Generalized Projective Synchronization of a Class of Neural Networks with Mixed Time Delays?

ZHU Ying-nian,JIANG Hai-jun,HU Cheng

(College of Mathematics and System Sciences,Xinjiang University,Urumqi,Xinjiang 830046,China)

Abstract: In this paper,the generalized projective synchronization of a class of neural networks with mixed time delays is discussed.Base on the Lyapunov stability theory combining with linear matrix inequalities,some sufficient criteria are derived to ensure the neural networks to be generalized projective synchronization.

Key words:generalized projective synchronization,neural network,linear matrix inequality,delay

0 Introduction

Since its introduction by Pecora and Carrol in 1990,chaotic synchronization has attracted considerable attention due to its important applications in nonlinear science,such as physics,secure communication,automatic control,and chemical reactions in[2]and biological systems in[3].Different types of synchronization techniques have been proposed in the literatures.These include complete synchronization,lag synchronization,phase synchronization,adaptive synchronization,etc.

In 1996,projective synchronization phenomenon wasfirst reported and discussed by Gonzalez-Miranda,In 1999,Mainieriand Rehacek proposed the concept of projective synchronization.Projective synchronization is interesting because of its proportionality between the synchronized dynamical states.The master and slave vectors synchronize up to a constant scaling factor α(a proportional relation)in partially linear systems.Complete synchronization can be regarded as a special case of projective synchronization at α =1 and anti-phase synchronization at α = ?1.It has received great recognition because of its proportional feather.Subsequently,In[7,8,9],some researchers extended the concept of projective synchronization as generalized projective synchronization.

Recently,most of works on the projective synchronization of chaotic neural networks withfinite distributed time delay have been extensively investigated.Most previous literature have mainly been devoted to the stability analysis and periodic oscillations of neural networks withfinite distributed time delay in[13].In Ref[14],several sufficient conditions were proposed to guarantee the projective synchronization of two chaotic delayed neural networks with parametermismatch.InRef[15],severalnewcriteriawereobtainedtoensure theprojectivesynchronization ofcoupled networks via active control.However,there are few works about the synchronization of the chaotic neural networks with time-varying delay and infinite distributed delay.In this paper,the generalized projective synchronization of a class of neural network with mixed time delayed is investigated by using the proposed controller.

The organization of this paper is as follows:In Section 2,model description and preliminary results are presented.In Section 3,sufficient conditions of the generalized projective synchronization for neural networks are established.

The notations used throughout this paper are fairly standard.Rnand Rn×ndenote the n-dimensional Euclidean space and the set of all n×n real matrices,respectively.The notation X>Y(X≥Y),where X and Y are symmetric matrices,means that X?Y is positive definite(semi-positive definite).The superscript T and?1 denote matrix transposition and matrix inversion.

1 Problem formulations and preliminaries

A class of neural networks with time-varying delay and infinite distributed delay is considered in this paper,which can be described by the following delayed differential equations:

or rewritten into the following vector form

where x(t)=(x1(t),x2(t),…,xn(t))T∈ Rnis the state vector of the network at time t;C=diag(c1,c2,…,cn)>0 is a diagonal matrix;A=(aij)n×n,B=(bij)n×n,and D=(dij)n×ndenote the connection weight matrix,the time-varying delayed connection weight matrix and the distributively delayed connection weight matrix,respectively;τ(t)is the time-varying delay;(x(t)),(x(t?τ(t)),(x(t))denote the activation function of neurons,where

In order to observe the synchronization behavior in this class of chaotic neural networks,we consider model(1)or(2)as the master system,the response neural networks is described by

or rewritten into the following vector form

where y(t)=(y1(t),y2(t),…,yn(t))T∈ Rn;C,A,B,and D are matrices which are the same as in(2),and U(t)=(u1(t),u2(t),…,un(t))T∈ Rnis a controller.

The following assumption are given for later study:

(A1) the activation function(x)andi(x)satisfy the Lipschitz condition,that is,there exist constants hi>0,li>0,ki>0,such that

for all u,v∈R,i=1,2,…,n,andi(0)=0.

(A2) ki:[0,∞)?→[0,∞)is a continuous integrable function for i=1,2,…,n;and

(A3)τ(t)is continuously differentiable function,and 0≤(t)≤μ<1.

The initial conditions of system(2)and(4)are of the from

Definition 1The generalized projective synchronization error between system(2)and(4)defined as

then systems(2)and(4)are said to be generalized projective synchronized,where||.||denotes the Euclidean norm of a vector.

Lemma 1For any vectors X,Y ∈Rnand a positive definite matrix Q∈Rn×n,the following inequality is satisfied:

Lemma 2[16]The following linear matrix inequality

is equivalent to one of the following conditions:

(1) A<0, C?BTA?1B<0,

(2) C<0, A?BC?1BT<0,

where A=AT,C=CT.

2 Generalize projective synchronization scheme

In this section,we will use Lyapunov stability theory to investigate the synchronization of systems(2)and(4).

Theorem 1Under assumptions if there is a positive definite matrix P=(pij)n×nand the positive definite diagonal matrix Q=diag(q1,q2,…,qn)∈ Rn×n,W=diag(w1,w2,…,wn)∈ Rn×n,N=diag(n1,n2,…,nn)∈Rn×n,such that

where f=2P(?C+ε)+LTQL+KTWK+HTNH and if the controller U is designed as

where ε =(εij)n×nis a constant matrix,then system(2)and(4)achieve the generalized projective synchronization.

ProofThe generalized projective synchronization error between system(2)and(4)is e(t)=y(t)?αx(t),then the synchronization error dynamical system can be described by

We construct the following Lyapunov function:

where P=(pij)n×nis a positive definite matrix,Q=diag(q1,q2,…,qn)∈ Rn×n,W=diag(w1,w2,…,wn)∈ Rn×n,h(e(t))=(e(t)+αx(t))?(αx(t)).

Evaluating the time derivative of V along the trajectories of system(10),we obtain

By assumption(A1)and(A2),one can obtain the following inequalities:

Furthermore,

where H=diag(h1,h2,…,hn)∈ Rn×n,L=diag(l1,l2,…,ln)∈ Rn×n,K=diag(k1,k2,…,kn)∈ Rn×n.Hence,we have

From the Lemma 1,we can obtain the following three inequalities:

From the Lemma 2,we know that the conditions for(8)is equivalent to the following inequality

Thus,it is obvious thatV˙(t)≤ 0,from Lyapunov stability theorem,we can get that limt→∞e(t)=0,it mean that zero solution of the error dynamics system(10)is globally asymptotically stable.Namely,the drive system(2)and the response system(4)can asymptotically achieve the generalized projective synchronization,this completes the proof of Theorem 1.

When the master system is the following the infinite distributed coupling matrix D=0 in system(2),and the response system has the from as follows

From Theorem 1 we have the following corollary.

Corollary 1Under assumptions(A1)?(A3),if there is a real positive definite matric P=(pij)n×nand the positive definite diagonal matrix Q=diag(q1,q2,…,qn)∈ Rn×n,N=diag(n1,n2,…,nn)∈ Rn×n,such that

where f=2P(?C+ε)+LTQL+HTNH,U(t)is controller which is the same as in(9),then system(19)and(20))achieve the generalized projective synchronization.

3 Conclusions

In this paper,we have deal with the generalized projective synchronization problem for a class of chaotic neural networks with time-varying delay and infinite distributed delay.We apply Lyapunov stability theory and linear matrix inequalities to research the generalized projective synchronization by using the proposed controller,This result can also be applied to other neural network(CNN)model.