Synchronization of stochastic perturbed chaotic neural networks with mixed delays

Journal of the Franklin Institute 347 (2010) 1266–1280

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doi:10.1016/j

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Synchronization of stochastic perturbed chaoticneural networks with mixed delays$

Xiaodi Lia,�, Changming Dinga, Quanxin Zhub

aSchool of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, PR ChinabDepartment of Mathematics, Ningbo University, Ningbo 315211, Zhejiang, PR China

Received 25 October 2009; received in revised form 8 May 2010; accepted 1 June 2010

Abstract

In this paper, we study the synchronization problem of a class of chaotic neural networks with

time-varying delays and unbounded distributed delays under stochastic perturbations. By using

Lyapunov–Krasovskii functional, drive-response concept, output coupling with delay feedback and

linear matrix inequality (LMI) approach, we obtain some sufficient conditions in terms of LMIs

ensuring the exponential synchronization of the addressed neural networks. The feedback controllers

can be easily obtained by solving the derived LMIs. Moreover, the main results are generalizations of

some recent results reported in the literature. A numerical example is also provided to demonstrate

the effectiveness and applicability of the obtained results.

& 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

In 1990, Aihara [1] firstly introduced chaotic neural network models to simulate thechaotic behavior of biological neurons. Consequently, chaotic neural networks have drawnconsiderable attention and have been successfully applied in combinational optimization,secure communication, information science, and so on [2–4]. Recently, the synchronizationof chaotic neural networks has been intensively investigated due to their potentialapplication in secure communication, parallel recognition, etc. [5–7]. It is especially worth

2.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.jfranklin.2010.06.001

k was jointly supported by the National Natural Science Foundation of China (10801056) and K.C.

Fund in Ningbo University.

nding author.

dress: [email protected] (X. Li).

dx.doi.org/10.1016/j.jfranklin.2010.06.001

www.elsevier.com/locate/jfranklin

dx.doi.org/10.1016/j.jfranklin.2010.06.001

mailto:[email protected]

X. Li et al. / Journal of the Franklin Institute 347 (2010) 1266–1280 1267

noting that delayed neural networks such as delayed Hopfield neural networks and delayedcellular neural networks can exist some complicated dynamics and even chaotic behaviorsif the networks’ parameters and time delays are appropriately chosen [8,9], in addition tothe stability and periodic oscillations considered previously in [10–18]. Hence, thesynchronization issue of chaotic neural networks with time delays has drawn particularresearch interests and many interesting synchronization schemes have been obtained viadifferent approaches, see [19–26] and the references therein.

On the other hand, stochastic phenomenon usually appears in the electrical circuit design ofneural networks, a neural network could be stabilized or destabilized by certain stochastic inputs[27,28]. Therefore, it is significant and of prime importance to consider stochastic effects to thechaos synchronization of neural networks with delays. To date, many researchers have studiedthe synchronization problem for delayed neural networks with environmental noise and a greatnumber of results on this topic have been reported in the literature [29–38,43,44]. For instance,Sun and Cao [29,30] investigated the global asymptotical synchronization and exponentialsynchronization for stochastic perturbed chaotic neural networks with constant delays viaadaptive feedback control techniques and Halanay inequalities for stochastic differentialequations. In [33–35], the authors studied the synchronization problem of stochastic perturbedneural networks with time-varying delays via different approaches. In [36], Tang et al. consideredthe lag synchronization of stochastic perturbed chaotic neural networks with both the discrete

delays and bounded distributed delays via adaptive feedback technique. However, thesesynchronization results in [29–36] cannot be applied to stochastic perturbed chaotic neuralnetworks with unbounded distributed delays. Very recently, Liu et al. [37] investigated the pthmoment exponential synchronization of a class of stochastic perturbed chaotic neural networkswith time-varying delays and unbounded distributed delays by establishing two new integro-differential inequalities. However, the obtained results in [37] are highly dependent on thedimensions n. In other words, these results cannot be applied to stochastic network models withhigh-dimensions. In fact, in real applications there exist many neural network models with high-dimensions which have their unique advantages in computing technology, solving linear andnonlinear algebraic equations, especially in solving some difficult optimization problems [39,40].In addition, the obtained results in [37] are not expressed in terms of LMIs, which make themchecked inconveniently by the developed algorithms.

Motivated by the above discussions, in this paper, we consider a class of stochasticperturbed chaotic neural networks with time-varying delays and unbounded distributed

delays. The main purpose of this paper is to study the exponential synchronization for theaddressed chaotic neural networks under stochastic perturbations. To the best of theauthors’ knowledge, there are few studies on the synchronization issue in terms of LMIsfor stochastic perturbed chaotic neural networks with time-varying delays and unboundeddistributed delays. By using Lyapunov–Krasovskii functional, drive-response concept,output coupling with delay feedback and linear matrix inequality (LMI) approach, weobtain the synchronization schemes in terms of LMIs, which can be easily calculated byMATLAB LMI toolbox [38]. We also provide a numerical example to demonstrate theeffectiveness and applicability of the proposed synchronization schemes.

2. Notations and preliminaries

Notations: Let R denotes the set of real numbers, Zþ denotes the set of positive integers, Rn

and Rn�m denote the n-dimensional and n�m�dimensional real spaces equipped with the

X. Li et al. / Journal of the Franklin Institute 347 (2010) 1266–12801268

Euclidean norm j�j, respectively. Eð�Þ stands for the mathematical expectation of a stochasticprocess. jj�jj denotes a vector or a matrix norm. A40 or Ao0 denotes that the matrix A is asymmetric and positive definite or negative definite matrix. The notation AT and A�1 meanthe transpose of A and the inverse of a square matrix. If A;B are symmetric matrices, A4Bmeans that A�B is positive definite matrix. C denotes the set of real-valued boundedcontinuous function defined on ð�1; 0�. oðtÞ ¼ ðo1ðtÞ; . . . ;omðtÞÞ

T is an m-dimensionalBrownian motion defined on a complete probability space ðO;F ;PÞ with a natural filtrationfF tgtZ0 generated by foðsÞ : 0rsrtg, where we associate O with the canonical spacegenerated by oðtÞ, and denote by F the associated s�algebra generated by oðtÞ with theprobability measure P. C2

F 06C2

F 0ðð�1; 0�;RnÞ denotes the family of all bounded

F 0�measurable, Cðð-1; 0�;RnÞ�valued random variables c, satisfying supsr0EjcðsÞj2o1.I denotes the identity matrix with appropriate dimensions and L ¼ f1; 2; . . . ; ng. In addition,the notation % always denotes the symmetric block in one symmetric matrix.In this paper, we consider a class of neural networks represented by the compact form as

follows:

dxðtÞ ¼ ½�CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt�tðtÞÞÞ þWR t

�1hðt�sÞf ðxðsÞÞ dsþ J� dt; t40;

xðsÞ ¼ jðsÞ; s 2 ð�1; 0�;

(

ð1Þ

where the initial value function jð�Þ 2 C; nZ2 corresponds to the number of neurons in thenetworks; x(t)=(x1(t),y,xn(t))

T is the neuron state vector of the neural network;C=diag(c1,y,cn) is a diagonal matrix with ci40; i ¼ 1; . . . ; n; A, B, W are the connectionweight matrix, the delayed weight matrix and the distributively delayed connection weightmatrix, respectively; J is the external input; tðtÞ is the transmission delay with 0rtðtÞrtand _tðtÞrr, t; r are two real constants; f ðxð�ÞÞ ¼ ðf1ðx1ð�ÞÞ; . . . ; fnðxnð�ÞÞÞ

T represents theneuron activation function and hð�Þ ¼ diagðh1ð�Þ; . . . ; hnð�ÞÞ is the delay kernel function.Based on the drive-response concept for synchronization of coupled chaotic systems, we

construct the response system as follows:

dyðtÞ ¼ �CyðtÞ þ Af ðyðtÞÞ þ Bf ðyðt�tðtÞÞÞ þWR t

�1hðt�sÞf ðyðsÞÞ dsþ J þ uðtÞ

� �dt

þ sðt; eðtÞ; eðt�tðtÞÞÞ doðtÞ; t40;

yðsÞ ¼ cðsÞ; s 2 ð�1; 0�;

8><>:

ð2Þ

where the initial value function cð�Þ 2 C2F 0; sð�Þ : Rþ � Rn � Rn-Rn�m is the noise

intensity matrix; u(t) is the controller. Define the synchronization error as e(t)=y(t)�x(t)and the control input in the response system (2) is designed as follows:

uðtÞ ¼ K1½f ðyðtÞÞ�f ðxðtÞÞ� þ K2½f ðyðt�tðtÞÞÞ�f ðxðt�tðtÞÞÞ�; ð3Þ

where K1, K2 are the gain matrices to be scheduled. With the above control law, the errordynamics between system (1) and (2) can be expressed by

deðtÞ ¼ �CeðtÞ þ A%gðeðtÞÞ þ B%gðeðt�tðtÞÞÞ þWR t

�1hðt�sÞgðeðsÞÞ ds

� �dt


eðsÞ ¼ cðsÞ�jðsÞ; s 2 ð�1; 0�;

8><>: ð4Þ

where A% ¼ Aþ K1, B% ¼ Bþ K2, gðeð�ÞÞ ¼ f ðeð�Þ þ xð�ÞÞ�f ðxð�ÞÞ.


Remark 2.1. In many applications, we are interested in designing the state-feedbackcontroller or time-delay feedback controller as u(t)=K1e(t) or uðtÞ ¼ K1eðtÞ þ K2eðt�tðtÞÞ.However, in many real networks only output signals can be measured. Hence, it isnecessary to consider the controller (3) in the response system. We refer to this as outputcoupling with delay feedback.

Let C1;2ðRþ � Rn-RþÞ denote the family of all nonnegative functions V(t,x) on Rþ �

Rn which are continuous once differentiable in t and twice differentiable in x. For eachsuch V, we define an operator LV associated with Eq. (4) as

LV ¼ Vtðt; eðtÞÞ þ Veðt; eðtÞÞ �CeðtÞ þ A%gðeðtÞÞ þ B%gðeðt�tðtÞÞÞ�

þW

Z t

�1

hðt�sÞgðeðsÞÞ ds

�þ

1

2trace ½sT Veeðt; eðtÞÞs�;

where

Vtðt; eðtÞÞ ¼@V ðt; eðtÞÞ

@t; Veðt; eðtÞÞ ¼

@V ðt; eðtÞÞ

@e1; . . . ;

@V ðt; eðtÞÞ

@en

� �1�n

;

Veeðt; eðtÞÞ ¼@2V ðt; eðtÞÞ

@eiej

� �n�n

:

Furthermore, we make the following assumptions in this paper.(H1) The neuron activation functions fjð�Þ; j 2 L, are bounded and satisfy:

d�j rfjðuÞ�fjðvÞ

u�vrdþj ; j 2 L

for any u; v 2 R, uav, where d�j ; dþj ; j 2 L are some real constants.

(H2) The delay kernels hj ; j 2 L, are some real value non-negative continuous functionsdefined in ½0;1Þ and satisfyZ 1

0

hjðsÞ ds ¼ 1;

Z 10

hjðsÞeZs ds6h%

j o1; j 2 L

in which Z;h%

j are some positive constants.(H3) The diffusion matrix sð�Þ is local Lipschitz continuous and satisfies the linear

growth condition as well. Moreover, there exist two constant n� n matrices G140;G240such that

trace½sT ðt; u; vÞsðt; u; vÞ�ruTG1uþ vTG2v

for all u; v 2 Rn.

Remark 2.2. It should be noted that as pointed out in [25,26,41,42], the constantsd�j ; d

þj ; j 2 L in assumption (H1) are allowed to be positive, negative or zero. Then, those

previously used Lipschitz conditions (e.g., [22–24] ) are just the special cases of assumption(H1).


In addition, the following definitions are needed:

S1 ¼ diagðd�1 dþ1 ; . . . ; d

�n dþn Þ; S2 ¼ diag

d�1 þ dþ12

; . . . ;d�n þ dþn

2

� �:

Definition 2.1. Systems (1) and (2) are said to be exponentially synchronized if there existconstants l40 andM40 such that EjjeðtÞjj2rMEjjc�jjj2we�lt, t40, where l is called theconvergence rate (or degree) of exponential synchronization.

3. Main results

In the section, we will design the desirable schemes for controller feedback gain matricesK1, K2, which guarantee the exponential synchronization between the drive system (1) andthe response one (2).

Theorem 3.1. Assume that assumptions (H1)–(H3) hold. Then systems (1) and (2) are

exponentially synchronized if there exist two constants a 2 ð0; ZÞ;b40, two n� n matrices

Q140;Q240 and four n� n diagonal matrices P40;Q340;U140;U240, such that PrbI

and

C11 0 C13 PBþ PK2 PW

% C22 0 U2S2 0

% % C33 0 0

% % % C44 0

% % % % �Q3

0BBBBBB@

1CCCCCCAo0; ð5Þ

where C11 ¼ aP�PC�CPþQ1 þ bG1�U1S1, C13 ¼ PAþ PK1 þU1S2, C22 ¼ bG2�

ð1�rÞe�atQ1�U2S1, C33 ¼ Q2 þQ3H�U1, C44 ¼ �ð1�rÞe�atQ2�U2, H ¼ diagðh%

1 ;. . . ;h%

n Þ.

Proof. For the error system (4), we define the following Lyapunov–Krasovskii functional:

V ðt; eðtÞÞ ¼ V1ðt; eðtÞÞ þ V2ðt; eðtÞÞ þ V3ðt; eðtÞÞ þ V4ðt; eðtÞÞ;

where

V1ðt; eðtÞÞ ¼ eateT ðtÞPeðtÞ;

V2ðt; eðtÞÞ ¼

Z t

t�tðtÞeaseT ðsÞQ1eðsÞ ds;

V3ðt; eðtÞÞ ¼

Z t

t�tðtÞeasgT ðeðsÞÞQ2gðeðsÞÞ ds;

V4ðt; eðtÞÞ ¼Xn

j¼1

qð3Þj

Z 10

hjðuÞ

Z t

t�u

eaðsþuÞg2j ðejðsÞÞ ds du; Q36diagðq

ð3Þ1 ; q

ð3Þ2 ; . . . ; q

ð3Þn Þ:


By the Ito’s formula, we can calculate LV1;LV2;LV3 and LV4 along the trajectories of thesystem (4). Then we have

LV1ðt; eðtÞÞ

¼ aeateT ðtÞPeðtÞ þ 2eateT ðtÞP_eðtÞ

¼ aeateT ðtÞPeðtÞ þ 2eateT ðtÞP �CeðtÞ þ A%gðeðtÞÞ þ B%gðeðt�tðtÞÞÞ�

þW

Z t

�1


�þ eattrace½sT Ps�

reat eT ðtÞ½aP�2PC þ lmaxðPÞG1�eðtÞ þ 2eT ðtÞPA%gðeðtÞÞ�þ2eT ðtÞPB%gðeðt�tðtÞÞÞ þ 2eT ðtÞPW

Z t

�1


þlmaxðPÞeT ðt�tðtÞÞG2eðt�tðtÞÞ

reat eT ðtÞ½aP�2PC þ bG1�eðtÞ þ 2eT ðtÞPA%gðeðtÞÞ

�þ2eT ðtÞPB%gðeðt�tðtÞÞÞ þ 2eT ðtÞPW

Z t

�1


þbeT ðt�tðtÞÞG2eðt�tðtÞÞ

ð6Þ

LV2ðt; eðtÞÞ ¼ eateT ðtÞQ1eðtÞ�eaðt�tðtÞÞeT ðt�tðtÞÞQ1eðt�tðtÞÞð1�_tðtÞÞreatfeT ðtÞQ1eðtÞ�e�ateT ðt�tðtÞÞQ1eðt�tðtÞÞð1�rÞg ð7Þ

LV3ðt; eðtÞÞ ¼ eatgT ðeðtÞÞQ2gT ðeðtÞÞ�eaðt�tðtÞÞgT ðeðt�tðtÞÞÞQ2gðeðt�tðtÞÞÞð1�_tðtÞÞreatfgT ðeðtÞÞQ2gðeðtÞÞ�e�atgT ðeðt�tðtÞÞÞQ2gðeðt�tðtÞÞÞð1�rÞg: ð8Þ

By the well-known Cauchy–Schwarz inequality and assumption (H2), we also have

LV4ðt; eðtÞÞ ¼Xn

j¼1

qð3Þj

Z 10

hjðuÞeaðtþuÞg2

j ðejðtÞÞ du�Xn

j¼1

qð3Þj

Z10

hjðuÞeatg2

j ðejðt�uÞÞ du

reat gT ðeðtÞÞQ3HgðeðtÞÞ�Xn

j¼1

qð3Þj

Z 10

hjðuÞ du

Z 10

hjðuÞg2j ðejðt�uÞÞ du

( )

reat gT ðeðtÞÞQ3HgðeðtÞÞ�Xn

j¼1

qð3Þj

Z 10

hjðuÞgjðejðt�uÞÞ du

� �2( )

¼ eat gT ðeðtÞÞQ3HgðeðtÞÞ�

Z t

�1


� �T

Q3

Z t

�1


� �( ):

ð9Þ

In addition, according to [19], for any n� n diagonal matrices U140;U240, it follows that

eateðtÞ

gðeðtÞÞ

!T�U1S1 U1S2

% �U1

!eðtÞ

gðeðtÞÞ

!8<:þ

eðt�tðtÞÞ

gðeðt�tðtÞÞÞ

!T�U2S1 U2S2

% �U2

!�

eðt�tðtÞÞ


!9=;Z0: ð10Þ


Therefore, combining Eqs. (6)–(10) we obtain

e�atLVreT ðtÞ½aP�PC�CPþQ1 þ bG1�eðtÞ þ 2eT ðtÞPA%gðeðtÞÞ

þ2eT ðtÞPB%gðeðt�tðtÞÞÞ þ 2eT ðtÞPW

Z t

�1


þeT ðt�tðtÞÞ½bG2�ð1�rÞe�atQ1�eðt�tðtÞÞþgT ðeðtÞÞ½Q2 þQ3H�gðeðtÞÞ�ð1�rÞe�atgT ðeðt�tðtÞÞÞQ2gðeðt�tðtÞÞÞ

�

Z t

�1


� �T

Q3

Z t

�1


� �

þeðtÞ

gðeðtÞÞ

!T�U1S1 U1S2

% �U1

!eðtÞ

gðeðtÞÞ

!

þeðt�tðtÞÞ


!T

��U2S1 U2S2

% �U2

!eðt�tðtÞÞ


!

rxTðtÞCxðtÞ;

where

C ¼

C11 0 PA% þU1S2 PB% PW

% C22 0 U2S2 0

% % C33 0 0

% % % C44 0

% % % % �Q3

0BBBBBB@

1CCCCCCA;

xðtÞ ¼ eðtÞ; eðt�tðtÞÞ; gðeðtÞÞ; gðeðt�tðtÞÞÞ;Z t

�1


� �T

:

It follows from condition (5) and Ito’s formula that

EV ðt; eðtÞÞ�EV ð0; eð0ÞÞ ¼

Z t

0

ELVr0; t40;

which implies that

eatlminðPÞEjjeðtÞjj2rEV ðt; eðtÞÞrEV ð0; eð0ÞÞ; t40: ð11Þ

Furthermore, we note that

EV ð0; eð0ÞÞ ¼ EeT ð0ÞPeð0Þ þ

Z 0

�tð0ÞeasEeT ðsÞQ1eðsÞ dsþ

Z 0

�tð0ÞeasEgT ðeðsÞÞQ2gðeðsÞÞ ds

þXn

j¼1

qð3Þj

Z 10

hjðuÞ

Z 0

�u

eaðsþuÞEg2j ðejðsÞÞ ds du

rEeT ð0ÞPeð0Þ þ lmaxðQ1Þ

Z 0

�tEeT ðsÞeðsÞ dsþ lmaxðQ2Þ

Z 0

�tEgT ðeðsÞÞgðeðsÞÞ ds

þXn

j¼1

qð3Þj d2j

Z 10

hjðuÞeau

Z 0

�u

easEe2j ðsÞ ds du


rbEjjc�jjj2w þ tlmaxðQ1ÞEjjc�jjj2w þ td2lmaxðQ2ÞEjjc�jjj2w

þ1

a

Xn

i¼1

qð3Þj d2j h

%

j Ejjc�jjj2w

r bþ tlmaxðQ1Þ þ td2lmaxðQ2Þ þ1

a

Xn

i¼1

qð3Þj d2j h

%

j

( )Ejjc�jjj2w; ð12Þ

where d ¼ maxj2Ldj, dj ¼ maxfjd�j j; jdþj jg.

Substituting Eq. (12) into Eq. (11), we finally obtain

EjjeðtÞjj2rMEjjc�jjj2we�at; t40;

where

M ¼bþ tlmaxðQ1Þ þ td2lmaxðQ2Þ þ ð1=aÞ

Pni¼1 q

ð3Þj d2j h

%

j

lminðPÞ40:

Using Definition 2.1, we can conclude that systems (1) and (2) can be exponentiallysynchronized. This completes the proof. &

Remark 3.1. From Theorem 3.1, one may find that the constant a is the convergence rateof exponential synchronization between systems (1) and (2).

In order to estimate the gain matrices K1 and K2, we give the following desirable result:

Corollary 3.1. Assume that assumptions (H1)–(H3) hold. Then systems (1) and (2) are

exponentially synchronized if there exist two constants a 2 ð0; ZÞ; b40, four n� n matrices

Q140;Q240;Y1;Y2, and four n� n diagonal matrices P40;Q340;U140;U240, such

that PrbI and

C11 0 C13 PBþ Y2 PW

% C22 0 U2S2 0

% % C33 0 0

% % % C44 0

% % % % �Q3

0BBBBBB@

1CCCCCCAo0;

where C11 ¼ aP�PC�CPþQ1 þ bG1�U1S1, C13 ¼ PAþ Y1 þU1S2, C22 ¼

bG2�ð1�rÞe�atQ1�U2S1, C33 ¼ Q2 þQ3H�U1, C44 ¼ �ð1�rÞe�atQ2�U2, H ¼diagðh%

1 ; . . . ; h%

n Þ.

Proof. Let K1=P�1Y1, K2=P�1Y2 in Theorem 3.1, then we can obtain the resultimmediately. &

Remark 3.2. Recently, Liu et al. [37] have investigated the pth moment exponentialsynchronization of systems (1) and (2) by establishing two new integro-differentialinequalities. Moreover, some strict constraints on time delays and kernel functions areremoved. Unfortunately, the obtained results in [37] cannot be applied to stochasticnetwork models with high-dimensions. Moreover, it is not convenient to apply thosecriteria to real networks. Hence, the present paper makes up the gap and improves those inRef. [37]. In Section 4, a numerical example will be given to show the advantages of ourresults.


Remark 3.3. When W=0, systems (1) and (2) become the following simple neuralnetworks [30,34]:

dxðtÞ ¼ ½�CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt�tðtÞÞÞ þ J� dt; t40;

xðsÞ ¼ jðsÞ; s 2 ð�1; 0�

(ð13Þ

and

dyðtÞ ¼ ½�CyðtÞ þ Af ðyðtÞÞ þ Bf ðyðt�tðtÞÞÞ þ J þ uðtÞ� dtþ sðt; eðtÞ; eðt�tðtÞÞÞ doðtÞ; t40;

yðsÞ ¼ cðsÞ; s 2 ð�1; 0�:

(

ð14Þ

By Corollary 3.1, we have the following result:

Corollary 3.2. Assume that assumptions (H1) and (H3) hold. Then systems (13) and (14) are

exponentially synchronized if there exist two constants a40;b40, four n� n matrices

Q140;Q240;Y1;Y2, and three n� n diagonal matrices P40;U140;U240, such that

PrbI and

C11 0 C13 PBþ Y2

% C22 0 U2S2

% % Q2�U1 0

% % % C44

0BBB@

1CCCAo0;

where C11 ¼ aP�PC�CPþQ1 þ bG1�U1S1, C13 ¼ PAþ Y1 þU1S2, C22 ¼ bG2�

ð1�rÞe�atQ1�U2S1, C44 ¼ �ð1�rÞe�atQ2�U2.

Remark 3.4. In [30,34], the authors have presented some exponential synchronizationschemes for systems (13) and (14) via the adaptive feedback controller or time-delayfeedback controller. In this paper, we present a new exponential synchronization schemefor systems (13) and (14) via output coupling with delay feedback. Therefore, our resultsand those established in [30,34] are complementary each other.

4. A numerical example

In this section, a numerical example is given to show the effectiveness and advantages ofthe obtained results.

Example 4.1. Consider the following three-dimensional chaotic neural network models:

dxðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt�tðtÞÞÞ þWR t

�1hðt�sÞf ðxðsÞÞ dsþ J

� �dt; t40;

xðsÞ ¼ jðsÞ; s 2 ð�1; 0�;

(

ð15Þ

where the initial condition fðsÞ ¼ ð2;�1;�1:5ÞT , s 2 ð�1; 0�, f ðxÞ ¼ tanhðxÞ, tðtÞ ¼ 0:3,J ¼ ð0; 0; 0ÞT . For the simplicity of our computer simulations, the delay kernel h(s) is usedas follows: h(s)=e�s for s 2 ½0; 20�, and h(s)=0 for s420. In addition, the parameter


matrices C, A, B and W are given as follows:

C ¼

1 0 0

0 1 0

0 0 1

0B@

1CA; A ¼

1:25 �3:2 �3:2

�3:2 1:1 �4:4

�3:2 4:4 1

0B@

1CA; B ¼

6:3 �8:5 �3

�3 1:2 �5:5

�3:2 4:5 �2:3

0B@

1CA;

W ¼

0:6000 �3:99 �6:03

�0:99 3:15 �3:111

�0:945 0:969 0:285

0B@

1CA:

The corresponding response system is designed as follows:

dyðtÞ ¼ �CyðtÞ þ Af ðyðtÞÞ þ Bf ðyðt�tðtÞÞÞ þWR t

�1hðt�sÞf ðyðsÞÞ dsþ J þ uðtÞ

� �dt


yðsÞ ¼ cðsÞ; s 2 ð�1; 0�;

8><>:

ð16Þ

where the initial condition jðsÞ ¼ ð�3; 6; 3ÞT , s 2 ð�1; 0�, uðtÞ ¼ K1½f ðyðtÞÞ�f ðxðtÞÞ�þ

K2½f ðyðt�tðtÞÞÞ�f ðxðt�tðtÞÞÞ�, and

sðt; eðtÞ; eðt�tðtÞÞÞ ¼

0:6e1ðt�tðtÞÞ

0:8e2ðt�tðtÞÞ

0:9e3ðt�tðtÞÞ

0B@

1CA:

Clearly, we have G1 ¼ 0, G2 ¼ diagð0:36; 0:64; 0:81Þ and H ¼ diagð1:25; 1:25; 1:25Þ. LetZ ¼ 0:2; a ¼ 0:1, using Matlab LMI toolbox, we can obtain the following feasible solutionsto LMIs in Corollary 3.1: b ¼ 53:4554,

P ¼

39:0420 0 0

0 39:0420 0

0 0 39:0420

0B@

1CA; Q1 ¼

28:2645 �2:5008 2:7672

�2:5008 49:1886 �1:4911

2:7672 �1:4911 61:5409

0B@

1CA;

Q2 ¼

664:8512 19:0098 �16:7751

19:0098 614:3913 1:5378

�16:7751 1:5378 604:6317

0B@

1CA; Q3 ¼

1902:8 0 0

0 1902:8 0

0 0 1902:8

0B@

1CA;

U1 ¼

3436:5 0 0

0 3436:5 0

0 0 3436:5

0B@

1CA; U2 ¼

95:8793 0 0

0 95:8793 0

0 0 95:8793

0B@

1CA;

Y1 ¼ 103 �

�1:7671 0:1249 0:1249

0:1249 �1:7612 0:1718

0:1249 �0:1718 �1:7573

0B@

1CA;

Y2 ¼

�245:9646 331:8570 117:1260

117:1260 �46:8504 214:7310

124:9344 �175:6890 89:7966

0B@

1CA:


Thus, the controller gain matrices K1 and K2 are designed as follows:

K1 ¼ P�1Y1 ¼

�45:2606 3:2 3:2

3:2 �45:1106 4:4

3:2 �4:4 �45:0106

0B@

1CA;

K2 ¼ P�1Y2 ¼

�6:3 8:5 3:0

3:0 �1:2 5:5

3:2 �4:5 2:3

0B@

1CA: ð17Þ

It follows from Corollary 3.1 and Remark 3.1 that systems (15) and (16) are exponentiallysynchronized with a convergence rate 0.1. The simulation results are illustrated inFigs. 2(c) and 3(a)–(d) in which the controller designed in Eq. (17) is applied.

Remark 4.1. In the simulations, we choose the time step size h=0.005 and time segmentT=40. The initial condition of drive system (15): fðsÞ ¼ ð2;�1;�1:5ÞT , s 2 ð�1; 0�; theinitial condition of response system (16): jðsÞ ¼ ð�3; 6; 3ÞT , s 2 ð�1; 0�. The simulationresults can be described as follows. Figs. 1(a)–(d) show the state trajectories and the errortrajectories of drive system (15) and response system (16) without control input. It isobvious that the synchronization error between drive system and response one does notapproach to zero when the control input does not apply. Figs. 2(a)–(c) depict the chaotic

0 5 10 15 20 25 30 35 40

−40

−20

0

20

40

60

t

x 1 (t

), y 1

(t)

x1 (t)y1 (t)

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

15

20

t

0 5 10 15 20 25 30 35 40t

0 5 10 15 20 25 30 35 40t

x 2 (t

), y 2

(t)

−60

−40

−20

0

20

40

x 3 (t

), y 3

(t)

−60

−40

−20

0

20

40

60

e 1 (t

), e 2

(t),

e 3 (t

)

e1 (t) e2 (t) e3 (t)x3 (t)y3 (t)

x2 (t)y2 (t)

Fig. 1. State trajectories and error trajectories of drive system (15) and response system (16) without control

input.

−20−10 0

1020

−10−5

05

10−10

−5

0

5

10

x1 (t)

x2 (t)

x 3 (t

)

−100−50

050

−40−200

2040

−20

−10

0

10

20

y2 (t)

y 3 (t

)

−20 −10 0 1020

−10−5

05

10−10

−5

0

5

10

y 3 (t

)

y1 (t)

y2 (t) y1 (t)

Fig. 2. (a) The chaotic behavior of drive system (15) in phase space with the initial condition

fðsÞ ¼ ð2;�1;�1:5ÞT , s 2 ð�1; 0�. (b) The chaotic behavior of response system (16) in phase space without

control input with the initial condition jðsÞ ¼ ð�3; 6; 3ÞT , s 2 ð�1; 0�. (c) The chaotic behavior of response system(16) in phase space with control input (17) and the initial condition jðsÞ ¼ ð�3; 6; 3ÞT , s 2 ð�1; 0�.


behavior in phase space of system (15), system (16) without control input and system (16)with control input designed in Eq. (17), respectively. Figs. 3(a)–(d) show the statestrajectories and error trajectories of drive system (15) and response one (16) with controlinput designed in Eq. (17). From the simulations, we can see that the exponentialsynchronization of system (15) is realized via the feedback gain matrices K1,K2, and thosesimulations match the obtained results perfectly.

Remark 4.2. It is easy to check that the results in [37] are not applicable to ascertain theexponential synchronization between drive system (15) and response one (16). In fact, wenotice that for three-dimensional system (15)

g2Znp � 0:81p=2 � 5p�1Z36:4541

for any pZ2. Therefore, the conditions of Theorems 3.3–3.5 in [37] are not satisfied.

5. Conclusions

In this paper, we investigate the synchronization problem of stochasticperturbed chaotic neural networks with time-varying delays and unboundeddistributed delays. We have proposed a novel control scheme for exponential chaossynchronization by using Lyapunov–Krasovskii functional, drive-response concept,

0 5 10 15 20 25 30 35 40

−20

−10

0

10

20

30

t

x 1 (t

), y 1

(t)

x1 (t)y1 (t)

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

t

x 2 (t

), y 2

(t)

0 5 10 15 20 25 30 35 40

−10

−5

0

5

10

15

t

x 3 (t

), y 3

(t)

0 2 4 6 8 10

−10

−5

0

5

10

t

e 1 (t

), e 2

(t),

e 3 (t

)e1 (t)e2 (t)e3 (t)

x3 (t)y3 (t)

x2 (t)y2 (t)

Fig. 3. State trajectories and error trajectories of drive system (15) and response system (16) with control input

(17).


output coupling with delay feedback and linear matrix inequality approach. The gainedresults generalize and improve some of the existing results mentioned in the literature.Finally, a numerical simulation is provided to show the effectiveness and applicability ofthe obtained results.

Acknowledgements

The authors would like to thank the associate editor and the referees for their detailedcomments and valuable suggestions which considerably improved the presentation of thepaper.

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Synchronization of stochastic perturbed chaotic neural networks with mixed delays