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Abstract and Applied Analysis
Volume 2012, Article ID 298095, 18 pages
http://dx.doi.org/10.1155/2012/298095
Research Article

Mean-Square Exponential Synchronization of Markovian Switching Stochastic Complex Networks with Time-Varying Delays by Pinning Control

1College of Information and Engineering, Shenzhen University, Shenzhen 518060, China
2College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, China
3Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

Received 19 December 2011; Accepted 2 March 2012

Academic Editor: Márcia Federson

Copyright © 2012 Jingyi Wang et al. 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 investigates the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method. The switching parameters are modeled by a continuous-time, finite-state Markov chain, and the complex network is subject to noise perturbations, Markovian switching, and internal and outer time-varying delays. Sufficient conditions for mean-square exponential synchronization are obtained by using the Lyapunov-Krasovskii functional, Itö’s formula, and the linear matrix inequality (LMI), and numerical examples are given to demonstrate the validity of the theoretical results.

1. Introduction

A complex network is a structure that is made up of a large set of nodes (also called vertices) that are interconnected to varying extents by a set of links (also called edges). Coupled biological systems such as neural networks and socially interacting animal species are simple examples of complex networks and so too is the Internet and the World Wide Web [1]. Complex networks, indeed, are so ubiquitously found in nature and in the modern world that it is absolutely essential for us to have a thorough understanding of their dynamical behavior, and complex networks synchronization holds particular promise for applications to many fields (e.g., population dynamics, power systems and automatic control [26]).

Chaos synchronization is a phenomenon that has been widely investigated since it was first discovered by Pecora and Carroll in 1990, and it is a process in which two or more dynamical systems seek to adjust a certain prescribed property of their motion to a common behavior in the limit as time tends to infinity [7]. Many synchronization patterns have been explored (e.g., complete synchronization [8], cluster synchronization [9], phase synchronization [10], and partial synchronization [11]), and synchronization can be achieved by the use of adaptive control [12], feedback control [13], intermittent control [14], fuzzy control [15], impulsive control [16], or pinning control [1720].

Stochastic perturbations and time delays are important considerations when simulating realistic complex networks because signals traveling along real physical systems are usually randomly perturbed by the environmental elements [12] and time-delayed by chaotic behavior (consider, e.g., a delayed neural network or a delayed Chua's circuit). Although some results have recently appeared on the synchronization of complex networks with coupling delays ([2127]), most of the stochastically perturbed networks that have been investigated were one dimensional in the sense that the same noise impacted all the transmitted signals ([2830]). Results on the more realistic vector-formed perturbations (in which different nodes are subject different types of noise disturbances) are scanty with [12, 31] being the only such results that have been reported on the stochastic synchronization of coupled neural networks. One popular model for stochastics in the sciences and industries is the Markovian switching model driven by continuous-time Markov chains ([3237]), and Mao [32, 33] considered the stability of stochastic delayed differential equations using this model while others ([3436]) discussed the exponential stability of stochastic delayed neural networks. Liu et al. [37], on the other hand, investigated the synchronization of discrete-time stochastic complex networks with Markovian jumping and mode-dependent mixed time delays.

Pinning control is a technique that applies controllers to only a small fraction of the nodes in a network, and the technique is important because it greatly reduces the number of controlled nodes for real-world complex networks (which, in most cases, is huge). In fact, pinning control can be so effective for some networks that a single pinning controller is required for synchronization, namely, for complex networks that have either a symmetric or an asymmetric coupling matrix (Chen et al. [17]). Other pinning schemes, on the other hand, are capable of globally and exponentially stabilizing a network onto a homogeneous state by using an optimal combination of the number of pinned nodes and the feedback control gain (Zhao et al. [18]). The dependence of the number of pinned nodes on the coupling strength, indeed, is also known for networks with a fixed network structure (Zhou et al. [19] and Zhao et al. [20]).

In this paper, we study the mean-square exponential synchronization of stochastic time-varying delayed complex networks with Markovian switching by using the pinning control method. We consider a stochastic complex network with internal time-varying delayed couplings, Markovian switching, and Wiener processes. We prove some sufficient conditions for mean-square exponential synchronization of these networks by applying the Lyapunov-Krasovskii functional method and the linear matrix inequality (LMI).

This paper is organized as follows. In Section 2, we introduce the general model for a stochastic complex network with time-varying delayed dynamical nodes and Markovian switching coupling. We also write down some preliminary definitions and theorems that will be needed for the rest of the paper. In Section 3, we establish some exponential synchronization criteria for such complex dynamical networks, and, in Section 4, we discuss a numerical example of the theoretical results. The paper concludes in Section 5.

2. Preliminaries

2.1. Notations

Throughout this paper, 𝑛 shall denote the 𝑛-dimensional Euclidean space and 𝑛×𝑛 the set of all 𝑛×𝑛 real matrices. The superscript 𝑇 shall denote the transpose of a matrix or a vector, Tr() the trace of the corresponding matrix 𝐴𝑠=(𝐴+𝐴𝑇)/2 and 𝟏𝑛=(1,1,,1)𝑇𝑛 and 𝐼𝑛 the 𝑛-dimensional identity matrix. For square matrices 𝑀, the notation 𝑀>0 (resp., <0) shall mean that 𝑀 is a positive-definite (resp., negative-definite) matrix and 𝜆max(𝐴), and 𝜆min(𝐴) shall denote the greatest and least eigenvalues of a symmetric matrix, respectively.

Let (Ω,,{𝑡}𝑡0,𝒫) be a complete probability space with a filtration {𝑡}𝑡0 that is right continuous with 0 containing all the 𝒫-null sets. 𝐶([𝜏,0];𝑛) shall denote the family of continuous functions 𝜙 from [𝜏,0] to 𝑛 with the uniform norm 𝜙=sup𝜏𝑠0|𝜙(𝑠)| and 𝐶20([𝜏,0];𝑛) the family of all 0 measurable, 𝐶([𝜏,0];𝑛)-valued stochastic variables 𝜉={𝜉(𝜃)𝜏𝜃0} such that 0𝜏𝔼|𝜉(𝑠)|2𝑑𝑠, where 𝔼 stands for the correspondent expectation operator with respect to the given probability measure 𝒫.

Consider a complex network consisting of 𝑁 identical nodes with nondelayed and time-varying delayed linear coupling and Markovian switching𝑑𝑥𝑖𝑓(𝑡)=𝑡,𝑥𝑖(𝑡),𝑥𝑖+(𝑡𝜏(𝑡))𝑁𝑗=1,𝑖𝑗𝑎𝑖𝑗𝑟𝑎𝑖𝑗Σ𝑥(𝑡)𝑗(𝑡)𝑥𝑖+(𝑡)𝑁𝑗=1,𝑖𝑗𝑏𝑖𝑗𝑟𝑏𝑖𝑗Σ𝑥(𝑡)𝑗𝑡𝜏𝑐(𝑡)𝑥𝑖𝑡𝜏𝑐(𝑡)𝑑𝑡+𝜎𝑖𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐(𝑡),𝑟𝜎𝑖(𝑡)𝑑𝑤𝑖(𝑡),𝑖=1,2,,𝑁,(2.1) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),,𝑥𝑖𝑛(𝑡))𝑇𝑛 is the state vector of the 𝑖th node of the network, 𝑓(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏(𝑡)))=[𝑓1(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏(𝑡))),𝑓2(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏(𝑡))),,𝑓𝑛(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏(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 𝑟𝑎𝑖𝑗(𝑡), 𝑟𝑏𝑖𝑗(𝑡) and 𝑟𝜎𝑖(𝑡) are the continuous-time Markov processes that describe the evolution of the modes at time 𝑡. Here, 𝐴(𝑟𝑎(𝑡))=[𝑎𝑖𝑗(𝑟𝑎𝑖𝑗(𝑡))]𝑛×𝑛 and 𝐵(𝑟𝑏(𝑡))=[𝑏𝑖𝑗(𝑟𝑏𝑖𝑗(𝑡))]𝑛×𝑛 are the outer coupling matrices of the network at time 𝑡 at nodes 𝑟𝑎𝑖𝑗(𝑡), 𝑡𝜏𝑐(𝑡) and 𝑟𝑏𝑖𝑗(𝑡), respectively, such that 𝑎𝑖𝑗(𝑟𝑎𝑖𝑗(𝑡))0 for 𝑖𝑗, 𝑎𝑖𝑖(𝑟𝑎𝑖𝑖(𝑡))=𝑁𝑗=𝑖,𝑗𝑖𝑎𝑖𝑗(𝑟𝑎𝑖𝑗(𝑡)), 𝑏𝑖𝑗(𝑟𝑏𝑖𝑗(𝑡))0 for 𝑖𝑗, and 𝑏𝑖𝑖(𝑟𝑏𝑖𝑖(𝑡))=𝑁𝑗=𝑖,𝑗𝑖𝑏𝑖𝑗(𝑟𝑏𝑖𝑗(𝑡)). 𝜏(𝑡) is the inner time-varying delay satisfying 𝜏𝜏(𝑡)0 and 𝜏𝑐(𝑡) is the coupling time-varying delay satisfying 𝜏𝑐𝜏𝑐(𝑡)0. Finally, 𝜎𝑖(𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥(𝑡𝜏𝑐(𝑡)),𝑟𝜎𝑖(𝑡))=𝜎𝑖(𝑡,𝑥1(𝑡),,𝑥𝑛(𝑡),𝑥1(𝑡𝜏(𝑡)),,𝑥𝑛(𝑡𝜏(𝑡)),𝑥1(𝑡𝜏𝑐(𝑡)),,𝑥𝑛𝜏(𝑡𝑐(𝑡)),𝑟𝜎𝑖(𝑡))𝑛×𝑛 and 𝑤𝑖(𝑡)=(𝑤𝑖1(𝑡),𝑤𝑖2(𝑡),,𝑤𝑖𝑛(𝑡))𝑇𝑛 is a bounded vector-form Weiner process, satisfying𝔼𝑤𝑖𝑗(𝑡)=0,𝔼𝑤2𝑖𝑗(𝑡)=1,𝔼𝑤𝑖𝑗(𝑡)𝑤𝑖𝑗(𝑠)=0(𝑠𝑡).(2.2) In this paper, 𝐴(𝑟𝑎(𝑡)) is assumed to be irreducible in the sense that there are no isolated nodes.

The initial conditions associated with (2.1) are𝑥𝑖(𝑠)=𝜉𝑖(𝑠),̌𝜏𝑠0,𝑖=1,2,,𝑁,(2.3) where ̌𝜏=max{𝜏(𝑡),𝜏𝑐(𝑡)}, 𝜉𝑖𝐶𝑏0([̌𝜏,0],𝑛) with the norm 𝜉𝑖2=sup̌𝜏𝑠0𝜉𝑖(𝑠)𝑇𝜉𝑖(𝑠), and our objective is to control system (2.1) so that it stays in the trajectory 𝑠(𝑡)𝑛 of the system𝑑𝑠(𝑡)=𝑓(𝑡,𝑠(𝑡),𝑠(𝑡𝜏(𝑡)))𝑑𝑡(2.4) by adding pinning controllers to some of the nodes. Without loss of generality, let the first 𝑙 nodes be controlled. Then the network is described by𝑑𝑥𝑖𝑓(𝑡)=𝑡,𝑥𝑖(𝑡),𝑥𝑖+(𝑡𝜏(𝑡))𝑁𝑗=1,𝑖𝑗𝑎𝑖𝑗𝑟𝑎𝑖𝑗Σ𝑥(𝑡)𝑗(𝑡)𝑥𝑖+(𝑡)𝑁𝑗=1,𝑖𝑗𝑏𝑖𝑗𝑟𝑏𝑖𝑗Σ𝑥(𝑡)𝑗𝑡𝜏𝑐(𝑡)𝑥𝑖𝑡𝜏𝑐(𝑡)+𝑢𝑖(𝑡)𝑑𝑡+𝜎𝑖𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐(𝑡),𝑟𝜎𝑖(𝑡)𝑑𝑤𝑖(𝑡),𝑖=1,2,,𝑁,(2.5) where 𝑢𝑖(𝑡)  (𝑖=1,2,,𝑁) are the linear state feedback controllers that are defined by𝑢𝑖(𝑡)=𝜀𝑖𝑥𝑖(𝑡)𝑠(𝑡),𝑖=1,2,,𝑙,0,𝑖=𝑙+1,𝑙+2,,𝑁,(2.6) where 𝜀𝑖>0 (𝑖=1,2,,𝑙) are the control gains, denoted by Ξ=diag{𝜀1,𝜀2,,𝜀𝑙,0,,0}𝑛×𝑛. Define 𝑒𝑖(𝑡)=𝑥𝑖(𝑡)𝑠(𝑡) (𝑖=1,2,,𝑁) as the synchronization error. Then, according to the controller (2.6), the error system is𝑑𝑒𝑖𝑓(𝑡)=𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏(𝑡))𝑓𝑡,𝑠𝑖(𝑡),𝑠𝑖+(𝑡𝜏(𝑡))𝑁𝑗=1,𝑖𝑗𝑎𝑖𝑗𝑟𝑎𝑖𝑗Σ𝑒(𝑡)𝑗(𝑡)𝑒𝑖+(𝑡)𝑁𝑗=1,𝑖𝑗𝑏𝑖𝑗𝑟𝑏𝑖𝑗Σ𝑒(𝑡)𝑗𝑡𝜏𝑐(𝑡)𝑒𝑖𝑡𝜏𝑐(𝑡)+𝑢𝑖(𝑡)𝑑𝑡+𝜎𝑖𝑡,𝑒(𝑡),𝑒(𝑡𝜏(𝑡)),𝑒𝑡𝜏𝑐(𝑡),𝑟𝜎𝑖(𝑡)𝑑𝑤𝑖(𝑡),𝑖=1,2,,𝑁.(2.7)

Remark 2.1. Since the Markov chains 𝑟𝑎𝑖𝑗(𝑡), 𝑟𝑏𝑖𝑗(𝑡), and 𝑟𝜎𝑖(𝑡) are independent, we have an equivalent system as follows.
Let 𝑟(𝑡), 𝑡>0 be a right-continuous Markov chain on a probability space that takes values in a finite state space 𝑆=1,2,,𝑀 with a generator Γ=[𝛾𝑖𝑗]𝑀×𝑀 given by 𝛾𝑃{𝑟(𝑡+Δ)=𝑗𝑟(𝑡)=𝑖}=𝑖𝑗Δ+𝑜(Δ)if𝑖𝑗,1+𝛾𝑖𝑖Δ+𝑜(Δ)if𝑖=𝑗,(2.8) for some Δ>0. Here 𝛾𝑖𝑗=0 is the transition rate from 𝑖 to 𝑗 if 𝑖𝑗 and 𝛾𝑖𝑖=𝑖𝑗𝛾𝑖𝑗, 𝑑𝑒𝑖𝑓(𝑡)=𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏(𝑡))𝑓𝑡,𝑠𝑖(𝑡),𝑠𝑖+(𝑡𝜏(𝑡))𝑁𝑗=1𝑎𝑖𝑗(𝑟(𝑡))Σ𝑒𝑗+(𝑡)𝑁𝑗=1𝑏𝑖𝑗(𝑟(𝑡))Σ𝑒𝑗𝑡𝜏𝑐(𝑡)+𝑢𝑖(𝑡)𝑑𝑡+𝜎𝑖𝑡,𝑒(𝑡),𝑒(𝑡𝜏(𝑡)),𝑒𝑡𝜏𝑐(𝑡),𝑟(𝑡)𝑑𝑤𝑖(𝑡),𝑖=1,2,,𝑁.(2.9)

Definition 2.2. The complex network (2.5) is said to be exponentially synchronized in mean square if the trivial solution of system (2.9) is such that 𝑁𝑖=1𝔼𝑒𝑖𝑡,𝑡0,𝜉𝑖2𝐾𝑒𝜅𝑡,(2.10) for some 𝐾>0 and some 𝜅>0 for any initial data 𝜉𝑖𝒞𝑏0([𝜏,0];𝑛).

Definition 2.3 (see [12]). A continuous function 𝑓(𝑡,𝑥,𝑦)[0,+]×𝑛×𝑛𝑛 is said to belong to the function class QUAD, denoted by 𝑓QUAD(𝑃,Δ,𝜂,𝜃) for some given matrix Σ=diag{𝜚1,𝜚2,,𝜚𝑛} if there exists a positive definite diagonal matrix 𝑃=diag{𝑝1,𝑝2,,𝑝𝑛}, a diagonal matrix Δ=diag{𝛿1,𝛿2,,𝛿𝑛}, and a constant 𝜂>0,𝜃>0 such that 𝑓() satisfies the condition (𝑥𝑦)𝑇𝑃((𝑓(𝑡,𝑥,𝑧)𝑓(𝑡,𝑦,𝑤))ΔΣ(𝑥𝑦))𝜂(𝑥𝑦)𝑇(𝑥𝑦)+𝜃(𝑧𝑤)𝑇(𝑧𝑤)(2.11) for all 𝑥,𝑦,𝑧,𝑤𝑛.

Remark 2.4. 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 𝑝̌𝑝=max1,𝑝2,,𝑝𝑛𝑝,̂𝑝=min1,𝑝2,,𝑝𝑛,̌𝛿𝛿=max1,𝛿2,,𝛿𝑛.(2.12)

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 .𝜏𝑐(𝑡)<𝜏𝑐<1. Let ̌𝜏=max{𝜏,𝜏𝑐}.(H2) Let 𝜎(𝑡,𝑒(𝑡),𝑒(𝑡𝜏(𝑡)),𝑒(𝑡𝜏𝑐(𝑡)),𝑟) = 𝜎(𝑡,𝑒1(𝑡),,𝑒𝑁(𝑡),𝑒1(𝑡𝜏(𝑡)),,𝑒𝑁(𝑡𝜏(𝑡)),𝑒1(𝑡𝜏𝑐(𝑡)),,𝑒𝑁(𝑡𝜏𝑐(𝑡)),𝑟). Then there exist positive definite constant matrices Υ𝑟𝑖1, Υ𝑟𝑖2, and Υ𝑟𝑖3 for 𝑖=1,2,,𝑁 and 𝑟=1,2,,𝑀 such that𝜎Tr𝑖𝑡,𝑒(𝑡),𝑒(𝑡𝜏(𝑡)),𝑒𝑡𝜏𝑐(𝑡),𝑟𝑇𝜎𝑖𝑡,𝑒(𝑡),𝑒(𝑡𝜏(𝑡)),𝑒𝑡𝜏𝑐(𝑡),𝑟𝑁𝑗=1𝑒𝑗(𝑡)𝑇Υ𝑟𝑖1𝑒𝑗(𝑡)+𝑁𝑗=1𝑒𝑗(𝑡𝜏(𝑡))𝑇Υ𝑟𝑖2𝑒𝑗(𝑡𝜏(𝑡))+𝑁𝑗=1𝑒𝑗𝑡𝜏𝑐(𝑡)𝑇Υ𝑟𝑖3𝑒𝑗𝑡𝜏𝑐.(𝑡)(2.13)

Lemma 2.5 (see [32, 33, the generalized Itô formula]). Consider a stochastic delayed differential equation with Markovian switching of the form 𝑑𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡𝜏),𝑟(𝑡))𝑑𝑡+𝜎(𝑡,𝑥(𝑡),𝑥(𝑡𝜏),𝑟(𝑡))𝑑𝜔(𝑡)(2.14) on 𝑡0 with initial value 𝑥0=𝜉𝐶𝑏𝐹0([𝜏,0];𝑛), where 𝑓𝑛×+×𝑆𝑛,𝜎𝑛×+×𝑆𝑛×𝑚.(2.15) Let 𝐶2,1(+×𝑛;+) be the family of all the nonnegative functions 𝑉(𝑡,𝑥,𝑟) on +×𝑛×𝑆 that are twice continuously differentiable in 𝑥 and once differentiable in 𝑡. Let 𝑉𝐶2,1(+×𝑛×𝑆;+). Define an operator 𝑉 from 𝑛×+×𝑆 to 𝑛 by 𝑉(𝑡,𝑥,𝑟)=𝑉𝑡(𝑡,𝑥,𝑟)+𝑉𝑥1(𝑡,𝑥,𝑟)𝑓(𝑡,𝑥,𝑟)+2Tr𝜎(𝑡,𝑥,𝑟)𝑇𝑉𝑥𝑥+𝜎(𝑡,𝑥,𝑟)𝑀𝑗=1𝛾𝑖𝑗𝑉(𝑡,𝑥,𝑗),(2.16) where 𝑉𝑡(𝑡,𝑥,𝑟)=𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑡, 𝑉𝑥(𝑡,𝑥,𝑟)=(𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑥1,,𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑥𝑛), 𝑉𝑥𝑥(𝑡,𝑥,𝑟)=(𝜕2𝑉(𝑡,𝑥,𝑟)/𝜕𝑥𝑖𝑥𝑗)𝑛×𝑛. If 𝑉𝐶2,1(+×𝑛×𝑆;+), then 𝑡𝔼𝑉(𝑡,𝑥(𝑡),𝑟)=𝔼𝑉0𝑡,𝑥0,𝑟+𝔼𝑡𝑡0𝑉(𝑠,𝑥(𝑠),𝑟)𝑑𝑠,(2.17) for all >𝑡>𝑡00, as long as the expectations of the integrals exist.

3. Main Result

Theorem 3.1. Let assumptions (H1) and (H2) be true and let 𝑓QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼𝑟 and 𝛽𝑟 such that 𝐴(𝑟)𝑠+̌𝛿𝐼𝑁Ξ+𝛼𝑟𝐼𝑁𝐵(𝑟)2𝐵(𝑟)𝑇2𝛽𝑟𝐼𝑁̌0,for𝑟=1,2,,𝑀,(3.1)̌𝜏𝜃𝑇,̌𝜏(1𝜃)𝑇,0̌𝜏1̌𝑞𝑏+̌𝑐,11+𝜃(3.2)𝑏1+𝑐1,1𝑏2+𝑐21,,𝑏𝑀+𝑐𝑀>Γ1𝟏𝑀,(3.3) where 𝛾>0 is the greatest root of the equation ̌̌𝑞𝛾(1+𝜃)+𝑏̌𝑞1𝜏𝑒𝛾𝜏+̌𝑐̌𝑞1𝜏𝑐𝑒𝛾𝜏𝑐𝑎=0,(3.4)Γ=diag1,𝑎2,,𝑎𝑀𝑎Γ,𝑟=𝜆min2𝜂𝐼𝑛̌𝑝𝑁𝑖=1Υ𝑟𝑖1+2𝛼1𝑃Σ̌𝑝,̌𝑎=max𝑟𝑆𝑎𝑟,𝑏𝑟=𝜆max𝑁𝑖=1𝑃Υ𝑟𝑖2+2𝜁𝐼𝑁,̌̂𝑝𝑏=max𝑟𝑆𝑏𝑟,𝑐𝑟=𝜆max𝑁𝑖=1𝑃Υ𝑟𝑖3+2𝛽2𝑃Σ̂𝑝,̌𝑐=max𝑟𝑆𝑐𝑟.(3.5) Then the solutions 𝑥1(𝑡), 𝑥1(𝑡), and 𝑥𝑁(𝑡) of system (2.9) are globally and exponentially stable.

Proof. By (3.3), there exists a sufficiently small constant 𝜃>0 such that 1𝑏1+𝑐1,1𝑏2+𝑐21,,𝑏𝑀+𝑐𝑀Γ(1+𝜃)1𝟏.(3.6) Set Γ(1+𝜃)1𝟏=𝑞=(𝑞1,𝑞2,,𝑞𝑀)𝑇. Then Γ𝑞=(1+𝜃)𝟏𝑀,(3.7) that is, 𝑏𝑟+𝑐𝑟𝑞𝑟1,𝑎𝑟𝑞𝑟𝑀𝑠=1𝛾𝑠𝑟𝑞𝑟=1+𝜃.(3.8) For 1𝑟𝑀, define the Lyapunov-Krasovskii function 𝑉(𝑡,𝑒(𝑡),𝑟)=𝑞𝑟12𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)(3.9) and let ̃𝑒𝑘(𝑡)=(𝑒1𝑘(𝑡),𝑒2𝑘(𝑡),,𝑒𝑁𝑘(𝑡))𝑇, 𝑘=1,2,,𝑛. By Lemma 2.5, we have 𝑉(𝑡,𝑒(𝑡),𝑟)=𝑞𝑟𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑓𝑡,𝑥𝑖(𝑡),𝑥𝑖+(𝑡𝜏(𝑡))𝑓(𝑡,𝑠(𝑡),𝑠(𝑡𝜏(𝑡)))𝑁𝑗=1𝑎𝑖𝑗(𝑟)Σ𝑒𝑗(𝑡)+𝑁𝑗=1𝑏𝑖𝑗(𝑟)Σ𝑒𝑗𝑡𝜏𝑐(𝑡)+𝑢𝑖+1(𝑡)2𝑞𝑟𝑁𝑖=1𝜎Tr𝑖𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐(𝑡),𝑟𝑇𝑃𝜎𝑖×𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐+(𝑡),𝑟𝑀𝑠=1𝛾𝑟𝑠𝑞𝑠12𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)=𝑞𝑟𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑓𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏(𝑡))𝑓(𝑡,𝑠(𝑡),𝑠(𝑡𝜏(𝑡)))ΔΣ𝑒𝑖(𝑡)+𝑞𝑟𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃ΔΣ𝑒𝑖(𝑡)+𝑞𝑟𝑁𝑁𝑖=1𝑗=1𝑎𝑖𝑗(𝑟)𝑒𝑖(𝑡)𝑇𝑃Σ𝑒𝑗(𝑡)+𝑞𝑟𝑁𝑁𝑖=1𝑗=1𝑏𝑖𝑗(𝑟)𝑒𝑖(𝑡)𝑇𝑃Σ𝑒𝑗𝑡𝜏𝑐(𝑡)𝑞𝑟𝑙𝑖=1𝜀𝑖𝑒𝑖(𝑡)𝑇𝑃Σ𝑒𝑖+1(𝑡)2𝑞𝑟𝜎Tr𝑖𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐(𝑡),𝑟𝑇𝑃𝜎𝑖×𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐+(𝑡),𝑟𝑀𝑠=1𝛾𝑟𝑠𝑞𝑠12𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)𝑞𝑟𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝜃𝑁𝑖=1𝑒𝑖(𝑡𝜏(𝑡))𝑇𝑒𝑖(𝑡𝜏(𝑡))+𝑛𝑘=1𝑝𝑘𝜚𝑘𝛿𝑘̃𝑒𝑘(𝑡)𝑇̃𝑒𝑘+(𝑡)𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇𝐴(𝑟)̃𝑒𝑘(𝑡)+𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇𝐵(𝑟)̃𝑒𝑘𝑡𝜏𝑐(𝑡)𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇Ξ̃𝑒𝑘+1(𝑡)2̌𝑝𝑁𝑗=1𝑁𝑖=1𝑒𝑖(𝑡)𝑇Υ𝑟𝑗1𝑒𝑖(𝑡)+𝑁𝑖=1𝑒𝑖(𝑡𝜏(𝑡))𝑇Υ𝑟𝑗2𝑒𝑖(+𝑡𝜏(𝑡))𝑁𝑖=1𝑒𝑖𝑡𝜏𝑐(𝑡)𝑇Υ𝑟𝑗3𝑒𝑖𝑡𝜏𝑐+(𝑡)𝑀𝑠=1𝛾𝑟𝑠𝑞𝑠12𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)=𝑞𝑟𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝜃𝑁𝑖=1𝑒𝑖(𝑡𝜏(𝑡))𝑇𝑒𝑖1(𝑡𝜏(𝑡))+2̌𝑝𝑁𝑁𝑖=1𝑗=1𝑒𝑖𝑡𝜏𝑐(𝑡)𝑇×Υ𝑟𝑗3𝑒𝑖𝑡𝜏𝑐+(𝑡)𝑀𝑠=1𝛾𝑟𝑠𝑞𝑠12𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+𝑞𝑟𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇𝐴(𝑟)̃𝑒𝑘(𝑡)+𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇𝐵(𝑟)̃𝑒𝑘𝑡𝜏𝑐(𝑡)𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇Ξ̃𝑒𝑘+1(𝑡)2̌𝑝𝑁𝑁𝑖=1𝑗=1𝑒𝑖(𝑡)𝑇Υ𝑟𝑗1𝑒𝑖(𝑡)+𝑁𝑁𝑖=1𝑗=1𝑒𝑖(𝑡𝜏(𝑡))𝑇Υ𝑟𝑗2𝑒𝑖+(𝑡𝜏(𝑡))𝑛𝑘=1𝑝𝑘𝜚𝑘𝛿𝑘̃𝑒𝑘(𝑡)𝑇̃𝑒𝑘(𝑡)=𝑞𝑟𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝜂𝐼𝑁+12̌𝑝𝑁𝑗=1Υ𝑟𝑗1𝛼𝑟𝑒𝑃Σ𝑖(𝑡)+𝑁𝑖=1𝑒𝑖(𝑡𝜏(𝑡))𝑇×1𝜃+2̌𝑝𝑁𝑗=1Υ𝑟𝑗2𝑒𝑖(𝑡𝜏(𝑡))+𝑁𝑖=1𝑒𝑖𝑡𝜏𝑐(𝑡)𝑇12̌𝑝𝑁𝑗=1Υ𝑟𝑗3+𝛽𝑟𝑃Σ×𝑒𝑖𝑡𝜏𝑐+(𝑡)𝑀𝑠=1𝛾𝑟𝑠𝑞𝑠12𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+𝑞𝑟𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇𝐴̌(𝑟)Ξ+𝛿+𝛼𝑟𝐼𝑁̃𝑒𝑘+(𝑡)𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘(𝑡)𝑇𝐵(𝑟)̃𝑒𝑘𝑡𝜏𝑐(𝑡)𝑛𝑘=1𝑝𝑘𝜚𝑘̃𝑒𝑘𝑡𝜏𝑐(𝑡)𝑇𝛽𝑟̃𝑒𝑘𝑡𝜏𝑐(𝑡)𝑞𝑟𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝜂𝐼𝑁+12̌𝑝𝑁𝑗=1Υ𝑟𝑗1𝛼𝑟𝑒𝑃Σ𝑖(𝑡)+𝑁𝑖=1𝑒𝑖(𝑡𝜏(𝑡))𝑇×𝛽𝑟𝐼𝑁+12̌𝑝𝑁𝑗=1Υ𝑟𝑗2𝑒𝑖(𝑡𝜏(𝑡))+𝑁𝑖=1𝑒𝑖𝑡𝜏𝑐(𝑡)𝑇12̌𝑝𝑁𝑗=1Υ𝑟𝑗3+𝛽𝑟𝑃Σ×𝑒𝑖𝑡𝜏𝑐+(𝑡)𝑀𝑠=1𝛾𝑟𝑠𝑞𝑠12𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡).(3.10) Let 1𝐸(𝑡)=2𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡).(3.11) Then we have 𝑉(𝑡)𝑎𝑟𝑞𝑟𝐸(𝑡)+𝑏𝑟𝑞𝑟𝐸(𝑡𝜏(𝑡))+𝑐𝑟𝑞𝑟𝐸𝑡𝜏𝑐+(𝑡)𝑀𝑠=1𝛾𝑟𝑠𝑞𝑠𝐸(𝑡),(3.12) and by (3.8), we have ̌𝑉(𝑡)(1+𝜃)𝐸(𝑡)+𝑏̌𝑞𝐸(𝑡𝜏(𝑡))+̌𝑐̌𝑞𝐸𝑡𝜏𝑐.(𝑡)(3.13) Define 𝑊(𝑡)=𝑒𝛾𝑡𝑉(𝑡)(3.14) and use (3.13) to compute the operator 𝑊(𝑡)=𝑒𝛾𝑡[]𝛾𝑉(𝑡)+𝑉(𝑡)𝑒𝛾𝑡̌𝛾̌𝑝𝐸(𝑡)(1+𝜃)𝐸(𝑡)+𝑏̌𝑞𝐸(𝑡𝜏(𝑡))+̌𝑐̌𝑞𝐸𝑡𝜏𝑐,(𝑡)(3.15) which, after applying the generalized Itô's formula, gives 𝑞𝑟𝑒𝛾𝑡𝔼𝐸(𝑡)=𝑞𝑟𝑒𝛾𝑡0𝑡𝔼𝐸0+𝔼𝑡𝑡0𝑊(𝑠)𝑑𝑠(3.16) for any 𝑡>𝑡00. Hence we have 𝑞𝑟𝑒𝛾𝑡𝔼𝐸(𝑡)̌𝑞𝔼𝐸(0)+𝔼𝑡0𝑒𝛾𝑠̌𝛾𝐸(𝑠)(1𝜃)𝐸(𝑠)+𝑏̌𝑞𝐸(𝑠𝜏(𝑠))+̌𝑐̌𝑞𝐸𝑠𝜏𝑐(𝑠)𝑑𝑠̌𝑝̌𝑞2𝑁𝑖=1𝔼𝜉𝑖2+(𝛾(1+𝜃))𝑡0𝑒𝛾𝑠̌𝔼𝐸(𝑠)𝑑𝑠+𝑏̌𝑞𝑒𝛾𝜏𝑡0𝑒𝛾(𝑠𝜏(𝑠))𝔼𝐸(𝑠𝜏(𝑠))𝑑𝑠+̌𝑐̌𝑞𝑒𝛾𝜏𝑐𝑡0𝑒𝛾(𝑠𝜏𝑐(𝑠))𝔼𝐸𝑠𝜏𝑐(𝑠)𝑑𝑠,(3.17) which, by using the change of variables 𝑠𝜏(𝑠)=𝑢, gives 𝑡0𝑒𝛾(𝑠𝜏(𝑠))𝔼𝐸(𝑠𝜏(𝑠))𝑑𝑠=𝑡𝜏(𝑡)𝜏(0)𝑒𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢.𝜏1(𝑡)̌𝑝𝜏21𝜏𝑁𝑖=1𝔼𝜉𝑖2+11𝜏𝑡0𝑒𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢,(3.18) and a further change of variables 𝑠𝜏𝑐(𝑠)=𝑢 gives 𝑡0𝑒𝑠𝜏𝑐(𝑠)𝔼𝐸𝑠𝜏𝑐(𝑠)𝑑𝑠=𝑡𝜏𝑐(𝑡)𝜏𝑐(0)𝑒𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢.𝜏1𝑐(𝑡)̌𝑝𝜏𝑐21𝜏𝑐𝑁𝑖=1𝔼𝜉𝑖2+11𝜏𝑐𝑡0𝑒𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢.(3.19) By Condition (3.4), we obtain 𝔼𝐸(𝑡)̌𝑝̌𝑞2̌1+𝑏̌𝑞𝜏1𝜏𝑒𝛾𝜏+̌𝑐̌𝑞𝜏𝑐1𝜏𝑐𝑒𝛾𝜏𝑐𝑁𝑖=1𝔼𝜉𝑖2𝑒𝛾𝑡(3.20) so that 𝔼𝑒(𝑡)2̌𝑝̌𝑞̌2̂𝑝1+𝑏̌𝑞𝜏1𝜏𝑒𝛾𝜏+̌𝑐̌𝑞𝜏𝑐1𝜏𝑐𝑒𝛾𝜏𝑐𝑁𝑖=1𝔼𝜉𝑖2𝑒𝛾𝑡.(3.21) The proof is hence complete.

When the time-varying delays are constant (i.e., 𝜏(𝑡)=𝜏, 𝜏𝑐(𝑡)=𝜏𝑐), we obtain the following corollary.

Corollary 3.2. Let assumptions (H1) and (H2) be true and let 𝑓QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼𝑟 and 𝛽𝑟 such that 𝐴(𝑟)𝑠+̌𝛿𝐼𝑁Ξ+𝛼𝑟𝐼𝑁𝐵(𝑟)2𝐵(𝑟)𝑇2𝛽𝑟𝐼𝑁10,for𝑟=1,2,,𝑀,̌𝜏𝜃𝑇,̌𝜏(1𝜃)𝑇,𝑏1+𝑐1,1𝑏2+𝑐21,,𝑏𝑀+𝑐𝑀>Γ1𝟏𝑀,(3.22) where 𝛾>0 is the greatest root of the equation ̌̌𝑞𝛾(1+𝜃)+𝑏̌𝑞𝑒𝛾𝜏+̌𝑐̌𝑞𝑒𝛾𝜏𝑐Γ𝑎=0,=diag1,𝑎2,,𝑎𝑀𝑎Γ,𝑟=𝜆min2𝜂𝐼𝑛̌𝑝𝑁𝑖=1Υ𝑟𝑖1+2𝛼1𝑃Σ̌𝑝,̌𝑎=max𝑟𝑆𝑎𝑟,𝑏𝑟=𝜆max𝑁𝑖=1𝑃Υ𝑟𝑖2+2𝜁𝐼𝑁,̌̂𝑝𝑏=max𝑟𝑆𝑏𝑟,𝑐𝑟=𝜆max𝑁𝑖=1𝑃Υ𝑟𝑖3+2𝛽2𝑃Σ̂𝑝,̌𝑐=max𝑟𝑆𝑐𝑟,(3.23) then the solutions 𝑥1(𝑡), 𝑥1(𝑡), and 𝑥𝑁(𝑡) of system (2.9) are globally and exponentially stable.

When 𝐴(𝑟(𝑡))=𝐴, 𝐵(𝑟(𝑡))=𝐵, and 𝜎𝑖(𝑡,𝑒(𝑡),𝑒(𝑡𝜏(𝑡)),𝑒(𝑡𝜏𝑐(𝑡)),𝑟(𝑡))=𝜎𝑖(𝑡,𝑒(𝑡),𝑒(𝑡𝜏(𝑡)),𝑒(𝑡𝜏𝑐(𝑡))), we can get the following corollary.

Corollary 3.3. Let assumptions (H1) and (H2) be true, and let 𝑓QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼𝑟 and 𝛽𝑟 such that 𝐴𝑠+̌𝛿𝐼𝑁Ξ+𝛼𝑟𝐼𝑁𝐵2𝐵𝑇2𝛽𝑟𝐼𝑁̌0,̌𝜏𝜃𝑇,̌𝜏(1𝜃)𝑇,0̌𝜏1𝑏+̌𝑐,1+𝜃(3.24) where 𝑎𝑟=𝜆min2𝜂𝐼𝑛̌𝑝𝑁𝑖=1Υ𝑟𝑖1+2𝛼1𝑃Σ̌𝑝,̌𝑎=max𝑟𝑆𝑎𝑟,𝑏𝑟=𝜆max𝑁𝑖=1𝑃Υ𝑟𝑖2+2𝜁𝐼𝑁,̌̂𝑝𝑏=max𝑟𝑆𝑏𝑟,𝑐𝑟=𝜆max𝑁𝑖=1𝑃Υ𝑟𝑖3+2𝛽2𝑃Σ̂𝑝,̌𝑐=max𝑟𝑆𝑐𝑟,(3.25) then the solutions 𝑥1(𝑡), 𝑥1(𝑡), and 𝑥𝑁(𝑡) of system (2.9) are globally and exponentially stable.

4. Numerical Simulation

In this section, we present some numerical simulation results that validate the theorem of the previous section.

Consider the chaotic delayed neural network[]𝑑𝑠(𝑡)=𝐶𝑠(𝑡)+𝐴𝑓(𝑠(𝑡))+𝐵𝑔(𝑠(𝑡𝜏(𝑡)))𝑑𝑡,(4.1) where 𝑓(𝑠)=𝑔(𝑠)=tanh(𝑠), 𝜏(𝑡)=1,𝐶=1001,𝐴=20.154.5,𝐵=1.50.10.24.(4.2) Taking 𝑃=diag{1,2} and Δ=diag{5,11,5}, we have 𝜂=0.15 and 𝜃=3.25 so that Condition (2.11) is satisfied. Thus𝑑𝑥𝑖𝑓(𝑡)=𝑡,𝑥𝑖(𝑡),𝑥𝑖+(𝑡𝜏(𝑡))5𝑗=1𝑎𝑟𝑖𝑗Σ𝑥𝑗(𝑡)+5𝑗=1𝑏𝑟𝑖𝑗Σ𝑥𝑗𝑡𝜏𝑐(𝑡)𝜀𝑖𝑥𝑖(𝑡)𝑠(𝑡)𝑑𝑡+𝜎𝑟𝑖𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐(𝑡)𝑑𝑤𝑖(𝑡),𝑖=1,2,,5,𝑟=1,2,3,(4.3)𝐴1=𝑎1𝑖𝑗=203111014111114111114101113,𝐴2=𝑎2𝑖𝑗,𝐴=1541111131101141111141101133=𝑎3𝑖𝑗=372110013110113010101000101,𝐵1=𝑏1𝑖𝑗𝐵=0.121100021100121001120011022=𝑏2𝑖𝑗=0.21001001010011000101000011,𝐵3=𝑏3𝑖𝑗=0.71010011000001100011010001Ξ=1001000001000000000000000000,Γ=312242235,𝜏𝑐𝑒(𝑡)=0.1𝑡1+𝑒𝑡𝜎1𝑖𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐𝑥(𝑡)=0.1diag𝑖1(𝑡)𝑥𝑖+1,1(𝑡),𝑥𝑖2(𝑡)𝑥𝑖+1,2,𝜎(𝑡)2𝑖𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥𝑡𝜏𝑐𝑥(𝑡)=0.1diag𝑖1(𝑡𝜏(𝑡))𝑥𝑖+1,1(𝑡𝜏(𝑡)),𝑥𝑖2(𝑡𝜏(𝑡))𝑥𝑖+1,2(,𝜎𝑡𝜏(𝑡))3𝑖𝑥(𝑡,𝑥(𝑡),𝑥(𝑡𝜏(𝑡)),𝑥(𝑡𝜏(𝑡)))=0.1diag𝑖1𝑡𝜏𝑐(𝑡)𝑥𝑖+1,1𝑡𝜏𝑐(𝑡),𝑥𝑖2𝑡𝜏𝑐(𝑡)𝑥𝑖+1,2𝑡𝜏𝑐.(𝑡)(4.4) Computations then yield 𝜏=1, 𝜏=0, 𝜏𝑐=0.1, 𝜏𝑐=0.1, Υ𝑖𝑗=0.01𝐼2 for 𝑖=1,2,,𝑁, and 𝑗=1,2. Let 𝐽={1,2} and the control strength 𝜀𝑖=100 for 𝑖=1,2. Then the solutions of (3.1)–(3.3) are (by using the MATLAB LMI toolbox): 𝛼1=3.5000, 𝛽1=0.0020, 𝑎1=7.1351, 𝑏1=6.5300, 𝑐1=0.0379; 𝛼2=3.6000, 𝛽2=0.0088, 𝑎2=7.3351, 𝑏2=6.5300, 𝑐2=0.0651; 𝛼3=3.7000, 𝛽3=0.0852, 𝑎3=7.5351, 𝑏3=6.5300, 𝑐3=0.3709. So 𝜃=0.003, andΓ1=0.105260.0143370.0190820.0224860.0951740.0187730.0221760.0250650.087313.(4.5) Therefore[]𝑞=0.1391,0.1368,0.1350𝑇,(4.6) and, after solving (3.4), we obtain 𝛾=0.0349.

The initial conditions for this simulation are 𝑥𝑖𝑗(𝑡0)=6+2𝑖+𝑗, 𝑖=1,2,,5, 𝑗=1,2 and 𝑠(𝑡0)=[5,4]𝑇 for all 𝑡0[1,0] and the trajectories of the periodically intermittent pinning control gains are shown in Figure 1. Figure 2 shows the time evolution of the synchronization errors with periodically intermittent pinning control.

fig1
Figure 1: The trajectories of the state variables of 𝑥𝑖1 and 𝑥𝑖2 (𝑖=1,2,,5) in system (4.3) under pinning control.
fig2
Figure 2: The time evolution of 𝑒𝑖1 and 𝑒𝑖2 (𝑖=1,2,,5) in system (4.3) under pinning control.

Next, let 𝑡𝑐(𝑡)=0.1, we get the 𝛾=0.0402. The initial conditions for this simulation are 𝑥𝑖𝑗(𝑡0)=6+2𝑖+𝑗, 𝑖=1,2,,5, 𝑗=1,2 and 𝑠(𝑡0)=[5,4]𝑇 for all 𝑡0[1,0] and the trajectories of the periodically intermittent pinning control gains are shown in Figure 3. Figure 4 shows the time evolution of the synchronization errors with periodically intermittent pinning control.

fig3
Figure 3: The trajectories of the state variables of 𝑥𝑖1 and 𝑥𝑖2 (𝑖=1,2,,5) in system (4.3) under pinning control.
fig4
Figure 4: The time evolution of 𝑒𝑖1 and 𝑒𝑖2 (𝑖=1,2,,5) in system (4.3) under pinning control.

5. Conclusion

In this paper, we investigated the synchronization problem for stochastic complex networks with Markovian switching and nondelayed and time-varying delayed couplings. Specifically, we achieved global exponential synchronization by applying a pinning control scheme to a small fraction of the nodes and derived sufficient conditions for global exponential stability of synchronization in mean square. Finally, we considered some numerical examples that illustrate the theoretical analysis.

Acknowledgments

This work was supported by the National Science Foundation of China under Grant no. 61070087, the Guangdong Education University Industry Cooperation Project (2009B090300355), and the Shenzhen Basic Research Project (JC201006010743A, JC200903120040A). The research of the authors is partially supported by the Hong Kong Polytechnic University Grant G-U996 and Hong Kong Government GRF Grant B-Q21F.

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