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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 232794, 20 pages
http://dx.doi.org/10.1155/2012/232794
Research Article

Exponential Synchronization for Impulsive Dynamical Networks

1Department of Mathematics, Southeast University, Nanjing 210096, China
2School of Mathematics, Jiaying University, Meizhou, Guangdong 514015, China

Received 29 November 2011; Accepted 20 June 2012

Academic Editor: Cengiz Çinar

Copyright © 2012 Lijun Pan and Jinde Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is devoted to exponential synchronization for complex dynamical networks with delay and impulsive effects. The coupling configuration matrix is assumed to be irreducible. By using impulsive differential inequality and the Kronecker product techniques, some criteria are obtained to guarantee the exponential synchronization for dynamical networks. We also extend the delay fractioning approach to the dynamical networks by constructing a Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical examples are given to demonstrate the theoretical results.

1. Introduction

In the past two decades, complex dynamical networks have attracted lot of attention in different areas, such as physical science, engineering, mathematics, biology, and sociology [13]. The synchronization of all dynamical nodes is an important and interesting phenomena mostly because the synchronization can well explain many natural phenomena. Consequently, the synchronization has been actively investigated due to past physics and potential engineering applications. Recently, there has been an increasing interest in the investigation of synchronization of complex dynamical networks, then many synchronization results have been derived for complex dynamical networks [49].

Impulsive effects widely exist in the networks. Such systems are described by impulsive differential systems which have been used efficiently in modelling many practical problems that arise in the fields of engineering, physics, and science as well. So the theory of impulsive differential equations is also attracting much attention in recent years [1013]. Correspondingly, based on the theory of impulsive differential equations, a lot of synchronization results of dynamical networks with impulsive effects have been obtained [1320].

As is well known, two kinds of impulses in terms of synchronization in complex dynamical networks are considered. One is desynchronizing impulse, the other is synchronizing impulse. An impulsive sequence is said to be desynchronizing if the impulsive effect can suppress the synchronization of complex dynamical networks. An impulsive sequence is said to be synchronizing if a corresponding impulsive effect can enhance the synchronization of the complex dynamical networks. According to the previous literature, complex dynamical networks with delay and impulses can reach synchronization provided that delayed dynamical networks are synchronized. In this paper, by impulsive differential inequality [21], the Lyapunov functional method and the Kronecker product techniques, some sufficient conditions are derived for the globally exponential synchronization of dynamical networks. We also extend the delay fractioning method [22, 23] to dynamical networks by constructing Lyapunov-Krasovskii functional and comparing to a linear discrete system. Meanwhile, numerical simulations are given to show that our derived criteria can easily be used to make judgements on synchronization for the delayed dynamical networks with impulsive effects and show that impulsive effects play an important role in the delay dynamical networks. The rest of this paper is organized as follows. In Section 2, the network model is presented, together with some definitions and lemmas. In Section 3, some synchronization criteria are derived for general dynamical networks with delay and impulsive effects. In Section 4, two numerical examples are given to demonstrate that our results are relevant to not only linear coupling but also delay and impulsive effects. Finally, some conclusions are given in Section 5.

Notations. Throughout this paper, the superscript 𝑇 represents the transpose. 𝐼𝑛 stands for the identity matrix of order 𝑛. For 𝑥=(𝑥1,𝑥2,,𝑥𝑛)𝑇𝑅𝑛, the norm is defined as 𝑥=(𝑛𝑖=1𝑥2𝑖)1/2. For matrix 𝐴, 𝜆max(𝐴) and 𝜆min(𝐴) denote the maximum and minimum eigenvalues of matrix 𝐴, respectively. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋𝑌 (resp., 𝑋<𝑌) means that the matrix 𝑋𝑌 is negative semidefinite (resp., negative definite). For a sequence {𝑡𝑘,𝑘=0,1,} satisfying 0=𝑡0<𝑡1<<𝑡𝑘<𝑡𝑘+1<, let Δ𝑘𝑡𝑘+1𝑡𝑘, Δsup=sup𝑘0{Δ𝑘}, Δinf=inf𝑘0{Δ𝑘}.

2. Model Description and Preliminaries

We consider a delayed complex dynamical network consisting of 𝑁-coupled identical nodes. Each node is an 𝑛-dimensional dynamical system composed of linear term and nonlinear term. The 𝑖th node can be described as follows: ̇𝑥𝑖=𝐶𝑥𝑖+𝐵1𝑓𝑥𝑖(𝑡)+𝐵2𝑔𝑥𝑖(𝑡𝜏(𝑡)),𝑖=1,2,,𝑁,(2.1) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),,𝑥𝑖𝑛(𝑡))𝑇 is the state vector of the 𝑖th node at time 𝑡, 𝐶, 𝐵1, 𝐵2𝑅𝑛×𝑛; 0<𝜏(𝑡)𝜏, 𝜏(𝑡)𝜎<1, 𝜏>0, 𝑓(𝑥),𝑔(𝑥)𝐶(𝑅𝑛,𝑅𝑛), 𝑓(𝑥𝑖(𝑡))=(𝑓1(𝑥𝑖1(𝑡)),𝑓2(𝑥𝑖2(𝑡)),,𝑓𝑛(𝑥𝑖𝑛(𝑡)))𝑇, 𝑔(𝑥𝑖(𝑡𝜏(𝑡)))=(𝑔1(𝑥𝑖1(𝑡𝜏(𝑡))),𝑔2(𝑥𝑖2(𝑡𝜏(𝑡))),,𝑔𝑛(𝑥𝑖𝑛(𝑡𝜏(𝑡))))𝑇.

The dynamical behavior of the dynamical network with delay can be described by the following linearly coupled systems: ̇𝑥𝑖=𝐶𝑥𝑖+𝐵1𝑓𝑥𝑖(𝑡)+𝐵2𝑔𝑥𝑖(𝑡𝜏(𝑡))+𝑐𝑁𝑗=1,𝑗𝑖𝑎𝑖𝑗Γ𝑥𝑗(𝑡)𝑥𝑖(𝑡),𝑖=1,2,,𝑁,(2.2) where Γ=diag{𝛾1,𝛾2,,𝛾𝑛} is the inner coupling positive definite matrix between two connected nodes 𝑖 and 𝑗, 𝑐 is the coupling strength, and 𝑎𝑖𝑗 is defined as follows: if there is a connection from node 𝑗 to node 𝑖(𝑗𝑖), then 𝑎𝑖𝑗>0; otherwise, 𝑎𝑖𝑗=0.

In the process of signal transmission, due to the impulsive effects, the states 𝑥𝑖(𝑡),𝑖=1,2,,𝑁 are suddenly changed in the form of impulses at discrete times 𝑡𝑘. That is, 𝑥𝑖(𝑡+𝑘)=𝑑𝑘𝑥𝑖(𝑡𝑘). Let 𝑎𝑖𝑖=𝑁𝑗=1,𝑗𝑖𝑎𝑖𝑗. Thus, the dynamical network with delay and impulsive effects can be obtained by the following form: ̇𝑥𝑖=𝐶𝑥𝑖+𝐵1𝑓𝑥𝑖(𝑡)+𝐵2𝑔𝑥𝑖(𝑡𝜏(𝑡))+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ𝑥𝑗(𝑡),𝑡𝑡0,𝑡𝑡𝑘,𝑥𝑖𝑡+𝑘=𝑑𝑘𝑥𝑖𝑡𝑘𝑥,𝑘=1,2,,𝑖(𝑡)=𝜑𝑖𝑡(𝑡),𝑡0𝜏,𝑡0,𝑖=1,2,,𝑁,(2.3) where 𝑥𝑖(𝑡+𝑘)=lim0+𝑥𝑖(𝑡𝑘+),𝑥𝑖(𝑡𝑘)=lim0𝑥𝑖(𝑡𝑘+),𝑡𝑘0 are impulsive moments satisfying 𝑡𝑘<𝑡𝑘+1 and lim𝑘+𝑡𝑘=+, 𝑑𝑘,𝑘=1,2, are the impulsive gains at 𝑡𝑘 for 𝑖th unit, 𝐴=(𝑎𝑖𝑗)𝑁×𝑁 is the Laplacian matrix of the corresponding network.

By a solution 𝑥𝑖=𝑥𝑖(𝑡) of system (2.3), we mean a real function on [𝑡0𝜏,) such that 𝑥𝑖(𝑡0)=𝜑𝑖(𝑡) for 𝑡[𝑡0𝜏,𝑡0], and 𝑥𝑖(𝑡) satisfies system (2.3) for 𝑡𝑡0, and 𝑥𝑖(𝑡) is continuous everywhere except for some 𝑡𝑘 and left continuous at 𝑡=𝑡𝑘, and the right limit 𝑥(𝑡+𝑘)  𝑘=1,2, exists. Here, we always assume that system (2.3) has a unique solution.

Remark 2.1. If |𝑑𝑘|<1, the impulsive sequence is of synchronizing impulse, which may enhance the synchronization of the networks. But if |𝑑𝑘|>1, the impulsive sequence can suppress the synchronization, which is said to be desynchronizing impulse.

Definition 2.2. The dynamical networks (2.3) are said to be globally exponentially synchronized if there exist 𝜂>0 and 𝑀>0 such that for any initial values 𝜑𝑖(𝑡)  (𝑖=1,2,,𝑁): 𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑀𝑒𝜂(𝑡𝑡0)(2.4) hold all 𝑡>𝑡0, and for any 𝑖,𝑗=1,2,,𝑁.

Definition 2.3. For 𝐴=(𝑎𝑖𝑗)𝑚×𝑛𝑅𝑚×𝑛, 𝐵=(𝑏𝑖𝑗)𝑝×𝑞𝑅𝑝×𝑞, the Kronecker product between two matrices is defined by 𝑎𝐴𝐵=11𝐵𝑎12𝐵𝑎1𝑛𝐵𝑎21𝐵𝑎22𝐵𝑎2𝑛𝐵𝑎𝑚1𝐵𝑎𝑚2𝐵𝑎𝑚𝑛𝐵𝑅𝑚𝑝×𝑛𝑞.(2.5)

Assumption 2.4. There exist constants 𝑙𝑖,𝑙𝑖>0(𝑖=1,2,,𝑁) such that |𝑓𝑖(𝑥1)𝑓𝑖(𝑥2)|𝑙𝑖|𝑥1𝑥2| and |𝑔𝑖(𝑥1)𝑔𝑖(𝑥2)|𝑙𝑖|𝑥1𝑥2| hold for any 𝑥1,𝑥2𝑅.

Assumption 2.5. The coupling configuration matrix 𝐴 is irreducible, and the real parts of the eigenvalues of 𝐴 are all negative except an eigenvalue 0 with multiplicity 1.

To derive our main results, we need the following lemmas.

Lemma 2.6 (see [24]). If an irreducible matrix 𝐴 with nonnegative offdiagonal elements satisfies 𝑎𝑖𝑖=𝑁𝑗=1,𝑗𝑖𝑎𝑖𝑗,𝑖=1,2,,𝑁, then the following propositions are obtained:(1)if 𝜆 is an eigenvalue of 𝐴 and 𝜆0, then Re(𝜆)<0;(2)𝐴 has an eigenvalue 0 with multiplicity 1 and the right eigenvector (1,1,,1)𝑇;(3)suppose that 𝜉=(𝜉1,𝜉2,,𝜉𝑁)𝑇𝑅𝑁 satisfying 𝑁𝑖=1𝜉𝑖=1 is the normalized left eigenvector of A corresponding to eigenvalue 0. Then, 𝜉𝑖>0 hold for all 𝑖=1,2,,𝑁;(4)furthermore, if 𝐴 is symmetric, then we have 𝜉𝑖=1/𝑁 for 𝑖=1,2,,𝑁.

Lemma 2.7 (see [21]). Let 𝑝, 𝑞, 𝜏, 𝑑𝑘,𝑘=1,2, be constants and 𝑞0, 𝜏>0, 𝑑𝑘0 and assume that 𝑢(𝑡) is a piece continuous nonnegative function satisfying: 𝐷+𝑢(𝑡)𝑝𝑢(𝑡)+𝑞𝑢(𝑡)𝑡𝑡0,𝑡𝑡𝑘,𝑢𝑡+𝑘𝑑𝑘𝑢𝑡𝑘𝑡,𝑘=1,2,,𝑢(𝑡)=𝜙(𝑡),𝑡0𝜏,𝑡0.(2.6) If there exist 𝛼 such that for 𝑘=1,2,ln𝑑𝑘𝑡𝑘𝑡𝑘1𝛼,𝑝+𝑑𝑞+𝛼<0.(2.7) Then 𝑢(𝑡)𝑑sup𝑡0𝜏𝑡𝑡0||𝜙||𝑒𝜆(𝑡𝑡0),(2.8) where 𝑢(𝑡)=sup𝑡𝜏𝜎𝑡𝑥(𝜎), 𝑑=sup1𝑘<+{𝑒𝛼(𝑡𝑘𝑡𝑘1),1/𝑒𝛼(𝑡𝑘𝑡𝑘1)}, 𝜆 is an unique positive solution of 𝜆+𝑝+𝑑𝑞𝑒𝜆𝜏+𝛼=0.

Remark 2.8. The condition of Lemma 2.7 does not need 𝑝>𝑞 due to the effects 𝛼, which implies that the above inequality is less conservative than the results in [25].

Lemma 2.9. For any vectors 𝑥,𝑦𝑅𝑛, scalar 𝜖>0, and positive definite matrix 𝑄𝑅𝑛×𝑛, the following inequality holds: 2𝑥𝑇𝑦𝜖𝑥𝑇𝑄𝑥+𝜖1𝑦𝑇𝑄1𝑦.(2.9)

Lemma 2.10. Let 𝐴𝑅𝑛×𝑛 be a positive definite matrix, then for 𝑥𝑅𝑛, 𝜆min(𝐴)𝑥𝑇𝑥𝑥𝑇𝐴𝑥𝜆max(𝐴)𝑥𝑇𝑥.(2.10)

3. Synchronization Analysis

In this section, the globally exponential synchronization will be analyzed for delayed dynamical networks with impulsive effects. We assume that the network topology is strongly connected, then the corresponding Laplacian coupling matrix 𝐴 is irreducible.

Let 𝑥(𝑡)=(𝑥𝑇1(𝑡),𝑥𝑇2(𝑡),,𝑥𝑇𝑁(𝑡))𝑇, 𝐹(𝑥(𝑡))=(𝑓𝑇(𝑥1(𝑡)),𝑓𝑇(𝑥2(𝑡)),,𝑓𝑇(𝑥𝑁(𝑡)))𝑇, 𝐺(𝑥(𝑡))=(𝑔𝑇(𝑥1(𝑡)),𝑔𝑇(𝑥2(𝑡)),,𝑔𝑇(𝑥𝑁(𝑡)))𝑇 and 𝜑(𝑡)=(𝜑1(𝑡),𝜑2(𝑡),,𝜑𝑁(𝑡))𝑇. Then, the delayed dynamical network (2.3) can be rewritten in the following Kronecker product form: 𝐼̇𝑥=𝑁𝐼𝐶𝑥(𝑡)+𝑁𝐵1+𝐼𝐹(𝑥(𝑡))𝑁𝐵2𝐺(𝑥(𝑡𝜏(𝑡)))+𝑐(𝐴Γ)𝑥(𝑡),𝑡𝑡0,𝑡𝑡𝑘,𝑥𝑡+𝑘=𝑑𝑘𝑥𝑡𝑘𝑡,𝑘=1,2,,𝑥(𝑡)=𝜑(𝑡),𝑡0𝜏,𝑡0.(3.1)

Suppose that 𝜉=(𝜉1,𝜉2,,𝜉𝑁)𝑇 is the left eigenvector of the configuration coupling matrix 𝐴 with respect to eigenvalue 0 satisfying 𝑁𝑖=1𝜉𝑖=1. Since the coupling configuration matrix 𝐴 is irreducible, by Lemma 2.6, we can see that 𝜉𝑖>0 for 𝑖=1,2,,𝑁. Let Ξ=diag{𝜉1,𝜉2,,𝜉𝑁}>0, 𝐿=diag{𝑙1,𝑙2,,𝑙𝑛}, 𝐿=diag{𝑙1,𝑙2,,𝑙𝑛}, 𝑊=Ξ𝜉𝜉𝑇 and 𝐴=Ξ𝐴+𝐴𝑇Ξ.

Theorem 3.1. Suppose that Assumptions 2.4 and 2.5 hold. Also suppose that there exist a diagonal positive-definite matrix 𝑃 and scalars 𝜂>0, 𝜀>0, 𝛾>0, 𝜇20, 𝜇1, 𝛿 such that(𝐻1)Θ1=𝑃𝐶+𝐶𝑇𝑃+𝜀𝑃𝐵1𝐵𝑇1𝑃+𝛾𝑃𝐵2𝐵𝑇2𝑃+𝜀1𝐿2𝑐𝜂𝑃Γ𝜇1𝑃0;(𝐻2)Θ2=𝛾1𝐿2𝜇2𝑃0;(𝐻3)forall  𝑘=1,2,,  2ln|𝑑𝑘|/(𝑡𝑘𝑡𝑘1)𝛿;(𝐻4)𝜇1+𝑑𝜇2+𝛿<0;(𝐻5)𝜂𝜆max(𝑊)+𝜆2(𝐴)0.Then the complex dynamical networks (3.1) are exponentially synchronized, where 𝑑=sup1𝑘<+{𝑒𝛿(𝑡𝑘𝑡𝑘1),1/𝑒𝛿(𝑡𝑘𝑡𝑘1)}, 𝜆2(𝐴) is defined to be the second largest eigenvalue of 𝐴.

Proof. We define a Lyapunov function 𝑉(𝑡)=𝑥𝑇(𝑡)(𝑊𝑃)𝑥(𝑡). Since 𝑊=Ξ𝜉𝜉𝑇, we have 𝑤𝑖𝑗=𝜉𝑖𝜉𝑗 for 𝑖𝑗 and 𝑤𝑖𝑖=𝜉𝑖𝜉2𝑖. In view of 𝑁𝑗=1𝜉𝑗=1, it follows that 𝑁𝑗=1𝑤𝑖𝑗=𝜉𝑖𝑁𝑗=1𝜉𝑖𝜉𝑗=0. Therefore, we can conclude that 𝑉(𝑡)=𝑁𝑖=1𝑁𝑗=1,𝑗𝑖(1/2)𝑤𝑖𝑗(𝑥𝑖(𝑡)𝑥𝑗(𝑡))𝑇𝑃(𝑥𝑖(𝑡)𝑥𝑗(𝑡)). Calculating the Dini derivative of 𝑉(𝑡) along the trajectories of the systems (3.1), we have for 𝑡𝑡𝑘,𝑘=1,2,: 𝐷+𝑉(𝑡)=2𝑥𝑇𝐼(𝑡)(𝑊𝑃)×𝑁𝑥𝐶(𝑡)+2𝑥𝑇(𝐼𝑊𝑃)×𝑁𝐵1𝐹(𝑥(𝑡))+2𝑥𝑇𝐼(𝑊𝑃)×𝑁𝐵2𝐺(𝑥(𝑡𝜏(𝑡)))+2𝑐𝑥𝑇(𝑡)(𝑊𝑃)×(𝐴Γ)𝑥(𝑡).(3.2) By adding 𝑐𝑥𝑇(𝑡)(𝑊𝜂𝑃Γ)𝑥(𝑡)+𝑐𝑥𝑇(𝑡)(𝑊𝜂𝑃Γ)𝑥(𝑡) to (3.2) and noting that 𝑊𝐴=(Ξ𝜉𝜉𝑇)𝐴=Ξ𝐴𝜉(𝜉𝑇𝐴)=Ξ𝐴, we can obtain that 𝐷+𝑉(𝑡)𝑁𝑁𝑖=1𝑗=1,𝑗𝑖𝑤𝑖𝑗𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇1𝑃𝐶2𝑥𝑐𝜂𝑃Γ𝑖(𝑡)𝑥𝑗+𝑥(𝑡)𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝐵1𝑓𝑥𝑖𝑥(𝑡)𝑓𝑗+𝑥(𝑡)𝑖𝑥𝑗𝑇𝑃𝐵2𝑔𝑥𝑖𝑥(𝑡𝜏(𝑡))𝑔𝑗(𝑡𝜏(𝑡))+𝑐𝑥𝑇(𝑡)×Ξ𝐴+𝐴𝑇Ξ𝑃Γ+𝑊𝜂𝑃Γ𝑥(𝑡).(3.3) Since the matrix 𝐴=Ξ𝐴+𝐴𝑇Ξ has the following property: 𝐴=𝐴𝑖𝑗𝑁×𝑁,𝐴𝑖𝑖=2𝜉𝑖𝐴𝑖𝑖<0,𝑖=1,2,,𝑁,𝐴𝑖𝑗=𝜉𝑖𝐴𝑖𝑗+𝜉𝑗𝐴𝑗𝑖=𝐴𝑗𝑖,𝑖𝑗,𝑁𝑗=1𝐴𝑖𝑗=𝜉𝑖𝑁𝑗=1𝐴𝑖𝑗+𝑁𝑗=1𝜉𝑗𝐴𝑗𝑖=0.(3.4) By Perron-Frobenius theorem (see [24]), we can arrange the eigenvalues of matrix 𝐴 as follows: 0=𝜆1(𝐴)>𝜆2(𝐴)𝜆𝑁(𝐴). Applying matrix decomposition theory (see [24]), there exists unitary matrix 𝑈, such that 𝐴=𝑈Λ𝑈𝑇, where Λ=diag{0,𝜆2(𝐴),,𝜆𝑁(𝐴)} and 𝑈={𝑢1,𝑢2,,𝑢𝑁} with 𝑢1=(1/𝑁,1/𝑁,,1/𝑁)𝑇 and 𝑈𝑇𝑈=𝐼𝑁.
Let 𝑦(𝑡)=(𝑈𝑇𝐼𝑛)𝑥(𝑡), where 𝑦(𝑡)=(𝑦𝑇1(𝑡),𝑦𝑇2(𝑡),,𝑦𝑇𝑁(𝑡))𝑇, 𝑦𝑖(𝑡)𝑅𝑛,𝑖=1,2,,𝑁. Then we have 𝑥(𝑡)=(𝑈𝐼𝑛)𝑦(𝑡). Thus, we have 𝑥𝑇(𝑡)Ξ𝐴+𝐴𝑇Ξ𝑃Γ𝑥(𝑡)=𝑦𝑇𝑈(𝑡)𝑇𝐼𝑛𝐴𝑃Γ𝑈𝐼𝑛=𝑦(𝑡)𝑁𝑖=2𝜆𝑖𝐴𝑦𝑇𝑖(𝑡)𝑃Γ𝑦𝑖(𝑡)𝜆2𝐴𝑁𝑖=2𝑦𝑇𝑖(𝑡)𝑃Γ𝑦𝑖(𝑡).(3.5) In view of matrix 𝑊 is a zero row sum irreducible symmetric matrix with negative off-diagonal elements, we see that 𝜆max(𝑊)>0 and 𝑊𝑢1=(0,0,,0))𝑇. Hence by Lemma 2.10, we have 𝑥𝑇(𝑡)(𝑊𝜂𝑃Γ)𝑥(𝑡)=𝜂𝑦𝑇𝑈(𝑡)𝑇𝑦𝑊𝑈𝑃Γ(𝑡)𝜂𝜆max(𝑊)𝑁𝑖=2𝑦𝑇𝑖(𝑡)𝑃Γ𝑦𝑖(𝑡).(3.6) It follows from condition 𝜆2(𝐴)+𝜂𝜆max(𝑊)0 that 𝑐𝑥𝑇(𝑡)Ξ𝐴+𝐴𝑇Ξ𝑥𝜆𝑃Γ+𝑊𝜂𝑃Γ(𝑡)𝑐2𝐴+𝜂𝜆max(𝑊)𝑁𝑖=2𝑦𝑇𝑖(𝑡)𝑃𝑦𝑖(𝑡)0.(3.7) By Assumption 2.4 and Lemma 2.10, there exists 𝜀>0 such that 2𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝐵1𝑓𝑥𝑖𝑥(𝑡)𝑓𝑗𝑥(𝑡)𝜀𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝐵1𝐵𝑇1𝑃𝑥𝑖(𝑡)𝑥𝑗(𝑡)+𝜀1𝑓𝑥𝑖𝑥(𝑡)𝑓𝑗(𝑡)𝑇𝑓𝑥𝑖𝑥(𝑡)𝑓𝑗𝑥(𝑡)𝜀𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝐵1𝐵𝑇1𝑃𝑥𝑖(𝑡)𝑥𝑗(𝑡)+𝜀1𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝐿2𝑥𝑖(𝑡)𝑥𝑗=𝑥(𝑡)𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝜀𝑃𝐵1𝐵𝑇1𝑃+𝜀1𝐿2𝑥𝑖(𝑡)𝑥𝑗.(𝑡)(3.8) Similarly, we have the following estimation: 2𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝐵2𝑔𝑥𝑖𝑥(𝑡𝜏(𝑡))𝑔𝑗𝑥(𝑡𝜏(𝑡))𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝛾𝑃𝐵2𝐵𝑇2𝑃𝑥𝑖(𝑡)𝑥𝑗+𝑥(𝑡)𝑖(𝑡𝜏(𝑡))𝑥𝑗(𝑡𝜏(𝑡))𝑇𝛾1𝐿2𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑗,(𝑡𝜏(𝑡))(3.9) where 𝛾>0. Substituting these into (3.3), we have for 𝑡𝑡𝑘̇𝑉(𝑡)𝑁𝑁𝑖=1𝑗=1,𝑗𝑖12𝑤𝑖𝑗×𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝐶+𝐶𝑇𝑃+𝜀𝑃𝐵1𝐵𝑇1𝑃+𝛾𝑃𝐵2𝐵𝑇2𝑃+𝜀1𝐿2𝑥𝑐𝜂𝑃Γ𝑖(𝑡)𝑥𝑗(𝑡)𝑁𝑁𝑖=1𝑗=1,𝑗𝑖12𝑤𝑖𝑗×𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑗(𝑡𝜏(𝑡))𝑇𝛾1𝐿2𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑗(𝑡𝜏(𝑡))=𝑁𝑁𝑖=1𝑗=1,𝑗𝑖12𝑤𝑖𝑗×𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝐶+𝐶𝑇𝑃+𝜀𝑃𝐵1𝐵𝑇1𝑃+𝛾𝑃𝐵2𝐵𝑇2𝑃+𝜀1𝐿2𝑐𝜂𝑃Γ𝜇1𝑃×𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝜇1𝑁𝑁𝑖=1𝑗=1,𝑗𝑖12𝑤𝑖𝑗𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑁𝑁𝑖=1𝑗=1,𝑗𝑖12𝑤𝑖𝑗×𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑗(𝑡𝜏(𝑡))𝑇𝛾1𝐿2𝜇2𝑃𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑗(𝑡𝜏(𝑡))𝜇2𝑁𝑁𝑖=1𝑗=1,𝑗𝑖12𝑤𝑖𝑗𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑗(𝑡𝜏(𝑡))𝑇𝑃𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑗(𝑡𝜏(𝑡))𝜇1𝑉(𝑡)+𝜇2𝑉(𝑡𝜏(𝑡)).(3.10) For 𝑡=𝑡𝑘, we have 𝑉𝑡+𝑘1=2𝑁𝑁𝑖=1𝑗=1,𝑗𝑖𝑤𝑖𝑗𝑥𝑖𝑡+𝑘𝑥𝑗𝑡+𝑘𝑇𝑃𝑥𝑖𝑡+𝑘𝑥𝑗𝑡+𝑘𝑑=2𝑘2𝑁𝑁𝑖=1𝑗=1,𝑗𝑖𝑤𝑖𝑗𝑥𝑖𝑡𝑘𝑥𝑗𝑡𝑘𝑇𝑃𝑥𝑖𝑡𝑘𝑥𝑗𝑡𝑘=𝑑2𝑘𝑉𝑡𝑘.(3.11) By Lemma 2.7, there exist 𝑀>0 such that 𝑉(𝑡)𝑀sup𝜏𝑠0𝑉𝑡0𝑒+𝑠𝜂(𝑡𝑡0),(3.12) which implies that 12𝜉𝑖𝜉𝑗𝜆min𝑥(𝑃)𝑖(𝑡)𝑥𝑗(𝑡)212𝑁𝑖=1,𝑗=1𝜉𝑖𝜉𝑗𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝑥𝑖(𝑡)𝑥𝑗𝑒(𝑡)=𝑉(𝑡)=𝑂𝜂(𝑡𝑡0).(3.13) Consequently, the complex dynamical network (3.1) can reach globally exponential synchronization.

Remark 3.2. When the impulsive effects are desynchronizing, that is, |𝑑𝑘|>1, the condition (𝐻4) in Theorem 3.1 yields 𝜇1>𝜇2, which means that the delayed complex networks without impulsive effects of (2.2) is exponentially synchronized. But when the impulsive effects are synchronizing, that is, |𝑑𝑘|<1, we do not need the condition 𝜇1>𝜇2 due to the effect of impulses.

Theorem 3.3. Suppose that Assumptions 2.4 and 2.5 hold and Δsup<. Also suppose that there exist a diagonal positive definite matrix 𝑃 and scalars 𝜂>0, 𝜀>0, 𝛾>0, 𝜇1>0, 𝜇20 such that(𝐻1)Θ1=𝑃𝐶+𝐶𝑇𝑃+𝜀𝑃𝐵1𝐵𝑇1𝑃+𝛾𝑃𝐵2𝐵𝑇2𝑃+𝜀1𝐿2𝑐𝜂𝑃Γ𝜇1𝑃0;(𝐻2)Θ2=𝛾1𝐿2𝜇2𝑃0;(𝐻3) for all 𝑘=1,2,, |𝑑𝑘|<1;(𝐻4) there exists a integer 𝑚1 such that 𝑡𝑘𝑚𝑡𝑘𝜏𝑡𝑘+1𝑚 for all 𝑘𝑚, and the discrete system: 𝜃(𝑘+1)=𝐽𝑘(𝑚)𝜃(𝑘)(3.14)is globally exponentially stable with decay 𝜆>0, where 𝐽𝑘(𝛽𝑚)010000010000001𝑘+1𝑚𝛽𝑘+2𝑚𝛽𝑘+3𝑚𝛽𝑘1𝛼𝑘1,(3.15)𝜁=𝜇1+𝜇2/(1𝜎), 𝛼𝑘=𝑑2𝑘𝑒𝜁Δ𝑘1+𝛽𝑘1, 𝛽𝑘𝑗=(𝛽/(1𝜎))Δ𝑘𝑗𝑒𝜁Δ𝑘𝑗,𝑗=1,2,,𝑚1; (𝐻5) there exists a constant 𝒯0 such that the average dwell time 𝒯𝑎 satisfies 𝒩𝑡0,𝑡𝒯0+𝑡𝑡0𝒯𝑎,𝑡𝑡0,(3.16)where 𝒩[𝑡0,𝑡] is the number of impulsive times of the impulsive sequence on the interval [𝑡0,𝑡];(𝐻6)𝜂𝜆max(𝑊)+𝜆2(𝐴)0.
Then the complex dynamical networks (3.1) are exponentially synchronized with decay rate 𝜆/2𝒯𝑎.

Proof. Consider a Lyapunov-Krasovskii functional: 𝑉(𝑡)=𝑉1(𝑡)+𝑉2(𝑡),(3.17) with 𝑉1(𝑡)=𝑥𝑇(𝑡)(𝑊𝑃)𝑥(𝑡),𝑉2(𝑡)=𝜇21𝜎𝑡𝑡𝜏(𝑡)𝑥𝑇(𝑠)(𝑊𝑃)𝑥(𝑠)𝑑𝑠.(3.18) Similar to the proof of Theorem 3.1, for 𝑡(𝑡𝑘,𝑡𝑘+1], we get 𝐷+𝑉1(𝑡)𝜇1𝑥𝑇(𝑡)(𝑊𝑃)𝑥(𝑡)+𝜇2𝑥𝑇(𝑡𝜏(𝑡))(𝑊𝑃)𝑥(𝑡𝜏(𝑡)).(3.19) For 𝑡(𝑡𝑘,𝑡𝑘+1], we have 𝐷+𝑉2(𝑡)𝜇2𝑥1𝜎𝑇(𝑡)(𝑊𝑃)𝑥(𝑡)𝜇2𝑥𝑇(𝑡𝜏(𝑡))(𝑊𝑃)𝑥(𝑡𝜏(𝑡)).(3.20) Then 𝐷+𝑉(𝑡)=𝐷+𝑉1(𝑡,𝑥(𝑡))+𝐷+𝑉2(𝑡)𝜇1+𝜇2𝑉1𝜎1(𝑡)𝜁𝑉(𝑡).(3.21) Thus 𝑡𝑉(𝑡)𝑉+𝑘𝑒𝜁(𝑡𝑡𝑘)𝑡,𝑡𝑘,𝑡𝑘+1.(3.22) By (3.11), for 𝑡=𝑡𝑘, we have 𝑉1𝑡+𝑘𝑑2𝑘𝑉1𝑡𝑘.(3.23) It follows from condition (iii) that there exists some ̂𝑡𝑘𝑗+1(𝑡𝑘𝑗,𝑡𝑘𝑗+1] such that 𝑉2𝑡+𝑘𝜇21𝜎𝑡𝑘𝑡𝑘𝑡𝜏𝑘𝑉1(𝑠)𝑑𝑠𝜇21𝜎𝑡𝑘𝑡𝑘𝑚𝑉1(=𝑠)𝑑𝑠𝜇21𝜎𝑚𝑗=1𝑡𝑘𝑗+1𝑡+𝑘𝑗𝑉1(𝑠)𝑑𝑠=𝜇21𝜎𝑚𝑗=1Δ𝑘𝑗𝑉1̂𝑡𝑘𝑗+1.(3.24) Then from (3.22), we have 𝑉2𝑡+𝑘𝜇21𝜎𝑚𝑗=1Δ𝑘𝑗𝑉̂𝑡𝑘𝑗+1𝜇21𝜎𝑚𝑗=1Δ𝑘𝑗𝑒𝜁Δ𝑘𝑗𝑉𝑡+𝑘𝑗.(3.25) Together with (3.22), (3.23) and the above inequality, we have 𝑉𝑡+𝑘𝑑2𝑘+𝜇2Δ1𝜎𝑘1𝑒𝜁Δ𝑘1𝑉𝑡+𝑘1+𝜇21𝜎𝑚𝑗=2Δ𝑘𝑗𝑒𝜁Δ𝑘𝑗𝑉𝑡+𝑘𝑗𝛼𝑘1𝑉𝑡+𝑘1+𝑚1𝑗=1𝛽𝑘𝑗1𝑉𝑡+𝑘𝑗1.(3.26) Set 𝑍(𝑘)=(𝑧1(𝑘),𝑧2(𝑘),,𝑧𝑚(𝑘))𝑇 and 𝑧1(𝑘)=𝑉(𝑡+𝑘+1), 𝑧2(𝑘)=𝑉(𝑡+𝑘+2),,𝑧𝑚(𝑘)=𝑉(𝑡+𝑘+𝑚). Then 𝑧1𝑧(𝑘+1𝑚)2(𝑧𝑘+1𝑚)𝑚(𝑘+1𝑚)𝐽𝑘𝑧(𝑚)1𝑧(𝑘𝑚)2(𝑧𝑘𝑚)𝑚(𝑘𝑚),(3.27) that is, 𝑍(𝑘𝑚+1)𝐽𝑘(𝑚)𝑍(𝑘𝑚).(3.28) We consider the discrete system: 𝜃(𝑘+1)=𝐽𝑘(𝑚)𝜃(𝑘),𝜃(𝑚1)=𝑍(1).(3.29) Then, by the comparison principle, we see that for 𝑘𝑚1𝑍(𝑘𝑚)𝜃(𝑘).(3.30) Note that the system (3.29) is globally exponential stable with decay 𝜆>0, then there exists constant 𝑀>0 such that 𝑍(𝑘𝑚)𝑀𝑒𝜆(𝑘𝑚+1)𝑍(1),𝑘𝑚1,(3.31) where 𝑍(1)=[𝑚1𝑗=0𝑉2(𝑡𝑗)]1/2,𝑍(𝑘𝑚)=[𝑚𝑗=1𝑉2(𝑡𝑗+𝑘𝑚)]1/2. From (3.17) and (3.22), we have 𝑉2𝑡+𝑗=𝜇21𝜎𝑡𝑗𝑡𝑗𝑡𝜏𝑗𝑥𝑇(𝑠)(𝑊𝑃)𝑥(𝑠)𝑑𝑠𝜇21𝜎𝑡𝑗𝑡0𝜏𝑥𝑇(=𝑠)(𝑊𝑃)𝑥(𝑠)𝑑𝑠𝜇21𝜎𝑡0𝑡0𝜏𝑥𝑇(𝑠)(𝑊𝑃)𝑥(𝑠)𝑑𝑠+𝜇21𝜎𝑗1𝑠=0𝑡𝑠+1𝑡+𝑠𝑥𝑇(𝑠)(𝑊𝑃)𝑥(𝑠)𝑑𝑠𝜇2𝜏1𝜎sup𝜏𝑠0𝑉𝑡0++𝑠𝜇21𝜎𝑗1𝑠=0Δ𝑠𝑉𝑡+𝑠𝑒𝜁Δ𝑠,𝑗=0,1,,𝑚1.(3.32) Furthermore, it follows that 𝑉𝑡+𝑗=𝑉1𝑡+𝑗+𝑉2𝑡+𝑗𝜇2𝜏1𝜎sup𝜏𝑠0𝑉𝑡0+𝑠+𝑑2𝑗𝑒𝜁Δ𝑗1𝑉𝑡+𝑗1+𝜇21𝜎𝑗1𝑠=0Δ𝑠𝑉𝑡+𝑠𝑒𝜁Δ𝑠1=𝜇2𝜏1𝜎sup𝜏𝑠0𝑉𝑡0+𝑠+𝛼𝑗𝑉𝑡+𝑗1+𝑗2𝑠=0𝛽𝑠𝑉𝑡+𝑠𝑉𝑡,𝑗=1,2,,𝑚1,01+𝜇2𝜏1𝜎sup𝜏𝑠0𝑉𝑡0.+𝑠(3.33) By induction, there exists a constant 𝜗>0, which is dependent on 𝜏, 𝜎, 𝜇1, 𝜇2,  Δ𝑗,  𝑗=0,1,,𝑚1 such that 𝑉𝑡+𝑗𝜗𝜉2,(3.34) which yields that 𝑍(1)=𝑚1𝑗=0𝑉2𝑡+𝑗1/2𝑚𝜗sup𝜏𝑠0𝑉𝑡0.+𝑠(3.35) From (3.31) and the above inequality, we see that for all 𝑘=0,1,, 𝑉𝑡+𝑘𝑍(𝑘𝑚)𝑀𝑚𝜗𝑒𝜆(𝑘𝑚+1)sup𝜏𝑠0𝑉𝑡0.+𝑠(3.36) Therefore, by (3.17), (3.22), and (3.36), we conclude that for 𝑡(𝑡𝑘,𝑡𝑘+1],𝑘=0,1,, 𝑉(𝑡)𝑒𝜁(𝑡𝑡𝑘)𝑉𝑡+𝑘Υ𝑒𝜆𝑘sup𝜏𝑠0𝑉𝑡0,+𝑠(3.37) where Υ=𝑀𝑚𝜗𝑒𝜆(𝑚1)+𝜁Δ𝑘. For all 𝑡(𝑡𝑘,𝑡𝑘+1],𝑘=0,1,, we obtain that 𝒩[𝑡0,𝑡]=𝑘. Then 𝑉(𝑡)Υ𝑒𝜆𝒯0𝑒(𝜆/𝒯𝑎)(𝑡𝑡0)sup𝜏𝑠0𝑉𝑡0,+𝑠(3.38) which means that 12𝜉𝑖𝜉𝑗𝜆min𝑥(𝑃)𝑖(𝑡)𝑥𝑗(𝑡)212𝑁𝑖=1,𝑗=1𝜉𝑖𝜉𝑗𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑇𝑃𝑥𝑖(𝑡)𝑥𝑗𝑒(𝑡)=𝑉(𝑡)=𝑂(𝜆/𝒯𝑎)(𝑡𝑡0).(3.39) This completes the proof of the theorem.

Remark 3.4. Theorem 3.3 presents a new delay-dependent exponential synchronization criterion for complex dynamical networks by using the Lyapunov-Krasovskii functional. Note that, for 𝑑𝑘<1, 𝜎=0, the proposed result demonstrates its superiority to Theorem 3.1, which will be well illustrated via an example in the next section.

Corollary 3.5. Suppose that Assumptions 2.4 and 2.5 hold and Δsup< and 𝜏<𝑡𝑘𝑡𝑘1 for all 𝑘=1,2,. If there exist positive definite matrix 𝑃 and scalars 𝜂>0, 𝜀>0, 𝛾>0,  𝜇1>0, 𝜇20 such that (𝐻1)(𝐻3) and (𝐻5)-(𝐻6) of Theorem 3.3 hold, and condition (𝐻4) of Theorem 3.3 is replaced by the following condition:(𝐻4) there exists a constant 𝜆>0 such that 𝑑ln2𝑘+𝜇2𝜏1𝜎+𝜁Δ𝑘1𝜆,(3.40)where 𝜁=𝜇1+𝜇2/(1𝜎), then the complex dynamical networks (3.1) are exponentially synchronized with decay rate 𝜆/2𝒯𝑎.

Proof. Choose a Lyapunov-Krasovskii functional candidate 𝑉(𝑥(𝑡)) as 𝑉(𝑥(𝑡))=𝑉1(𝑥(𝑡))+𝑉2(𝑥(𝑡)),(3.41) with 𝑉1(𝑥(𝑡))=𝑥𝑇(𝑡)(𝑊𝑃)𝑥(𝑡),𝑉2(𝑥(𝑡))=𝜇21𝜎𝑡𝑡𝜏(𝑡)𝑥𝑇(𝑠)(𝑊𝑃)𝑥(𝑠)𝑑𝑠.(3.42) By the proof of Theorem 3.3, for 𝑡(𝑡𝑘,𝑡𝑘+1], we have 𝐷+𝑡𝑉(𝑡)𝜁𝑉(𝑡),𝑉(𝑡)𝑉+𝑘𝑒𝜁(𝑡𝑡𝑘)𝑡,𝑡𝑘,𝑡𝑘+1.(3.43) Note that 𝜏<𝑡𝑘𝑡𝑘1, then there exists some ̂𝑡𝑘[𝑡𝑘𝜏,𝑡𝑘] such that 𝑉2𝑡+𝑘𝜇21𝜎𝑡𝑘𝑡𝑘𝜏𝑉1(𝑠)𝑑𝑠=𝜇2𝜏𝑉1𝜎1̂𝑡𝑘.(3.44) Thus 𝑉𝑡+𝑘𝑑2𝑘𝑉1𝑡𝑘+𝜇2𝜏𝑉1𝜎1̂𝑡𝑘𝑑2𝑘𝑒𝜁Δ𝑘1𝑉1𝑡+𝑘1+𝜇2𝜏𝑒1𝜎𝜁Δ𝑘1𝑉1𝑡+𝑘1=𝑒ln(𝑑2𝑘+𝜇2𝜏/(1𝜎))+𝜁Δ𝑘1𝑉𝑡+𝑘1.(3.45) Then from condition (𝐻4), we obtain 𝑉𝑡+𝑘𝑒𝜆𝑉𝑡+𝑘1𝑒𝜆𝑘𝑉𝑡0,(3.46) for all 𝑘=1,2,. The remainder proof of the theorem is similar to Theorem 3.3.

Remark 3.6. By Corollary 3.5, under the case that 𝜏<𝑡𝑘𝑡𝑘1 for all 𝑘=1,2,, we see that the estimations of maximal time-delay 𝜏 and maximal dwell time Δsup as 𝜏<sup𝑘1(1𝜎)𝑒𝜁Δ𝑘1𝜆𝑑2𝑘𝜁,Δsup<sup𝑘1𝑑𝜆ln2𝑘+𝜇2𝜏/(1𝜎)𝜁.(3.47)

Remark 3.7. By Corollary 3.5, if we take the impulsive gains 𝑑𝑘 as 0<𝑑𝑘<𝑒𝜁Δ𝑘1𝜆𝜇2𝜏1𝜎,𝑘=1,2,,(3.48) then network (3.1) achieves exponential synchronization.

Corollary 3.8. Suppose that Assumptions 2.4 and 2.5 hold and Δsup<. If there exist positive definite matrix 𝑃 and scalars 𝜂>0, 𝜀>0, 𝛾>0, 𝜇1>0, 𝜇20 such that (𝐻1)(𝐻3) and (𝐻5)-(𝐻6) of Theorem 3.3 hold and condition (𝐻4) of Theorem 3.3 is replaced by one of the following two conditions:(𝐻4) there exists a constant 𝑚>1 such that 𝑡𝑘𝑚<𝑡𝑘𝜏𝑡𝑘+1𝑚  for all  𝑘𝑚, and the matrix 𝐽(𝑚) satisfies the spectral radius condition for some 𝜆>0𝜌(𝐽(𝑚))<𝑒𝜆,(3.49)where 𝜖𝐽(𝑚)01000001000000011𝜖1𝜖1𝜖1𝜖1+𝜖2,(3.50)𝜖1=(𝜇2/(1𝜎))Δsup𝑒𝜁Δsup, 𝜖2=𝑑𝑒𝜁Δsup, 𝑑=sup𝑘1{𝑑2𝑘}, 𝜁=𝜇1+𝜇2/(1𝜎); (𝑖𝑖𝑖) there exists a positive integer 𝑚1 such that 𝑡𝑘𝑚<𝑡𝑘𝜏𝑡𝑘+1𝑚 for all 𝑘𝑚, and there exists a constant 0<𝜚<1 such that all roots 𝜆𝑗(𝑗=1,2,,𝑚) of the characteristic polynomial: Ψ𝑘(𝜆)𝜆𝑚𝜇𝑘1𝜆𝑚1𝜈𝑘1𝜆𝑚2𝜈𝑘+2𝑚𝜆𝜈𝑘+1𝑚(3.51)satisfy that |𝜆𝑗|𝜚<1,then the complex dynamical networks (3.1) are exponentially synchronized.

Remark 3.9. From Theorems 3.1 and 3.3, when the delayed network dynamics are desynchronizing and the impulsive effects are synchronizing, in order to ensure synchronization, it should be naturally assumed that the frequency of impulses should not be too low. Usually, we always use condition 𝑡𝑘𝑡𝑘1𝑇1  (𝑇1>0) to ensure that the frequency of impulses should not be too low. Conversely, when the delayed network dynamics are synchronizing but the impulsive effects are desynchronizing, the impulses should not occur too frequently in order to guarantee synchronization. To ensure that the impulses do not occur too frequently, we always assume that 𝑡𝑘𝑡𝑘1𝑇2  (𝑇2>0).

4. Examples and Simulations

In this section, some examples and numerical simulations are provided to illustrate our results.

Example 4.1. Consider the following delayed neural networks [26]: 𝑥(𝑡)=𝐶𝑥(𝑡)+𝐵1𝑓(𝑥(𝑡))+𝐵2𝑔(𝑥(𝑡1)),(4.1) where 𝑥=(𝑥1,𝑥2)𝑇, 𝑓(𝑥)=(𝑓(𝑥1),𝑓(𝑥2))𝑇,𝑔(𝑥)=(𝑔(𝑥1),𝑔(𝑥2))𝑇, 𝑓(𝑥)=𝑔(𝑥)=tanh(𝑥), 𝐶=1001, 𝐵1=20.15.01.5, and 𝐵2=1.50.10.21. These neural networks (4.1) are chaotic, and chaotic attractor is shown in Figure 1.
We consider the following linear coupled delayed networks: 𝑥𝑖(𝑡)=𝐶𝑥𝑖(𝑡)+𝐵1𝑓𝑥𝑖(𝑡)+𝐵2𝑔𝑥𝑖(𝑡1)+𝑐4𝑗=1𝑎𝑖𝑗Γ𝑥𝑗(𝑡),𝑖=1,2,3,4,(4.2) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡))𝑇,  𝑐=1.4, Γ=4.18004.9 and 𝐴=20.410.60.4302.6102.41.40.62.61.44.6.

232794.fig.001
Figure 1: Chaotic and chaotic attractor.

Figure 2 shows the synchronization of networks of (4.3).

232794.fig.002
Figure 2: The state variables 𝑥𝑖1(𝑡) and 𝑥𝑖2(𝑡) without impulsive effects.

At last, we consider the following linear coupled delayed networks with impulsive effects: 𝑥𝑖(𝑡)=𝐶𝑥𝑖(𝑡)+𝐵1𝑓𝑥𝑖(𝑡)+𝐵2𝑔𝑥𝑖(𝑡1)+𝑐4𝑗=1𝑎𝑖𝑗Γ𝑥𝑗𝑥(𝑡),𝑡0,𝑡𝑘,𝑘=1,2,,𝑖𝑡+𝑘=𝑑𝑘𝑥𝑖𝑡𝑘,𝑡=𝑘,𝑖=1,2,3,4,(4.3) where 𝑑𝑘=1.2,𝜆max(𝑊)=0.25,and𝜆2(𝐴)=1.0439. Letting 𝐿=𝐿=(1/2)𝐼,𝜂=4.1756,𝜀=𝛾=𝜇2=1 and solving the LMIs in (i), (ii) in Theorem 3.1, we get that 𝜇1=6.4461 and 𝑃=diag{0.8432,0.8774}. By Theorem 3.1, we see that the complex dynamical networks (4.3) are exponentially synchronized. Figure 3 shows the synchronization of networks with delay and impulsive effects.

232794.fig.003
Figure 3: The state variables 𝑥𝑖1(𝑡) and 𝑥𝑖2(𝑡) with impulsive effects.

Example 4.2. Consider the following neural networks with delay and impulse: 𝑥𝑖(𝑡)=𝐶𝑥𝑖(𝑡)+𝐵1𝑓𝑥𝑖(𝑡)+𝐵2𝑔𝑥𝑖(𝑡𝜏(𝑡))+𝑐5𝑗=1𝑎𝑖𝑗Γ𝑥𝑗(𝑥𝑡),𝑡0,𝑡𝑘,𝑘=1,2,,𝑖𝑡+𝑘=𝑑𝑘𝑥𝑖𝑡𝑘,𝑡=𝑘,𝑖=1,2,,5,(4.4) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),𝑥𝑖3(𝑡))𝑇,  𝜏(𝑡)=0.01,𝑑𝑘=0.25,𝑐=0.8, 𝐶=diag{0.3,0.6,1}, 𝐵1=0.41.20.40.310.41.40.50.8,𝐵2=,,0.60.70.20.30.30.41.10.50.4𝐴=0.60.300.20.10.30.500.10.1000.30.10.20.20.10.10.400.10.10.200.4,Γ=1.10001.20002.1(4.5)𝑓(𝑥𝑖(𝑡))=(𝑓1(𝑥𝑖1(𝑡)),𝑓2(𝑥𝑖2(𝑡)),𝑓3(𝑥𝑖3(𝑡)))𝑇,𝑔(𝑥𝑖(𝑡))=(𝑔1(𝑥𝑖1(𝑡)),𝑔2(𝑥𝑖2(𝑡)),𝑔3(𝑥𝑖3(𝑡)))𝑇, 𝑓𝑗(𝑥)=𝑥,𝑔𝑗(𝑥)=(1/10)(|𝑥+1||𝑥1|),𝑗=1,2,3. 𝜆max(𝑊)=0.56,𝜆2(𝐴)=0.4693. Letting 𝐿=𝐼,𝐿=(1/10)𝐼,𝜂=0.838,𝜀=𝛾=𝜇2=1 and solving the LMIs in (i), (ii) in Theorem 3.1, we get that 𝜇1=1.5626 and 𝑃=diag{1.3782,0.9991,1.4307}. We can verify that the synchronization criteria proposed by Theorem 3.1 are not satisfied. However, we conclude that the complex dynamical networks (4.4) are exponentially synchronized by Corollary 3.5. Figure 4 depicts the synchronization state variables 𝑥𝑖1(𝑡), 𝑥𝑖2(𝑡), and 𝑥𝑖3(𝑡) with impulsive effects. Figure 5 depicts the synchronization state variables 𝑥𝑖1(𝑡), 𝑥𝑖2(𝑡), and 𝑥𝑖3(𝑡) without impulsive effects.

232794.fig.004
Figure 4: The state variables 𝑥𝑖1(𝑡), 𝑥𝑖2(𝑡), and 𝑥𝑖3(𝑡) with impulses.
232794.fig.005
Figure 5: The state variables 𝑥𝑖1(𝑡), 𝑥𝑖2(𝑡), and 𝑥𝑖3(𝑡) without impulsive effects.

Remark 4.3. In Example 4.2, if we take 𝑡𝑘𝑡𝑘1=0.1 and 𝑑𝑘=0.2,𝜏(𝑡)=0.9sin𝑡, it is easy to see that the synchronization criteria proposed by Corollary 3.8 are not satisfied. However, we conclude that the networks (4.4) are exponentially synchronized by Theorem 3.1.

5. Conclusions

In this paper, by establishing some lemmas of new impulsive differential inequality and by using the Lyapunov functional method and the Kronecker product techniques, exponential synchronization for impulsive dynamical networks with irreducible coupling matrix is derived. Some criteria are obtained not only relevant to delay but also to impulsive effects. In particular, the results can be extended to the case of one reducible coupling matrix 𝐴, which implies that the network topology may be a weakly connected graph containing a rooted spanning tree.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 60874088, the Natural Science Foundation of Jiangsu Province of China under Grant BK2009271, and JSPS Innovation program under Grant CXZZ11_0132.

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