Exponential Synchronization for Impulsive Dynamical Networks

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 [1–3]. 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 [4–9].

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 [10–13]. Correspondingly, based on the theory of impulsive differential equations, a lot of synchronization results of dynamical networks with impulsive effects have been obtained [13–20].

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. 𝐼_𝑛 stands for the identity matrix of order 𝑛. For 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛, the norm is defined as ‖𝑥‖=(∑^𝑛𝑖=1𝑥^2𝑖)^1/2. For matrix 𝐴, 𝜆_max(𝐴) and 𝜆_min(𝐴) denote the maximum and minimum eigenvalues of matrix 𝐴, respectively. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≤𝑌 (resp., 𝑋<𝑌) means that the matrix 𝑋−𝑌 is negative semidefinite (resp., negative definite). For a sequence {𝑡_𝑘,𝑘=0,1,…} satisfying 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<𝑡_𝑘+1<⋯, let Δ_𝑘≜𝑡_𝑘+1−𝑡_𝑘, Δ_sup=sup_𝑘≥0{Δ_𝑘}, Δ_inf=inf_𝑘≥0{Δ_𝑘}.

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: 𝑥̇_𝑖=𝐶𝑥_𝑖+𝐵₁𝑓𝑥_𝑖(𝑡)+𝐵₂𝑔𝑥_𝑖(𝑡−𝜏(𝑡)),𝑖=1,2,…,𝑁,(2.1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇 is the state vector of the 𝑖th node at time 𝑡, 𝐶, 𝐵₁, 𝐵₂∈𝑅^𝑛×𝑛; 0<𝜏(𝑡)≤𝜏, 𝜏^(𝑡)≤𝜎<1, 𝜏>0, 𝑓(𝑥),𝑔(𝑥)∈𝐶(𝑅^𝑛,𝑅^𝑛), 𝑓(𝑥_𝑖(𝑡))=(𝑓₁(𝑥_𝑖1(𝑡)),𝑓₂(𝑥_𝑖2(𝑡)),…,𝑓_𝑛(𝑥_𝑖𝑛(𝑡)))^𝑇, 𝑔(𝑥_𝑖(𝑡−𝜏(𝑡)))=(𝑔₁(𝑥_𝑖1(𝑡−𝜏(𝑡))),𝑔₂(𝑥_𝑖2(𝑡−𝜏(𝑡))),…,𝑔_𝑛(𝑥_𝑖𝑛(𝑡−𝜏(𝑡))))^𝑇.

The dynamical behavior of the dynamical network with delay can be described by the following linearly coupled systems: 𝑥̇_𝑖=𝐶+𝑥𝑐_𝑖+_𝑁𝐵₁𝑓𝑥_𝑖(𝑡)+𝐵₂𝑔𝑥_𝑖(𝑡−𝜏(𝑡))_𝑗=_{1,𝑗≠𝑖}𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡),𝑖=1,2,…,𝑁,(2.2) where Γ=diag{𝛾₁,𝛾₂,…,𝛾_𝑛} is the inner coupling positive definite matrix between two connected nodes 𝑖 and 𝑗, 𝑐 is the coupling strength, and 𝑎_𝑖𝑗 is defined as follows: if there is a connection from node 𝑗 to node 𝑖(𝑗≠𝑖), then 𝑎_𝑖𝑗>0; otherwise, 𝑎_𝑖𝑗=0.

In the process of signal transmission, due to the impulsive effects, the states 𝑥_𝑖(𝑡),𝑖=1,2,…,𝑁 are suddenly changed in the form of impulses at discrete times 𝑡_𝑘. That is, 𝑥_𝑖(𝑡^+𝑘)=𝑑_𝑘𝑥_𝑖(𝑡_𝑘). Let 𝑎_𝑖𝑖=−∑^{𝑁𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗. Thus, the dynamical network with delay and impulsive effects can be obtained by the following form: 𝑥̇_𝑖=𝐶𝑥_𝑖+𝐵₁𝑓𝑥_𝑖(𝑡)+𝐵₂𝑔𝑥_𝑖(𝑡−𝜏(𝑡))+𝑐^𝑁_𝑗=1𝑥_𝑖𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥𝑡^+𝑘=𝑑_𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,…,_𝑖(𝑡)=𝜑_𝑖(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀,𝑖=1,2,…,𝑁,(2.3) where 𝑥_𝑖(𝑡^+𝑘)=lim_ℎ→0+𝑥_𝑖(𝑡_𝑘+ℎ),𝑥_𝑖(𝑡_𝑘)=lim_ℎ→0−𝑥_𝑖(𝑡_𝑘+ℎ),𝑡_𝑘≥0 are impulsive moments satisfying 𝑡_𝑘<𝑡_𝑘+1 and lim_𝑘→+∞𝑡_𝑘=+∞, 𝑑_𝑘,𝑘=1,2,… are the impulsive gains at 𝑡_𝑘 for 𝑖th unit, 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 is the Laplacian matrix of the corresponding network.

By a solution 𝑥_𝑖=𝑥_𝑖(𝑡) of system (2.3), we mean a real function on [𝑡₀−𝜏,∞) such that 𝑥_𝑖(𝑡₀)=𝜑_𝑖(𝑡) for 𝑡∈[𝑡₀−𝜏,𝑡₀], and 𝑥_𝑖(𝑡) satisfies system (2.3) for 𝑡≥𝑡₀, and 𝑥_𝑖(𝑡) is continuous everywhere except for some 𝑡_𝑘 and left continuous at 𝑡=𝑡_𝑘, 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 |𝑑_𝑘|<1, the impulsive sequence is of synchronizing impulse, which may enhance the synchronization of the networks. But if |𝑑_𝑘|>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 𝜑_𝑖(𝑡)  (𝑖=1,2,…,𝑁): ‖‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖‖≤𝑀𝑒^{−𝜂(𝑡−𝑡0)}(2.4) hold all 𝑡>𝑡₀, and for any 𝑖,𝑗=1,2,…,𝑁.

Definition 2.3. For 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑛∈𝑅^𝑚×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑝×𝑞∈𝑅^𝑝×𝑞, the Kronecker product between two matrices is defined by 𝐴⊗𝐵=⎛⎜⎜⎜⎜⎜⎜⎝𝑎¹¹𝑎₂₁𝐵𝑎₁₂𝐵⋯𝑎_1𝑛𝐵𝑎⋮𝐵𝑎⋮₂₂𝐵⋯⋯𝑎⋮_2𝑛𝑚1𝐵𝐵𝑎_𝑚2𝐵⋯𝑎_𝑚𝑛𝐵⎞⎟⎟⎟⎟⎟⎟⎠∈𝑅^{𝑚𝑝×𝑛𝑞}.(2.5)

Assumption 2.4. There exist constants 𝑙_𝑖,𝑙^𝑖>0(𝑖=1,2,…,𝑁) such that |𝑓_𝑖(𝑥₁)−𝑓_𝑖(𝑥₂)|≤𝑙_𝑖|𝑥₁−𝑥₂| and |𝑔_𝑖(𝑥₁)−𝑔_𝑖(𝑥₂)|≤𝑙^𝑖|𝑥₁−𝑥₂| hold for any 𝑥₁,𝑥₂∈𝑅.

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 𝑎_𝑖𝑖=−∑^{𝑁𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗,𝑖=1,2,…,𝑁, then the following propositions are obtained:(1)if 𝜆 is an eigenvalue of 𝐴 and 𝜆≠0, 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,…,𝜉_𝑁)^𝑇∈𝑅^𝑁 satisfying ∑^𝑁𝑖=1𝜉_𝑖=1 is the normalized left eigenvector of A corresponding to eigenvalue 0. Then, 𝜉_𝑖>0 hold for all 𝑖=1,2,…,𝑁;(4)furthermore, if 𝐴 is symmetric, then we have 𝜉_𝑖=1/𝑁 for 𝑖=1,2,…,𝑁.

Lemma 2.7 (see [21]). Let 𝑝, 𝑞, 𝜏, 𝑑_𝑘,𝑘=1,2,… be constants and 𝑞≥0, 𝜏>0, 𝑑_𝑘≥0 and assume that 𝑢(𝑡) is a piece continuous nonnegative function satisfying: 𝐷𝑢⁺𝑢𝑡(_+𝑘𝑡)≤𝑝𝑢(𝑡)+𝑞𝑢(𝑡)𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑢(𝑡)≤=𝑑𝜙_𝑘(𝑡)𝑢,𝑡_𝑘𝑡∈,𝑡𝑘₀=1,2,…,−𝜏,𝑡₀.(2.6) If there exist 𝛼 such that for 𝑘=1,2,…ln𝑑_𝑘𝑡_𝑘−𝑡_𝑘−1𝑝+𝑑𝑞+𝛼≤<𝛼0,.(2.7) Then 𝑢(𝑡)≤𝑑sup_{𝑡0−𝜏≤𝑡≤𝑡0}||𝜙||𝑒^{−𝜆(𝑡−𝑡0)},(2.8) where 𝑢(𝑡)=sup_{𝑡−𝜏≤𝜎≤𝑡}𝑥(𝜎), 𝑑=sup_{1≤𝑘<+∞}{𝑒^{𝛼(𝑡𝑘−𝑡𝑘−1)},1/𝑒^{𝛼(𝑡𝑘−𝑡𝑘−1)}}, 𝜆 is an unique positive solution of 𝜆+𝑝+𝑑𝑞𝑒^𝜆𝜏+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,𝑦∈𝑅^𝑛, scalar 𝜖>0, and positive definite matrix 𝑄∈𝑅^𝑛×𝑛, the following inequality holds: 2𝑥^𝑇𝑦≤𝜖𝑥^𝑇𝑄𝑥+𝜖⁻¹𝑦^𝑇𝑄⁻¹𝑦.(2.9)

Lemma 2.10. Let 𝐴∈𝑅^𝑛×𝑛 be a positive definite matrix, then for 𝑥∈𝑅^𝑛, 𝜆_min(𝐴)𝑥^𝑇𝑥≤𝑥^𝑇𝐴𝑥≤𝜆_max(𝐴)𝑥^𝑇𝑥.(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)=(𝑥^𝑇1(𝑡),𝑥^𝑇2(𝑡),…,𝑥^𝑇𝑁(𝑡))^𝑇, 𝐹(𝑥(𝑡))=(𝑓^𝑇(𝑥₁(𝑡)),𝑓^𝑇(𝑥₂(𝑡)),…,𝑓^𝑇(𝑥_𝑁(𝑡)))^𝑇, 𝐺(𝑥(𝑡))=(𝑔^𝑇(𝑥₁(𝑡)),𝑔^𝑇(𝑥₂(𝑡)),…,𝑔^𝑇(𝑥_𝑁(𝑡)))^𝑇 and 𝜑(𝑡)=(𝜑₁(𝑡),𝜑₂(𝑡),…,𝜑_𝑁(𝑡))^𝑇. Then, the delayed dynamical network (2.3) can be rewritten in the following Kronecker product form: 𝑥̇=+𝐼_𝑁𝐼⊗𝐶𝑥(𝑡)+𝐼_𝑁⊗𝐵₁𝐹(𝑥(𝑡))_𝑁⊗𝐵₂𝑥𝐺𝑡_+𝑘(𝑥(𝑡−𝜏(𝑡)))+𝑐(𝐴⊗Γ)𝑥(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥(𝑡)==𝜑𝑑_𝑘(𝑡𝑥),𝑡_𝑘𝑡∈,𝑘𝑡₀=1,2,…,−𝜏,𝑡₀.(3.1)

Suppose that 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑁)^𝑇 is the left eigenvector of the configuration coupling matrix 𝐴 with respect to eigenvalue 0 satisfying ∑^𝑁𝑖=1𝜉_𝑖=1. Since the coupling configuration matrix 𝐴 is irreducible, by Lemma 2.6, we can see that 𝜉_𝑖>0 for 𝑖=1,2,…,𝑁. Let Ξ=diag{𝜉₁,𝜉₂,…,𝜉_𝑁}>0, 𝐿=diag{𝑙₁,𝑙₂,…,𝑙_𝑛}, 𝐿^=diag{𝑙^1,𝑙^2,…,𝑙^𝑛}, 𝑊=Ξ−𝜉𝜉^𝑇 and 𝐴=Ξ𝐴+𝐴^𝑇Ξ.

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, 𝜇₂≥0, 𝜇₁, 𝛿 such that(𝐻₁)Θ₁=𝑃𝐶+𝐶^𝑇𝑃+𝜀𝑃𝐵₁𝐵^𝑇1𝑃+𝛾𝑃𝐵₂𝐵^𝑇2𝑃+𝜀⁻¹𝐿²−𝑐𝜂𝑃Γ−𝜇₁𝑃≤0;(𝐻₂)Θ₂=𝛾⁻¹𝐿^2−𝜇₂𝑃≤0;(𝐻₃)forall  𝑘=1,2,…,  2ln|𝑑_𝑘|/(𝑡_𝑘−𝑡_𝑘−1)≤𝛿;(𝐻₄)𝜇₁+𝑑𝜇₂+𝛿<0;(𝐻₅)𝜂𝜆_max(𝑊)+𝜆₂(𝐴)≤0.Then the complex dynamical networks (3.1) are exponentially synchronized, where 𝑑=sup_{1≤𝑘<+∞}{𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)},1/𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)}}, 𝜆₂(𝐴) is defined to be the second largest eigenvalue of 𝐴.

Proof. We define a Lyapunov function 𝑉(𝑡)=𝑥^𝑇(𝑡)(𝑊⊗𝑃)𝑥(𝑡). Since 𝑊=Ξ−𝜉𝜉^𝑇, we have 𝑤_𝑖𝑗=−𝜉_𝑖𝜉_𝑗 for 𝑖≠𝑗 and 𝑤_𝑖𝑖=𝜉_𝑖−𝜉^2𝑖. In view of ∑^𝑁𝑗=1𝜉_𝑗=1, it follows that ∑^𝑁𝑗=1𝑤_𝑖𝑗=𝜉_𝑖−∑^𝑁𝑗=1𝜉_𝑖𝜉_𝑗=0. Therefore, we can conclude that 𝑉(𝑡)=∑^𝑁𝑖=1∑^{𝑁𝑗=1,𝑗≠𝑖}−(1/2)𝑤_𝑖𝑗(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡))^𝑇𝑃(𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)). Calculating the Dini derivative of 𝑉(𝑡) along the trajectories of the systems (3.1), we have for 𝑡≠𝑡_𝑘,𝑘=1,2,…: 𝐷⁺𝑉(𝑡)=2+𝑥^𝑇2𝑥(𝑡_𝑇)(𝑊⊗𝑃)×𝐼_𝑁⊗𝐶𝑥(𝑡)+2𝑥_𝑇(𝑊⊗𝑃)×𝐼_𝑁⊗𝐵₁𝐹(𝑥(𝑡))+2𝑐𝑥_𝑇(𝑊⊗𝑃)×𝐼_𝑁⊗𝐵₂𝐺(𝑥(𝑡−𝜏(𝑡)))(𝑡)(𝑊⊗𝑃)×(𝐴⊗Γ)𝑥(𝑡).(3.2) By adding −𝑐𝑥^𝑇(𝑡)(𝑊⊗𝜂𝑃Γ)𝑥(𝑡)+𝑐𝑥^𝑇(𝑡)(𝑊⊗𝜂𝑃Γ)𝑥(𝑡) to (3.2) and noting that 𝑊𝐴=(Ξ−𝜉𝜉^𝑇)𝐴=Ξ𝐴−𝜉(𝜉^𝑇𝐴)=Ξ𝐴, we can obtain that 𝐷⁺𝑉(𝑡)≤−^𝑁_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}𝑤_𝑖𝑗𝑥^𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐶−12𝑐𝜂𝑃Γ𝑥^𝑖+𝑥(𝑡)−𝑥_𝑗(𝑡)_𝑖+𝑥(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐵₁𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)_𝑖+𝑐𝑥_𝑇−𝑥_𝑗^𝑇𝑃𝐵₂𝑔𝑥_𝑖(𝑡−𝜏(𝑡))−𝑔𝑥_𝑗(𝑡−𝜏(𝑡))(𝑡)×Ξ𝐴+𝐴^𝑇Ξ⊗𝑃Γ+𝑊⊗𝜂𝑃Γ𝑥(𝑡).(3.3) Since the matrix 𝐴=Ξ𝐴+𝐴^𝑇Ξ has the following property: 𝐴=𝐴_𝑖𝑗_𝑁×𝑁,𝐴_𝑖𝑖=2𝜉_𝑖𝐴_𝑖𝑖<0,𝑖=1,2,…,𝑁,𝐴_𝑖𝑗=𝜉_𝑖𝐴_𝑖𝑗+𝜉_𝑗𝐴_𝑗𝑖=𝐴_𝑗𝑖,𝑖≠𝑗,^𝑁_𝑗=1𝐴_𝑖𝑗=𝜉_𝑖𝑁_𝑗=1𝐴_𝑖𝑗+^𝑁_𝑗=1𝜉_𝑗𝐴_𝑗𝑖=0.(3.4) By Perron-Frobenius theorem (see [24]), we can arrange the eigenvalues of matrix 𝐴 as follows: 0=𝜆₁(𝐴)>𝜆₂(𝐴)≥⋯≥𝜆_𝑁(𝐴). Applying matrix decomposition theory (see [24]), there exists unitary matrix 𝑈, such that 𝐴=𝑈Λ𝑈^𝑇, where Λ=diag{0,𝜆₂(𝐴),…,𝜆_𝑁(𝐴)} and 𝑈={𝑢₁,𝑢₂,…,𝑢_𝑁} with 𝑢₁=(1/√𝑁,1/√𝑁,…,1/√𝑁)^𝑇 and 𝑈^𝑇𝑈=𝐼_𝑁.
Let 𝑦(𝑡)=(𝑈^𝑇⊗𝐼_𝑛)𝑥(𝑡), where 𝑦(𝑡)=(𝑦^𝑇1(𝑡),𝑦^𝑇2(𝑡),…,𝑦^𝑇𝑁(𝑡))^𝑇, 𝑦_𝑖(𝑡)∈𝑅^𝑛,𝑖=1,2,…,𝑁. Then we have 𝑥(𝑡)=(𝑈⊗𝐼_𝑛)𝑦(𝑡). Thus, we have 𝑥^𝑇(𝑡)Ξ𝐴+𝐴^𝑇Ξ⊗𝑃Γ𝑥(𝑡)==𝑦^𝑇(𝑡)𝑈^𝑇⊗𝐼_𝑛𝐴⊗𝑃Γ𝑈⊗𝐼_𝑛𝑦(𝑡)_𝑁_𝑖=2𝜆_𝑖𝐴𝑦^𝑇𝑖(𝑡)𝑃Γ𝑦_𝑖(𝑡)≤𝜆₂𝐴^𝑁_𝑖=2𝑦^𝑇𝑖(𝑡)𝑃Γ𝑦_𝑖(𝑡).(3.5) In view of matrix 𝑊 is a zero row sum irreducible symmetric matrix with negative off-diagonal elements, we see that 𝜆_max(𝑊)>0 and 𝑊𝑢₁=(0,0,…,0))^𝑇. Hence by Lemma 2.10, we have 𝑥^𝑇(𝑡)(𝑊⊗𝜂𝑃Γ)𝑥(𝑡)≤=𝜂𝜂𝑦𝜆^𝑇(_max𝑡)𝑈^𝑇𝑊𝑈⊗𝑃Γ𝑦(𝑡)(𝑊)^𝑁_𝑖=2𝑦^𝑇𝑖(𝑡)𝑃Γ𝑦_𝑖(𝑡).(3.6) It follows from condition 𝜆₂(𝐴)+𝜂𝜆_max(𝑊)≤0 that 𝑐𝑥^𝑇(𝑡)≤Ξ𝑐𝐴𝜆+₂𝐴^𝑇Ξ⊗𝑃Γ+𝑊⊗𝜂𝑃Γ𝑥(𝑡)𝐴+𝜂𝜆_max(𝑊)^𝑁_𝑖=2𝑦^𝑇𝑖(𝑡)𝑃𝑦_𝑖(𝑡)≤0.(3.7) By Assumption 2.4 and Lemma 2.10, there exists 𝜀>0 such that 2𝑥_𝑖(𝑡≤)−𝜀𝑥𝑥_𝑗𝑖(𝑡)^𝑇𝑃𝐵₁𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)+𝜀₋₁(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐵₁𝐵^𝑇1𝑃𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)≤𝜀𝑥_𝑖𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)^𝑇𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)+𝜀₋₁(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐵₁𝐵^𝑇1𝑃𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=𝑥𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝐿²𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝜀𝑃𝐵₁𝐵^𝑇1𝑃+𝜀⁻¹𝐿²𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡).(3.8) Similarly, we have the following estimation: 2𝑥_𝑖(𝑡≤)−𝑥𝑥_𝑗(𝑡)^𝑇𝐵₂𝑔𝑥_𝑖(𝑡−𝜏(𝑡))−𝑔𝑥_𝑗(𝑡−𝜏(𝑡))_𝑖+𝑥(𝑡)−𝑥_𝑗(𝑡)^𝑇𝛾𝑃𝐵₂𝐵^𝑇2𝑃𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))^𝑇𝛾⁻¹𝐿^2𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡)),(3.9) where 𝛾>0. Substituting these into (3.3), we have for 𝑡≠𝑡_𝑘𝑉̇(𝑡)≤−^𝑁_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}12𝑤^𝑖𝑗×−𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐶+𝐶^𝑇𝑃+𝜀𝑃𝐵₁𝐵^𝑇1𝑃+𝛾𝑃𝐵₂𝐵^𝑇2𝑃+𝜀⁻¹𝐿²−𝑐𝜂𝑃Γ𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)_𝑁_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}12𝑤^𝑖𝑗=−×_𝑁𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))^𝑇𝛾⁻¹𝐿^2𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}12𝑤^𝑖𝑗××𝑥_𝑖𝑥(_𝑖𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝐶+𝐶^𝑇𝑃+𝜀𝑃𝐵₁𝐵^𝑇1𝑃+𝛾𝑃𝐵₂𝐵^𝑇2𝑃+𝜀⁻¹𝐿²−𝑐𝜂𝑃Γ−𝜇₁𝑃−𝜇_1𝑁(𝑡)−𝑥_𝑗(𝑡)_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}12𝑤^𝑖𝑗𝑥_𝑖−(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑃𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)_𝑁_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}12𝑤^𝑖𝑗×−𝜇𝑥_𝑖2𝑁(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))^𝑇𝛾⁻¹𝐿^2−𝜇₂𝑃𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}12𝑤^𝑖𝑗𝑥^𝑖≤𝜇₁(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))^𝑇𝑃𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑗(𝑡−𝜏(𝑡))𝑉(𝑡)+𝜇₂𝑉(𝑡−𝜏(𝑡)).(3.10) For 𝑡=𝑡_𝑘, we have 𝑉𝑡^+𝑘=−12^𝑁_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}=−𝑑_2𝑘𝑤_𝑖𝑗𝑥_𝑖𝑡^+𝑘−𝑥_𝑗𝑡^+𝑘^𝑇𝑃𝑥_𝑖𝑡^+𝑘−𝑥_𝑗𝑡^+𝑘2^𝑁_𝑁𝑖=1_{𝑗=1,𝑗≠𝑖}=𝑑_2𝑘𝑤_𝑖𝑗𝑥_𝑖𝑡_𝑘−𝑥_𝑗𝑡_𝑘^𝑇𝑃𝑥_𝑖𝑡_𝑘−𝑥_𝑗𝑡_𝑘𝑉𝑡_𝑘.(3.11) By Lemma 2.7, there exist 𝑀>0 such that 𝑉(𝑡)≤𝑀sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠𝑒^{−𝜂(𝑡−𝑡0)},(3.12) which implies that 12𝜉^𝑖𝜉_𝑗𝜆_min(𝑃)‖‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖‖²≤12^𝑁_{𝑖=1,𝑗=1}=𝑉(𝑡)=𝜉𝑂_𝑖𝜉_𝑗𝑒𝑥_𝑖(𝑡)−_{−𝜂(𝑡−𝑡0)}𝑥_𝑗(𝑡)^𝑇𝑃𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡).(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, |𝑑_𝑘|>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, |𝑑_𝑘|<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 Δ_sup<∞. Also suppose that there exist a diagonal positive definite matrix 𝑃 and scalars 𝜂>0, 𝜀>0, 𝛾>0, 𝜇₁>0, 𝜇₂≥0 such that(𝐻₁)Θ₁=𝑃𝐶+𝐶^𝑇𝑃+𝜀𝑃𝐵₁𝐵^𝑇1𝑃+𝛾𝑃𝐵₂𝐵^𝑇2𝑃+𝜀⁻¹𝐿²−𝑐𝜂𝑃Γ−𝜇₁𝑃≤0;(𝐻₂)Θ₂=𝛾⁻¹𝐿^2−𝜇₂𝑃≤0;(𝐻₃) for all 𝑘=1,2,…, |𝑑_𝑘|<1;(𝐻₄) there exists a integer 𝑚≥1 such that 𝑡_𝑘−𝑚≤𝑡_𝑘−𝜏≤𝑡_{𝑘+1−𝑚} for all 𝑘≥𝑚, and the discrete system: 𝜃(𝑘+1)=𝐽_𝑘(𝑚)𝜃(𝑘)(3.14) is globally exponentially stable with decay 𝜆>0, where 𝐽_𝑘(𝑚)≜⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝛽⋮000⋮100⋮010⋯⋯⋯⋯⋮000⋮001_{𝑘+1−𝑚}𝛽_{𝑘+2−𝑚}𝛽_{𝑘+3−𝑚}⋯𝛽_𝑘−1𝛼_𝑘−1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.15) 𝜁=𝜇₁+𝜇₂/(1−𝜎), 𝛼_𝑘=𝑑^2𝑘𝑒^{𝜁Δ𝑘−1}+𝛽_𝑘−1, 𝛽_𝑘−𝑗=(𝛽/(1−𝜎))Δ_𝑘−𝑗𝑒^{𝜁Δ𝑘−𝑗},𝑗=1,2,…,𝑚−1; (𝐻₅) there exists a constant 𝒯₀ such that the average dwell time 𝒯_𝑎 satisfies 𝒩𝑡₀,𝑡≥−𝒯₀+𝑡−𝑡⁰𝒯_𝑎,𝑡≥𝑡₀,(3.16) where 𝒩[𝑡₀,𝑡] is the number of impulsive times of the impulsive sequence on the interval [𝑡₀,𝑡];(𝐻₆)𝜂𝜆_max(𝑊)+𝜆₂(𝐴)≤0.
Then the complex dynamical networks (3.1) are exponentially synchronized with decay rate 𝜆/2𝒯_𝑎.

Proof. Consider a Lyapunov-Krasovskii functional: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡),(3.17) with 𝑉₁(𝑡)=𝑥^𝑇(𝑡)(𝑊⊗𝑃)𝑥(𝑡),𝑉₂(𝑡)=𝜇₂1−𝜎^{𝑡𝑡−𝜏(𝑡)}𝑥^𝑇(𝑠)(𝑊⊗𝑃)𝑥(𝑠)𝑑𝑠.(3.18) Similar to the proof of Theorem 3.1, for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], we get 𝐷⁺𝑉₁(𝑡)≤𝜇₁𝑥^𝑇(𝑡)(𝑊⊗𝑃)𝑥(𝑡)+𝜇₂𝑥^𝑇(𝑡−𝜏(𝑡))(𝑊⊗𝑃)𝑥(𝑡−𝜏(𝑡)).(3.19) For 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], we have 𝐷⁺𝑉₂(𝑡)≤𝜇₂1−𝜎𝑥^𝑇(𝑡)(𝑊⊗𝑃)𝑥(𝑡)−𝜇₂𝑥^𝑇(𝑡−𝜏(𝑡))(𝑊⊗𝑃)𝑥(𝑡−𝜏(𝑡)).(3.20) Then 𝐷⁺𝑉(𝑡)=𝐷⁺𝑉₁(𝑡,𝑥(𝑡))+𝐷⁺𝑉₂(𝑡)≤𝜇₁+𝜇₂1−𝜎𝑉¹(𝑡)≤𝜁𝑉(𝑡).(3.21) Thus 𝑉(𝑡)≤𝑉𝑡^+𝑘𝑒^{𝜁(𝑡−𝑡𝑘)},𝑡∈𝑡_𝑘,𝑡_𝑘+1.(3.22) By (3.11), for 𝑡=𝑡_𝑘, we have 𝑉₁𝑡^+𝑘≤𝑑^2𝑘𝑉₁𝑡_𝑘.(3.23) It follows from condition (iii) that there exists some 𝑡̂_{𝑘−𝑗+1}∈(𝑡_𝑘−𝑗,𝑡_{𝑘−𝑗+1}] such that 𝑉₂𝑡^+𝑘≤𝜇₂1−𝜎^𝑡𝑘_{𝑡𝑘𝑡−𝜏𝑘}𝑉₁(𝑠)𝑑𝑠≤𝜇₂1−𝜎^𝑡𝑘_{𝑡𝑘−𝑚}=𝑉₁(𝑠)𝑑𝑠𝜇₂1−𝜎^𝑚_𝑗=1^{𝑡𝑘−𝑗+1}_{𝑡+𝑘−𝑗}𝑉₁(𝑠)𝑑𝑠=𝜇₂1−𝜎^𝑚_𝑗=1Δ_𝑘−𝑗𝑉₁𝑡̂_{𝑘−𝑗+1}.(3.24) Then from (3.22), we have 𝑉₂𝑡^+𝑘≤𝜇₂1−𝜎^𝑚_𝑗=1Δ_𝑘−𝑗𝑉𝑡̂_{𝑘−𝑗+1}≤𝜇₂1−𝜎^𝑚_𝑗=1Δ_𝑘−𝑗𝑒^{𝜁Δ𝑘−𝑗}𝑉𝑡^+𝑘−𝑗.(3.25) Together with (3.22), (3.23) and the above inequality, we have 𝑉𝑡^+𝑘≤𝑑^2𝑘+𝜇₂1−𝜎Δ^𝑘−1𝑒^{𝜁Δ𝑘−1}𝑉𝑡^+𝑘−1+𝜇₂1−𝜎^𝑚_𝑗=2≜𝛼_𝑘−1Δ_𝑘−𝑗𝑒^{𝜁Δ𝑘−𝑗}𝑉𝑡^+𝑘−𝑗𝑉𝑡^+𝑘−1+^𝑚−1_𝑗=1𝛽_{𝑘−𝑗−1}𝑉𝑡^{+𝑘−𝑗−1}.(3.26) Set 𝑍(𝑘)=(𝑧₁(𝑘),𝑧₂(𝑘),…,𝑧_𝑚(𝑘))^𝑇 and 𝑧₁(𝑘)=𝑉(𝑡^+𝑘+1), 𝑧₂(𝑘)=𝑉(𝑡^+𝑘+2),…,𝑧_𝑚(𝑘)=𝑉(𝑡^+𝑘+𝑚). Then ⎛⎜⎜⎜⎜⎜⎜⎝𝑧𝑧₁(𝑘+1−𝑚)₂𝑧(𝑘+⋮1−𝑚)_𝑚(𝑘+1−𝑚)⎞⎟⎟⎟⎟⎟⎟⎠≤𝐽^𝑘(𝑚)⎛⎜⎜⎜⎜⎜⎜⎝𝑧¹𝑧(𝑘−𝑚)₂𝑧(𝑘⋮−𝑚)_𝑚(𝑘−𝑚)⎞⎟⎟⎟⎟⎟⎟⎠,(3.27) that is, 𝑍(𝑘−𝑚+1)≤𝐽_𝑘(𝑚)𝑍(𝑘−𝑚).(3.28) We consider the discrete system: 𝜃(𝑘+1)=𝐽_𝑘(𝑚)𝜃(𝑘),𝜃(𝑚−1)=𝑍(−1).(3.29) Then, by the comparison principle, we see that for 𝑘≥𝑚−1𝑍(𝑘−𝑚)≤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(𝑘−𝑚)‖≤𝑀𝑒^{−𝜆(𝑘−𝑚+1)}‖𝑍(−1)‖,𝑘≥𝑚−1,(3.31) where ‖𝑍(−1)‖=[∑^{𝑚−1𝑗=0}𝑉²(𝑡_𝑗)]^1/2,‖𝑍(𝑘−𝑚)‖=[∑^𝑚𝑗=1𝑉²(𝑡_{𝑗+𝑘−𝑚})]^1/2. From (3.17) and (3.22), we have 𝑉₂𝑡^+𝑗=𝜇₂1−𝜎^𝑡𝑗_{𝑡𝑗𝑡−𝜏𝑗}𝑥^𝑇(𝑠)(𝑊⊗𝑃)𝑥(𝑠)𝑑𝑠≤𝜇₂1−𝜎^𝑡𝑗_𝑡0−𝜏=𝑥^𝑇(𝑠)(𝑊⊗𝑃)𝑥(𝑠)𝑑𝑠𝜇₂1−𝜎^𝑡0_𝑡0−𝜏𝑥^𝑇(𝑠)(𝑊⊗𝑃)𝑥(𝑠)𝑑𝑠+𝜇₂1−𝜎^𝑗−1_𝑠=0^𝑡𝑠+1_𝑡+𝑠≤𝑥^𝑇(𝑠)(𝑊⊗𝑃)𝑥(𝑠)𝑑𝑠𝜇₂𝜏1−𝜎sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠+𝜇₂1−𝜎^𝑗−1_𝑠=0Δ_𝑠𝑉𝑡^+𝑠𝑒^𝜁Δ𝑠,𝑗=0,1,…,𝑚−1.(3.32) Furthermore, it follows that 𝑉𝑡^+𝑗=≤𝑉¹𝑡^+𝑗+𝑉²𝑡^+𝑗𝜇₂𝜏1−𝜎sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠+𝑑^2𝑗𝑒^{𝜁Δ𝑗−1}𝑉𝑡^+𝑗−1+𝜇₂1−𝜎^𝑗−1_𝑠∑₌₀Δ_𝑠𝑉𝑡^+𝑠𝑒^{𝜁Δ𝑠−1}=𝜇₂𝜏1−𝜎sup_{−𝜏≤𝑠≤0}𝑉𝑉𝑡𝑡₀+𝑠+𝛼_𝑗𝑉𝑡^+𝑗−1+∑^𝑗−2_𝑠=0𝛽_𝑠𝑉𝑡^+𝑠,𝑗=1,2,…,𝑚−1,₀≤1+𝜇₂𝜏1−𝜎sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠.(3.33) By induction, there exists a constant 𝜗>0, which is dependent on 𝜏, 𝜎, 𝜇₁, 𝜇₂,  Δ_𝑗,  𝑗=0,1,…,𝑚−1 such that 𝑉𝑡^+𝑗≤𝜗‖𝜉‖²,(3.34) which yields that ‖𝑍(−1)‖=^𝑚−1_𝑗=0𝑉²𝑡^+𝑗^1/2≤√𝑚𝜗sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠.(3.35) From (3.31) and the above inequality, we see that for all 𝑘=0,1,…, 𝑉𝑡^+𝑘≤‖𝑍(𝑘−𝑚)‖≤𝑀√𝑚𝜗𝑒^{−𝜆(𝑘−𝑚+1)}sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠.(3.36) Therefore, by (3.17), (3.22), and (3.36), we conclude that for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1],𝑘=0,1,…, 𝑉(𝑡)≤𝑒^{𝜁(𝑡−𝑡𝑘)}𝑉𝑡^+𝑘≤Υ𝑒^−𝜆𝑘sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠,(3.37) where Υ=𝑀√𝑚𝜗𝑒^{𝜆(𝑚−1)+𝜁Δ𝑘}. For all 𝑡∈(𝑡_𝑘,𝑡_𝑘+1],𝑘=0,1,…, we obtain that 𝒩[𝑡₀,𝑡]=𝑘. Then 𝑉(𝑡)≤Υ𝑒^𝜆𝒯0𝑒^{−(𝜆/𝒯𝑎)(𝑡−𝑡0)}sup_{−𝜏≤𝑠≤0}𝑉𝑡₀+𝑠,(3.38) which means that 12𝜉^𝑖𝜉_𝑗𝜆_min(𝑃)‖‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖‖²≤12^𝑁_{𝑖=1,𝑗=1}=𝑉(𝑡)=𝜉𝑂_𝑖𝜉_𝑗𝑒𝑥_𝑖(𝑡)−𝑥_{𝑗−(𝜆/𝒯𝑎)(𝑡−𝑡0)}(𝑡)^𝑇𝑃𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡).(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 𝑑_𝑘<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 Δ_sup<∞ and 𝜏<𝑡_𝑘−𝑡_𝑘−1 for all 𝑘=1,2,…. If there exist positive definite matrix 𝑃 and scalars 𝜂>0, 𝜀>0, 𝛾>0,  𝜇₁>0, 𝜇₂≥0 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:(𝐻^4) there exists a constant 𝜆>0 such that ln𝑑^2𝑘+𝜇₂𝜏1−𝜎+𝜁Δ^𝑘−1≤−𝜆,(3.40) where 𝜁=𝜇1+𝜇₂/(1−𝜎), then the complex dynamical networks (3.1) are exponentially synchronized with decay rate 𝜆/2𝒯_𝑎.

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))=𝑥^𝑇(𝑡)(𝑊⊗𝑃)𝑥(𝑡),𝑉₂(𝑥(𝑡))=𝜇₂1−𝜎^{𝑡𝑡−𝜏(𝑡)}𝑥^𝑇(𝑠)(𝑊⊗𝑃)𝑥(𝑠)𝑑𝑠.(3.42) By the proof of Theorem 3.3, for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], we have 𝑉(𝑡)≤𝑉𝐷𝑡⁺_+𝑘𝑉(𝑡)≤𝜁𝑉(𝑡),𝑒^{𝜁(𝑡−𝑡𝑘)},𝑡∈𝑡_𝑘,𝑡_𝑘+1.(3.43) Note that 𝜏<𝑡_𝑘−𝑡_𝑘−1, then there exists some 𝑡_𝑘̂∈[𝑡_𝑘−𝜏,𝑡_𝑘] such that 𝑉₂𝑡^+𝑘≤𝜇₂1−𝜎^𝑡𝑘_{𝑡𝑘−𝜏}𝑉₁(𝑠)𝑑𝑠=𝜇₂𝜏1−𝜎𝑉¹𝑡_𝑘̂.(3.44) Thus 𝑉𝑡^+𝑘≤𝑑^2𝑘𝑉₁𝑡_𝑘+𝜇₂𝜏1−𝜎𝑉¹𝑡_𝑘̂≤𝑑^2𝑘𝑒^{𝜁Δ𝑘−1}𝑉₁𝑡^+𝑘−1+𝜇₂𝜏1−𝜎𝑒^{𝜁Δ𝑘−1}=𝑒_{ln(𝑑2𝑘+𝜇2𝜏/(1−𝜎))+𝜁Δ𝑘−1}𝑉₁𝑡^+𝑘−1𝑉𝑡^+𝑘−1.(3.45) Then from condition (𝐻^4), we obtain 𝑉𝑡^+𝑘≤𝑒^−𝜆𝑉𝑡^+𝑘−1≤⋯≤𝑒^−𝜆𝑘𝑉𝑡₀,(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<𝑡_𝑘−𝑡_𝑘−1 for all 𝑘=1,2,…, we see that the estimations of maximal time-delay 𝜏^ and maximal dwell time Δ_sup as 𝜏^<sup_𝑘≥1(1−𝜎)𝑒^{−𝜁Δ𝑘−1}−𝜆−𝑑^2𝑘𝜁,Δ^sup<sup_𝑘≥1−𝜆−ln𝑑^2𝑘+𝜇₂𝜏/(1−𝜎)𝜁.(3.47)

Remark 3.7. By Corollary 3.5, if we take the impulsive gains 𝑑_𝑘 as 0<𝑑_𝑘<𝑒^{−𝜁Δ𝑘−1−𝜆}−𝜇₂𝜏1−𝜎,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 Δ_sup<∞. If there exist positive definite matrix 𝑃 and scalars 𝜂>0, 𝜀>0, 𝛾>0, 𝜇₁>0, 𝜇₂≥0 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:(𝐻^4) there exists a constant 𝑚>1 such that 𝑡_𝑘−𝑚<𝑡_𝑘−𝜏≤𝑡_{𝑘+1−𝑚}  for all  𝑘≥𝑚, and the matrix 𝐽(𝑚) satisfies the spectral radius condition for some 𝜆>0𝜌(𝐽(𝑚))<𝑒^−𝜆,(3.49) where 𝐽(𝑚)≜⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜖⋮000⋮100⋮010⋯⋯⋯0⋮000⋮001₁𝜖₁𝜖₁⋯𝜖₁𝜖₁+𝜖₂⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.50) 𝜖₁=(𝜇₂/(1−𝜎))Δ_sup𝑒^𝜁Δsup, 𝜖₂=𝑑𝑒^𝜁Δsup, 𝑑=sup_𝑘≥1{𝑑^2𝑘}, 𝜁=𝜇₁+𝜇₂/(1−𝜎); (𝑖𝑖𝑖^) there exists a positive integer 𝑚≥1 such that 𝑡_𝑘−𝑚<𝑡_𝑘−𝜏≤𝑡_{𝑘+1−𝑚} for all 𝑘≥𝑚, and there exists a constant 0<𝜚<1 such that all roots 𝜆_𝑗(𝑗=1,2,…,𝑚) of the characteristic polynomial: Ψ_𝑘(𝜆)≜𝜆^𝑚−𝜇_𝑘−1𝜆^𝑚−1−𝜈_𝑘−1𝜆^𝑚−2−⋯−𝜈_{𝑘+2−𝑚}𝜆−𝜈_{𝑘+1−𝑚}(3.51) satisfy that |𝜆_𝑗|≤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 𝑡_𝑘−𝑡_𝑘−1≤𝑇₁  (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 𝑡_𝑘−𝑡_𝑘−1≥𝑇₂  (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(𝑡−1)),(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=⎛⎜⎜⎝−01−01⎞⎟⎟⎠, 𝐵₁=⎛⎜⎜⎝−52.0−10.5.1⎞⎟⎟⎠, and 𝐵₂=⎛⎜⎜⎝−−10..52−−01.1⎞⎟⎟⎠. These neural networks (4.1) are chaotic, and chaotic attractor is shown in Figure 1.
We consider the following linear coupled delayed networks: 𝑥^𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)+𝐵₁𝑓𝑥_𝑖(𝑡)+𝐵₂𝑔𝑥_𝑖(𝑡−1)+𝑐⁴_𝑗=1𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=1,2,3,4,(4.2) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡))^𝑇,  𝑐=1.4, Γ=⎛⎜⎜⎝4.01840.9⎞⎟⎟⎠ and 𝐴=⎛⎜⎜⎜⎜⎜⎜⎝00−1..24602−0..346−1210.4.4−2104...646.6⎞⎟⎟⎟⎟⎟⎟⎠.

Figure 1 

Chaotic and chaotic attractor.

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

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: 𝑥^𝑖(𝑡)=+𝐶𝑐𝑥_𝑖4(𝑡)+𝐵₁𝑓𝑥_𝑖(𝑡)+𝐵₂𝑔𝑥_𝑖(𝑡−1)𝑥∑_𝑗=1𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑡≥0,𝑡≠𝑘,𝑘=1,2,…,_𝑖𝑡^+𝑘=𝑑_𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑘,𝑖=1,2,3,4,(4.3) where 𝑑_𝑘=1.2,𝜆_max(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.

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: 𝑥^𝑖(𝑡)=+𝐶𝑐𝑥_𝑖5(𝑡)+𝐵₁𝑓𝑥_𝑖(𝑡)+𝐵₂𝑔𝑥_𝑖(𝑡−𝜏(𝑡))𝑥∑_𝑗=1𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑡≥0,𝑡≠𝑘,𝑘=1,2,…,_𝑖𝑡^+𝑘=𝑑_𝑘𝑥_𝑖𝑡_𝑘,𝑡=𝑘,𝑖=1,2,…,5,(4.4) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),𝑥_𝑖3(𝑡))^𝑇,  𝜏(𝑡)=0.01,𝑑_𝑘=0.25,𝑐=0.8, 𝐶=diag{−0.3,−0.6,−1}, 𝐴=𝐵⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝₁−=00000...321.⎛⎜⎜⎜⎜⎝6001...−43400000...−311.0−51.15.−200000..−−12.0300.4..−48000⎞⎟⎟⎟⎟⎠00...211,.4−0000𝐵0...112.₂4⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠=,⎛⎜⎜⎜⎜⎝−010..Γ61.3=−00⎛⎜⎜⎜⎜⎝0..175.003.1−100000...224.4⎞⎟⎟⎟⎟⎠200.,1⎞⎟⎟⎟⎟⎠,(4.5)𝑓(𝑥_𝑖(𝑡))=(𝑓₁(𝑥_𝑖1(𝑡)),𝑓₂(𝑥_𝑖2(𝑡)),𝑓₃(𝑥_𝑖3(𝑡)))^𝑇,𝑔(𝑥_𝑖(𝑡))=(𝑔₁(𝑥_𝑖1(𝑡)),𝑔₂(𝑥_𝑖2(𝑡)),𝑔₃(𝑥_𝑖3(𝑡)))^𝑇, 𝑓_𝑗(𝑥)=𝑥,𝑔_𝑗(𝑥)=(1/10)(|𝑥+1|−|𝑥−1|),𝑗=1,2,3. 𝜆_max(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 𝑥_𝑖1(𝑡), 𝑥_𝑖2(𝑡), and 𝑥_𝑖3(𝑡) with impulsive effects. Figure 5 depicts the synchronization state variables 𝑥_𝑖1(𝑡), 𝑥_𝑖2(𝑡), and 𝑥_𝑖3(𝑡) without impulsive effects.

Figure 4 

The state variables 𝑥𝑖1(𝑡), 𝑥𝑖2(𝑡), and 𝑥𝑖3(𝑡) with impulses.

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 𝑡_𝑘−𝑡_𝑘−1=0.1 and 𝑑_𝑘=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.