# Mean-Square Exponential Synchronization of Stochastic Complex Dynamical Networks with Switching Topology by Impulsive Control

Authors: Xuefei Wu; Chen Xu
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper investigates the mean-square exponential synchronization issues of delayed stochastic complex dynamical networks with switching topology and impulsive control. By using the Lyapunov functional method, impulsive control theory, and linear matrix inequality (LMI) approaches, some sufficient conditions are derived to guarantee the mean-square exponential synchronization of delay complex dynamical network with switch topology, which are independent of the network size and switch topology. Numerical simulations are given to illustrate the effectiveness of the obtained results in the end.


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## Body

## 1. Introduction

Complex networks are everywhere in nature and our daily life, such as the Internet, ecosystems, social networks, World Wide Web, and neural networks. A complex network can be described by a set of nodes and edges interconnecting these nodes together. During the last two decades, complex networks have been focused on by scientists from various fields, such as mathematical, engineering, and social and economic science. There are many literatures concerning the collective behaviors of complex networks [1–11]. Among them, synchronization is the most interesting phenomenon [3–11], because the synchronization is a kind of typical collective behaviors exhibited in many natural systems.Due to many complex systems may experience abrupt changes in their connection caused by some phenomena such as link failures, component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbance, and so forth. Thus, the synchronization of complex network with switching topology has been studied in [12–17]. In [12], Yao et al. studied the synchronization of a general complex dynamical network with switching topology and the time-varying coupling is unknown but bounded. Wang et al. [13] investigated the synchronization issues of complex dynamical networks with switching topology. Yu et al. [14] explored the synchronization of switched linearly coupled neural networks with delay. Some sufficient conditions were given to guarantee the global synchronization. In [15], Liu et al. studied the local and global exponential synchronization of complex dynamical network with switching topology and time-varying coupling delays. In [16], some sufficient conditions were given to guarantee the synchronization of leader following issues with switching connective network and coupling delay. In [17], authors investigated the consensus problem in mean square for uncertain multiagent systems with stochastic measurement noises and symmetric or asymmetric time-varying delays.Uncertainties commonly exist in the real world, such as stochastic forces on the physical systems and noisy measurements caused by environmental uncertainties. Thus, a stochastic behavior should be produced instead of a deterministic one [18]. In fact, signals transmitted between nodes of complex networks are unavoidably subject to stochastic perturbations from environment, which may cause information contained in these signals to be lost [19]. Therefore, stochastic perturbations should be considered [18–23]. In [19–21], stochastic perturbations are all one-dimensional, which means that the signal transmitted by nodes is influenced by the same noise. In [18, 22], the authors considered stochastic synchronization of coupled neural networks, in which noise perturbations are vector forms. Vector-form perturbation means that different nodes are influenced by different noise, which is more practical in the real world. In [23], authors investigated the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method.In many systems, the impulsive effects are common phenomena due to instantaneous perturbations at certain moments [24–28]. In the past several years, impulsive control strategies have been widely used to stabilize and synchronize coupled complex dynamical system, such as signal processing system, computer networks, automatic control systems, and telecommunications. In [24], Cai et al. investigated the robust impulsive synchronization of complex delayed dynamical networks. Yang et al. [25] studied the exponential synchronization of complex dynamical network with a coupling delay and impulsive control. Zhu et al. gave some global impulsive exponential synchronization criteria of time-delayed coupled chaotic systems [27]. In [28], some sufficient conditions were given to guarantee the consensus of nonlinear multiagent systems with switching topology.Based on the above discussion, studying the synchronization problem of the complex dynamical network with switch topology and impulsive effects is very useful and meaningful. It should be pointed out that exponential synchronization of the coupling delay complex dynamical networks with switch topology and impulsive effects has received very little research attention.In this paper, we investigate the problem of exponential synchronization of coupling delay complex dynamical network with switching topology and impulsive effects, basing on the Lyapunov theory and impulsive control theory and by assuming that there exists finite connective topology of the complex dynamical network, and the connective topology may be switched (or jumped) from one to another at different moments. For controlling, the synchronization state can be any weighting average of the network states. It means that each node of the dynamical network can contribute to synchronization of the network in its weight. Finally, some sufficient conditions are given to guarantee the synchronization of the complex dynamical network, which are independent of the network size and switch topology.


## 2. Model and Preliminaries

The switching complex dynamical networks investigated in this paper consist ofN nodes, whose state is described as follows:
(1)dxi(t)={∑j=1N‍f(t,xi(t),xi(t-τ(t)))+∑j=1N‍aij(r(t))Σxj(t)+∑j=1N‍bij(r(t))Σxj(t-τc(t))}dt+σi(t,x(t),x(t-τ(t)),x(t-τc(t)),r(t))dwi(t),t≠tk,Δxi(tk)=xi(tk+)-xi(tk-)=ϵikxi(tk-),t=tk,k∈Z+,i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈ℝn is the state vector of the ith node of the network, f(t,xi(t),xi(t-τ(t)))=[f1(t,xi(t),xi(t-τ(t))),f2(t,xi(t),xi(t-τ(t))),…,fn(t,xi(t),xi(t-τ(t)))]T is a continuous vector-valued function, Σ=diag(ϱ1,ϱ2,…,ϱn) is an inner coupling of the networks that satisfies ϱj>0, j=1,2,…,n, and r(t):[0,∞)→ℵ={1,2,…,M} is a switching signal, which is a piecewise constant function. Here, A(r(t))=[aij(r(t))]∈ℝn×n and B(r(t))=[bij(r(t))]∈ℝn×n are the outer-coupling matrices of the network at time t at nodes r(t), t-τc(t), and r(t), respectively, such that aij(r(t))≥0 for i≠j, aii(r(t))=-∑j=i,j≠iN‍aij(r(t)), bij(r(t))≥0 for i≠j, and bii(r(t))=-∑j=i,j≠iN‍bij(r(t)). τ(t) is the inner time-varying delay satisfying τ≥τ(t)≥0 and τc(t) is the coupling time-varying delay satisfying τc≥τc(t)≥0. Finally, σi(t,x(t),x(t-τ(t)),x(t-τc(t)),r(t))=σi(t,x1(t),…,xn(t),x1(t-τ(t)),…,xn(t-τ(t)),x1(t-τc(t)),…,xn(t-τc(t)),r(t))∈ℝn×n and wi(t)=(wi1(t),wi2(t),…,win(t))T∈ℝn is a bounded vector-form Weiner process, satisfying
(2)𝔼wij(t)=0,𝔼wij2(t)=1,𝔼wij(t)wij(s)=0(s≠t).ϵik is the ith node impulsive gain at t=tk. The discrete set {tk} satisfies tk=tk-1+T, t0≥0, tk→+∞ as k→+∞, note x(tk-)=limt→tk-x(t), and x(tk+)=limt→tk+x(t)=x(tk). In this paper, A(r(t)) is assumed to be irreducible in the sense that there are no isolated nodes.The initial conditions associated with (1) are
(3)xi(s)=ξi(s),-τˇ≤s≤0,i=1,2,…,N,
where τˇ=max{τ(t),τc(t)}, ξi∈Cℱ0b([-τˇ,0],ℝn) with the norm ∥ξi∥2=sup-τˇ≤s≤0ξi(s)Tξi(s) and our objective is to control system (1) so that it stays in the trajectory s(t)∈ℝn of the system
(4)ds(t)=f(t,s(t),s(t-τ(t)))dt.Remark 1.
In general, the synchronization states(t) may be an equilibrium point, a periodic orbit, or a chaotic attractor.
The following assumptions will be used throughout this paper for establishing the synchronization conditions.(H1)τ(t) and τc(t) are bounded and continuously differentiable functions such that 0<τ(t)≤τ, τ˙(t)<τ-<1, 0<τc(t)≤τc, and τ˙c(t)<τ-c<1. Let τˇ=max{τ,τc} and τ-ˇ=max{τ-,τ-c}.(H2) Letσ(t,e(t),e(t-τ(t)),e(t-τc(t)),r)=σ(t,e1(t),…,eN(t),e1(t-τ(t)),…,eN(t-τ(t)),e1(t-τc(t)),…,eN(t-τc(t)),r). Then there exist positive definite constant matrices Υi1r, Υi2r, and Υi3r for i=1,2,…,N and r=1,2,…,M such that
(5)Tr[σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r)TTr×σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r)(t,e(t),e(t-τ(t)),e(t-τc(t)),r)T]≤∑j=1N‍ej(t)TΥi1rej(t)+∑j=1N‍ej(t-τ(t))TΥi2rej(t-τ(t))+∑j=1N‍ej(t-τc(t))TΥi3rej(t-τc(t)).(H3)σi(t,s(t),…,s(t),s(t-τ(t)),…,s(t-τ(t)),s(t-τc(t)),…,s(t-τc(t)),r(t))=0.Define error stateei(t)=xi(t)-s(t),
(6)dei(t)={+∑j=1N‍f(t,xi(t),xi(t-τ(t)))-f(t,s(t),s(t-τ(t)))dei(t)=+∑j=1N‍aij(r(t))Σej(t)+∑j=1N‍bij(r(t))Σej(t-τc(t))dei(t)=+ui(t)+∑j=1N‍}dtdei(t)=+σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r(t))dwi(t),t≠tk,Δei(tk)=ei(tk+)-ei(tk-)=ϵikei(tk-),t=tk,k∈Z+,i=1,2,…,N.Definition 2.
The complex network (1) is said to be exponentially synchronized in mean square if the trivial solution of system (6) is such that
(7)∑i=1N‍𝔼∥ei(t,t0,ξi)∥2≤Ke-κt,
for some K>0 and some κ>0 for any initial data ξi∈𝒞ℱ0b([-τ,0];ℝn).Definition 3.
A continuous functionf(t,x,y):[0,+∞]×ℝn×ℝn→ℝn is said to belong to the function class QUAD, denoted by f∈QUAD(P,Δ,η,θ) for some given matrix Σ=diag{ϱ1,ϱ2,…,ϱn} if there exist a positive definite diagonal matrix P=diag{p1,p2,…,pn}, a diagonal matrix Δ=diag{δ1,δ2,…,δn}, and constants η>0,θ>0 such that f(·) satisfies the condition
(8)(x-y)TP((f(t,x,z)-f(t,y,w))-ΔΣ(x-y))≤-η(x-y)T(x-y)+θ(z-w)T(z-w)
for all x,y,z,w∈ℝn.Remark 4.
The function class QUAD includes almost all the well-known chaotic systems with or without delays such as the Lorenz system, the Rössler system, the Chen system, the delayed Chua’s circuit, the logistic delayed differential system, the delayed Hopfield neural network, and the delayed CNNs. We shall simply write(9)pˇ=max{p1,p2,…,pn},p^=min{p1,p2,…,pn},δˇ=max{δ1,δ2,…,δn}.In order to derive the main results, it is necessary to propose the following lemmas.Lemma 5 (see [16]).
The following linear matrix inequality(10)[QSSTR]>0,
where Q=QT,R=RT, is equivalent to one of the following conditions: (i)
Q>0,R-STQ-1S>0.(ii)
R>0,Q-STR-1S>0.


## 3. Main Result

In this section, we investigate the exponential stability condition of the error system (6). Some new criteria are presented for the exponential synchronization of network (1) based on the Lyapunov functional method, linear matrix inequality approach, and impulsive control theory.Theorem 6.
Let assumptions (H1) and (H2) be true and letf∈QUAD(P,Δ,η,θ). If there exist positive constants αr and βr such that
(11)[A(r)s+δˇIN-αrINB(r)2B(r)T2-βrIN]≤0,forr=1,2,…,M,(12)(γ+aˇ+b1-τ-eγτ+cˇ1-τ-ceγτc)×(1+Tτˇ)T+2ln|1+ϵ|T-γ≤-η,
where
(13)ar=λmin(-2ηIn+pˇ∑i=1N‍Υi1r+2αrPΣ)pˇ,aˇ=maxr∈Sar,br=λmax(∑i=1NPΥi2r+2θIN)p^,bˇ=maxr∈Sbr,cr=λmax(∑i=1N‍PΥi3r+2βrPΣ)p^,cˇ=maxr∈Scr,|1+ϵ|=maxi=1,2,…,N,k∈Z+|1+ϵik|,
then the solutions x1(t),x1(t),…,xN(t) of system (6) are globally and exponentially stable.Proof.
Define the Lyapunov-Krasovskii function(14)V(t,e(t))=12∑i=1N‍ei(t)TPei(t)
and let e~k(t)=(e1k(t),e2k(t),…,eNk(t))T, k=1,2,…,n. For r(t)=r, we have
(15)ℒV(t,e(t),r)=∑i=1N‍ei(t)TP{+∑j=1N‍f(t,xi(t),xi(t-τ(t)))=∑i=1N‍ei(t)TP-f(t,s(t),s(t-τ(t)))+∑j=1N‍aij(r)Σej(t)=∑i=1N‍ei(t)TP+∑j=1N‍bij(r)Σej(t-τc(t))}+12∑i=1N‍Tr{σi(t,x(t),x(t-τ(t)),x(t-τc(t)),r)T+12∑i=1N‍Tr×Pσi(t,x(t),x(t-τ(t)),x(t-τc(t)),r)(t,x(t),x(t-τ(t)),x(t-τc(t)),r)T}=∑i=1N‍ei(t)TP{f(t,xi(t),xi(t-τ(t)))=∑i=1N‍ei(t)TP-f(t,s(t),s(t-τ(t)))-ΔΣei(t)}+∑i=1N‍ei(t)TPΔΣei(t)+∑i=1N‍∑j=1N‍aij(r)ei(t)TPΣej(t)+∑i=1N‍∑j=1N‍bij(r)ei(t)TPΣej(t-τc(t))+12∑i=1N‍Tr{σi(t,x(t),x(t-τ(t)),x(t-τc(t)),r)T+12∑i=1N‍Tr×Pσi(t,x(t),x(t-τ(t)),x(t-τc(t)),r)(t,x(t),x(t-τ(t)),x(t-τc(t)),r)T}≤{-η∑i=1N‍ei(t)Tei(t)+θ∑i=1N‍ei(t-τ(t))Tei(t-τ(t))≤+∑k=1n‍pkϱkδke~k(t)Te~k(t)+∑k=1n‍pkϱke~k(t)TA(r)e~k(t)≤+∑k=1n‍pkϱke~k(t)TB(r)e~k(t-τc(t))≤+12pˇ∑j=1N‍[∑i=1N‍ei(t)TΥj1rei(t)≤+12pˇ∑j=1N‍+∑i=1N‍ei(t-τ(t))TΥj2rei(t-τ(t))≤+12pˇ∑j=1N‍+∑i=1N‍ei(t-τc(t))TΥj3rei(t-τc(t))]}={+12pˇ∑i=1N‍∑j=1N‍ei(t-τc(t))TΥj3rei(t-τc(t))-η∑i=1N‍ei(t)Tei(t)+θ∑i=1N‍ei(t-τ(t))Tei(t-τ(t))=+12pˇ∑i=1N‍∑j=1N‍ei(t-τc(t))TΥj3rei(t-τc(t))}+{[∑i=1N‍∑j=1N‍ei(t)TΥj1rei(t)∑k=1n‍pkϱke~k(t)TA(r)e~k(t)++∑k=1n‍pkϱke~k(t)TB(r)e~k(t-τc(t))++12pˇ[∑i=1N‍∑j=1N‍ei(t)TΥj1rei(t)++12pˇ+∑i=1N‍∑j=1N‍ei(t-τ(t))TΥj2rei(t-τ(t))]++∑k=1n‍pkϱkδke~k(t)Te~k(t)[∑i=1N‍∑j=1N‍ei(t)TΥj1rei(t)}={∑i=1N‍ei(t)T(-ηIN+12pˇ∑j=1N‍Υj1r+αrPΣ)ei(t)=+∑i=1N‍ei(t-τ(t))T(θ+12pˇ∑j=1N‍Υj2r)ei(t-τ(t))=+∑i=1N‍ei(t-τc(t))T(12pˇ∑j=1N‍Υj3r+βrPΣ)=×ei(t-τc(t))(-ηIN+12pˇ∑j=1N‍Υj1r+αrPΣ)}+{∑k=1n‍pkϱke~k(t)T[A(r)+(δˇ-αr)IN]e~k(t)++∑k=1n‍pkϱke~k(t)TB(r)e~k(t-τc(t))+-∑k=1n‍pkϱke~k(t-τc(t))Tβre~k(t-τc(t))∑k=1n‍}≤{∑i=1N‍ei(t)T(-ηIN+12pˇ∑j=1N‍Υj1r+αrPΣ)ei(t)≤+∑i=1N‍ei(t-τ(t))T(θIN+12pˇ∑j=1N‍Υj2r)ei(t-τ(t))≤+∑i=1N‍ei(t-τc(t))T(12pˇ∑j=1N‍Υj3r+βrPΣ)≤×ei(t-τc(t))(-ηIN+12pˇ∑j=1N‍Υj1r+αrPΣ)}.
Then we have
(16)ℒV(t,e(t))≤arV(t)+brV(t-τ(t))+crV(t-τc(t)).
So we have
(17)ℒV(t)≤aˇV(t)+bˇV(t-τ(t))+cˇV(t-τc(t)).
Define
(18)W(t)=eγtV(t)
and use (17) to compute the operator
(19)ℒW(t)=eγt[γV(t)+ℒV(t)]≤eγt[γV(t)+aˇV(t)+bˇV(t-τ(t))≤eγt+cˇV(t-τc(t))γV(t)+aˇV(t)+bˇV(t-τ(t))],
which after applying the generalized Itô’s formula, gives
(20)eγt𝔼V(t)=eγt0𝔼V(t0)+𝔼∫t0t‍ℒW(s)ds
for any tk>t>t0>tk-1≥0. Hence we have
(21)eγt𝔼V(t)≤eγt0𝔼V(t0)+∫t0t‍eγs[+bˇV(s-τ(s))+cˇV(s-τc(s))γV(s)+aˇV(s)+∫t0t‍eγs+bˇV(s-τ(s))+cˇV(s-τc(s))]ds≤eγt0𝔼V(t0)+(γ+aˇ)∫t0t‍eγs𝔼V(s)ds+bˇeγτ∫t0t‍eγ(s-τ(s))𝔼V(s-τ(s))ds+cˇeγτc∫t0t‍eγ(s-τc(s))𝔼V(s-τc(s))ds.
By changing variable s-τ(s)=u, we have
(22)∫t0t‍eγ(s-τ(s))𝔼V(s-τ(s))ds=∫t0-τ(t0)t-τ(t)‍eγu𝔼V(u)du1-τ˙(t)≤∫t0-τt‍eγu𝔼V(u)du1-τ-≤11-τ-∫t0-τt‍eγu𝔼V(u)du.
Similarly, we have
(23)∫t0t‍eγ(s-τc(s))𝔼V(s-τ(s))ds≤11-τ-c∫t0-τct‍eγu𝔼V(u)du.
Substituting (22) and (23) into (21), we get
(24)eγt𝔼V(t)≤eγt0𝔼V(t0)+(γ+aˇ+bˇeγτ1-τ-+cˇeγτc1-τ-c)∫t0-τˇt‍eγu𝔼V(u)du.
By using Gronwall’s inequality, we get
(25)eγt𝔼V(t)≤eγt0𝔼V(t0)e(γ+aˇ+(bˇ/(1-τ-))eγτ+(cˇ/(1-τ-c))eγτc)(t-t0+τˇ).
On the other hand, from the construction ofV(t), we have
(26)V(tk)≤(1+ϵk)2V(tk-),
where |1+ϵk|=maxi=1,2,…,N|1+ϵik|.
According to (8)–(11), for any t∈[tk-1,tk), we get
(27)eγt𝔼V(t)≤eγtk-1𝔼V(tk-1)e(γ+aˇ+(b/(1-τ-))eγτ+(cˇ/(1-τ-c))eγτc)(t-tk-1+τˇ)≤eγtk-1𝔼V(tk-1-)×e(γ+aˇ+(bˇ/(1-τ-))eγτ+(cˇ/(1-τ-c))eγτc)(t-tk-1+τˇ)+2ln|1+ϵk-1|≤⋯≤eγt0𝔼V(t0)×e(γ+aˇ+(bˇ/(1-τ-))eγτ+(cˇ/(1-τ-c))eγτc)(t-t0+kτˇ)+∑v=1k-12ln|1+ϵv|.
Let|1+ϵ|=maxv∈Z+|1+ϵv|. Because of k=⌊(t-t0)/T⌋, we have(28)𝔼V(t)≤𝔼V(t0)×e(γ+aˇ+(bˇ/(1-τ-))eγτ+(cˇ/(1-τ-c))eγτc)(t-t0+kτˇ)+2(k-1)ln|1+ϵv|-(t-t0)γ≤𝔼V(t0)×e(γ+aˇ+(bˇ/(1-τ-))eγτ+(cˇ/(1-τ-c))eγτc)(1+T)τˇ/T(t-t0)+(2(t-t0)/T-4)ln|1+ϵv|-(t-t0)γ.
Using Condition (12) of Theorem 6, we get 𝔼V(t)≤𝔼V(t0)e-η(t-t0). Hence, 𝔼||ei(t)||≤(V(t0)/pˇ)1/2e-(η/2)(t-t0). The proof of Theorem 6 is completed.Remark 7.
Theorem6 provides a sufficient condition for exponential synchronization of coupling delay switched stochastic dynamical networks with impulsive effects. If the time T is sufficiently small and the impulsive gains ϵik, then exponential synchronization of the network (1) could be achieved.If the switching signalσ(t)≡1, then the network (1) has only one coupling matrix G. Suppose G is irreducible and ξT=(ξ1,ξ2,…,ξN) is the left eigenvector of coupling matrix G corresponding to eigenvalue 0. Let impulsive gains are ϵik=bk. By the proof of Theorem 6, we can derive the exponential synchronization criteria of the network (1) with only one topology, which is given as follows.Corollary 8.
Let assumptions (H1) and (H2) be true and letf∈QUAD(P,Δ,η,θ). If there exist positive constants α and β such that
(29)[As+δˇIN-αINB2BT2-βIN]≤0,(γ+a+b1-τ-eγτ+c1-τ-ceγτc)×(1+Tτˇ)T+2ln|1+ϵ|T-γ≤-η,
where
(30)a=λmin(-2ηIn+pˇ∑i=1N‍Υi1+2αPΣ)pˇ,b=λmax(∑i=1N‍PΥi2+2θIN)p^,c=λmax(∑i=1N‍PΥi3+2βPΣ)p^,|1+ϵ|=maxi=1,2,…,N,k∈Z+|1+ϵik|.
Then the solutions x1(t),x1(t),…,xN(t) of system (6) are globally and exponentially stable.Let impulsive gainsϵik=b, and choose the synchronization state s(t)=(1/N)∑i=1N‍xi(t). By the proof of Theorem 6, we can derive the exponential synchronization criteria of the network (1) with the fixed impulsive gain, which is given as follows.


## 4. Numerical Simulation

In this section, we give two numerical simulations to illustrate the feasibility and effectiveness of the theoretical results presented in the previous sections.Consider a three-order Chua’s circuit described as follows:(31)x˙(t)=f(x(t)),
where x(t)=(x1(t),x2(t),x3(t))T and function f(x(t)) was chosen below:
(32)f(x(t))=[m[x2-h(x1)]x1-x2+x3-nx2],
where h(x1)=(2/7)x1-(3/14)[|x1+1|-|x1-1|], m=9, and n=14(2/7).Consider a network model consists of 5 nodes and 2-connected topology. Each node in the network is a three-order Chua’s circuit described by(33)x˙i(t)=f(xi(t))+∑j=15‍gijσ(t)Γxj(t-τ)+σi(x(t-τc(t)),r(t)t,x(t),x(t-τ(t)),+σix(t-τc(t)),r(t))dwi(t),t≠tk,Δxi(tk)=xi(tk)-xi(tk-)=ϵikxi(tk-),t=tk,i=1,2,…,5,
where τ=1, Γ=I3, and σi(t,x(t),x(t-τ(t)),x(t-τc(t)))=0.1×diag{xi1(t)-xi+1,1(t),xi2(t)-xi+1,2(t)}.The coupling matrices are as follows:(34)G1=[-321000-220010-320000-222000-3],G2=[-211000-211000-211100-211100-2].If we chooseP=I3 and Δ=10I3, then function f(x) satisfies the condition of function class QUAD(Δ,P), where α=0.6218. Let β=23.7, Q=4.1I3, ϵik=-0.8, T=0.1, η=-3.488, a=4.1. a1=80.1500, a2=62.3271, b=6.5300, c1=0.0307, c2=80.0300, α1=40.0000, β1=0.0002, α2=15.6568, β2=20.0000, and the synchronization state s(t)=0.2x1+0.3x2+0.1x3+0.3x4+0.1x5; then all the conditions in Theorem 6 are satisfied (by using the Matlab LMI toolbox). The switch time is t=0.2s. The simulation results are given in Figures 1–3. It can be seen clearly from Figures 1, 2, and 3 that all states of the asymmetric coupled network (21) tend to the synchronization state s(t).Figure 1
Time evolution of synchronization errorsei1(t),(i=1,2,…,5) of the asymmetric coupled network (21) with random initial values.Figure 2
Time evolution of synchronization errorsei2(t),(i=1,2,…,5) of the asymmetric coupled network (21) with random initial values.Figure 3
Time evolution of synchronization errorsei3(t),(i=1,2,…,5) of the asymmetric coupled network (21) with random initial values.


## 5. Conclusion

In this paper, the exponential synchronization of the coupling delay stochastic complex networks with switch topology and impulsive effects has been investigated. Based on the Lyapunov stability theory, LMI, and impulsive control theory, some simple, yet generic, criteria for exponential synchronization have been derived. It has shown that criteria can provide an effective control scheme to synchronize for a given coupled delay, the network size, and switch topology. Furthermore, the effectiveness of the presented method has been verified by numerical simulations.



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*Source: 932058-2013-04-22.xml*