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Journal of the Franklin Institute 349 (2012) 1699–1720

Synchronization criteria for coupled stochasticneural networks with time-varying delays

and leakage delay

M.J. Parka, O.M. Kwona,n, Ju H. Parkb, S.M. Leec, E.J. Chad

aSchool of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro,

Heungduk-gu, Cheongju 361-763, Republic of KoreabDepartment of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong,

Kyongsan 712-749, Republic of KoreacSchool of Electronic Engineering, Daegu University, Gyungsan 712-714, Republic of KoreadDepartment of Biomedical Engineering, School of Medicine, Chungbuk National University,

52 Naesudong-ro, Heungduk-gu, Cheongju 361-763, Republic of Korea

Received 7 November 2011; received in revised form 12 January 2012; accepted 5 February 2012

Available online 13 February 2012

Abstract

This paper proposes new delay-dependent synchronization criteria for coupled stochastic neural

networks with time-varying delays and leakage delay. By constructing a suitable Lyapunov–

Krasovskii’s functional and utilizing Finsler’s lemma, novel synchronization criteria for the networks

are established in terms of linear matrix inequalities (LMIs) which can be easily solved by using the

LMI toolbox in MATLAB. Three numerical examples are given to illustrate the effectiveness of the

proposed methods.

& 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Complex networks have received increasing attention of researches from various fieldsof science and engineering such as the World Wide Web, social networks, electrical powergrids, global economic markets, and so on [1,2]. Also, in the real applications of systems,there exists naturally time-delay due to the finite information processing speed and the

www.elsevier.com/locate/jfranklin

0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfranklin.2012.02.002

nCorresponding author. Tel.: þ82 43 261 2422; fax: þ82 43 263 2419.

E-mail address: [email protected] (O.M. Kwon).


finite switching speed of amplifiers. It is well known that time-delay often causesundesirable dynamic behaviors such as performance degradation, and instability of thesystems. Therefore, recently, the problem of synchronization of coupled neural networkswith time-delay which is one of the hot research fields of complex networks has been achallenging issue due to its potential applications such as information science, biologicalsystems and so on [3,4].On the other hand, in implementation of many practical systems such as aircraft,

chemical and biological systems and electric circuits, there exist occasionally stochasticperturbations. It is no less important than the time-delay as a considerable factor affectingdynamics in the systems. Therefore, stochastic modeling with time-delay plays animportant role in many fields of science and engineering applications. For this reason,various approaches to stability criteria for stochastic systems with time-delay have beeninvestigated in the literature [5–8]. Xu et al. [5] studied the problem of stability analysisfor stochastic systems with parameter uncertainties and a class of nonlinearities. In [6],by the Lyapunov–Krasovskii’s functional based on the delay fractioning approach, anexponential stability criterion for stochastic systems with time-delay was presented. Kwon[7] derived the delay-dependent stability criteria for uncertain stochastic dynamic systemswith time-varying delays via the Lyapunov–Krasovskii’s functional approach with twodelay fraction numbers. Moreover, by choosing the Lyapunov matrices in the decomposedintegral intervals, stability analysis for stochastic neural networks with time-varying delaywere proposed in [8]. Above this, the study on the problems for various forms of complexnetworks or stochastic systems have been addressed. For more details, see the literature[9–16] and references therein.Recently, the synchronization stability problem for a class of complex dynamical

networks with Markovian jumping parameters and mixed time delays had been studied in[17]. The considered model in [17] has stochastic coupling term and stochastic disturbanceto reflect more realistic dynamical behaviors of the complex networks that are affected bynoisy environment. Very recently, a leakage delay, which is the time delay in leakage termof the systems and a considerable factor affecting dynamics for the worse in the systems, isbeing put to use in the problem of stability for neural networks [18,19]. Balasubramaniamet al. [18] investigated the problem of passivity analysis for neutral type neural networkswith Markovian jumping parameters and time delay in the leakage term. By use of thetopological degree theory, delay-dependent stability conditions of neural networks ofneutral type with time delay in the leakage term was proposed in [19]. However, to the bestof authors’ knowledge, delay-dependent synchronization analysis of coupled stochasticneural networks with time-varying delay and leakage delay has not been investigated yet.Here, delay-dependent analysis has been paid more attention than delay-independent onebecause the sufficient conditions for delay-dependent analysis make use of the informationon the size of time delay [20]. That is, the former is generally less conservative than thelatter.Motivated by the above discussions, the problem of new delay-dependent synchroniza-

tion criteria for coupled stochastic neural networks with time-varying delays in networkcoupling and leakage delay is considered. The coupled stochastic neural networks arerepresented as a simple mathematical model by use of Kronecker product technique. Then,by construction of a suitable Lyapunov–Krasovskii’s functional and utilization of Finsler’slemma, new synchronization criteria are derived in terms of LMIs which can be solvedefficiently by standard convex optimization algorithms [21]. In order to utilize Finsler’s

M.J. Park et al. / Journal of the Franklin Institute 349 (2012) 1699–17201700


lemma as a tool of getting less conservative synchronization criteria, it should be notedthat a new zero equality from the constructed mathematical model is devised. The conceptof scaling transformation matrix will be utilized in deriving zero equality of the method.Finally, three numerical examples are included to show the effectiveness of the proposedmethod.

Notation: Rn is the n-dimensional Euclidean space, and Rm�n denotes the set of m� n

real matrix. For symmetric matrices X and Y, X4Y (respectively, XZY ) means that thematrix X�Y is positive definite (respectively, nonnegative). In, 0n and 0m�n denote n� n

identity matrix, n� n and m� n zero matrices, respectively. En denotes the n� n matrixwhich all elements are 1. J � J refers to the Euclidean vector norm and the induced matrixnorm. lmaxð�Þ means the maximum eigenvalue of a given square matrix. diagf� � �g denotesthe block diagonal matrix. % represents the elements below the main diagonal of asymmetric matrix. For a given matrix X 2 Rm�n, such that rankðX Þ ¼ r, we define X? 2

Rn�ðn�rÞ as the right orthogonal complement of X; i.e., XX? ¼ 0. Let ðO,F,fFtgtZ0,PÞ becomplete probability space with a filtration fFtgtZ0 satisfying the usual conditions (i.e., it isright continuous and F0 contains all P-pull sets). Ef�g stands for the mathematicalexpectation operator with respect to the given probability measure P.

2. Problem statements

Consider the following neural networks with time-varying delays and leakage delay

_yðtÞ ¼�Ayðt�tÞ þW1gðyðtÞÞ þW2gðyðt�hðtÞÞÞ þ b, ð1Þ

where yðtÞ ¼ ½y1ðtÞ, . . . ,ynðtÞ�T 2 Rn is the neuron state vector, n denotes the number of

neurons in a neural network, gð�Þ ¼ ½g1ð�Þ, . . . ,gnð�Þ�T 2 Rn denotes the neuron activation

function vector with gð0Þ ¼ 0, b¼ ½b1, . . . ,bn�T 2 Rn means the constant external input

vector, A¼ diagfa1, . . . ,ang 2 Rn�nðak40,k¼ 1, . . . ,nÞ is the self-feedback matrix, Wk 2

Rn�nðk¼ 1; 2Þ are the connection weight matrices, and h(t) and t are time-varying delaysand leakage delay, respectively, satisfying

0rhðtÞrhM , _hðtÞrhD and 0ot, ð2Þ

where hM and t are positive constant scalars and hD is any constant one.In this paper, it is assumed that the activation functions satisfy the following

assumption:

Assumption 1. The neurons activation functions, gkð�Þ, are assumed to be nondecreasing,bounded and globally Lipschiz; that is

jgkðx1Þ�gkðx2Þjjx1�x2j

rlk, x1,x2 2 R, x1ax2, k¼ 1, . . . ,n, ð3Þ

where lk are positive constants.For simplicity, in stability analysis of the network (1), the equilibrium point

yn ¼ ½yn1, . . . ,y

nn �

T is shifted to the origin by utilization of the transformationxð�Þ ¼ yð�Þ�yn, which leads the system (1) to the following form:

_xðtÞ ¼ �Axðt�tÞ þW1f ðxðtÞÞ þW2f ðxðt�hðtÞÞÞ, ð4Þ

where xðtÞ ¼ ½x1ðtÞ, . . . ,xnðtÞ�T 2 Rn is the state vector of the transformed network.

M.J. Park et al. / Journal of the Franklin Institute 349 (2012) 1699–1720 1701


Also, the function f ð�Þ ¼ ½f1ð�Þ, . . . ,fnð�Þ�T with fqð�Þ ¼ gqð� þ yn

qÞ�gqðynqÞ and fqð0Þ ¼ 0

satisfies Assumption 1.In this paper, a model of coupled stochastic neural networks with time-varying delays in

network coupling and leakage delay is considered as

dxiðtÞ ¼ ½�Axiðt�tÞ þW1f ðxiðtÞÞ þW2f ðxiðt�hðtÞÞÞ� dtþXN

j ¼ 1

gijGxjðt�hðtÞÞðdtþ do1ðtÞÞ

þsiðt,xiðtÞ,xiðt�hðtÞÞÞ do2ðtÞ, i¼ 1; 2, . . . ,N, ð5Þ

where N is the number of couple nodes, xiðtÞ ¼ ½xi1ðtÞ, . . . ,xinðtÞ�T 2 Rn is the state vector of

the ith node, G 2 Rn�n is the constant inner-coupling matrix of nodes, which represent theindividual coupling between the subnetworks, G¼ ½gij �N�N is the outer-coupling matrixrepresenting the coupling strength, the topological structure of the networks satisfy thediffusive coupling connections

gij ¼ gjiZ0 ðiajÞ, gii ¼�XN

j ¼ 1,iaj

gij ði,j ¼ 1; 2, . . . ,NÞ,

and okðtÞðk¼ 1; 2Þ are m-dimensional Wiener processes (Brownian motion) onðO,F,fFtgtZ0,PÞ which satisfies the following definition.

Definition 1 (Morters and Peres [22]). A real-valued stochastic process oðtÞ is called aBrownian motion or Wiener process if the following holds:

(i) oð0Þ ¼ 0 a.s.,(ii) oðtÞ has independent increment, i.e., for all 0rt1rt2r � � �rtn, the increments

oðt2Þ�oðt1Þ, . . . ,oðtnÞ�oðtn�1Þ are independent random variables,(iii) the increments oðtÞ�oðsÞ are N ð0,t�sÞ for all 0rsrt, where N ðm,n2Þ denote the

normal distribution with expected value m and variance n2,(iv) oðtÞ is almost surely continuous.

It should be noted that

EfdokðtÞg ¼ 0, Efdo2kðtÞg ¼ dt:

Here, o1ðtÞ and o2ðtÞ, which are mutually independent, are the coupling strengthdisturbance and the system noise, respectively. And the nonlinear uncertainties sið�, � ,�Þ 2Rn�mði¼ 1, . . . ,NÞ are the noise intensity functions satisfying the Lipschitz condition andthe following assumption:

sTi ðt,xiðtÞ,xiðt�hðtÞÞÞsiðt,xiðtÞ,xiðt�hðtÞÞÞrJH1xiðtÞJ

2þ JH2xiðt�hðtÞÞJ2, ð6Þ

where Hkðk¼ 1; 2Þ are constant matrices with appropriate dimensions.

Remark 1. In this paper, the activation functions and the noise intensity functions areassumed to be satisfied the conditions (3) and (6) which are Lipschitz continuous andsatisfy the linear growth condition as well. Under the assumptions, according to the workin Chapter 5 of [23], it is easy that the ith stochastic neural networks (5) expressed bydelayed stochastic differential equation has a unique continuous solution. One can alsoconfirm this fact in [24].



For the convenience of stability analysis for the networks (5), the following Kroneckerproduct and its properties are used.

Lemma 1 (Graham [25]). Let � denote the notation of Kronecker product. Then, the

following properties of the Kronecker product are easily established:

(i) ðaAÞ � B¼A� ðaBÞ.(ii) ðAþ BÞ � C ¼A� C þ B� C.(iii) ðA� BÞðC �DÞ ¼ ðACÞ � ðBDÞ.(iv) ðA� BÞT ¼AT � BT .

Let us define

xðtÞ ¼ ½x1ðtÞ, . . . ,xNðtÞ�T , fðxðtÞÞ ¼ ½f ðx1ðtÞÞ, . . . ,f ðxN ðtÞÞ�

T ,

rðtÞ ¼ ½s1ð�, � ,�Þ, . . . ,sN ð�, � ,�Þ�T :

Then, with Kronecker product in Lemma 1, the network (5) can be represented as

dxðtÞ ¼ ½�ðIN � AÞxðt�tÞ þ ðIN �W1ÞfðxðtÞÞ þ ðIN �W2Þfðxðt�hðtÞÞÞ� dt

þðG � GÞxðt�hðtÞÞðdtþ do1ðtÞÞ þ rðtÞ do2ðtÞ: ð7Þ

For stability analysis, Eq. (7) can be rewritten as

dxðtÞ ¼ gðtÞ dtþ .ðtÞ dxðtÞ, ð8Þ

where

gðtÞ ¼ �ðIN � AÞxðt�tÞ þ ðIN �W1ÞfðxðtÞÞ þ ðIN �W2Þfðxðt�hðtÞÞÞ

þðG � GÞxðt�hðtÞÞ,

.ðtÞ ¼ ½ðG � GÞxðt�hðtÞÞ,rðtÞ�, xT ðtÞ ¼ ½oT1 ðtÞ,o

T2 ðtÞ�:

The aim of this paper is to investigate the delay-dependent synchronization stabilityanalysis of network (8) with time-varying delays in network coupling and leakage delay.In order to do this, the following definition and lemmas are needed.

Definition 2 (Liu and Chen [26]). The networks (5) are said to be asymptoticallysynchronized if the following condition holds:

limt-1

JxiðtÞ�xjðtÞJ¼ 0, i,j ¼ 1; 2, . . . ,N:

Lemma 2 (Cao et al. [3]). Let U ¼ ½uij �N�N , P 2 Rn�n, xT ¼ ½x1,x2, . . . ,xn�T , and

yT ¼ ½y1,y2, . . . ,yn�T . If U ¼UT and each row sum of U is zero, then

xT ðU � PÞy¼�X

1riojrN

uijðxi�xjÞT Pðyi�yjÞ:

Lemma 3 (Gu [27]). For any constant matrix M ¼MT40, the following inequality holds:

�hðtÞ

Z t

t�hðtÞ

_xT ðsÞM _xðsÞ dsrxðtÞ

xðt�hðtÞÞ

" #T�M M

% �M

� �xðtÞ

xðt�hðtÞÞ

" #: ð9Þ



Lemma 4 (Finsler’s lemma [28]). Let z 2 Rn, F¼FT 2 Rn�n, and B 2 Rm�n such that

rankðBÞon. The following statements are equivalent:

(i) zTFzo0,8Bz¼ 0,za0,(ii) B?

TFB?o0,(iii) (X 2 Rn�m : Fþ XBþ BT X To0.

3. Main results

In this section, new synchronization criteria for network (8) will be proposed. For thesake of simplicity on matrix representation, eiði¼ 1, . . . ,7Þ 2 R7n�n and gij are defined asblock entry matrices (For example, e2 ¼ ½0n,In,0n,0n,0n,0n,0n�

T ) and the (i,j)th entry of amatrix GT G¼ ½gij�N�N , respectively. The notations of several matrices are defined as:

zTðtÞ ¼ ½xT ðtÞ,xT ðt�tÞ,xT ðt�hðtÞÞ,xT ðt�hM Þ,f

T ðxðtÞÞ,fT ðxðt�hðtÞÞÞ,gT ðtÞ�,

U¼ ½0nN ,�ðIN � AÞ,ðG � GÞ,0nN ,ðIN �W1Þ,ðIN �W2Þ,�InN �,

zijðtÞ ¼ xiðtÞ�xjðtÞ,f ðzijðtÞÞ ¼ f ðxiðtÞÞ�f ðxjðtÞÞ,ZijðtÞ ¼ ZiðtÞ�ZjðtÞ,

zTij ðtÞ ¼ ½z

Tij ðtÞ,z

Tij ðt�tÞ,z

Tij ðt�hðtÞÞ,zT

ij ðt�hM Þ,fT ðzijðtÞÞ,f

T ðzijðt�hðtÞÞÞ,ZTij ðtÞ�,

Uij ¼ ½0n,�A,�ðNgijGÞ,0n,W1,W2,�In�,

X1 ¼ e1P1eT7 þ e7P1e

T1 þ e3ðNgijG

T P1GÞeT3 þ e1ðrHT

1 H1ÞeT1 þ e3ðrHT

2 H2ÞeT3 ,

X2 ¼ e1ðQ1 þQ2ÞeT1�e3ð1�hDÞQ1e

T3�e4Q2e

T4 ,

X3 ¼ e7ðh2MR1Þe

T7 �ðe1�e3ÞRðe1�e3Þ

T�ðe3�e4ÞRðe3�e4Þ

T

�ðe1�e3ÞTðe3�e4ÞT�ðe3�e4ÞT

T ðe1�e3ÞT ,

X4 ¼ e1S1eT1 �e2S1e

T2 þ e7ðt2S2Þe

T7 �e1S2e

T1 þ e1S2eT

2 þ e2S2eT1�e2S2e

T2 ,

X5 ¼ e1LT1 D1L1e

T1�e5D1eT

5 ,

X6 ¼ e3LT2 D2L2e

T3�e6D2eT

6 ,

F¼X6l ¼ 1

Xl : ð10Þ

Then, the main results of this paper are presented as follows:

Theorem 1. For given scalars 0ohM , hD and 0ot, the network (8) is asymptotically

synchronized for 0rhðtÞrhM and _hðtÞrhD, if there exist a positive scalar r, positive definite

matrices P1, Q1, Q2, R, S1, S2, positive diagonal matrices D1, D2, and any matrix T satisfying

the following LMIs for 1riojrN:

P1�rInr0, ð11Þ



R T

% R

� �Z0, ð12Þ

fððj�iÞUijÞ?gTFfððj�iÞUijÞ

?go0, ð13Þ

where Uij and F are defined in Eq. (10).

Proof. Define a matrix U as

U ¼ ½uij �N�N ¼NIN�EN : ð14Þ

Consider the following Lyapunov–Krasovskii’s functional candidate as

V ¼V1 þ V2 þ V3 þ V4, ð15Þ

where

V1 ¼ xT ðtÞðU � P1ÞxðtÞ,

V2 ¼

Z t

t�hðtÞ

xT ðsÞðU �Q1ÞxðsÞ dsþ

Z t

t�hM

xT ðsÞðU �Q2ÞxðsÞ ds,

V3 ¼ hM

Z t

t�hM

Z t

s

gT ðuÞðU � RÞgðuÞ du ds,

V4 ¼

Z t

t�txT ðsÞðU � S1ÞxðsÞ dsþ t

Z t

t�t

Z t

s

gT ðuÞðU � S2ÞgðuÞ du ds:

By the weak infinitesimal operator L in [29] and Lemma 2, the upper bound of LV1 can beobtained as

LV1 ¼ 2xT ðtÞðU � P1ÞgðtÞ þ .T ðtÞðU � P1Þ.ðtÞ

¼ 2xT ðtÞðU � P1ÞgðtÞ þ xT ðt�hðtÞÞðG � GÞT ðU � P1ÞðG � GÞxðt�hðtÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}C1

þ rT ðtÞðU � P1ÞrðtÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}C2

r2xT ðtÞðU � P1ÞgðtÞ þ xT ðt�hðtÞÞðGT NG � GT P1GÞxðt�hðtÞÞ

þr xT ðtÞðU �HT1 H1ÞxðtÞ þ xT ðt�hðtÞÞðU �HT

2 H2Þxðt�hðtÞÞ� �

¼X

1riojrN

zTij ðtÞX1zijðtÞ, ð16Þ

where X1 is defined in Eq. (10).Here, by properties of Kronecker product in Lemma 1 and UG ¼GU ¼NG, the term

C1 of LV1 is calculated as

C1 ¼ xT ðt�hðtÞÞðG � GÞT ðUG � P1GÞxðt�hðtÞÞ

¼ xT ðt�hðtÞÞðGT � GT ÞðNG � P1GÞxðt�hðtÞÞ

¼ xT ðt�hðtÞÞðGT NG � GT P1GÞxðt�hðtÞÞ, ð17Þ

and, if P1rrIn, then the upper bound of term C2 of LV1 is calculated as

C2rlmaxðP1ÞfrT ðtÞðU � InÞrðtÞg

rrfxT ðtÞðU �HT1 H1ÞxðtÞ þ xT ðt�hðtÞÞðU �HT

2 H2Þxðt�hðtÞÞg: ð18Þ



Calculating LV2 gives

LV2rxT ðtÞðU �Q1ÞxðtÞ�ð1�hDÞxT ðt�hðtÞÞðU �Q1Þxðt�hðtÞÞ

þxT ðtÞðU �Q2ÞxðtÞ�xT ðt�hMÞðU �Q2Þxðt�hMÞ

¼X

1riojrN


where X2 is defined in Eq. (10).From Lemmas 2 and 3 and Theorem 1 in [30], the calculation of LV3 leads to

LV3 ¼ h2MgT ðtÞðU � RÞgðtÞ

�hM

Z t

t�hðtÞ

gT ðsÞðU � RÞgðsÞ dsþ

Z t�hðtÞ

t�hM

gT ðsÞðU � RÞgðsÞ ds

� �rh2

MgT ðtÞðU � RÞgðtÞ

�

R t

t�hðtÞgðsÞ dsR t�hðtÞ

t�hMgðsÞ ds

24

35T

1

hðtÞh�1M

ðU � RÞ 0nN

%

1

1�hðtÞh�1M

ðU � RÞ

26664

37775R t


t�hMgðsÞ ds

24

35

¼ h2MgT ðtÞðU � RÞgðtÞ

�X

1riojrN

R t

t�hðtÞZijðsÞ dsR t�hðtÞ

t�hMZijðsÞ ds

24

35T

1

hðtÞh�1M

� �R 0n

%

1

1�hðtÞh�1M

� �R

26664

37775R t


t�hMZijðsÞ ds

24

35

rh2MgT ðtÞðU � RÞgðtÞ

�X

1riojrN

R t


t�hMZijðsÞ ds

24

35T

R T

% R

� � R t


t�hMZijðsÞ ds

24

35

¼ h2MgT ðtÞðU � RÞgðtÞ

�

R t


t�hMgðsÞ ds

24

35T

ðU � RÞ ðU � TÞ

% ðU � RÞ

" # R t


t�hMgðsÞ ds

24

35: ð20Þ

Since the following two zero equalities hold from [31]

xðtÞ�xðt�hðtÞÞ�

Z t

t�hðtÞ

gðsÞ ds�

Z t

t�hðtÞ

.ðsÞ dxðsÞ ¼ 0, ð21Þ

xðt�hðtÞÞ�xðt�hM Þ�

Z t�hðtÞ

t�hM

gðsÞ ds�

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ ¼ 0, ð22Þ

we have

�

Z t

t�hðtÞ

gðsÞ ds

� �T

ðU � RÞ

Z t

t�hðtÞ

gðsÞ ds

� �

r�ðxðtÞ�xðt�hðtÞÞÞT ðU � RÞðxðtÞ�xðt�hðtÞÞÞ



þ2ðxðtÞ�xðt�hðtÞÞÞT ðU � RÞ

Z t

t�hðtÞ

.ðsÞ dxðsÞ

�

Z t

t�hðtÞ

.ðsÞ dxðsÞ

� �T

ðU � RÞ

Z t

t�hðtÞ

.ðsÞ dxðsÞ

� �, ð23Þ

�

Z t�hðtÞ

t�hM

gðsÞ ds

� �T

ðU � RÞ

Z t�hðtÞ

t�hM

gðsÞ ds

� �r�ðxðt�hðtÞÞ�xðt�hMÞÞ

TðU � RÞðxðt�hðtÞÞ

�xðt�hMÞÞ þ 2 xðt�hðtÞÞ�xðt�hM Þð ÞTðU � RÞ

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

�

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

� �T

ðU � RÞ

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

� �, ð24Þ

�2

Z t

t�hðtÞ

gðsÞ ds

� �T

ðU � TÞ

Z t�hðtÞ

t�hM

gðsÞ ds

� �r�2ðxðtÞ�xðt�hðtÞÞÞT ðU � TÞðxðt�hðtÞÞ

�xðt�hMÞÞ þ 2ðxðtÞ�xðt�hðtÞÞÞT ðU � TÞ

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

þ 2ðxðt�hðtÞÞ�xðt�hM ÞÞTðU � TÞ

Z t

t�hðtÞ

.ðsÞ dxðsÞ

� xðtÞ�xðt�hðtÞÞ�

Z t

t�hðtÞ

gðsÞ ds

� �T

ðU � TÞ

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

� xðt�hðtÞÞ�xðt�hMÞ�

Z t�hðtÞ

t�hM

gðsÞ ds

� �T

ðU � TÞ

Z t

t�hðtÞ

.ðsÞ dxðsÞ: ð25Þ

It should be noted that

�

Z t

t�hðtÞ

.ðsÞ dxðsÞ

� �T

ðU � RÞ

Z t

t�hðtÞ

.ðsÞ dxðsÞ

� �r0, ð26Þ

�

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

� �T

ðU � RÞ

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

� �r0: ð27Þ

Then, an upper bound of LV3 can be rewritten as

LV3rgT ðtÞðU � h2MRÞgðtÞ�ðxðtÞ�xðt�hðtÞÞÞT ðU � RÞðxðtÞ�xðt�hðtÞÞÞ

�ðxðt�hðtÞÞ�xðt�hMÞÞTðU � RÞðxðt�hðtÞÞ�xðt�hMÞÞ

�2ðxðtÞ�xðt�hðtÞÞÞT ðU � TÞðxðt�hðtÞÞ�xðt�hMÞÞ þ x1ðdxðtÞÞ,

¼X

1riojrN

zTij ðtÞX3zijðtÞ þ x1ðdxðtÞÞ, ð28Þ

where X3 is defined in Eq. (10), and

x1ðdxðtÞÞ ¼ 2ðxðtÞ�xðt�hðtÞÞÞT ðU � RÞ

Z t

t�hðtÞ

.ðsÞ dxðsÞ þ 2ðxðt�hðtÞÞ



�xðt�hMÞÞTðU � RÞ

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ

þ xðt�hðtÞÞ�xðt�hMÞ þ

Z t�hðtÞ

t�hM

gðsÞ ds

� �T

ðU � TÞ

Z t

t�hðtÞ

.ðsÞ dxðsÞ

þ xðtÞ�xðt�hðtÞÞ þ

Z t

t�hðtÞ

gðsÞ ds

� �T

ðU � TÞ

Z t�hðtÞ

t�hM

.ðsÞ dxðsÞ:

Calculating LV4 obtains

LV4rxT ðtÞðU � S1ÞxðtÞ�xT ðt�tÞðU � S1Þxðt�tÞ

þt2gT ðtÞðU � S2ÞgðtÞ�

Z t

t�tgðsÞ ds

� �T

ðU � S2Þ

Z t

t�tgðsÞ ds

� �rxT ðtÞðU � S1ÞxðtÞ�x

T ðt�tÞðU � S1Þxðt�tÞþt2gT ðtÞðU � S2ÞgðtÞ�ðxðtÞ�xðt�tÞÞ

TðU � S2ÞðxðtÞ�xðt�tÞÞ þ x2ðdxðtÞÞ

¼X

1riojrN

zTij ðtÞX4zijðtÞ þ x2ðdxðtÞÞ, ð29Þ

where X4 is defined in Eq. (10)

x2ðdxðtÞÞ ¼ 2ðxðtÞ�xðt�tÞÞT ðU � S2Þ

Z t

t�t.ðsÞ dxðsÞ,

and the similar process of obtaining the upper bound of LV3 were utilized in calculation ofan upper bound of LV4.For any diagonal matrices Dk40ðk¼ 1; 2Þ and from Assumption 1, the following

inequalities hold:

0rX

1riojrN

fzTij ðtÞL

T1 D1L1zijðtÞ�f T ðzijðtÞÞD1f ðzijðtÞÞg

¼X

1riojrN


0rX

1riojrN

fzTij ðt�hðtÞÞLT

2 D2L2zijðt�hðtÞÞ�f T ðzijðt�hðtÞÞÞD2f ðzijðt�hðtÞÞÞg

¼X

1riojrN


where Xkðk¼ 5; 6Þ are defined in Eq. (10).From Eqs. (16)–(31) and by application of S-procedure [21], LV has a new upper

bound as

LVrX

1riojrN

zTij ðtÞFzijðtÞ þ x1ðdxðtÞÞ þ x2ðdxðtÞÞ, ð32Þ

where F, zijðtÞ and xkðdxðtÞÞðk¼ 1; 2Þ are defined in Eqs. (10), (28), and (29).Also, the network (8) with the augmented vector zijðtÞ can be rewritten asX

1riojrN

ðj�iÞUijzijðtÞ ¼ 0, ð33Þ

where Uij is defined as in Eq. (10).



Here, in order to illustrate the process of obtaining Eq. (33), let us define

L¼ ½L1,L2, . . . ,LN � ¼ ½N,N�1, . . . ,1� � In 2 Rn�nN , ð34Þ

where Lkðk¼ 1, . . . ,NÞ 2 Rn�n.By Eqs. (8) and (14) and properties of Kronecker product in Lemma 1, we have the

following zero equality:

0¼LðU � InÞUzðtÞ

¼LðU � InÞ½0nN ,�ðIN � AÞ,ðG � GÞ,0nN ,ðIN �W1Þ,ðIN �W2Þ,�InN �zðtÞ¼L½0nN ,�ðUIN � InAÞ,ðUG � InGÞ,0nN ,ðUIN � InW1Þ,ðUIN � InW2Þ,�ðU � InÞ�zðtÞ

¼L½0nN ,�ðU � AÞ,ðNG � GÞ,0nN ,ðU �W1Þ,ðU �W2Þ,�ðU � InÞ�zðtÞ¼�LðU � AÞxðt�tÞ þ LðNG � GÞxðt�hðtÞÞ þ LðU �W1ÞfðxðtÞÞ

þLðU �W2Þfðxðt�hðtÞÞÞ�LðU � InÞgðtÞ, ð35Þ

where U and zðtÞ are defined in Eq. (10).By Lemma 2, the first term of Eq. (35) can be obtained as

�LðU � AÞxðt�tÞ ¼ �½NIn,ðN�1ÞIn, . . . ,In�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n�nN

ðU � AÞ|fflfflfflfflffl{zfflfflfflfflffl}nN�nN

½x1ðt�tÞ, . . . ,xN ðt�tÞ�T|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}nN�1

¼X

1riojrN

uijðLi�LjÞAðxiðt�tÞ�xjðt�tÞÞ

¼�X

1riojrN

ðLi�LjÞAðxiðt�tÞ�xjðt�tÞÞ

¼�X

1riojrN

ððN þ 1�iÞInÞ�ðN þ 1�jÞInÞÞAðxiðt�tÞ�xjðt�tÞÞ

¼�X

1riojrN

ðj�iÞInAðxiðt�tÞ�xjðt�tÞÞ,

¼�X

1riojrN

ðj�iÞAðxiðt�tÞ�xjðt�tÞÞ: ð36Þ

Similarly, the other terms of Eq. (35) are calculated as

LðNG � GÞxðt�hðtÞÞ ¼ ½NIn, . . . ,In�ðNG � GÞ½x1ðt�hðtÞÞ, . . . ,xNðt�hðtÞÞ�T

¼�X

1riojrN

NgijðLi�LjÞGðxiðt�hðtÞÞ�xjðt�hðtÞÞÞ

¼ �X

1riojrN

ðj�iÞðNgijGÞðxiðt�hðtÞÞ�xjðt�hðtÞÞÞ, ð37Þ

LðU �W1ÞfðxðtÞÞ ¼ ½NIn, . . . ,In�ðU �W1Þ½f ðx1ðtÞÞ, . . . ,f ðxNðtÞÞ�T

¼�X

1riojrN

uijðLi�LjÞW1ðf ðxiðtÞÞ�f ðxjðtÞÞÞ

¼X

1riojrN

ðj�iÞW1ðf ðxiÞÞ�f ðxjðtÞÞÞ, ð38Þ

LðU �W2Þfðxðt�hðtÞÞÞ ¼ ½NIn, . . . ,In�ðU �W2Þ½f ðx1ðt�hðtÞÞÞ, . . . ,f ðxN ðt�hðtÞÞÞ�T

¼�X

1riojrN

uijðLi�LjÞW2ðf ðxiðt�hðtÞÞÞ�f ðxjðt�hðtÞÞÞÞ



¼X

1riojrN

ðj�iÞW2ðf ðxiðt�hðtÞÞÞ�f ðxjðt�hðtÞÞÞÞ, ð39Þ

�LðU � InÞgðtÞ ¼�½NIn, . . . ,In�ðU � InÞ½Z1ðtÞ, . . . ,ZN ðtÞ�T

¼X

1riojrN

uijðLi�LjÞInðZiðtÞÞ�ZjðtÞÞ

¼ �X

1riojrN

ðj�iÞInðZiðtÞ�ZjðtÞÞ: ð40Þ

Then, Eq. (35) can be rewritten as

0¼LðU � InÞUzðtÞ¼ �LðU � AÞxðt�tÞ þ LðNG � GÞxðt�hðtÞÞ þ LðU �W1ÞfðxðtÞÞ

þLðU �W2Þfðxðt�hðtÞÞÞ�LðU � InÞgðtÞ

¼ �X

1riojrN

ðj�iÞAðxiðt�tÞ�xjðt�tÞÞ

�X

1riojrN

ðj�iÞðNgijGÞðxiðt�hðtÞÞ�xjðt�hðtÞÞÞ

þX

1riojrN

ðj�iÞW1ðf ðxiðtÞÞ�f ðxjðtÞÞÞ

þX

1riojrN

ðj�iÞW2ðf ðxiðt�hðtÞÞÞ�f ðxjðt�hðtÞÞÞÞ

�X

1riojrN

ðj�iÞInðZiðtÞ�ZjðtÞÞ

¼X

1riojrN

ðj�iÞUijzijðtÞ, ð41Þ

where Uij and zijðtÞ are defined in Eq. (10).Therefore, if Eq. (33) holds, then, a synchronization condition for network (8) isX

1riojrN

zTij ðtÞFzijðtÞo0 subject to

X1riojrN

ðj�iÞUijzijðtÞ ¼ 0, ð42Þ

where F is defined in Eq. (10).Here, if the inequality (42) holds, then there exist a positive scalar E such that

Fo�EI7n: ð43Þ

From Eqs. (32), (42), and (43), by utilization of the mathematical expectation on Eq. (32),we have

EfLVgrEX

1riojrN

zTij ðtÞFzijðtÞ þ x1ðdxðtÞÞ þ x2ðdxðtÞÞ

( )

oEX

1riojrN

zTij ðtÞð�EI7nÞzijðtÞ

( )

rEX

1riojrN

zTij ðtÞð�EInÞzijðtÞ

( )



¼ EX

1riojrN

ð�EJzijðtÞJ2Þ

( )

¼ EX

1riojrN

ð�EJxiðtÞ�xjðtÞJ2Þ

( ), ð44Þ

where EfxkðdxðtÞÞg ¼ 0ðk¼ 1; 2Þ according to work [5].By Lyapunov theorem and Definition 1, it can be guaranteed that the subnetworks in the

coupled stochastic neural networks (8) are asymptotically synchronized.Finally, by use of Lemma 4, the condition (42) is equivalent to the following inequality:X

1riojrN

fððj�iÞUijÞ?gTFfððj�iÞUijÞ

?go0: ð45Þ

From Eq. (45), if the LMIs (13) satisfy, then stability condition (42) holds. This completesour proof. &

When the information about _hðtÞ is unknown, then by setting Q1 ¼ 0n in the Lyapunov–Krasovskii’s functional (15), the following corollary can be obtained.

Corollary 1. For given scalars 0ohM and 0ot, the network (8) is asymptotically

synchronized for 0rhðtÞrhM , if there exist a positive scalar r, positive definite matrices

P1, Q2, R, S1, S2, positive diagonal matrices D1, D2, and any matrix T satisfying the

following LMIs for 1riojrN:

P1�rInr0, ð46Þ

R T

% R

� �Z0, ð47Þ

fððj�iÞUijÞ?gT Ffððj�iÞUijÞ

?go0, ð48Þ

where Uij is defined in Eq. (10), and F ¼F�e1Q1eT1 þ ð1�hDÞe3Q1eT

3 .

Proof. The proof of Corollary 1 is very similar to the proof of Theorem 1, so it is omitted. &

Remark 2. In order to induce a new zero equality (33), the matrix L in Eq. (34) wasdefined. It is inspired by the concept of scaling transformation matrix. To reduce decisionvariable, Finsler’s lemma (ii) B?

TFB?o0 without free-weighting matrices was used. Atthis time, a zero equality is required. If the matrix L is not considered, then the followingdescription (see only Eq. (36) as example):

�f gðU � AÞxðt�tÞ ¼ �f gðU � AÞ½x1ðt�tÞ, . . . ,xN ðt�tÞ�T

¼X

1riojrN

f�gAðxiðt�tÞ�xjðt�tÞÞ ð49Þ

does not hold. Thus, the derivation of zero equality in Eq. (33) is impossible. Here, in orderto use Lemma 2, a suitable vector or matrix in the empty parentheses f g is needed.Therefore, by defining the matrix L, the induction of the zero equality (33) is possible.

Remark 3. The equality LðU � InÞUzðtÞ ¼ 0 in Eq. (35) is similar to the concept ofeigenvectors for the square matrix. For example, the nonzero vector x is an eigenvetor ofthe square matrix A if there is a scale factor l such that Ax¼ lx. The eigenvector of A are



not changed by the matrix A except for the scale factor l. That is, the equalityUzðtÞ ¼LðU � InÞUzðtÞ ¼ 0 with zðtÞa0 can be considered as an operator in the augmentedvector zðtÞ. The augmented vector zðtÞ is also not changed by the operator U except for thematrix LðU � InÞ.

Remark 4. In this paper, the problem of new delay-dependent synchronization criteria forcoupled stochastic neural networks with leakage delay is considered. By using Finsler’slemma without free-weighting matrices, the proposed synchronization criteria for thenetwork are formulated in terms of LMIs. Here, as mentioned in Introduction, the leakagedelay is the time delay in leakage or forgetting term of the systems and a considerablefactor affecting dynamics for the worse in the systems. The effect of the leakage delaywhich cannot be negligible is shown in Fig. 1. In addition, in the field of delay-dependentstability or synchronization analysis, one of major concerns is to get maximum delaybounds with fewer decision variables [7,32,33]. By utilizing Finsler lemma, one caneliminate free variables which were used in zero equalities in the works [7,33]. FromLemma 4, one can check that the B?

TFB?o0 is equivalent to the existence of X such thatFþ XBþ BT X To0 holds. Insertion of such an additional matrix X does not play a role toreduce the conservatism of B?

TFB?o0. It only increases the number of decision variables.Therefore, our proposed synchronization criteria are derived in the form of (ii) in Lemma4. To do this, the new zero equality (33) with Kronecker product technique was devised,which has not been deduced yet in the literature.

4. Numerical examples

In this section, we provide three numerical examples to illustrate the effectiveness of theproposed synchronization criteria in this paper.

Example 1. Consider the following coupled stochastic neural networks with three nodes:

dxiðtÞ ¼ ½�Axiðt�tÞ þW1f ðxiðtÞÞ þW2f ðxiðt�hðtÞÞÞ� dt

þX3j ¼ 1

gijGxjðt�hðtÞÞðdtþ do1ðtÞÞ þ siðt,xiðtÞ,xiðt�hðtÞÞÞ do2ðtÞ, ð50Þ

where

A¼3:5 0

0 1:5

� �, W1 ¼

�1 0:4

0 �0:1

� �, W2 ¼

0:4 �0:6

0:2 0:4

� �,

G¼0:2 0

0 0:2

� �, G¼

�2 1 1

1 �2 1

1 1 �2

264

375,

with

f ðxÞ ¼ 14ðjxþ 1j�jx�1jÞ, L1 ¼ L2 ¼ diagf0:5,0:5g,

sTi ðt,xiðtÞ, xiðt�hðtÞÞÞsiðt,xiðtÞ,xiðt�hðtÞÞÞrJxiðtÞJ

2þ Jxiðt�hðtÞÞJ2:

For the networks above, the results of maximum bounds of time-delay with different hD

and t by Theorem 1 are listed in Table 1. Also, when the value of the time-derivative of



time-delay is unknown, then by applying Corollary 1 to the above system (50), maximumdelay bounds for guaranteeing the asymptotic stability can be obtained as listed in Table 1.In order to confirm the obtained results with the time-varying delay conditions in Table 2,the simulation results for the synchronization errors trajectories, zi1ðtÞ ¼ xiðtÞ�x1ðtÞ

(i¼ 2; 3), of the networks (50) are shown in Figs. 1–3. These figures show that the networkswith the errors converge to zero for given initial values of the state by randomly choosing.Specially, the simulation results in Fig. 1 show synchronization errors trajectories for the

0 1 2 3 4 5−3

−2

−1

0

1

2

3

Time (Seconds)

z(t)

0 1 2 3 4 5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (Seconds)

z(t)

0 50 100 150−3

−2

−1

0

1

2

3

Time (Seconds)

z(t)

0 50 100 150−1.5

−1

−0.5

0

0.5

1

1.5

Time (Seconds)

z(t)

0 10 20 30 40 50−150

−100

−50

0

50

100

150

200

Time (Seconds)

z(t)

0 10 20 30 40 50−250

−200

−150

−100

−50

0

50

100

150

200

Time (Seconds)

z(t)

Fig. 1. Synchronization errors trajectories with C1-1: (top) t¼ 0:05, (middle) t¼ 0:48 and (bottom) t¼ 0:49(Example 1).



values of leakage delay, t, by 0.05, 0.48 and 0.49 with fixed values hM¼1.4092 and hD¼0.2.It is easy to illustrate that the larger value of leakage delay gives the worse dynamicbehaviors of the networks.

Example 2. Consider the following coupled stochastic neural networks with five nodes:

dxiðtÞ ¼ ½�Axiðt�tÞ þW1f ðxiðtÞÞ þW2f ðxiðt�hðtÞÞÞ� dt

þX5j ¼ 1

gijGxjðt�hðtÞÞðdtþ do1ðtÞÞ þ siðt,xiðtÞ,xiðt�hðtÞÞÞ do2ðtÞ, ð51Þ

Table 1

Maximum bounds hM of time-delay with different hD and t (Example 1).

t hD

0 0.2 0.5 0.8 Unknown

0.05 1.6602 1.4092 1.1565 0.9849 0.8826

0.1 1.4844 1.2572 1.0300 0.8411 0.7662

0.2 1.1043 0.8767 0.4820 0.4243 0.4243

0.29 0.0504 0.0436 0.0436 0.0436 0.0436

0.3 Infeasible

Table 2

The conditions of simulation in Example 1.

No. t hM hD h(t)

0.05

C1-1 0.48 1.4092 0.2 1:4092 sin2ð0:14tÞ

0.49

C1-2 0.2 0.4820 0.5 0:4820 sin2ð1:03tÞ

C1-3 0.29 0.0436 Unknown 0:0436jsinðtÞj

0 1 2 3 4 5−3

−2

−1

0

1

2

3

Time (Seconds)

z(t)

0 1 2 3 4 5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (Seconds)

z(t)

Fig. 2. Synchronization errors trajectories with C1-2 (Example 1).



where

A¼1:4 0

0 1

� �, W1 ¼

�1:5 0:4

0:1 �0:2

� �, W2 ¼

0:5 �0:2

0:1 �0:5

� �,

G¼0:1 0:1

0 0:2

� �, G¼

�2 1 0 0 1

1 �3 1 1 0

0 1 �2 1 0

0 1 1 �3 1

1 0 0 1 �2

26666664

37777775,

with

f ðxÞ ¼ 110ðjxþ 1j�jx�1jÞ,L1 ¼ L2 ¼ diagf0:2,0:2g,

sTi ðt,xiðtÞ,xiðt�hðtÞÞÞsiðt,xiðtÞ,xiðt�hðtÞÞÞrJxiðtÞJ

2þ Jxiðt�hðtÞÞJ2:

The results of maximum bounds of time-delay with different hD and fixed t¼ 0:01 byTheorem 1 and Corollary 1 are listed in Table 3. Figs. 4–7 show for the state responses,xiðtÞði¼ 1; 2,3; 4,5Þ, and the synchronization errors trajectories, zi1ðtÞ ¼ xiðtÞ�x1ðtÞ

(i¼ 2; 3,4; 5) of system (51) with the conditions of time-delay in Table 4 when initialvalues of the state are xT

1 ð0Þ ¼ ½1,�1�, xT2 ð0Þ ¼ ½�1:5,1:2�, xT

3 ð0Þ ¼ ½2:0,�1:6�, xT4 ð0Þ ¼

½0:7,1:1� and xT5 ð0Þ ¼ ½�1:2,�1:8�. Figures show that the system (51) with the errors

converge to zero. This implies the synchronization of the network (51).

0 1 2 3 4 5−3

−2

−1

0

1

2

3

Time (Seconds)

z(t)

0 1 2 3 4 5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (Seconds)

z(t)


Table 3

Maximum bounds hM of time-delay with different hD and fixed t¼ 0:01 (Example 2).

hD 0 0.2 0.5 0.8 Unknown

hM 0.5847 0.4827 0.3712 0.3712 0.3712



0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (Seconds)

x(t)

Fig. 4. State responses with C2-1 (Example 2).

0 0.5 1 1.5 2 2.5 3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time (Seconds)

z(t)

0 0.5 1 1.5 2 2.5 3−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (Seconds)

z(t)

0 0.5 1 1.5 2 2.5 3−0.5

0

0.5

1

1.5

2

2.5

Time (Seconds)

z(t)

0 0.5 1 1.5 2 2.5 3−2.5

−2

−1.5

−1

−0.5

0

0.5

Time (Seconds)

z(t)




0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time (Seconds)

x(t)

Fig. 6. State responses with C2-2 (Example 2).

0 0.5 1 1.5 2 2.5 3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time (Seconds)

z(t)

0 0.5 1 1.5 2 2.5 3−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (Seconds)

z(t)

0 0.5 1 1.5 2 2.5 3−0.5

0

0.5

1

1.5

2

2.5

Time (Seconds)

z(t)

0 0.5 1 1.5 2 2.5 3−2.5

−2

−1.5

−1

−0.5

0

Time (Seconds)

z(t)




Example 3. Recall the coupled stochastic neural networks in Example 1 with the followingchanged parameters:

G¼ 2In, H1 ¼ 4In, H2 ¼H1:

The result of maximum bound of time-delay with hD¼0.2 and t¼ 0:05 by Theorem 1 is0.0874. With the condition of time-varying delay C3 as hðtÞ ¼ 0:0874 sin2ð2:28tÞ and

Table 4

The conditions of simulation in Example 2.

No. t hM hD h(t)

C2-1 0.01 0.4827 0.2 0:2 sinðtÞ þ 0:2827C2-2 0.01 0.3712 Unknown 0:3712jsinðtÞj

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (Seconds)

x(t)

Fig. 8. State responses with C3 (Example 3).

0 0.2 0.4 0.6 0.8 1−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Time (Seconds)

z(t)

0 0.2 0.4 0.6 0.8 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (Seconds)

z(t)

Fig. 9. Synchronization errors trajectories with C3 (Example 3).



t¼ 0:05, the simulation results for the state responses, xi(t) (i¼ 1; 2,3), and thesynchronization errors trajectories, zi1ðtÞ ¼ xiðtÞ�x1ðtÞ (i¼ 2; 3), of the networks areshown in Figs. 8 and 9.

5. Conclusions

In this paper, the delay-dependent synchronization criteria for the coupled stochastic neuralnetworks with time-varying delays in network coupling and leakage delay have been proposed.To do this, a suitable Lyapunov–Krasovskii’s functional was used to investigate the feasibleregion of stability criteria. By establishment of a new zero equality and utilization of Finsler’slemma, sufficient conditions for guaranteeing asymptotic synchronization for the concernednetworks have been derived in terms of LMIs. Three numerical examples have been given toshow the effectiveness and usefulness of the presented criteria.

Acknowledgments

This research was supported by Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (2011-0001045), and by a grant of the Korea Healthcare Technology R&DProject, Ministry of Health & Welfare, Republic of Korea (A100054).

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