# Synchronization of Switched Complex Bipartite Neural Networks with Infinite Distributed Delays and Derivative Coupling

Authors: Qiuxiang Bian; Jinde Cao; Jie Wu; Hongxing Yao; Tingfang Zhang; Xiaoxu Ling
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A new model of switched complex bipartite neural network (SCBNN) with infinite distributed delays and derivative coupling is established. Using linear matrix inequality (LMI) approach, some synchronization criteria are proposed to ensure the synchronization between two SCBNNs by constructing effective controllers. Some numerical simulations are provided to illustrate the effectiveness of the theoretical results obtained in this paper.


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## Body

## 1. Introduction

In recent years, neural networks have been intensively studied due to their potential applications in many different areas such as signal and image processing, content-addressable memory, optimization, and parallel computation [1–3]. Bidirectional associative memory (BAM) neural networks were first proposed by Kosko in [4, 5]. This class of networks has good applications in pattern recognition, solving optimization problems, and automatic control engineering. A large number of results on the dynamical behavior of BAM neural networks have been reported [6–9].Switched systems, as an important kind of hybrid systems, have drawn considerable attention of researchers because of their theoretical significance and practical applications [10–12]. Switched systems are composed of a family of continuous-time or discrete-time subsystems and a rule that specifies the switching among them [13, 14]. Recently, the switched neural networks, whose individual subsystems are a set of neural networks, have found applications in the field of high speed signal processing, artificial intelligence, and biology, so there are many theoretical results about the switched neural networks [15–17].Complex networks, which are a set of interconnected nodes with specific dynamics, have sparked the interest of many researchers from various fields of science and engineering such as the World Wide Web, electrical power grids, global economic markets, sensor networks; for example, see [18–20] and references therein. Bipartite networks are an important kind of complex networks, whose nodes can be divided into two disjoint nonempty sets such that every edge only connects a pair of nodes, which belong to different sets. Many real-world networks are naturally bipartite networks, such as the papers-scientists networks [21] and producer-consumer networks [22]. Recently, authors [23] have introduced a bipartite-graph complex dynamical network model that is only linearly coupled and has no delays. It is well known that time delays exist commonly in real-world systems. Therefore, many models of coupled networks with coupling delays are proposed, for example, constant single time delay [24], time-varying delays [25], and mix-time delays [26]. On the other hand, the coupled network often occurs in other forms, for example, nonlinearly coupled networks [27] and linearly derivative coupled networks [28]. In [29], a general model of bipartite dynamical network (BDN) with distributed delays and nonlinear derivative coupling was introduced. Synchronization of complex networks has been intensively investigated since they can be applied in power system control, secure communication, automatic control, chemical reaction, and so on [30–32]. The study of synchronization of coupled neural networks is an important step for both understanding brain science and designing coupled neural networks for practical use. Yu et al. [33] consider the synchronization of switched linearly coupled neural networks with constant delays, but the controllers are complex and changed with the switched rule. Synchronization of two coupled BDNs was investigated by adaptive method [29], but the controllers are complicated and the model does not include infinite distributed delays coupling and switching. Extending BAM neural networks to complex networks, we get complex bipartite dynamical networks (CBDNs). The dynamics of individual node in CBDNs is switched system and the switched coupling is considered; switched complex bipartite neural network (SCBNN) can be obtained. To the best of our knowledge, up to now, there is not any work that discusses the synchronization problem in SCBNN.Motivated by the previous discussion, we first proposed a model of SCBNN, and then investigated the synchronization between two SCBNNs with infinite distributed delays and derivative coupling. Using adaptive controllers and linear matrix inequality (LMI) approach, some synchronization criteria are proposed to ensure the synchronization between two coupled SCBNNs. In our paper, the proposed controllers are simpler and do not change with the switched rule, which can be realize more easily.The paper is organized as follows. In Section2, a model of SCBNN with infinite distributed delays and derivative coupling is presented, and some hypotheses and lemmas are given too. In Section 3, several synchronization criteria on the SCBNNs are deduced. In Section 4, numerical examples are given to demonstrate the effectiveness of the proposed controller design methods in Section 3. Finally, conclusions are given in Section 5.Notations. Throughout this paper, ρmax(·) and ρmin(·) denote the maximum eigenvalue and minimum eigenvalue of a real symmetric matrix, respectively. The notation * denotes the symmetric block.


## 2. Model Description, Assumptions, and Lemmas

Consider a complex bipartite dynamical network (CBDN) consisting of two disjoint nonempty node setsV1 and V2. Suppose that V1={μ1,μ2,…,μl} and V2={ν1,ν2,…,νm}, l, m are integer. The coupled network is described as follows:
(1)x˙i(t)=-Dxi+R1f1(xi(t))+R2f2(xi(t-τ(t)))+I+∑j=1maijyj(t-τ1(t))+∑j=1mbijg(y˙j(t-τ2(t)))+∑j=1mcij∫-∞th(t-s)k(yj(s))ds,i=1,2,…,l,y˙j(t)=-D-yj+R-1f-1(yj(t))+R-2f-2(yj(t-σ(t)))+J+∑i=1la-jixi(t-σ1(t))+∑i=1lb-jig-(x˙i(t-σ2(t)))+∑i=1lc-ji∫-∞th-(t-s)k-(xi(s))ds,j=1,2,…,m,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T, yj(t)=(yj1(t),yj2(t),…,yjn(t))T∈Rn denotes the state variables of nodes μi and νj, respectively. D=diag(d1,d2,…,dn) and D-=diag(d-1,d-2,…,d-n) are diagonal matrices with di,d-i>0. R1,R-1∈Rn×n are weight matrices, R2,R-2∈Rn×n are delayed weight matrices, fk(xi)=(fk1(xi1),fk2(xi2),…,fkn(xin))T, f-k(yj)=(f-k1(yj1),f-k2(yj2),…,f-kn(yjn))T∈Rn, k=1,2, g(y˙j)=(g1(y˙j1),g2(y˙j2),…,gn(y˙jn))T, g-(x˙i)=(g-1(x˙i1),g-2(x˙i2),…,g-n(x˙in))T, k(yj)=(k1(yj1),k2(yj2),…,kn(yjn))T, k-(xi)=(k-1(xi1),k-2(xi2),…,k-n(xin))T∈Rn corresponds to the boundedness activation functions of neurons. h(t)=diag(h1(t),h2(t),…,hn(t)), h-(t)=diag(h-1(t),h-2(t),…,h-n(t))∈Rn×n are the delay kernel functions. τ(t),τ1(t),τ2(t),σ(t),σ1(t), and σ2(t)>0 are time delays. ∫-∞th(t-s)k(yj(s))ds and ∫-∞th-(t-s)k-(xi(s))ds express infinite distributed delays. I=(I1,I2,…,In)T and J=(J1,J2,…,Jn)T∈Rn are the constant external input vectors. The matrix A=(aij)l×m is the delayed weight coupling matrix denoting coupling strength between nodes. If there is a connection from node μi to νj, then aij≠0; otherwise, aij=0 and the matrix A satisfies the sum of every row being zero. The definitions of the other coupling matrixes B=(bij)l×m, C=(cij)l×m, A-=(a-ji)m×l,B-=(b-ji)m×l, and C-=(c-ji)m×l are similar to that of matrix A; hence, they are omitted here.In this paper, we consider a class of switched complex bipartite neural network with infinite distributed delays and derivative coupling, which is described as follows:(2)x˙i(t)=-Dλxi+Rλ1f1(xi(t))+Rλ2f2(xi(t-τ(t)))+Iλ+∑j=1maλijyj(t-τ1(t))+∑j=1mbλijg(y˙j(t-τ2(t)))+∑j=1mcλij∫-∞th(t-s)k(yj(s))ds,i=1,2,…,l,y˙j(t)=-D-λyj+R-λ1f-1(yj(t))+R-λ2f-2(yj(t-σ(t)))+Jλ+∑i=1la-λjixi(t-σ1(t))+∑i=1lb-λjig-(x˙i(t-σ2(t)))+∑i=1lc-λji∫-∞th-(t-s)k-(xi(s))ds,j=1,2,…,m,
where switching signal λ is piecewise constant functions, which is a value in the finite set ℵ={1,2,…,N}. This means that the matrices {Dλ,Rλ1,Rλ2,Aλ=(aλij),Bλ=(bλij),Cλ=(cλij),Iλ,D-λ,R-λ1,R-λ2,A-λ=(a-λji),B-λ=(b-λji),andC-λ=(c-λji),Jλ} are allowed to take values at particular time, in a finite set {(Dr,Rr1,Rr2,Ar,Br,Cr,Ir,D-r,R-r1,R-r2,A-r,B-r,C-r,Jr)∣r=1,2,…,N}. We define the function as follows:
(3)ξr(t,λ)={1,whentheswitchedsystemisdescribedbytherthmode,thatis,λ=r,0,others.
It follows that under any switching rules ∑r=1Nξr(t,λ)=1. Model (2) can be written as
(4)x˙i(t)=∑r=1Nξr(t,λ)×[+∑j=1m∫-∞t(yj(s))ds-Drxi+Rr1f1(xi(t))+Rr2f2(xi(t-τ(t)))+Ir+∑j=1marijyj(t-τ1(t))+∑j=1mbrijg(y˙j(t-τ2(t)))+∑j=1mcrij∫-∞th(t-s)k(yj(s))ds],i=1,2,…,l,y˙j(t)=∑r=1Nξr(t,λ)×[+∑i=1l∫-∞tk-(xi(s))ds-D-ryj+R-r1f-1(yj(t))+R-r2f-2(yj(t-σ(t)))+Jr+∑i=1la-rjixi(t-σ1(t))+∑i=1lb-rjig-(x˙i(t-σ2(t)))+∑i=1lc-rji∫-∞th-(t-s)k-(xi(s))ds],j=1,2,…,m.
The response network of the drive network (4) is
(5)x^˙i(t)=∑r=1Nξr(t,λ)×[+∑j=1m∫-∞t(y^j(s))ds+ui(t)-Drx^i(t)+Rr1f1(x^i(t))+Rr2f2(x^i(t-τ(t)))+Ir+∑j=1marijy^j(t-τ1(t))+∑j=1mbrijg(y^˙j(t-τ2(t)))+∑j=1mcrij∫-∞th(t-s)k(y^j(s))ds+ui(t)],y^˙j(t)=∑r=1Nξr(t,λ)×[+∑i=1lc-rji∫-∞th-(t-s)k-(x^i(s))ds+vj(t)-D-ry^j(t)+R-r1f-1(y^j(t))+R-r2f-2(y^j(t-σ(t)))+Jr+∑i=1la-rjix^i(t-σ1(t))+∑i=1lb-rjig-(x^˙i(t-σ2(t)))+∑i=1lc-rji∫-∞th-(t-s)k-(x^i(s))ds+vj(t)],
where ui(t) and vj(t)∈Rn are the control inputs.Letei(t)=x^i(t)-xi(t), εj(t)=y^j(t)-yj(t), i=1,2,…,l, and j=1,2,…,m. The error dynamical system of (4) and (5) is given by
(6)e˙i(t)=∑r=1Nξr(t,λ)×[+∑j=1mcrij∫-∞th(t-s)K(εj(s))ds+ui(t)-Drei(t)+Rr1F1(ei(t))+Rr2F2(ei(t-τ(t)))+∑j=1marijεj(t-τ1(t))+∑j=1mbrijG(ε˙j(t-τ2(t)))+∑j=1mcrij∫-∞th(t-s)K(εj(s))ds+ui(t)],i=1,2,…,l,ε˙j(t)=∑r=1Nξr(t,λ)×[+∑i=1lc-rji∫-∞th-(t-s)K-(ei(s))ds+vj(t)-D-rεj(t)+R-r1F-1(εj(t))+R-r2F-2(εj(t-σ(t)))+∑i=1la-rjiei(t-σ1(t))+∑i=1lb-rjiG-(e˙i(t-σ2(t)))+∑i=1lc-rji∫-∞th-(t-s)K-(ei(s))ds+vj(t)],j=1,2,…,m,
where
(7)Fk(ei(t))=fk(x^i(t))-fk(xi(t)),F-k(εj(t))=f-k(y^j(t))-f-k(yj(t)),k=1,2,G(ε˙j(t))=g(y^˙j(t))-g(y˙j(t)),G-(e˙i(t))=g-(x^˙i(t))-g-(x˙i(t)),K(εj(s))=k(y^j(s))-k(yj(s))=(K1(εj1(s)),…,Kn(εjn(s)))T,K-(ei(s))=k-(x^i(s))-k-(xi(s))=(K-1(ei1(s)),…,K-n(ein(s)))T.In this paper, the following assumptions and lemmas are needed.(S1)
There exist diagonal matricesLi-=diag(li1-,li2-,…,lin-) and Li+=diag(li1+,li2+,…,lin+), such that
(8)lkj-≤fkj(x)-fkj(y)x-y≤lkj+,l3j-≤kj(x)-kj(y)x-y≤l3j+,l4j-≤gj(x)-gj(y)x-y≤l4j+,∀x,y∈R and x≠y, i=1,2,3,4,j=1,2,…,n, and k=1,2.(S2)
There exist diagonal matricesL-i-=diag(l-i1-,l-i2-,…,l-in-) and L-i+=diag(l-i1+,l-i2+,…,l-in+), such that
(9)l-kj-≤f-kj(x)-f-kj(y)x-y≤l-kj+,l-3j-≤k-j(x)-k-j(y)x-y≤l-3j+,l-4j-≤g-j(x)-g-j(y)x-y≤l-4j+,∀x,y∈R and x≠y, i=1,2,3,4, j=1,2,…,n, and  k=1,2.(S3)
τ(t), τ1(t), τ2(t), σ(t), σ1(t), and σ2(t) are differential functions with τ˙(t)<τ<1, σ˙(t)<σ<1, τ˙1(t)<τ1<1, σ˙1(t)<σ1<1, τ˙2(t)<τ2<1, and σ˙2(t)<σ2<1.(S4)
hi(t),h-i(t) are real-value nonnegative continuous functions defined in [0,∞) satisfying
(10)∫0∞hi(s)ds<∞,∫0∞h-i(s)ds<∞,i=1,2,…,n.Lemma 1 (see [34]).
Given any real matricesΣ1, Σ2, and Σ3 of appropriate dimensions and a scalar ε>0 such that 0<Σ3=Σ3T, then the following inequality holds:
(11)Σ1TΣ2+Σ2TΣ1≤εΣ1TΣ3Σ1+ε-1Σ2TΣ3-1Σ2.Lemma 2 (see [35]).
Given a positive definite matrixP∈Rn×nand a symmetric matrix Q∈Rn×n, then
(12)ρmin(P-1Q)xTPx≤xTQx≤ρmax(P-1Q)xTPx,∀x∈Rn.Lemma 3 (Schur complement).
Given constant symmetric matricesΣ1, Σ2, and Σ3, where Σ1=Σ1T and 0<Σ2=Σ2T, then Σ1+Σ3TΣ2-1Σ3<0 if and only if
(13)(Σ1Σ3TΣ3-Σ2)<0or(-Σ2Σ3Σ3TΣ1)<0.
For convenience, let(14)Li=diag(max{|li1-|,|li1+|},max{|li2-|,|li2+|},…,max{|lin-|,|lin+|}),i=1,2,3,4,L-i=diag(max{|l-i1-|,|l-i1+|},max{|l-i2-|,|l-i2+|},…,max{|l-in-|,|l-in+|}),i=1,2,3,4,H=diag(∫0∞h1(v)dv,∫0∞h2(v)dv,…,∫0∞hn(v)dv),H-=diag(∫0∞h-1(v)dv,∫0∞h-2(v)dv,…,∫0∞h-n(v)dv).


## 3. Main Results

Theorem 4.
Under assumptions(S1)–(S4), the two coupled SCBNNs (4) and (5) can be synchronized, if there exist positive constants, α,β,p,p-,γi,ηj(i=1,2,…,l,j=1,2,…,m), n×n positive matrices P, Q, U, P-, Q-, U- and n×n diagonal positive matrices W=diag(w1,w2,…,wn), W-=diag(w-1,w-2,…,w-n), Mi,M-i(i=1,2,3) such that
(15)[ZriPRr1PRr2(∑j=1marij2)1/2P(∑j=1mbrij2)1/2P(l∑j=1mcrij2)1/2P*-M10000**-M2000***-M300****-In0*****-W]<0,(16)[Z-rjP-R-r1P-R-r2(∑i=1la-rji2)1/2P-(∑i=1lb-rji2)1/2P-(m∑i=1lc¯rji2)1/2P-*-M-10000**-M-2000***-M-300****-In0*****-W-]<0,(17)m1-σ2-2αp≤0,P≥pIn,P-≥p-In,l1-τ2-2βp-≤0,(18)L2M2L2-(1-τ)Q≤0,M3-(1-τ1)U≤0,(19)L-2M-2L-2-(1-σ)Q-≤0,M-3-(1-σ1)U-≤0,
and the adaptive feedback controllers are designed as
(20)ui(t)=-[γi+αi(t)]ei(t),vj(t)=-[ηj+βj(t)]εj(t),αi(t)={∥G-(e˙i(t))∥2∥ei(t)∥2α,∥ei(t)∥2≠0,0,∥ei(t)∥2=0,βj(t)={∥G(ε˙j(t))∥2∥εj(t)∥2β,∥εj(t)∥2≠0,0,∥εj(t)∥2=0,
where Zri=-2PDr+L-3H-W-H-L-3+L1M1L1+mU--2γiP+Q, Z-rj=-2P-D-r+L3HWHL3+L-1M-1L-1+lU+Q--2ηjP-, r∈ℵ, i=1,2,…,l, and j=1,2,…,m.Proof.
For the error dynamical system (6), we design the following Lyapunov-Krasovskii function:
(21)V(t)=V1(E(t))+V2(E(t)),
where E(t)=(e1T(t),e2T(t),…,elT(t),ε1T(t),ε2T(t),…,εmT(t))T,(22)V1=∑i=1leiT(t)Pei(t)+∑i=1l∫t-τ(t)teiT(s)Qei(s)ds+∑j=1m∑i=1nwi∫0∞hi(v)dv∫0∞hi(θ)×∫t-θtKi2(εji(s))dsdθ+l∑j=1m∫t-τ1(t)tεjT(s)Uεj(s)ds+l1-τ2∑j=1m∫t-τ2(t)tGT(ε˙j(s))G(ε˙j(s))ds,(23)V2=∑j=1mεjT(t)P-εj(t)+∑j=1m∫t-μ(t)tεjT(s)Q-εj(s)ds+∑i=1l∑j=1nw-j∫0∞h-j(v)dv∫0∞h-j(θ)×∫t-θtK-j(eij(s))dsdθ+m∑i=1l∫t-μ1(t)teiT(s)U-ei(s)ds+m1-σ2∑i=1l∫t-μ2(t)tG-T(e˙i(s))G-(e˙i(s))ds.
Calculating the derivative of (22) along the trajectories of (6), we have
(24)V˙1=∑r=1Nξr(t,λ)×{∑j=1mGT(ε˙j(t-τ2(t)))G(ε˙j(t-τ2(t)))∑i=1l2eiT(t)P×[×∫-∞th(t-s)-Drei(t)+Rr1F1(ei(t))+Rr2F2(ei(t-τ(t)))+∑j=1marijεj(t-τ1(t))+∑j=1mbrijG(ε˙j(t-τ2(t)))+∑j=1mcrij×∫-∞th(t-s)K(εj(s))ds+ui(t)]+∑i=1leiT(t)Qei(t)-(1-τ˙(t))×∑i=1leiT(t-τ(t))Qei(t-τ(t))+∑j=1m∑i=1nwi(Ki(εji(t))∫0∞hi(v)dv)2-∑j=1m∑i=1nwi∫0∞hi(v)dv×∫0∞hi(θ)Ki2(εji(t-θ))dθ+l∑j=1mεjT(s)Uεj(s)-(1-τ˙1(t))l×∑j=1mεjT(t-τ1(t))Uεj(t-τ1(t))+l1-τ2∑j=1mGT(ε˙j(t))G(ε˙j(t))-1-τ˙2(t)1-τ2l×∑j=1mGT(ε˙j(t-τ2(t)))G(ε˙j(t-τ2(t)))}.
By Lemma 1, we can get from (S1)(25)2eiT(t)PRr1F1(ei(t))≤eiT(t)PRr1M1-1Rr1TPei(t)+F1T(ei(t))M1F1(ei(t))≤eiT(t)(PRr1M1-1Rr1TP+L1M1L1)ei(t),2eiT(t)PRr2F2(ei(t-τ(t)))≤eiT(t)PRr2M2-1Rr2TPei(t)+F2T(ei(t-τ(t)))M2F2(ei(t-τ(t)))≤eiT(t)PRr2M2-1Rr2TPei(t)+eiT(t-τ(t))L2M2L2ei(t-τ(t)),2arijeiT(t)Pεj(t-τ1(t))≤arij2eiT(t)PM3-1Pei(t)+εjT(t-τ1(t))M3εjT(t-τ1(t)).
By assumptions (S1) and (S4), it is obvious that
(26)∑j=1m∑i=1nwi(Ki(εji(t))∫0∞hi(v)dv)2=∑j=1mKT(εj(t))HWHK(εj(t))≤∑j=1mεjT(t)L3HWHL3εj(t),(27)2∑i=1lcrijeiT(t)P∫-∞th(t-s)K(εj(s))ds≤l∑i=1lcrij2eiT(t)PW-1Pei(t)+1l∑i=1l(∫-∞th(t-s)K(εj(s))ds)T×W∫-∞th(t-s)K(εj(s))ds=l∑i=1lcrij2eiT(t)PW-1Pei(t)+(∫-∞th(t-s)K(εj(s))ds)T×W∫-∞th(t-s)K(εj(s))ds.
Observe that
(28)-l(1-τ˙2(t))1-τ2∑j=1mGT(ε˙j(t-τ2(t)))G(ε˙j(t-τ2(t)))+2∑i=1l∑j=1meiT(t)PbrijG(ε˙j(t-τ2(t)))≤-∑i=1l∑j=1mGT(ε˙j(t-τ2(t)))G(ε˙j(t-τ2(t)))+2∑i=1l∑j=1meiT(t)PbrijG(ε˙j(t-τ2(t)))=-∑i=1l∑j=1m(brijPei(t)-G(ε˙j(t-τ2(t))))T×(brijPei(t)-G(ε˙j(t-τ2(t))))+∑i=1l∑j=1mbrij2eiT(t)P2ei(t),≤∑i=1l∑j=1mbrij2eiT(t)P2ei(t).
Using inequality
(29)∫0∞f2(s)ds∫0∞g2(s)ds≥(∫0∞f(s)g(s)ds)2,
we have
(30)∑i=1nwi∫0∞hi(v)dv∫0∞hi(θ)Ki2(εji(t-θ))dθ≥∑i=1nwi(∫0∞hi(θ)Ki(εji(t-θ))dθ)2=(∫-∞th(t-s)K(εj(s))ds)T×W∫-∞th(t-s)K(εj(s))ds.
Using Lemma 2 and condition (17), we get
(31)-eT(t)P∥G-(e˙i(t))∥2∥ei(t)∥2e(t)≤-pG-T(e˙i(t))G-(e˙i(t)).
Substituting (20) into (24) and combining (24)–(31), it can be derived by condition (18) that
(32)V˙1≤∑r=1Nξr(t,λ)×{∑i=1leiT(t)Ωriei(t)+∑i=1leiT(t-τ(t))[L2M2L2-(1-τ)Q]×ei(t-τ(t))+∑j=1mεjT(s)(lU+L3HWHL3)εj(s)+∑j=1mεjT(t-τ1(t))[lM3-l(1-τ1)U]×εj(t-τ1(t))+l1-τ2∑j=1mGT(ε˙j(t))G(ε˙j(t))-2αp∑i=1lG-T(e˙i(t))G-(e˙i(t))}≤∑r=1Nξr(t,λ)×{∑i=1leiT(t)Ωriei(t)+∑j=1mεjT(t)(lU+L3HWHL3)εj(t)+l1-τ2∑j=1mGT(ε˙j(t))G(ε˙j(t))-2αp∑i=1lG-T(e˙i(t))G-(e˙i(t))},
where Ωi=-2PDr+PRr1M1-1Rr1TP+L1M1L1+PRr2M2-1Rr2TP+∑j=1marij2PM3-1P+∑j=1mbrij2P2 + l∑j=1mcrij2PW-1P-2γiP+Q.
Meanwhile, by a similar process, the following inequality can be true:(33)V˙2≤∑r=1Nξr(t,λ)×{∑j=1mεjT(t)Ω-rjεj(t)+∑i=1leiT(t)(mU-+L-3H-W-H-L-3)ei(t)+m1-σ2∑i=1lG-T(e˙i(t))G-(e˙i(t))-2βp-∑j=1mGT(ε˙j(t))G(ε˙j(t))},
where Ω-j=-2P-D-r+P-R-r1M-1-1R-r1TP-+L-1M-1L-1+P-R-r2M-2-1R-r2TP-+∑i=1la-rij2P-M-3-1P-+∑i=1lb-rji2P-2+m∑i=1lc-rji2P-W--1P--2ηjP-+Q-.
By condition (17), we have
(34)V˙≤∑r=1Nξr(t,λ)×[+∑j=1mεjT(t)(Ω-rj+lU+L3HWHL3)εj(t)∑i=1leiT(t)(Ωri+mU-+L-3H-W-H-L-3)ei(t)+∑j=1mεjT(t)(Ω-rj+lU+L3HWHL3)εj(t)].
By (15)-(16) and Lemma 3 (Schur complement), it can be obtained that Ωri+mU-+L-3H-W-H-L-3<0,Ω-rj+lU+L3HWHL3<0. Set ρ=min{ρ1,ρ2}, where
(35)ρ1=-min{ρmin(Ωri+mU-+L-3H-W-H-L-3),r∈ℵ,1≤i≤lρmin(Ωri+mU-+L-3H-W-H-L-3),},ρ2=-min{ρmin(Ω-rj+lU+L3HWHL3),r∈ℵ,1≤j≤mρmin(Ω-rj+lU+L3HWHL3),},
then ρ>0, and
(36)V˙≤-ρ1∑i=1leiT(t)ei(t)-ρ2∑j=1mεjT(t)εj(t)V˙≤-ρET(t)E(t).
Therefore, V is nonincreasing in t≥0. One has V bounded since 0≤V(t,E(t))≤V(0,E(0)), so limt→+∞V(t,E(t)) exists and
(37)limt→+∞∫0tET(s)E(s)ds≤-1ρlimt→+∞∫0tdVdsds=1ρV(0,E(0))-1ρlimt→+∞V(t,E(t)).
From (22)-(23) and conditions P≥pI, P-≥p-I and we have 0≤ET(t)E(t)≤max{1/p,1/p-}V(t,E(t)), so ET(t)E(t) is bounded. According to error system (6), (d/dt)ET(t)E(t)=2ET(t)E˙(t) is bounded for t≥0 due to the boundedness of activation functions. From the above we can see that E(t)∈L2∩L∞ and (d/dt)ET(t)E(t)∈L∞. By using Barbǎlat lemma (see [36]), one has limt→+∞ET(t)E(t)=0, so the two SBNNs (4) and (5) can obtain synchronization under the controllers (20). This completes the proof.We take CBDN (1) as drive network. The response network of the drive network (1) is
(38)x^˙i(t)=-Dx^i(t)+R1f1(x^i(t))+R2f2(x^i(t-τ(t)))+I(t)+∑j=1maijy^j(t-τ1(t))+∑j=1mbijg(y^˙j(t-τ2(t)))+∑j=1mcij∫-∞th(t-s)k(y^j(s))ds+ui(t),y^˙j(t)=-D-y^j(t)+R-1f-1(y^j(t))+R-2f-2(y^j(t-σ(t)))+J(t)+∑i=1la-jix^i(t-σ1(t))+∑i=1lb-jig-(x^˙i(t-σ2(t)))+∑i=1lc-ji∫-∞th-(t-s)k-(x^i(s))ds+vj(t),
where ui(t),vj(t)∈Rn are the control inputs.From Theorem4, we can get the following corollary.Corollary 5.
Under assumptions(S1)–(S4), the two coupled CBDNs (1) and (38) can be synchronized, if there exist positive constants α,β,p,p-,γi,ηj(i=1,2,…,l,j=1,2,…,m), n×n positive matrices P, Q, U, P-, Q-, U- and n×n diagonal positive matrices W=diag(w1,w2,…,wn), W-=diag(w-1,w-2,…,w-n), Mi, M-i(i=1,2,3)such that
(39)[ZiPR1PR2(∑j=1maij2)1/2P(∑j=1mbij2)1/2P(l∑j=1mcij2)1/2P*-M10000**-M2000***-M300****-In0*****-W]<0,[Z-jP-R-1P-R-2(∑i=1la-ji2)1/2P-(∑i=1lb-ji2)1/2P-(m∑i=1lc¯ji2)1/2P-*-M-10000**-M-2000***-M-300****-In0*****-W-]<0m1-σ2-2αp≤0,P≥pIn,P-≥p-In,l1-τ2-2βp-≤0,L2M2L2-(1-τ)Q≤0,M3-(1-τ1)U≤0,L-2M-2L-2-(1-σ)Q-≤0,M-3-(1-σ1)U-≤0,
and the adaptive feedback controllers are designed as
(40)ui(t)=-[γi+αi(t)]ei(t),vj(t)=-[ηj+βj(t)]εj(t),αi(t)={∥G-(e˙i(t))∥2∥ei(t)∥2α,∥ei(t)∥2≠0,0,∥ei(t)∥2=0,βj(t)={∥G(ε˙j(t))∥2∥εj(t)∥2β,∥εj(t)∥2≠0,0,∥εj(t)∥2=0,whereZi=-2PD+L-3H-W-H-L-3+L1M1L1+mU--2γiP+Q,  Z-j=-2P-D-+L3HWHL3+L-1M-1L-1+lU-2ηjP-+Q-,i=1,2,…,l,j=1,2,…,m.Remark 6.
From Corollary5, we can easily get that the controllers in this paper are simpler than those of Theorem 1 in [29].Remark 7.
If the coupling matrix of the SCBNN is not a diffusive matrix satisfying the sum of every row being zero, we can still obtain the same result from the proof of Theorem4.
Theorem8 presents another sufficient condition to ascertain that the two networks (4) and (5) can be synchronized, using the following simple adaptive feedback controllers:
(41)ui(t)=-γiei(t),vj(t)=-γ-jεj(t),
where i=1,2,…,l,j=1,2,…,m, γi, and γ-j are positive constants.
Let(42)e(t)=(e1T(t),e2T(t),…,elT(t))T,ε(t)=(ε1T(t),ε2T(t),…,εmT(t))T,F~k(e(t))=(FkT(e1(t)),FkT(e2(t)),…,FkT(el(t)))T,k=1,2,F^k(e(t))=(F-kT(ε1(t)),F-kT(ε2(t)),…,F-kT(εm(t)))T,k=1,2,G~(ε˙(t))=(GT(ε˙1(t)),GT(ε˙2(t)),…,GT(ε˙m(t)))T,G^(e˙(t))=(G-T(e˙1(t)),G-T(e˙2(t)),…,G-T(e˙l(t)))T,K~(ε(t))=(KT(ε1(t)),KT(ε2(t)),…,KT(εm(t)))T,K^(e(t))=(K-T(e1(t)),K-T(e2(t)),…,K-T(el(t)))T,Γ~=-diag(γ1,γ2,…,γl),Γ^=-diag(γ-1,γ-2,…,γ-m),
then the error dynamical system of (6) becomes
(43)e˙(t)=∑r=1Nξr(t,λ)×[-(Il⊗Dr)e(t)+(Il⊗Rr1)F~1(e(t))+(Il⊗Rr2)F~2(e(t-τ(t)))+(Ar⊗In)ε(t-τ1(t))+(Br⊗In)G~(ε˙(t-τ2(t)))+(Cr⊗In)∫-∞t(Im⊗h(t-s))K~(ε(s))ds+(Γ~⊗In)e(t)],ε˙(t)=∑r=1Nξr(t,λ)×[-(Im⊗D-)ε(t)+(Im⊗R-r1)F^1(ε(t))+(Im⊗R-r2)F^2(ε(t-σ(t)))+(A-r⊗In)e(t-σ1(t))+(B-r⊗In)G^(e˙(t-σ2(t)))+(C-r⊗In)∫-∞t(Il⊗h(t-s))K^(e(s))ds+(Γ^⊗In)ε(t)],Theorem 8.
Under assumptions(S1)–(S4) and using the adaptive feedback controllers (41), the two coupled SCBNNs (4) and (5) can be synchronized, if there exist n×n positive matrices P,U,P-,U- and n×n diagonal positive matrices W=diag(w1,w2,…,wn), W-=diag(w-1,w-2,…,w-n), Q, V, M, Q-, V-, M- such that for r∈ℵ, the following matrix inequalities hold:(44)Ωr=(Ψr1Ψr2Ψr3Ψr4Ψr5Ψr6*Ψr7Il⊗(Rr1TV-Rr2)Ar⊗(Rr1TV-)Br⊗(Rr1TV-)Cr⊗(Rr1TV-)**Ψr8Ar⊗(Rr2TV-)Br⊗(Rr2TV-)Cr⊗(Rr2TV-)***Ψr9(ArTBr)⊗V-(ArTCr)⊗V-****Ψr10(BrTCr)⊗V-*****Ψr11)<0,Ω-r=(Ψ-r1Ψ-r2Ψ-r3Ψ-r4Ψ-r5Ψ-r6*Ψ-r7Il⊗(R-r1TVR-r2)A-r⊗(R-r1TV)B-r⊗(R-r1TV)C¯r⊗(R-r1TV)**Ψ-r8A-r⊗(R-r2TV)B-r⊗(R-r2TV)C-r⊗(R-r2TV)***Ψ-r9(B-r)⊗V(A-rTC-r)⊗V****Ψ-r10B-rTC-r⊗V*****Ψ-r11)<0,with
(45)Ψr1=Il⊗(-PDr-DrTP+Q+U-+L-3H-W-H-L-3+L1ML1+DrTV-Dr)+2Γ⊗P+Γ2⊗V--Γ⊗(DrTV-+V-Dr),Ψr2=Γ⊗(V-Rr1)+Il⊗(PRr1-DrTV-Rr1),Ψr3=Il⊗(PRr2-DrTV-Rr2)+Γ⊗V-Rr2,Ψr4=(ΓAr)⊗V-+Ar⊗(P-DrTV-),Ψr5=(ΓBr)⊗V-+Br⊗(P-DrTV-),Ψr6=(ΓCr)⊗V-+Cr⊗(P-DrTV-),Ψr7=Il⊗(Rr1TV-Rr1-M),Ψr8=Il⊗[Rr2TV-Rr2-(1-τ)L2-1QL2-1],Ψr9=(ArTAr)⊗V--(1-τ1)(Im⊗U),Ψr10=(BrTBr)⊗V--(1-τ2)(Im⊗(L4-1VL4-1)),Ψr11=(CrTCr)⊗V--Im⊗W-,Ψ-r1=Im⊗(-P-D-r-D-rTP-+Q-+U+L3HWHL3+L-1M-L-1+D-rTVD-r)+2Γ-⊗P-+Γ-2⊗V-Γ-⊗(D-rTV+VD-r),Ψ-r2=Γ-⊗(VR-r1)+Im⊗(P-R-r1-D-rTVR-r1),Ψ-3=Im⊗(P-R-r2-D-rTVR-r2)+Γ-⊗VR-r2,Ψ-r4=(Γ-A-r)⊗V+A-r⊗(P--D-rTV),Ψ-5=(Γ-B-r)⊗V+B-r⊗(P--D-rTV),Ψ-r6=(Γ-C-r)⊗V+C-r⊗(P--D-rTV),Ψ-r7=Im⊗(R-r1TVR-r1-M-),Ψ-8=Im⊗[R-r2TVR-r2-(1-σ)L-2-1Q-L-2-1],Ψ-r9=(A-rTA-r)⊗V-(1-σ1)(Il⊗U-),Ψ-r10=(B-rTB-r)⊗V-(1-σ2)(Il⊗(L-4-1V-L-4)),Ψ-r11=(C-rTC-r)⊗V-Il⊗W.Proof.
For the error dynamical system (43), we define the following Lyapunov-Krasovskii function:
(46)V(t,e(t),ε(t))=V1(t,e(t),ε(t))+V2(t,e(t),ε(t)),V1=eT(t)(Il⊗P)e(t)+∫t-τ(t)teT(s)(Il⊗Q)e(s)ds+∫t-τ1(t)tεT(s)(Im⊗U)ε(s)ds+∑j=1m∑i=1nwi∫0∞hi(v)dv∫0∞hi(θ)×∫t-θtKi2(εji(s))dsdθ+∫t-τ2(t)tε˙T(s)(Im⊗V)ε˙(s)ds,V2=εT(t)(Im⊗P-)ε(t)+∫t-σ(t)tεT(s)(Im⊗Q-)ε(s)ds+∫t-σ1(t)teT(s)(Il⊗U-)e(s)ds+∑i=1l∑j=1nw-j∫0∞h-j(v)dv∫0∞h-j(θ)×∫t-θtK-j(eij(s))dsdθ+∫t-σ2(t)te˙T(s)(Il⊗V-)e˙(s)ds,V˙1=eT(t)(Il⊗P)e˙(t)+e˙T(t)(Il⊗P)e(t)+eT(t)(Il⊗Q)e(t)-(1-τ˙(t))eT(t-τ(t))×(Il⊗Q)e(t-τ(t))+εT(t)(Im⊗U)ε(t)-(1-τ˙1(t))εT(t-τ1(t))(Im⊗U)ε(t-τ1(t))+∑j=1m∑i=1nwi(Ki(εji(t))∫0∞hi(v)dv)2-∑j=1m∑i=1nwi∫0∞hi(v)dv∫0∞hi(θ)Ki2(εji(t-θ))dθ+ε˙T(t)(Im⊗V)ε˙(t)-(1-τ˙2(t))ε˙T(t-τ2(t))(Im⊗V)ε˙(t-τ2(t)),V˙2=εT(t)(Im⊗P-)ε˙(t)+ε˙T(t)(Im⊗P-)ε(t)+ε(t)(Im⊗Q-)ε(t)-(1-σ˙(t))ε(t-σ(t))×(Im⊗Q-)ε(t-σ(t))+eT(t)(Il⊗U-)e(t)-(1-σ˙1(t))eT(t-σ1(t))(Il⊗U-)e(t-σ1(t))+∑i=1l∑j=1nw-j(K-j(eij(t))∫0∞h-j(v)dv)2-∑i=1l∑j=1nw-j∫0∞h-j(v)dv∫0∞h-j(θ)K-j2(eij(t-θ))dθ+e˙T(t)(Il⊗V-)e˙(t)-(1-σ˙2(t))e˙T(t-σ2(t))×(Il⊗V-)e˙(t-σ2(t)).
By (26), we have
(47)∑j=1m∑i=1nwi(Ki(εji(t))∫0∞hi(v)dv)2≤∑j=1mεjT(t)L3HWHL3εj(t)=εT(t)(Im⊗(L3HWHL3))ε(t).
Using (30), we get
(48)∑j=1m∑i=1nwi∫0∞hi(v)dv∫0∞hi(θ)Ki2(εji(s)(t-θ))dθ≥∑j=1m(∫-∞th(t-s)K(εj(s))ds)T×W∫-∞th(t-s)K(εj(s))ds=(∫-∞t(Im⊗h(t-s))K~(ε(s))ds)T(Im⊗W)×∫-∞t(Im⊗h(t-s))K~(ε(s))ds.
From (S3) and (46)–(48), we have
(49)V˙1≤eT(t)(Il⊗P)e˙(t)+e˙T(t)(Il⊗P)e(t)+eT(t)(Il⊗Q)e(t)-(1-τ)eT(t-τ(t))×(Il⊗Q)e(t-τ(t))+εT(t)[Im⊗(U+L3HWHL3)]ε(t)-(1-τ1)εT(t-τ1(t))(Im⊗U)ε(t-τ1(t))-(∫-∞t(Im⊗h(t-s))K~(ε(s))ds)T(Im⊗W)×∫-∞t(Im⊗h(t-s))K~(ε(s))ds+ε˙T(t)(Im⊗V)ε˙(t)-(1-τ2)ε˙T(t-τ2(t))×(Im⊗V)ε˙(t-τ2(t)).
In the same way, we have
(50)V˙2≤εT(t)(Im⊗P-)ε˙(t)+ε˙T(t)(Il⊗P-)ε(t)+εT(t)(Im⊗Q-)ε(t)-(1-σ)εT(t-σ(t))×(Im⊗Q)ε(t-σ(t))+eT(t)[Il⊗(U-+L-3H-W-H-L-3)]e(t)-(1-σ1)eT(t-σ1(t))(Il⊗U-)e(t-σ1(t))-(∫-∞t(Il⊗h-(t-s))K^(e(s))ds)T(Il⊗W-)×∫-∞t(Il⊗h-(t-s))K^(e(s))ds+e˙T(t)(Il⊗V-)e˙(t)-(1-σ2)e˙T(t-σ2(t))×(Il⊗V-)e˙(t-σ2(t)).
From (S1) and (S2),
(51)eT(t)[Il⊗(L1ML1)]e(t)-F~1T(e(t))(Il⊗M)F~1(e(t))≥0,εT(t)[Im⊗(L-1M-L-1)]ε(t)-F^1T(ε(t))(Im⊗M-)F^1(ε(t))≥0,eT(t-τ(t))(Il⊗Q)e(t-τ(t))≥F~2T(e(t-τ(t)))[Il⊗(L2-1QL2-1)]F~2(e(t-τ(t))),εT(t-σ(t))(Im⊗Q-)ε(t-σ(t))≥F^2T(ε(t-σ(t)))[Im⊗(L-2-1Q-L-2-1)]F^2(ε(t-σ(t))),ε˙T(t-τ2(t))(Im⊗V)ε˙(t-τ2(t))≥G~T(ε˙(t-τ2(t)))(Im⊗(L4-1VL4-1))G~(ε˙(t-τ2(t))),e˙T(t-σ2(t))(Il⊗V-)e˙(t-σ2(t))≥G^T(e˙(t-σ2(t)))(Il⊗(L-4-1V-L-4-1))G^(e˙(t-σ2(t))).
With the aid of (43) and (51), we have
(52)V˙≤∑r=1Nξr(t,λ)[ηT(t)Ωrη(t)+η-T(t)Ω-rη-(t)],
where
(53)η(t)=((∫-∞t(Im⊗h(t-s))K~(ε(s))ds)TeT(t),F~1T(e(t)),F~2T(e(t-τ(t))),εT(t-τ1(t)),G~T(ε˙(t-τ2(t))),(∫-∞t(Im⊗h(t-s))K~(ε(s))ds)T)T,η-(t)=((∫-∞t(Il⊗h-(t-s))K^(e(s))ds)TεT(t),F^1T(ε(t)),F^2T(ε(t-σ(t))),eT(t-σ1(t)),G^T(e˙(t-σ2(t))),(∫-∞t(Il⊗h-(t-s))K^(e(s))ds)T)T.
Letρ=min{ρ1,ρ2}, where ρ1=-min{ρmin(Ωr),r∈ℵ}, ρ2=-min{ρmin(Ω-r),r∈ℵ}, then ρ>0 and
(54)V˙≤-ρ1∑i=1leiT(t)ei(t)-ρ2∑j=1mεjT(t)εj(t)≤-ρ[∑i=1leiT(t)ei(t)+∑j=1mεjT(t)εj(t)].
The following proof is similar to that of Theorem 4 and is omitted here.


## 4. Simulations

In this section, numerical examples are provided to demonstrate the validity of the synchronization criteria obtained in the previous sections. Consider the following network as drive network:(55)x˙i(t)=∑r=1Nξr(t,λ)×[+∑j=1mcrij∫-∞th(t-s)k(yj(s))ds-Drxi+Rr1f1(xi(t))+Rr2f2(xi(t-τ(t)))+Ir+∑j=1marijyj(t-τ1(t))+∑j=1mbrijg(y˙j(t-τ2(t)))+∑j=1mcrij∫-∞th(t-s)k(yj(s))ds],i=12,…,l,y˙j(t)=∑r=1Nξr(t,λ)×[+∑i=1lc-rji-D-ryj+R-r1f-1(yj(t))+R-r2f-2(yj(t-σ(t)))+Jr+∑i=1la-rjixi(t-σ1(t))+∑i=1lb-rjig-(x˙i(t-σ2(t)))+∑i=1lc-rji∫-∞th-(t-s)k-(xi(s))ds],j=1,2,…,m,
where xi(t),yj(t)∈R2, l=3, and m=3. f1(z(t))=0.1(tanh(z1(t)),tanh(z2(t)))T, z(t)=(z1(t),z2(t))T, f-1=g=g-=k=k-=f2=f-2=f1, and h(t)=h-(t)=diag(e-t,e-t). Choose time delays τ(t)=1+0.4sint,τ1(t)=2+0.2arctan(t), τ2(t)=0.6+0.5cost, and σ(t)=1+0.8sint, σ1(t)=0.7+0.1cost, σ2(t)=0.5+(0.3et/(1+et)). We define a switching rule λ:t∈[0,+∞)→{1,2},λ(t)=int(t)mod2+1. The other parameters are as follows:
(56)D1=(1.8004),R11=(-1100.2),R12=(10.50.6-1),I1=J1=(1,2)T,A1=(a1ij)=(-2-2002-211-2),B1=(b1ij)=(-0.20-0.20.1-0.40.30.20.1-0.3),C1=(c1ij)=(1-1011-2-101),D-1=(2001),R-11=(-0.310.20.3),R-12=(0.30.40.6-0.5),A-1=(a-1ji)=(40-411-210-1),B-1=(b-1ji)=(0.10-0.10.2-0.30.10.10.2-0.2),C-1=(c-1ji)=(-31220-2-404),D2=(1.4001.4),R21=(1-1-53),R22=(-1.5-0.1-3-1),A2=(a2ij)=(-11002-21.21-2.2),B2=(b2ij)=(0.1-0.100.1-0.50.40.20-0.2),C2=(c2ij)=(2-203-1-20-11),D-2=(1.2001.2),R-21=(-0.31-41),R-22=(0.30.4-2-1),I2=J2=(3,4)T,A-2=(a-2ji)=(1-1012-311-2),B-2=(b-2ji)=(0.2-0.200.1-0.20.10.30.1-0.4),C-2=(c-2ji)=(-51411-2-101).
The response network of drive network (55) is
(57)x^˙i(t)=∑r=1Nξr(t,λ)×[+∑j=1mcrij∫-∞th(t-s)k(y^j(s))ds+ui(t)-Drx^i(t)+Rr1f1(x^i(t))+Rr2f2(x^i(t-τ(t)))+Ir+∑j=1marijy^j(t-τ1(t))+∑j=1mbrijg(y^˙j(t-τ2(t)))+∑j=1mcrij∫-∞th(t-s)k(y^j(s))ds+ui(t)],y^˙j(t)=∑r=1Nξr(t,λ)×[+∑i=1lc-rji∫-∞th-(t-s)k-(x^i(s))ds+vj(t)-D-ry^j(t)+R-r1f-1(y^j(t))+R-r2f-2(y^j(t-σ(t)))+Jr+∑i=1la-rjix^i(t-σ1(t))+∑i=1lb-rjig-(x^˙i(t-σ2(t)))+∑i=1lc-rji∫-∞th-(t-s)k-(x^i(s))ds+vj(t)],
where ui(t),vj(t)∈R2.Letγ1=γ2=γ3=15, η1=η2=η3=16, α=0.5, β=0.5, and the feasible solution of the matrix inequalities (15)–(19) by employing MATLAB LMI Toolbox be as follows:
(58)p=7.5267,p-=7.8951,P=(12.95080.28170.281710.7871),Q=(83.64053.66743.667468.8593),U=(119.10800.22160.221655.7869),W=(233.787000223.7801),M1=(227.115300202.1669),M2=(206.303400217.8341),M3=(86.97230041.4980),P-=(19.45480.03360.033611.6060),Q-=(26.19730.54110.54119.6902),U-=(30.91260.98500.985022.6634),W-=(316.694300287.9793),M-1=(183.041900162.4080),M-2=(250.278700409.0398),M-3=(13.87040010.1694).
The initial values are chosen as xi(s)=(-5,9), yj(s)=(-6,7)T, x^i(s)=2i(2,5)T, y^j(s)=3j(2,-1)T, and s∈[-2,0]. Clearly, the two coupled networks (55) and (57) satisfy the conditions of Theorem 4. Figure 1 presents the synchronization errors of the state variables between the two networks. The simulation result shows that the synchronization is achieved under the proposed controllers (20). Thus, the proposed synchronization control scheme in Theorem 4 is valid.Synchronization errors of BDN (55) and (57) with adaptive feedback controllers (20).
(a)
(b)
(c)
(d)
(e)
(f)Letγ1=γ2=γ3=12, γ-1=γ-2=γ-3=17, then the feasible solution of the matrix inequalities (44) in Theorem 8 by employing MATLAB LMI Toolbox is as follows:
(59)P=(0.02510.00050.00050.0214),U=(3.7826-0.0043-0.00433.5381),Q=(4.2792003.7789),V=(0.0116000.0079),M=(3.4521003.1482),W=(3.7071003.3433),P-=(0.0298000.0268),U-=(3.3712-0.0132-0.01323.1782),Q-=(7.2228006.3479),V-=(0.0077000.0052),M-=(3.3924003.0142),W-=(3.4120003.2474).
Using the controllers (41), the simulation result is given in Figure 2, which shows that the proposed synchronization control scheme in Theorem 8 is effective.Synchronization errors with adaptive feedback controllers (41).
(a)
(b)
(c)
(d)
(e)
(f)


## 5. Conclusions

In this paper, we have proposed a general SCBNN with distributed delays and derivative coupling and investigated the synchronization problem in the two coupled SCBNNs. Using linear matrix inequality (LMI) approach and Barbǎlat lemma, we have deviated some useful synchronization criteria to ensure the synchronization of these two SCBNNs by constructing effective controllers. Compared with relative previous jobs, the controllers proposed by us are more simple and feasible. Some simulation results have been presented to demonstrate our theoretical results. In our future work, we will consider using pinning control to realize the synchronization of SCBNNs and identify the network topology of the unknown SCBNNs.



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*Source: 728606-2013-05-30.xml*