2.2)

The admissible parameter uncertain terms are assumed to be the following form:

[ΔA(t),ΔB(t),ΔD(t),ΔH0(t),ΔH1(t)] = UF(t)[M1,M2,M3,M4,M5], (2.3)

where U,Mi, i = 1, . . . , 5 are known real constant matrices, F(t) is the time-varying uncertainmatrix satisfying

FTF ≤ I. (2.4)

Suppose that f(·) is bounded and satisfies the following condition:

‖f(x)‖ ≤ ‖Gx‖, (2.5)

where G ∈ Rn×n is a known constant matrix.Assume that the initial value ϕ : [−ρ, 0] → Rn is F0-measurable and continuously

differentiable, we introduce the following norm:

∥∥ϕ∥∥2ρ = max

⎧⎨⎩ sup

−α≤s≤0E∣∣ϕi(s)

∣∣2, sup−h≤s≤0

E∣∣ϕ′

i(s)∣∣2⎫⎬⎭ < ∞, (2.6)

where ρ = max{τ, h, r}, α = max{τ, r}.Under the above assumptions, it is easy to verify that there exists a unique equilibrium

point of system (2.1) (see [25]).

Definition 2.1. The equilibrium point of (2.1) is said to be globally robustly stochasticallyasymptotically stable in the mean square, if the following condition holds:

limt→+∞

E∣∣x(t, t0, ϕ)∣∣2 = 0, t ≥ t0, (2.7)

where x(t, t0, ϕ) is any solution of model (2.1) with initial value ϕ.

Lemma 2.2 (Schur complement [26]). Given constant matrices Ω1, Ω2, Ω3 with appropriatedimensions, whereΩT

1 = Ω1 and ΩT2 = Ω2 > 0, then

Ω1 +ΩT3Ω

−12 Ω3 < 0, (2.8)


if and only if

(Ω1 ΩT

3

∗ −Ω2

)< 0, or

(−Ω2 Ω3

∗ Ω1

)< 0. (2.9)

Lemma 2.3 (see [26]). Given matricesD, E, and F with FTF ≤ I and a scalar ε > 0, then

DFE + ETFTDT ≤ εDDT + ε−1ETE. (2.10)

Lemma 2.4 (see [27]). For any constant matrix M ∈ Rn×n, M = MT > 0, a scalar γ > 0, vectorfunction x(t) : [0, γ] → Rn such that the integrations are well defined, then

[∫ γ

0x(s)ds

]TM[∫ γ

0x(s)ds

]≤ γ

∫ γ

0xT (s)Mx(s)ds. (2.11)

3. Main Results

Theorem 3.1. System (2.1) is globally robustly stochastically asymptotically stable in the meansquare, if there exist symmetric positive definite matrices P, Q, R, S, U1, U2 and positive scalarsδ, ε1, ε2 > 0 such that LMI holds:

Λ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Γ1 ATPC PB − ε1MT

1M2 PD − ε1MT1M3 ε2MT

4M5 HT

0P −PU 0

∗ Γ2 −CTPB −CTPD 0 0 −CTPU 0

∗ ∗ Γ3 ε1MT2M3 0 0 0 0

∗ ∗ ∗ Γ4 0 0 0 0

∗ ∗ ∗ ∗ Γ5 HT

1P 0 0

∗ ∗ ∗ ∗ ∗ −P 0 PU

∗ ∗ ∗ ∗ ∗ ∗ −ε1I 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε2I

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

< 0, (3.1)

where Γ1 = −PA − ATP + Q + R + rGTSG + ε1MT

1M1 + ε2MT4M4, Γ2 = −U1 − (1 − η2)R, Γ3 =

−δI + ε1MT2M2, Γ4 = −r−1S + ε1MT

3M3, Γ5 = −U2 − (1 − η1)Q + δGTG + ε2MT5M5.


Proof. Using Lemma 2.2, the matrix Λ < 0 implies that

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Γ1 ATPC PB − ε1MT

1M2 PD − ε1MT1M3 ε2MT

4M5 HT

0P

∗ Γ2 −CTPB −CTPD 0 0

∗ ∗ Γ3 ε1MT2M3 0 0

∗ ∗ ∗ Γ4 0 0

∗ ∗ ∗ ∗ Γ5 HT

1P

∗ ∗ ∗ ∗ ∗ −P

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PU 0

−CTPU 0

0 0

0 0

0 0

0 PU

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎝ε−11 I 0

∗ ε−12 I

⎞⎠⎛⎝UTP −UTPC 0 0 0 0

0 0 0 0 0 UTP

⎞⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Φ1 ATPC PB PD 0 H

T

0P

∗ Γ2 −CTPB −CTPD 0 0

∗ ∗ −δI 0 0 0

∗ ∗ ∗ −r−1S 0 0

∗ ∗ ∗ ∗ Φ2 HT

1P

∗ ∗ ∗ ∗ ∗ −P

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ ε1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−MT1

0

MT2

MT3

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−MT1

0

MT2

MT3

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+ ε2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

MT4

0

0

0

MT5

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

MT4

0

0

0

MT5

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+ ε−11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PU

−CTPU

0

0

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PU

−CTPU

0

0

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+ ε−12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

0

0

0

PU

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

0

0

0

PU

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

< 0,

(3.2)

where Φ1 = −PA −ATP +Q + R + rGTSG, Φ2 = −U2 − (1 − η1)Q + δGTG.


From (2.3), (2.4), using Lemma 2.3, we have

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−PΔA(t) −ΔAT (t)P ΔAT (t)PC PΔB(t) PΔD(t) 0 ΔHT0 (t)P

∗ 0 −CTPΔB(t) −CTPΔD(t) 0 0

∗ ∗ 0 0 0 0

∗ ∗ ∗ 0 0 0

∗ ∗ ∗ ∗ 0 ΔHT1 (t)P

∗ ∗ ∗ ∗ ∗ 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−MT1

0

MT2

MT3

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

FT (t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PU

−CTPU

0

0

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PU

−CTPU

0

0

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−MT1

0

MT2

MT3

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

MT4

0

0

0

MT5

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

FT (t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

0

0

0

PU

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

0

0

0

PU

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

F(t)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

MT4

0

0

0

MT5

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

≤ ε1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−MT1

0

MT2

MT3

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−MT1

0

MT2

MT3

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+ ε−11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PU

−CTPU

0

0

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

PU

−CTPU

0

0

0

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+ ε2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

MT4

0

0

0

MT5

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

MT4

0

0

0

MT5

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

+ ε−12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

0

0

0

PU

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

0

0

0

PU

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

.

(3.3)


Together with (3.2), we get

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Ψ AT (t)PC PB(t) PD(t) 0 HT0 (t)P

∗ Γ2 −CTPB(t) −CTPD(t) 0 0

∗ ∗ −δI 0 0 0

∗ ∗ ∗ −r−1S 0 0

∗ ∗ ∗ ∗ Φ2 HT1 (t)P

∗ ∗ ∗ ∗ ∗ −P

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

< 0, (3.4)

where Ψ = −PA(t) −AT (t)P +Q + R + rGTSG.Utilizing Lemma 2.2 again, we obtain

Σ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

Ψ AT (t)PC PB(t) PD(t) 0

∗ Γ2 −CTPB(t) −CTPD(t) 0

∗ ∗ −δI 0 0

∗ ∗ ∗ −r−1S 0

∗ ∗ ∗ ∗ Φ2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

HT0 (t)

0

0

0

HT1 (t)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

P

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

HT0 (t)

0

0

0

HT1 (t)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

< 0. (3.5)

Constructing a positive definite Lyapunov-Krasovskii functional as follows:

V (t, x(t)) = yT (t)Py(t) +∫ t

t−τ(t)xT (s)Qx(s)ds +

∫ t

t−h(t)xT (s)Rx(s)ds

+∫0

−r(t)

∫ t

t+θfT (x(s))Sf(x(s))dsdθ +

∫T

t

xT (s − h(s))U1x(s − h(s))ds

+∫T

t

xT (s − τ(s))U2x(s − τ(s))ds,

(3.6)

where y(t) = x(t) − Cx(t − h(t)), T > t is a constant.By Ito’s differential formula, we get

dV (t, x(t)) ≤{2yT (t)P

[−A(t)x(t) + B(t)f(x(t − τ(t))) +D(t)

∫ t

t−r(t)f(x(s))ds

]

+ xT (t)Qx(t) − (1 − τ(t))xT (t − τ(t))Qx(t − τ(t)) + xT (t)Rx(t)

− (1 − h(t))xT (t − h(t))Rx(t − h(t)) + rfT (x(t))Sf(x(t))

−∫ t

t−r(t)fT (x(s))Sf(x(s))ds

− xT (t − h(t))U1x(t − h(t)) − xT (t − τ(t))U2x(t − τ(t))


+[H0(t)x(t) +H1(t)x(t − τ(t))]TP[H0(t)x(t) +H1(t)x(t − τ(t))]

}dt

+ 2yT (t)P[H0(t)x(t) +H1(t)x(t − τ(t))]dw(t)

≤{2[x(t) − Cx(t − h(t))]T

× P

[−A(t)x(t) + B(t)f(x(t − τ(t))) +D(t)

∫ t

t−r(t)f(x(s))ds

]

+ xT (t)Qx(t) − (1 − η1)xT (t − τ(t))Qx(t − τ(t)) + xT (t)Rx(t)

− (1 − η2)xT (t − h(t))Rx(t − h(t)) + rfT (x(t))Sf(x(t))

−∫ t

t−r(t)fT (x(s))Sf(x(s))ds

− xT (t − h(t))U1x(t − h(t)) − xT (t − τ(t))U2x(t − τ(t))

+[H0(t)x(t) +H1(t)x(t − τ(t))]TP[H0(t)x(t) +H1(t)x(t − τ(t))]

}dt

+ 2yT (t)P[H0(t)x(t) +H1(t)x(t − τ(t))]dw(t).

(3.7)

From (2.5), for a scalar δ > 0, we have

−δ[fT (x(t − τ(t)))f(x(t − τ(t))) − xT (t − τ(t))GTGx(t − τ(t))

]≥ 0. (3.8)

Using Lemma 2.4, we have

(∫ t

t−r(t)f(x(s))ds

)T

r−1S

(∫ t

t−r(t)f(x(s))ds

)≤∫ t

t−r(t)fT (x(s))Sf(x(s))ds. (3.9)

Together (3.8), (3.9) with dV (t, x(t)), we obtain

dV (t, x(t)) ≤{xT (t)

[−PA(t) −AT (t)P +Q + R + rGTSG

]x(t) + xT (t)AT (t)PCx(t − h(t))

+ xT (t − h(t))CTPA(t)x(t) + xT (t)PB(t)f(x(t − τ(t)))

+ fT (x(t − τ(t)))BT (t)Px(t)

+ xT (t)PD(t)∫ t

t−r(t)f(x(s))ds +

(∫ t

t−r(t)f(x(s))ds

)T

DT (t)Px(t)

+ xT (t − h(t))[−U1 −

(1 − η2

)R]x(t − h(t)) − xT (t − h(t))CTPB(t)f(x(t − τ(t)))


− fT (x(t − τ(t)))BT (t)PCx(t − h(t)) − xT (t − h(t))CTPD(t)∫ t

t−r(t)f(x(s))ds

−(∫ t

t−r(t)f(x(s))ds

)T

DT (t)PCx(t − h(t)) + xT (t − τ(t))

× [−U2 −(1 − η1

)Q]x(t − τ(t))

−(∫ t

t−r(t)f(x(s))ds

)T

r−1S

(∫ t

t−r(t)f(x(s))ds

)

+[H0(t)x(t) +H1(t)x(t − τ(t))]TP[H0(t)x(t) +H1(t)x(t − τ(t))]}dt

+ 2yT (t)P[H0(t)x(t) +H1(t)x(t − τ(t))]dw(t).

(3.10)

That is,

dV (t, x(t)) ≤ ξT (t)Σξ(t)dt + 2yT (t)P[H0(t)x(t) +H1(t)x(t − τ(t))]dw(t), (3.11)

where ξT (t) = (xT (t), xT (t − h(t)), fT (x(t − τ(t))), (∫ tt−r(t) f(x(s))ds)

T, xT (t − τ(t))), and the

matrix Σ is given in (3.5).Taking the mathematical expectation, we get

E(dV (t, x(t))

dt

)≤ E

(ξT (t)Σξ(t)

)≤ λmax(Σ)E‖x(t)‖2. (3.12)

From (3.5), we know Σ < 0, that is, λmax(Σ) < 0. By Lyapunov-Krasovskii stability theorems,the system (2.1) is globally robustly asymptotically stable. The proof is completed.

Remark 3.2. To the best of our knowledge, few authors have considered the stochasticallyasymptotic stability for uncertain neutral-type neural networks driven by Wiener process.We can find recent papers [18, 22–24]. However, it is assumed in [18] that the system is alinear model and all delays are constants. In [22], it is assumed that the time-varying delayssatisfying τ(t) ≤ ρτ < 1, h(t) ≤ ρh < 1, in this paper, we relax it to τ(t) ≤ ρτ < ∞, h(t) ≤ ρh < ∞.In [23, 24], the authors discussed the robust stability for uncertain stochastic neural networksof neutral-type with time-varying delays. However, the distributed delays were not takeninto account in the models. Hence, our results in this paper have wider adaptive range.

Remark 3.3. Suppose that C = 0, D(t) = 0 (i.e., without neutral-type and distributed delays),then the system (2.1) becomes the one investigated in [15].

Remark 3.4. In [17], the authors studied the global stability for uncertain stochastic neuralnetworks with time-varying delay by Lyapunov functional method and LMI technique.However, the neutral term and distributed delays were not taken into account in the models.Therefore, our developed results in this paper are more general than those reported in [17].


Remark 3.5. It should be noted that the condition is given as linear matrix inequalities LMIs,therefore, by using the MATLAB LMI Toolbox, it is straightforward to check the feasibility ofthe condition.

4. Numerical Example

Consider the following uncertain neutral-type delayed neural networks:

d[x(t) − Cx(t − h(t))] =

[− (A +UF(t)M1)x(t) + (B +UF(t)M2)f(x(t − τ(t)))

+(D +UF(t)M3)∫ t

t−r(t)f(x(s))ds

]dt

+ [UF(t)M4x(t) +UF(t)M5x(t − τ(t))]dw(t),

(4.1)

where n = 2, fi(xi) = sinxi, i = 1, 2, η1 = 0.7, η2 = 0.5, 0 < r(t) ≤ r = 3, FT (t)F(t) ≤ I.The constant matrices are

A =

(3 0

0 3

), B =

(0.2 0.16

0.04 0.08

), C =

(0.2 0

0 0.2

),

D =

(0.04 0.03

−0.02 0.05

), U =

(0.1 0.5

0.5 0.3

), M1 =

(0.6 0

0 0.6

),

M2 = M3 = M4 =

(0.2 0

0 0.2

), M5 =

(0.4 0

0 0.4

), G =

(1 0

0 1

).

(4.2)

By using the MATLAB LMI Control Toolbox, we obtain the feasible solution as follows: δ =2.0876, ε1 = 5.0486, ε2 = 8.0446,

P =

(6.8465 −0.7257−0.7257 6.6012

), Q =

(9.9371 −0.2792−0.2792 10.4388

), R =

(8.0104 −2.3936−2.3936 5.4991

),

S =

(3.4984 −0.8143−0.8143 1.9588

), U1 =

(1.1111 0.3889

0.3889 1.1060

), U2 =

(2.4193 −0.6561−0.6561 2.6270

).

(4.3)

That is the system (4.1) is globally robustly stochastically asymptotically stable in the meansquare.

5. Conclusion

In this paper, the stochastically asymptotic stability problem has been studied for a class ofuncertain neutral-type delayed neural networks driven by Wiener process by utilizing the


Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach. A numericalexample is given to illustrate the applicability of the result.

Acknowledgment

This paper was fully supported by the National Natural Science Foundation of China (no.10771199 and no. 10871117).

References

[1] A. Bouzerdoum and T. R. Pattison, “Neural network for quadratic optimization with boundconstraints,” IEEE Transactions on Neural Networks, vol. 4, no. 2, pp. 293–304, 1993.

[2] M. Forti and A. Tesi, “New conditions for global stability of neural networks with application to linearand quadratic programming problems,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 7, pp.354–366, 1995.

[3] M. P. Kennedy and L. O. Chua, “Neural networks for nonlinear programming,” IEEE Transactions onCircuits and Systems, vol. 35, no. 5, pp. 554–562, 1988.

[4] S. Xu, J. Lam, D. W. C. Ho, and Y. Zou, “Delay-dependent exponential stability for a class of neuralnetworkswith time delays,” Journal of Computational and AppliedMathematics, vol. 183, no. 1, pp. 16–28,2005.

[5] R. Zhang and L. Wang, “Global exponential robust stability of interval cellular neural networks withS-type distributed delays,”Mathematical and Computer Modelling, vol. 50, no. 3-4, pp. 380–385, 2009.

[6] J. H. Park, O. M. Kwon, and S. M. Lee, “LMI optimization approach on stability for delayed neuralnetworks of neutral-type,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 236–244, 2008.

[7] R. Samli and S. Arik, “New results for global stability of a class of neutral-type neural systems withtime delays,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 564–570, 2009.

[8] R. Rakkiyappan and P. Balasubramaniam, “LMI conditions for global asymptotic stability results forneutral-type neural networks with distributed time delays,”AppliedMathematics and Computation, vol.204, no. 1, pp. 317–324, 2008.

[9] R. Rakkiyappan and P. Balasubramaniam, “New global exponential stability results for neutral typeneural networks with distributed time delays,” Neurocomputing, vol. 71, no. 4–6, pp. 1039–1045, 2008.

[10] L. Liu, Z. Han, and W. Li, “Global stability analysis of interval neural networks with discrete anddistributed delays of neutral type,” Expert Systems with Applications, vol. 36, no. 3, pp. 7328–7331,2009.

[11] R. Samidurai, S. M. Anthoni, and K. Balachandran, “Global exponential stability of neutral-typeimpulsive neural networks with discrete and distributed delays,” Nonlinear Analysis: Hybrid Systems,vol. 4, no. 1, pp. 103–112, 2010.

[12] R. Rakkiyappan, P. Balasubramaniam, and J. Cao, “Global exponential stability results for neutral-type impulsive neural networks,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 122–130, 2010.

[13] J. H. Park and O. M. Kwon, “Further results on state estimation for neural networks of neutral-typewith time-varying delay,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 69–75, 2009.

[14] J. Qiu and J. Cao, “Delay-dependent robust stability of neutral-type neural networks with timedelays,” Journal of Mathematical Control Science and Applications, vol. 1, pp. 179–188, 2007.

[15] J. Zhang, P. Shi, and J. Qiu, “Novel robust stability criteria for uncertain stochastic Hopfield neuralnetworkswith time-varying delays,”Nonlinear Analysis: RealWorld Applications, vol. 8, no. 4, pp. 1349–1357, 2007.

[16] L. Wang, Z. Zhang, and Y. Wang, “Stochastic exponential stability of the delayed reaction-diffusionrecurrent neural networks with Markovian jumping parameters,” Physics Letters A, vol. 372, no. 18,pp. 3201–3209, 2008.

[17] Y.Wu, Y.Wu, and Y. Chen, “Mean square exponential stability of uncertain stochastic neural networkswith time-varying delay,” Neurocomputing, vol. 72, no. 10–12, pp. 2379–2384, 2009.

[18] M. H. Jiang, Y. Shen, and X. X. Liao, “Robust stability of uncertain neutral linear stochastic differentialdelay system,” Applied Mathematics and Mechanics, vol. 28, no. 6, pp. 741–748, 2007.


[19] H. Zhang, M. Dong, Y. Wang, and N. Sun, “Stochastic stability analysis of neutral-type impulsiveneural networks with mixed time-varying delays and Markovian jumping,” Neurocomputing, vol. 73,no. 13–15, pp. 2689–2695, 2010.

[20] Q. Zhu and J. Cao, “Stability analysis for stochastic neural networks of neutral type with bothMarkovian jump parameters and mixed time delays,” Neurocomputing, vol. 73, no. 13–15, pp. 2671–2680, 2010.

[21] H. Bao and J. Cao, “Stochastic global exponential stability for neutral-type impulsive neural networkswith mixed time-delays and Markovian jumping parameters,” Communications in Nonlinear Scienceand Numerical Simulation, vol. 16, no. 9, pp. 3786–3791, 2011.

[22] X. Li, “Global robust stability for stochastic interval neural networks with continuously distributeddelays of neutral type,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4370–4384, 2010.

[23] H. Chen, Y. Zhang, and P. Hu, “Novel delay-dependent robust stability criteria for neutral stochasticdelayed neural networks,” Neurocomputing, vol. 73, no. 13–15, pp. 2554–2561, 2010.

[24] G. Liu, S. X. Yang, Y. Chai,W. Feng, andW. Fu, “Robust stability criteria for uncertain stochastic neuralnetworks of neutral-type with interval time-varying delays,” Neural Computing and Applications. Inpress.

[25] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series inMathematics & Applications, Horwood Publishing, Chichester, UK, 1997.

[26] Y. Y. Wang, L. Xie, and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,”Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992.

[27] K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proceedings of the39th IEEE Confernce on Decision and Control, pp. 2805–2810, Sydney, Australia, December 2000.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014


Stochastic AnalysisInternational Journal of

robust stochastic stability analysis for uncertain neutral ...robust stochastic stability analysis...

Documents