# Synchronization for an Array of Coupled Cohen-Grossberg Neural Networks with Time-Varying Delay

Authors: Haitao Zhang; Tao Li; Shumin Fei
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

This paper makes some great attempts to investigate the global exponential synchronization
for arrays of coupled delayed Cohen-Grossberg neural networks with both delayed coupling and one single
delayed one. By resorting to free-weighting matrix and Kronecker product techniques, two novel synchronization criteria via linear matrix inequalities (LMIs) are presented based on convex combination, in
which these conditions are heavily dependent on the bounds of both the delay and its derivative. Owing
to that the addressed system can include some famous neural network models as the special cases, the
proposed methods can extend and improve those earlier reported ones. The efficiency and applicability
of the presented conditions can be demonstrated by two numerical examples with simulations.


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## Body

## 1. Introduction

In recent years, synchronization of various chaotic systems has gained considerable attention since the pioneering works of Pecora and Carroll [1, 2]. It is widely known that many benefits of having synchronization or chaos synchronization can be existent in various engineering fields such as secure communication, image processing, and harmonic oscillation generation. Thus recently, the problem on synchronization in chaotic systems has been extensively studied owing to its potential applications in many engineering areas. Especially, since chaos synchronization in arrays of linearly coupled dynamical systems was firstly studied in [3], arrays of coupled systems including delayed chaotic ones have received much attention as they can exhibit some interesting phenomena [4, 5], and a great number of elegant results have been derived [6–28].As a typical complex systems, delayed neural networks (DNNs) have been verified to exhibit some complex and unpredictable behaviors such as stable equilibria, periodic oscillations, bifurcation, and chaotic attractors. Thus recently, chaos synchronization for arrays of coupled DNNs have been discussed and some elegant results have been proposed [12–28]. The global synchronization of linearly coupled DNNs with delayed coupling was investigated in [12], in which the dynamical behavior of the uncoupled system could be chaotic or others. The authors in [13] have considered the robust synchronization of coupled DNNs under general impulsive control. In [14], this paper has proposed an adaptive procedure to the synchronization for coupled identical Yang-Yang fuzzy DNNs based on one simple adaptive controller. In [15], with all the parameters unknown, the authors focused on the robust synchronization between two coupled DNNs that were linearly and unidirectionally coupled, in which neither symmetry nor negative (positive) definiteness of the coupling matrix were required. However, those above-mentioned results were presented in terms of some complicated inequalities, which makes them uneasily checked and applied to real ceases by the most recently developed algorithms. By employing Kronecker product and LMI technique, the global synchronization and cluster one have been studied for DNNs with couplings, and some easy-to-test sufficient conditions have been obtained [16–25, 28]. Yet, the system forms addressed in [16–25] seemed simple and some improved techniques have not been utilized to reduce the conservatism, which make the above-mentioned results inapplicable to tackle DNNs of more general forms.The Cohen-Grossberg neural network (CGNN) model, first proposed by Cohen and Grossberg in 1983 [29], has recently gained particular research attention, since it is quite general to include many famous network models as its special case and has promising application potentials for tasks of associative memory, parallel computation, and nonlinear optimization problems. Meanwhile, owing to complexity of CGNNs themselves, there were few works studying the global synchronization for the coupled delayed CGNNs, except for that some researchers have studied the slave-master synchronization for continuous CGNNs in [26, 27] and synchronization for coupled discrete delayed CGNNs in [28]. Thus, it is urgent and challenging to establish some easy-to-check and less conservative results ensuring the global synchronization of coupled continuous-time delayed CGNNs, which constitutes the main focus of this presented work.In this paper, the global exponential synchronization ofN identical delayed CGNNs with both delayed coupling and one single delayed one is considered and two novel LMI-based conditions are derived by using Kronecker product technique, which has not been studied in the present literature. It shows that the chaos synchronization can be ensured by a suitable design of inner coupled linking matrix and the inner delayed linking ones. Moreover, some effective mathematical techniques are employed to reduce the conservatism. Finally, the efficiency of the derived criteria can be illustrated by utilizing two numerical examples.Notations. Rn denotes the n-dimensional Euclidean space, and Rn×m is the set of all n×m real matrices. For the symmetric matrices X,Y,X>Y (respectively, X≥Y) means that X-Y>0(X-Y≥0) is a positive-definite (respectively, positive-semidefinite) matrix; AT represents the transpose of the matrix A; λmax(A),λmin(A) denote the maximum eigenvalue and minimum one of matrix A, respectively; I represents the identity matrix of an appropriate dimension; [XYYTZ]=[XY*Z].


## 2. Problem Formulations and Preliminaries

Suppose the nodes are coupled with statesxi(t), i∈{1,…,N}, then the delayed Cohen-Grossberg neural network models can be formulated as follows:(2.1)ẋi(t)=-α(xi(t))[β(xi(t))-Af(xi(t))-Bf(xi(t-τ(t)))-I(t)]+∑j=1,j≠iNlijF[xj(t)-xi(t)]+∑j=1,j≠iNlijK[xj(t-τ(t))-xi(t)]+∑j=1,j≠iNlijJ[xj(t-τ(t))-xi(t-τ(t))],
in which xi(t)=[xi1(t),…,xin(t)]T∈Rn is the state vector of the ith network at time t, α(xi)=diag{α1(xi1),…,αn(xin)} represents the amplification function, β(xi)=[β1(xi1),…,βn(xin)]T is the behaved function, A=[aij]n×n, B=[bij]n×n, f(xi)=[f1(xi1),…,fn(xin)]T; I(t)=[I1(t),…,In(t)]T∈Rn is the external input vector; F=[fij]n×n, K=[kij]n×n, and J=[jij]n×n are respectively the inner coupling matrices between the connected nodes i and j at times  t and t-τ(t).For system (2.1), the following assumptions are introduced throughout this paper. (A1)
τ(t) denotes an interval time-varying delay satisfying(2.2)0≤τ0≤τ(t)≤τm,τ̇(t)≤μ<+∞,and we setτ¯m=τm-τ0. (A2)
L=[lij]N×N is the configuration matrix that is irreducible and satisfies(2.3)lij=lji,i≠j,lii=-∑j=1,j≠iNlij.
Here lij>0 if there is a connection between node i and the one j and otherwise, lij=0. (A3)
Fori∈{1,2,…,n}, each αi(·) is Lipschitz continuous and there exists the positive scalars a̲i,a¯i satisfying 0<a̲i≤αi(·)≤a¯i; and there exist the positive scalars πi,γi such that each function βi(·) satisfies 0<γi≤(βi(x)-βi(y)/x-y)≤πi, and(2.4)[β̇i(x)-β̇i(y)-ρi-(x-y)][β̇i(x)-β̇i(y)-ρi+(x-y)]≤0∀x,y∈R,i=1,…,n,
in which ρi-,ρi+ are given constants. Here we set Λ=diag{a̲1,…,a̲n}, Ψ=diag{a¯1,…,a¯n}, Γ=diag{γ1,…,γn}, Π=diag{π1,…,πn}, and(2.5)Λ1=diag⁡{γ1π1,…,γnπn},Λ2=diag⁡{π1+γ12,…,πn+γn2},Υ1=diag⁡{ρ1+ρ1-,…,ρn+ρn-},Υ2=diag⁡{ρ1++ρ1-2,…,ρn++ρn-2}.(A4)
For anyα,β∈R, and ρi-,ρi+ for i∈{1,2,…,n}, the activation function fi(·) satisfies(2.6)[fi(α)-fi(β)-σi+(α-β)][fi(α)-fi(β)-σi-(α-β)]≤0.
Here we denote Σ1=diag{σ1+σ1-,…,σn+σn-} and Σ2=diag{(σ1++σ1-)/2,…,(σn++σn-)/2}.Remark 2.1.
In  (A3), the assumption on the derivative ofβ(xi) in (2.4) is reasonable and does not result in the conservatism in many cases such as that, choosing the appropriate scalars a,b,c, the function β(xi) can be expressed as axi, axi+bsin(xi), axi+bsin2(xi), axi+ccos(xi), axi+ccos3(xi), axi+ctanh(xi), respectively. Moreover, the activation functions in system (2.1) can be of general description and those present ones in [22–26, 29] are just special cases of the system (2.1).
Based on assumption(A2), system (2.1) can be rewritten as the following forms:
(2.7)ẋi(t)=-α(xi(t))[β(xi(t))-Af(xi(t))-Bf(xi(t-τ(t)))-I(t)]-liiK[xi(t-τ(t))-xi(t)]+∑j=1NlijFxj(t)+∑j=1Nlij(K+J)xj(t-τ(t)).
To address the problem, we denote the set 𝒮={x(s)=[x1T(s),…,xNT(s)]:xi(s)∈𝒞([t0-τm,τ0],Rn),xi(s)=xj(s),i,j=1,2,…,N} as the synchronization manifold for system (2.7).Definition 2.2 (see [16]).
Dynamical network (2.7) is said to be asymptotically synchronized, if for any initial conditions ϕi(s),ϕj(s)∈C([t0-τm,t0],Rn), i,j=1,…,N, there exist M>0, ɛ>0 and sufficient large T>0 such that ∥xi(t)-xj(t)∥≤Me-ɛt for all t≥T, where ɛ and M are said to be the decay rate and the decay coefficient, respectively. Here ∥·∥ denotes the Euclidean norm.
Due to the communication delay, the array of coupled nodes cannot be decoupled, and the synchronized state is always not the trajectory of an isolated node but a modified one as (2.7). Furthermore, delayed coupling matrix and the degree of the node play the important roles in the synchronized state, which has been illustrated in [21]. In the paper, we give an improved discussion for such synchronization. In the case, system (2.7) reaches the synchronization, that is, x1(t)=⋯=xN(t)=s(t), we can deduce the synchronized state equation(2.8)ṡ(t)=-α(s(t))[β(s(t))-Af(s(t))-Bf(s(t-τ(t)))-I(t)]-liiK[s(t-τ(t))-s(t)],
where i=1,2,…,N. Obviously, the synchronization is invariant for the coupled system (2.7). Therefore, to realize complete synchronization, the assumption l11=⋯=lNN=l has to be imposed on the system (2.7).


## 3. Delay-Dependent Synchronization Criteria

Firstly, together with the Kronecker product in [16–21], we can reformulate the system (2.7) as(3.1)ẋ(t)=-a(x(t))[b(x(t))-(IN⊗A)f(x(t))-(IN⊗B)f(x(t-τ(t)))-I(t)]-l(IN⊗K)×[x(t-τ(t))-x(t)]+(L⊗F)x(t)+(L⊗(K+J))x(t-τ(t))
with x(t)=[x1T(t),…,xNT(t)]T, a(x(t))=diag{α(x1(t)),…,α(xN(t))}, b(x(t))=[βT(x1(t)),…,βT(xN(t))]T, f(x(·))=[fT(x1(·)),…,fT(xN(·))]T, and I(t)=[IT(t),…,IT(t)]T.In order to derive our results, the following lemmas are essential for obtaining the synchronization criteria.Lemma 3.1.
LetD,S, and  P>0 be real matrices of appropriate dimensions and  ɛ>0. Then for any vectors  x and  y with appropriate dimensions, one gets  2xTDTSy≤ɛ-1xTDTP-1Dx+ɛyTSTPSy.Lemma 3.2 (17)..
LetU=[uij]N×N,  P∈Rn×n,  x=[x1T,x2T,…,xNT]T, and  y=[y1T,y2T,…,yNT]T with  xi,yi∈Rn, i=1,…,N. If  U=UT and each row sum of  U is  0, then  xT(U⊗P)y=-∑1≤i<j≤Nuij(xi-xj)TP(yi-yj).Then by utilizing the most improved techniques for achieving the criteria in [30], we state and investigate the global exponential synchronization for the system (3.1).Theorem 3.3.
Supposing that assumptions  (A1)–(A4) hold, then the dynamical system (3.1) is globally exponentially synchronized, if there exist  n×n matrices  P>0,  S>0,  Z>0,  Li(i=1,2,3,4),  n×n matrices  Pl>0, Ql>0, Rl(l=1,2,3) making  [PlRl*Ql]≥0,  n×n diagonal matrices  R>0, Q>0, G>0, E>0, U>0, V>0,  W>0, H>0, Ti>0(i=1,2),  13n×n matrices  Ni(i=1,2,3), and one scalar  δ>0 such that, for  1≤i<j≤N, the LMIs in (3.2) hold
(3.2)[Ωij+$+$TH¯τ0N1τ¯mN2*-δI00**-S0***-Z]<0,[Ωij+$+$TH¯τ0N1τ¯mN3*-δI00**-S0***-Z]<0,
where $=[N1-N1+N2-N306n·13n-N2+N303n·13n], H¯=[0n·13nQ(Λ-1-Ψ-1)0n·n]T,
(3.3)Ωij=[Ξ1100Ξ1400Ξ17L1T0Ξ1,10lL1TKΞ1,12Ξ1,13*Ξ2200Ξ2500000000**Ξ3300Ξ360000000***Ξ44000ATL3ATL400ATQT0****Ξ5500000000*****Ξ660000000******Ξ77-L2T0Ξ7,10lL2TK-Ψ-1R0*******Ξ88-L3T0L3TB-L3T0********Ξ990L4TB-L4T0*********Ξ10,10Ξ10,11Ξ10,120**********Ξ11,11BTQT0***********Ξ12,120************-T2]
with Ξ11=P2+l(L1TK+KTL1)-lijN(L1TF+FTL1)-2ΓTG-UΣ1-T1Π1-T2Υ1, Ξ14=R2+UΣ2, Ξ17=P-L1T-lijNFTL2+2ΠTRΨ-1-2ΓTQΛ-1, Ξ1,10=-lL1TK-lijNL1T(K+J), Ξ1,12=lKTQΛ-1-lijNFTQΛ-1+GT+T1Π2, Ξ1,13=T2Υ2+QΛ-1-RΨ-1, Ξ22=-P2+P1+P3-WΣ1, Ξ25=-R2+R1+R3+WΣ2, Ξ33=-P3-HΣ1, Ξ36=HΣ2-R3, Ξ44=-U+Q2, Ξ55=-Q2+Q1+Q3-W, Ξ66=-Q3-H, Ξ77=-L2T-L2+τ0S+τ¯mZ, Ξ7,10=-lL2TK-lijNL2T(K+J), Ξ88=-E+δIn, Ξ99=-L4T-L4+ΨTEΨ, Ξ10,10=-(1-μ)P1-VΣ1, Ξ10,11=-(1-μ)R1+VΣ2, Ξ10,12=-lKTQΛ-1-lijN(K+J)TQΛ-1, Ξ11,11=-(1-μ)Q1-V, Ξ12,12=-2QT-T1.Proof.
Firstly, we can represent the system (3.1) as the following form:
(3.4)ẋ(t)=y(t),(3.5)y(t)=z(t)-l(IN⊗K)[x(t-τ(t))-x(t)]+(L⊗F)x(t)+(L⊗(K+J))x(t-τ(t)),(3.6)z(t)=a(x(t))w(t),(3.7)w(t)=-b(x(t))+(IN⊗A)f(x(t))+(IN⊗B)f(x(t-τ(t)))+I(t).
Based on assumptions  (A1) and  (A3), and
(3.8)U=[uij]N⋅N=[N-1⋯-1⋮⋱⋮-1⋯N-1],
we construct the following Lyapunov-Krasovskii functional:
(3.9)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t)),
where
(3.10)V1(x(t))=xT(t)(U⊗P)x(t)+2[Θx(t)-b(x(t))]T(U⊗RΨ-1)x(t)+2[b(x(t))-Υx(t)]T(U⊗QΛ-1)x(t),V2(x(t))=∫t-τ(t)t-τ0[x(s)f(x(s))]T(U⊗[P1R1*Q1])[x(s)f(x(s))]ds,V3(x(t))=∫t-τ0t[x(s)f(x(s))]T(U⊗[P2R2*Q2])[x(s)f(x(s))]ds,V4(x(t))=∫t-τmt-τ0[x(s)f(x(s))]T(U⊗[P3R3*Q3])[x(s)f(x(s))]ds,V5(x(t))=∫-τ00∫t+θtẋT(s)(U⊗S)ẋ(s)dsdθ+∫-τm-τ0∫t+θtẋT(s)(U⊗Z)ẋ(s)dsdθ
with two diagonal matrices R>0, Q>0, and setting Θ=diag{Π,Π,…,Π︸N}, Υ=diag{Γ,Γ,…,Γ︸N}. Based on  (A3) and Lemma 3.1, one can easily verify the definite positiveness of V1(x(t)).
Now, by directly calculatingV̇1(x(t)) along the trajectory of the system (3.1), we can deduce(3.11)V̇1(x(t))=2xT(t)(U⊗P)y(t)+2[Θx(t)-b(x(t))]T(U⊗RΨ-1)y(t)+2[Θy(t)-ḃ(x(t))]T×(U⊗RΨ-1)x(t)+2[b(x(t))-Υx(t)]T×(U⊗QΛ-1)y(t)+2[ḃ(x(t))-Υy(t)]T(U⊗QΛ-1)x(t)=2xT(t)(U⊗P)y(t)+2[Θx(t)-b(x(t))]T(U⊗RΨ-1)y(t)+2bT(x(t))×(U⊗QΛ-1)y(t)+2ḃT(x(t))(U⊗(QΛ-1-RΨ-1))x(t)+2yT(t)×(U⊗(ΠTRΨ-1-2ΓTQΛ-1))x(t).
Meanwhile by (3.5), it is easy to derive that
(3.12)2bT(x(t))(U⊗QΛ-1)y(t)=2bT(x(t))(U⊗QΛ-1)z(t)+2bT(x(t))(U⊗QΛ-1)×[∫t-τ(t)t-τ0-l(IN⊗K)[x(t-τ(t))-x(t)]+(L⊗F)x(t)+(L⊗(K+J))x(t-τ(t))∫t-τ(t)t-τ0].
Noting that U1/2≥0 does hold, then with Lemma 3.1 and δ>0, one can estimate 2bT(x(t))(U⊗QΛ-1)z(t) as
(3.13)2bT(x(t))(U⊗QΛ-1)z(t)=2bT(x(t))(U1/2⊗Q[Λ-1-α-1(x(t))])T(U1/2⊗I)z(t)+2bT(x(t))(U⊗Q)w(t)≤δzT(t)(U⊗In)z(t)+δ-1bT(x(t))×(U⊗(Λ-T-Ψ-T)QTQ(Λ-1-Ψ-1))b(x(t))+2bT(x(t))×(U⊗Q)[-b(x(t))+(IN⊗A)f(x(t))+(IN⊗B)f(x(t-τ(t)))+I(t)].
Now combining with terms (3.12) and (3.13) yields
(3.14)V̇1(x(t))≤2xT(t)(U⊗P)y(t)+2[Θx(t)-b(x(t))]T(U⊗RΨ-1)y(t)+δzT(t)(U⊗In)z(t)+δ-1bT(x(t))(U⊗(Λ-T-Ψ-T)QTQ(Λ-1-Ψ-1))b(x(t))+2bT(x(t))(U⊗Q)×[-b(x(t))+(IN⊗A)f(x(t))+(IN⊗B)f(x(t-τ(t)))+I(t)]+2bT(x(t))×(U⊗QΛ-1)[-l(IN⊗K)[x(t-τ(t))-x(t)]+(L⊗F)x(t)+(L⊗(K+J))x(t-τ(t))]+2ḃT(x(t))(U⊗(QΛ-1-RΨ-1))x(t)+2yT(t)(U⊗(ΠTRΨ-1-2ΓTQΛ-1))x(t),(3.15)V̇2(x(t))≤[∫t-τmt-τ0xT(t-τ0)(U⊗P1)x(t-τ0)+2xT(t-τ0)(U⊗R1)f(x(t-τ0))+fT(x(t-τ0))×(U⊗Q1)f(x(t-τ0))∫t-τmt-τ0]-(1-μ)[∫t-τ0txT(t-τ(t))(U⊗P1)x(t-τ(t))+2xT(t-τ(t))(U⊗R1)f(x(t-τ(t))),+fT(x(t-τ(t)))(U⊗Q1)f(x(t-τ(t)))∫t-τ0t],(3.16)V̇3(x(t))=[xT(t)(U⊗P2)x(t)+2xT(t)(U⊗R2)f(x(t))+fT(x(t))(U⊗Q2)f(x(t))]-[∫t-τmt-τ0xT(t-τ0)(U⊗P2)x(t-τ0)+2xT(t-τ0)(U⊗R2)f(x(t-τ0))+fT(x(t-τ0))(U⊗Q2)f(x(t-τ0))∫t-τmt-τ0],(3.17)V̇4(x(t))=[∫t-τmt-τ0xT(t-τ0)(U⊗P3)x(t-τ0)+2xT(t-τ0)(U⊗R3)f(x(t-τ0))+fT(x(t-τ0))(U⊗Q3)×f(x(t-τ0))∫t-τmt-τ0]-[∫t-τmt-τ0xT(t-τm)(U⊗P3)x(t-τm)+2xT(t-τm)(U⊗R3)f(x(t-τm))+fT(x(t-τm))(U⊗Q3)f(x(t-τm))∫t-τmt-τ0],(3.18)V̇5(x(t))=yT(t)[τ0(U⊗S)+τ¯m(U⊗Z)]y(t)-∫t-τ0tyT(s)(U⊗S)y(s)ds-∫t-τmt-τ0yT(s)(U⊗Z)y(s)ds.
For any n×n matrices Li(i=1,2,3,4), it follows from (3.5) and (3.7) that
(3.19)0=2[xT(t)(U⊗L1T)+yT(t)(U⊗L2T)]×[-y(t)+z(t)-l(IN⊗K)[x(t-τ(t))-x(t)]+(L⊗F)x(t)+(L⊗(K+J))x(t-τ(t))],(3.20)0=2[zT(t)(U⊗L3T)+wT(t)(U⊗L4T)]×[-w(t)-b(x(t))+(IN⊗A)f(x(t))+(IN⊗B)f(x(t-τ(t)))+I(t)].
By utilizing (A3) and (3.6), for any n×n diagonal matrices G≥0, E≥0, the following inequality holds
(3.21)0≤2[bT(x(t))(U⊗G)x(t)-xT(t)(U⊗ΓTG)x(t)]+[wT(t)(U⊗ΨTEΨ)w(t)-zT(t)(U⊗E)z(t)].
Meanwhile, based on (3.14) and (3.19), it is easy to check that UL=NL, and
(3.22)(U⊗QΛ-1)(L⊗F)=(NL)⊗(QΛ-1F),(U⊗QΛ-1)(L⊗(K+J))=(NL)⊗(QΛ-1(K+J)),(U⊗LiT)(L⊗F)=(NL)⊗(LiTF),(U⊗LiT)(L⊗(K+J))=(NL)⊗(LiT(K+J)),i=1,2.
Here we can employ the following notations to simplify the subsequent proof
(3.23)xij=xi-xj,yij=yi-yj,zij=zi-zj,wij=wi-wj,β(xij)=β(xi)-β(xj),f(xij)=f(xi)-f(xj).
Then together with (U⊗Q)I(t)=0 in (3.14) and (U⊗L3T)I(t)=(U⊗L4T)I(t)=0 in (3.19), it follows from Lemma 3.2 and (3.14)–(3.22) that
(3.24)V̇(x(t))≤-∑1≤i<j≤N{∑1≤i<j≤Nuij[2xijT(t)Pyij(t)+2[Πxij(t)-β(xij(t))]]TRΨ-1yij(t)+δzijT(t)Inzij(t)+δ-1βT(xij(t))(Λ-T-Ψ-T)QTQ(Λ-1-Ψ-1)β(xij(t))+2βT(xij(t))×Q[-β(xij(t))+Af(xij(t))+Bf(xij(t-τ(t)))]+2βT(xij(t))QΛ-1×[-lK[xij(t-τ(t))-xij(t)]]+2β̇T(xij(t))(QΛ-1-RΨ-1)xij(t)+2yijT(t)×(ΠTRΨ-1-2ΓTQΛ-1)xij(t)+[∑1≤i<j≤NxijT(t-τ0)(P1-P2+P3)xij(t-τ0)+2xijT(t-τ0)(R1-R2+R3)×f(xij(t-τ0))+fT(xij(t-τ0))(Q1-Q2+Q3)f(xij(t-τ0))∑1≤i<j≤N]×[∑1≤i<j≤NxijT(t-τ(t))P1xij(t-τ(t))+2xijT(t-τ(t))R1f(xij(t-τ(t)))+fT(xij(t-τ(t)))Q1f(xij(t-τ(t)))∑1≤i<j≤N]×[xijT(t)P2xij(t)+2xijT(t)R2f(xij(t))+fT(xij(t))Q2f(xij(t))]-[∑1≤i<j≤NxijT(t-τm)P3xij(t-τm)+2xijT(t-τm)R3f(xij(t-τm))+fT(xij(t-τm))Q3f(xij(t-τm))∑1≤i<j≤N]+yijT(t)(τ0S+τ¯mZ)yij(t)-∫t-τ0tyijT(s)Syij(s)ds-∫t-τmt-τ0yijT(s)Zyij(s)ds+2[xijT(t)L1T+yijT(t)L2T]×[-yij(t)+zij(t)-lK[xij(t-τ(t))-xij(t)]]+2[zijT(t)L3T+wijT(t)L4T][-wij(t)-β(xij(t))+Af(xij(t))+Bf(xij(t-τ(t)))]+2[βT(xij(t))Gxij(t)-xijT(t)ΓTGxij(t)]+[wijT(t)ΨTEΨwij(t)-zijT(t)Ezij(t)]+2Nlij[[βT(xij(t))QΛ-1+xijT(t)L1T+yijT(t)L2T]Fxij(t)+[βT(xij(t))QΛ-1+xijT(t)L1T+yijT(t)L2T](K+J)xij(t-τ(t))]∑1≤i<j≤N}.
For any n×n diagonal matrices U>0, V>0, W>0, H>0, Ti>0(i=1,2), and Σi, Πi, Υi(i=1,2) in (A3)-(A4), it can be deduced that
(3.25)0≤∑1≤i<j≤N{∑1≤i<j≤N-[xijT(t)UΣ1xij(t)-2xijT(t)UΣ2f(xij(t))+fT(xij(t))Uf(xij(t))]-[∑1≤i<j≤NxijT(t-τ(t))VΣ1×xij(t-τ(t))-2xijT(t-τ(t))VΣ2f(xij(t-τ(t)))+fT(xij(t-τ(t)))Vf(xij(t-τ(t)))∑1≤i<j≤N]-[xijT(t-τ0)WΣ1xij(t-τ0)-2xijT(t-τ0)WΣ2f(xij(t-τ0))+fT(xij(t-τ0))Wf(xij(t-τ0))]-[xijT(t-τm)HΣ1xij(t-τm)-2xijT(t-τm)HΣ2f(xij(t-τm))+fT(xij(t-τm))Hf(xij(t-τm))]-[xijT(t)T1Π1xij(t)-2xijT(t)T1Π2β(xij(t))+βT(xij(t))T1β(xij(t))]-[xijT(t)T2Υ1xij(t)-2xijT(t)T2Υ2β̇(xij(t))+β̇T(xij(t))T2β̇(xij(t))]∑1≤i<j≤N}.
For any 13  n×n matrices Ni(i=1,2,3), it follows from Newton-Leibniz formula that
(3.26)0=2∑1≤i<j≤NζijT(t){N1[xij(t)-xij(t-τ0)-∫t-τ(t)t-τ0yij(s)]+N2[xij(t-τ0)-xij(t-τ(t))-∫t-τ(t)t-τ0yij(s)]+N3[xij(t-τ(t))-xij(t-τm)-∫t-τmt-τ(t)yij(s)]},
where
(3.27)ζijT(t)=[xijT(t)xijT(t-τ0)xijT(t-τm)fT(xij(t))fT(xij(t-τ0))fT(xij(t-τm))yijT(t)zijT(t)×wijT(t)xijT(t-τ(t))fT(xij(t-τ(t)))βT(xij(t))β̇T(xij(t))].
Now together with the terms (3.24)–(3.26), and uij=-1, we can deduce that
(3.28)V̇(x(t))≤∑1≤i<j≤NζijT(t){∑1≤i<j≤NΩij+$+$T+δ-1H̅H̅T+τ0N1S-1N1T+[τ(t)-τ0]N2Z-1N2T+[τm-τ(t)]N3Z-1N3T∑1≤i<j≤N}ζij(t):=∑1≤i<j≤NζijT(t)Δij(t)ζij(t),
where Ωij,$, and H¯ are presented in (3.2). Through using Schur-complement and convex combination, the LMIs in (3.2) can guarantee Δij(t)<0 and thus, there must exist one scalar χ>0 such that Δij(t)≤-χI<0. Therefore, one can get
(3.29)V̇(x(t))≤∑1≤i<j≤NζijT(t)Δij(t)ζij(t)≤-χ[∑1≤i<j≤N‖xij(t)‖2+∑1≤i<j≤N‖xij(t-τ(t))‖2],
which indicates that the system (3.1) can reach the global asymptotical synchronization.
Based on  (A1)–(A4), (3.9), and direct computing, there must exist three scalars Θi>0(i=1,2,3) such that(3.30)V(x(t))≤∑1≤i<j≤N[Θ1‖xij(t)‖2+Θ2∫t-τmt‖xij(s)‖2ds+Θ3∫t-τmt‖xij(s-τ(s))‖2ds].
Letting V¯(x(t))=e2ktV(x(t)), one can deduce that  V¯̇(x(t))=2ke2ktV(x(t))+e2ktV̇(x(t)), and
(3.31)V¯(x(t))-V(x(0))=∫otV¯̇(x(s))ds≤∑1≤i<j≤N∫ote2ks{∫s-τms‖xij(θ)‖2-χ[‖xij(s)‖2+‖xij(s-τ(s))‖2]+2k[Θ1‖xij(s)‖2+Θ2∫s-τms‖xij(θ)‖2dθ+Θ3∫s-τms‖xij(θ-τ(θ))‖2dθ]}ds.
By 1≤i<j≤N and changing the integration sequences, we have
(3.32)∫0te2ks∫s-τms‖xij(θ)‖2dθds≤τme2kτm[∫-τm0‖xij(θ)‖2e2kθdθ+∫0t‖xij(θ)‖2e2kθdθ],∫0te2ks∫s-τms‖xij(θ-τ(θ))‖2dθds≤τme2kτm×[∫-2τm0‖xij(θ)‖2e2kθdθ+∫0t‖xij(θ-τ(θ))‖2e2kθdθ].
Substituting the terms (3.32) into the relevant ones in (3.31), it is easy to have
(3.33)V¯(x(t))≤V(x(0))+∑1≤i<j≤N{[2kΘ1+2kΘ2τme2kτm-χ]∫0t‖xij(θ)‖2e2kθdθ+[2kΘ3τme2kτm-χ]∫0t‖xij(θ-τ(θ))‖2e2kθdθ+h(k)}
in which h(k)=2kΘ2τme2kτm∫-τm0∥xij(θ)∥2dθ+2kΘ3τme2kτm∫-2τm0∥xij(θ)∥2dθ. Choose an appropriate scalar k0>0 such that 2k0Θ1+2k0Θ2τme2k0τm-χ≤0,2k0Θ3τme2k0τm-χ≤0, one has V¯(x(t))≤h(k0)+V(x(0)). By directly computing, there must exist a positive scalar ϖ>0 such that
(3.34)h(k0)+V(x(0))≤ϖ∑1≤i<j≤Nsup⁡-2τm≤s≤0‖ϕi(s)-ϕj(s)‖2.
Meanwhile, V¯(x(t))≥λmin(P)∑1≤i<j≤Ne2k0t∥xij(t)∥2. Therefore, it can be deduced that
(3.35)‖xij(t)‖≤λmin⁡-1(P)ϖ⋅∑1≤i<j≤Nsup⁡-2τm≤s≤0‖ϕi(s)-ϕj(s)‖e-k0t,∀t≥0.
By Definition 2.2, the system (3.1) is globally exponentially synchronized, and the proof is completed.Remark 3.4.
Theorem3.3 presents a novel delay-dependent criterion guaranteeing arrays of coupled Cohen-Grossberg neural networks (2.7) to be globally synchronized. In [16–22], the authors considered global synchronization of an array of coupled neural networks of simple forms and in the paper, we derive a more general delayed neural networks and extended the case to the time variable one, which generalizes the earlier ones. Moreover, the conditions are expressed in terms of LMIs, therefore, by using LMI in Matlab Toolbox, it is straightforward and convenient to check the feasibility of the proposed results without tuning any parameters.
If there does not exist one single delayed coupling in system (2.1), that is, K=0, which means that the restriction l11=l22=⋯=lNN=l in L=[lij]N×N is removed. Then together with the proof of Theorem 3.3, we can derive the following theorem.Theorem 3.5.
Supposing that assumptions (A1)–(A4) hold, then the dynamical system (3.1) is globally exponentially synchronized, if there exist n×n matrices P>0, S>0, Z>0, Li(i=1,2,3,4), n×n matrices Pl>0, Ql>0, Rl(l=1,2,3) making [PlRl*Ql]≥0, n×n diagonal matrices R>0, Q>0, G>0, E>0, U>0, V>0, W>0, H>0, Ti>0(i=1,2), 13n×n matrices Ni(i=1,2,3), and one scalar δ>0 such that, for 1≤i<j≤N, the LMIs in (3.36) hold
(3.36)[Ωij+$+$TH¯τ0N1τ¯mN2*-δI00**-S0***-Z]<0,[Ωij+$+$TH¯τ0N1τ¯mN3*-δI00**-S0***-Z]<0,
where $=[N1-N1+N2-N306n·13n-N2+N303n·13n], H¯=[0n·11nQ(Λ-1-Ψ-1)0n·n]T,
(3.37)Ωij=[Ξ1100Ξ1400Ξ17L1T0Ξ1,100Ξ1,12Ξ1,13*Ξ2200Ξ2500000000**Ξ3300Ξ360000000***Ξ44000ATL3ATL400ATQT0****Ξ5500000000*****Ξ660000000******Ξ77-L2T0Ξ7,100-Ψ-1R0*******Ξ88-L3T0L3TB-L3T0********Ξ990L4TB-L4T0*********Ξ10,10Ξ10,11Ξ10,120**********Ξ11,11BTQT0***********Ξ12,120************-T2]
with  Ξ11=P2-lijN(L1TF+FTL1)-ΓTG-GTΓ-UΣ1-T1Π1-T2Υ1, Ξ14=R2+UΣ2, Ξ17=P-L1T-lijNFTL2+2ΠTRΨ-1-2ΓTQΛ-1, Ξ1,10=-lijNL1TJ, Ξ1,12=-lijNFTQΛ-1+GT+T1Π2, Ξ1,13=T2Υ2+QΛ-1-RΨ-1, Ξ22=-P2+P1+P3-WΣ1, Ξ25=-R2+R1+R3+WΣ2, Ξ33=-P3-HΣ1, Ξ36=HΣ2-R3, Ξ44=-U+Q2, Ξ55=-Q2+Q1+Q3-W, Ξ66=-Q3-H, Ξ77=-L2T-L2+τ0S+τ¯mZ, Ξ7,10=-lijNL2TJ, Ξ88=-E+δIn, Ξ99=-L4T-L4+ΨTEΨ, Ξ10,10=-(1-μ)P1-VΣ1, Ξ10,11=-(1-μ)R1+VΣ2, Ξ10,12=-lijNJTQΛ-1, Ξ11,11=-(1-μ)Q1-V, Ξ12,12=-2QT-T1.Proof.
LettingK=0 in system (2.1) and employing similar methods of proving Theorem 3.3, we can easily derive the theorem and the detailed proof is omitted here.Remark 3.6.
Theorems3.3-3.5 require the upper bound μ of time-delay τ(t) to be known. If μ is unknown, by setting P1=R1=Q1 in (3.9), we can derive the delay-dependent and delay-derivative-independent synchronization criteria for the coupled systems (2.7) based on Theorems 3.3-3.5.Remark 3.7.
Together with the similar proof, we can deal with the global synchronization for arrays of coupled CGNNs with hybrid couplings described in [18]. Moreover, we still can investigate the global synchronization for arrays of coupled uncertain CGNNs with more general forms(3.38)ẋi(t)=-α(xi(t))[β(xi(t))-A(t)f(xi(t))-B(t)f(xi(t-τ(t)))-I(t)]+∑j=1,j≠iNlij1F[xj(t)-xi(t)]+∑j=1,j≠iNlij2K[xj(t-τ(t))-xi(t)]+∑j=1,j≠iNlij3J[xj(t-τ(t))-xi(t-τ(t))],
in which A(t)=A+ΔA(t), B(t)=B+ΔB(t), and we set Lh=[lijh]n×n for h=,1,2,3. Here ΔA(t) and ΔB(t) are unknown matrices representing variable parametric uncertainties satisfying
(3.39)[ΔA(t)ΔB(t)]=F0Δ(t)[E1E2],Δ(t)=Λ(t)(I-J0Λ(t))-1,I-J0TJ0>0,
in which F0,J0,Ei(i=1,2) are known appropriately as dimensional matrices and Λ(t) is an unknown variable matrix function satisfying ΛT(t)Λ(t)≤I. By utilizing the similar methods in Theorems 3.3-3.5, one can easily derive the more general results based on [30, Lemma 5].Remark 3.8.
It is worth pointing out that it is possible to extend our main results to more complex Cohen-Grossberg neural networks, such as CGNNs with distributed delay, stochastic perturbations, and Markovian jumping parameters. The corresponding results will appear in our future works.


## 4. Numerical Examples

In the section, two examples are provided to illustrate the effectiveness of the proposed results.Example 4.1.
We consider the delayed Cohen-Grossberg neural network models described by(4.1)ẋ(t)=-α(x(t))[β(x(t))-Af(x(t))-Bf(x(t-τ(t)))-I(t)]
with α(x)=diag{0.8+0.2|cos(2x1)|,1.0-0.2|cos(2x2)|,0.8+0.2|sin(2x3)|},
(4.2)A=[-101021010-100],B=[32-22-32-223],β(x)=[0.9x1+0.05sin(2x1)0.9x2+0.05cos⁡(2x2)0.9x3+0.05sin(2x3)],f(x)=[0.3(|x1+1|-|x1-1|)0.3(|x2+1|-|x2-1|)0.3(|x3+1|-|x3-1|)],I(t)=0,τ(t)=0.4+0.2sin(20t)+0.05cos⁡2(40t).
One can get τ0=0.2, τm=0.65, μ=8, and the functions βi(·),fi(·) satisfy the assumptions  (A3) and  (A4), respectively. Through setting the inner linking matrix L=[-2111-2111-2], we consider a dynamical networks comprised of three linearly coupled identical neural network models with time-delay couplings as follows:
(4.3)ẋi(t)=-α(xi(t))[β(xi(t))-Af(xi(t))-Bf(xi(t-τ(t)))]+∑j=13lijFxj(t)+∑j=13lij(K+J)xj(t-τ(t))-lK[xi(t-τ(t))-xi(t)],i=1,2,3,
and choose the coupling matrices as
(4.4)F=[100001000010],K=[0.050000.050000.05],J=[0.10000.10000.1].
Figure 1 shows that the system has a chaotic attractor. Together with Theorem 3.3 and LMI in Matlab Toolbox, it is easy to check that there exists the feasible solution to the LMIs in (3.2), which can guarantee the array of the system (4.3) to achieve the exponential synchronization. The total error is defined by
(4.5)error(t)=∑i=13[x1i(t)-x2i(t)]2+[x2i(t)-x3i(t)]2
and the synchronization error can be seen in Figure 1. During the process of simulation, the initial conditions of nodes are selected as x1=[-0.5,-0.3,0.3]T,  x2=[0.7,-0.5,-0.6]T, and x3=[1,0.5,0.3]T.The synchronized trajectory and total synchronous error of system (4.3).
(a)(b)Example 4.2.
Consider one 2-dimensional delayed Cohen-Grossberg neural networks as follows:(4.6)ẋ(t)=-α(x(t))[β(x(t))-Af(x(t))-Bf(x(t-τ(t)))-I(t)],
where α(x)=diag{0.9+0.1sin(x1),0.9+0.1cos(x1)},β(x)=[1.1x1+0.1sin(x1)1.1x2+0.1sin(x2)], A=[2-0.2-0.33], B=[-1.5-0.1-0.2-2.5], I(t)=[0.10.1], f(x)=[0.7tanh(x1)+0.15(|x1+1|-|x1-1|)0.7tanh(x2)+0.15(|x2+1|-|x2-1|)], and τ(t)=0.5+0.3cos(6t)+0.05sin2(40t). Choosing the following inner linking matrix and the coupling matrices, respectively,
(4.7)L=[-3121-2121-3],F=[100010],K=0,J=[0.1000.1],
we still consider a dynamical networks comprised of three coupled identical CGNNs with delayed couplings as
(4.8)ẋi(t)=-α(xi(t))[β(xi(t))-Af(xi(t))-Bf(xi(t-τ(t)))-I(t)]+∑j=13lijFxj(t)+∑j=13lijKxj(t-τ(t)).
Then based on Theorem 3.5 and Matlab LMI Toolbox, one can get part feasible solution to (3.36) as follows:
(4.9)P=[2.06350.02340.02342.2078],S=[0.0274-0.0036-0.00360.0032],Z=[0.07570.00160.00160.0870],R=[0.0415000.0415],Q=[0.0966000.0966],G=[1.2804001.2804],E=[0.0658000.0658],U=[9.4219009.4219],V=[0.6964000.6964],W=[1.1489001.1489],H=[0.6611000.6611],T1=[1.9626001.9626],T2=[1.6523001.6523],[P1R1R1TQ1]=[0.02010.0013-0.03160.00050.00130.02860.0003-0.0281-0.03160.00030.0621-0.00470.0005-0.0281-0.00470.0289],[P2R2R2TQ2]=[4.27300.8822-0.4717-0.00020.882210.22510.0277-0.3621-0.47170.02772.63800.0486-0.0002-0.36210.04860.7271],[P3R3R3TQ3]=[2.01310.3881-0.24550.00470.38814.65320.0180-0.2932-0.24550.01801.56190.01790.0047-0.29320.01790.9050]
which means that the global exponential synchronization is achieved for system (4.8). The total error of the array of the system (4.8) is defined by
(4.10)error(t)=∑i=12[x1i(t)-x2i(t)]2+[x2i(t)-x3i(t)]2,
and the synchronization state and total synchronous error can be depicted in Figure 2 with the initial conditions x1=[-0.5,-0.3]T, x2=[0.3,0.7]T, and x3=[-0.5,-0.6]T. Moreover, if we choose α(x)=diag{0.8+0.2(ex1/1+ex1),0.8+0.2(ex2/1+ex2)}, and τ(t)=0.5+0.3sin(20t)+0.05cos2(30t), then the synchronization state and total synchronous error can be described in Figure 3.The synchronized trajectory and total synchronous error of system (4.8).
(a)(b)The synchronized trajectory and total synchronous error of system (4.8).
(a)(b)


## 5. Conclusions

This paper has investigated the global exponential synchronization for the coupled Cohen-Grossberg neural networks with both delayed couplings and time-varying delay. Two novel conditions have been derived by employing Lyapunov-Krasovskii functional and convex combination. It is worth pointing out that, the addressed systems can include many neural network models as its special cases and some good mathematical techniques have been employed, which improve and extend those present results. The derived synchronization criteria are presented in terms of LMIs, which can be checked easily by resorting to Matlab LMI Toolbox. Finally, two numerical examples are utilized to illustrate the effectiveness of the derived methods based on the simulation results.



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*Source: 831695-2011-05-25.xml*