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

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 [2–6]).

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 [17–20].

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 ([21–27]), 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 ([28–30]). 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 ([32–37]), and Mao [32, 33] considered the stability of stochastic delayed differential equations using this model while others ([34–36]) 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, ℝ^𝑛 shall denote the 𝑛-dimensional Euclidean space and ℝ^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 𝐴^𝑠=(𝐴+𝐴^𝑇)/2 and 𝟏_𝑛=(1,1,…,1)^𝑇∈ℝ^𝑛 and 𝐼_𝑛 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 𝜆_max(𝐴), and 𝜆_min(𝐴) shall denote the greatest and least eigenvalues of a symmetric matrix, respectively.

Let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝒫) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 that is right continuous with ℱ₀ containing all the 𝒫-null sets. 𝐶([−𝜏,0];ℝ^𝑛) shall denote the family of continuous functions 𝜙 from [−𝜏,0] to ℝ^𝑛 with the uniform norm ‖𝜙‖=sup_{−𝜏≤𝑠≤0}|𝜙(𝑠)| and 𝐶^2ℱ₀([−𝜏,0];ℝ^𝑛) the family of all ℱ₀ measurable, 𝐶([−𝜏,0];ℝ^𝑛)-valued stochastic variables 𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0} such that ∫^0−𝜏𝔼|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 switching𝑑𝑥_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=_{1,𝑖≠𝑗}+𝑎_𝑖𝑗𝑟^𝑎_𝑖𝑗(𝑡)Σ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)_𝑁_𝑗=_{1,𝑖≠𝑗}+𝜎_𝑖𝑏_𝑖𝑗𝑟^𝑏_𝑖𝑗(𝑡)Σ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)−𝑥_𝑖𝑡−𝜏_𝑐(𝑡)𝑑𝑡𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟_𝜎𝑖(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,𝑁,(2.1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector of the 𝑖th node of the network, 𝑓(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))=[𝑓₁(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),𝑓₂(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),…,𝑓_𝑛(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))]^𝑇 is a continuous vector-valued function, Σ=diag(𝜚₁,𝜚₂,…,𝜚_𝑛) is an inner coupling of the networks that satisfies 𝜚_𝑗>0, 𝑗=1,2,…,𝑛, and 𝑟_𝑎𝑖𝑗(𝑡), 𝑟_𝑏𝑖𝑗(𝑡) and 𝑟_𝜎𝑖(𝑡) are the continuous-time Markov processes that describe the evolution of the modes at time 𝑡. Here, 𝐴(𝑟_𝑎(𝑡))=[𝑎_𝑖𝑗(𝑟_𝑎𝑖𝑗(𝑡))]∈ℝ^𝑛×𝑛 and 𝐵(𝑟_𝑏(𝑡))=[𝑏_𝑖𝑗(𝑟_𝑏𝑖𝑗(𝑡))]∈ℝ^𝑛×𝑛 are the outer coupling matrices of the network at time 𝑡 at nodes 𝑟_𝑎𝑖𝑗(𝑡), 𝑡−𝜏_𝑐(𝑡) and 𝑟_𝑏𝑖𝑗(𝑡), respectively, such that 𝑎_𝑖𝑗(𝑟_𝑎𝑖𝑗(𝑡))≥0 for 𝑖≠𝑗, 𝑎_𝑖𝑖(𝑟_𝑎𝑖𝑖(𝑡))=−∑^{𝑁𝑗=𝑖,𝑗≠𝑖}𝑎_𝑖𝑗(𝑟_𝑎𝑖𝑗(𝑡)), 𝑏_𝑖𝑗(𝑟_𝑏𝑖𝑗(𝑡))≥0 for 𝑖≠𝑗, and 𝑏_𝑖𝑖(𝑟_𝑏𝑖𝑖(𝑡))=−∑^{𝑁𝑗=𝑖,𝑗≠𝑖}𝑏_𝑖𝑗(𝑟_𝑏𝑖𝑗(𝑡)). 𝜏(𝑡) is the inner time-varying delay satisfying 𝜏≥𝜏(𝑡)≥0 and 𝜏_𝑐(𝑡) is the coupling time-varying delay satisfying 𝜏_𝑐≥𝜏_𝑐(𝑡)≥0. Finally, 𝜎𝜏_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏_𝑐(𝑡)),𝑟_𝜎𝑖(𝑡))=𝜎_𝑖(𝑡,𝑥₁(𝑡),…,𝑥_𝑛(𝑡),𝑥₁(𝑡−𝜏(𝑡)),…,𝑥_𝑛(𝑡−𝜏(𝑡)),𝑥₁(𝑡−𝜏_𝑐(𝑡)),…,𝑥_𝑛(𝑡−_𝑐(𝑡)),𝑟_𝜎𝑖(𝑡))∈ℝ^𝑛×𝑛 and 𝑤_𝑖(𝑡)=(𝑤_𝑖1(𝑡),𝑤_𝑖2(𝑡),…,𝑤_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 is a bounded vector-form Weiner process, satisfying𝔼𝑤_𝑖𝑗(𝑡)=0,𝔼𝑤^2𝑖𝑗(𝑡)=1,𝔼𝑤_𝑖𝑗(𝑡)𝑤_𝑖𝑗(𝑠)=0(𝑠≠𝑡).(2.2) In this paper, 𝐴(𝑟_𝑎(𝑡)) is assumed to be irreducible in the sense that there are no isolated nodes.

The initial conditions associated with (2.1) are𝑥_𝑖(𝑠)=𝜉_𝑖(𝑠),−𝜏̌≤𝑠≤0,𝑖=1,2,…,𝑁,(2.3) where 𝜏̌=max{𝜏(𝑡),𝜏_𝑐(𝑡)}, 𝜉_𝑖∈𝐶^𝑏ℱ₀([−𝜏̌,0],ℝ^𝑛) with the norm ‖𝜉_𝑖‖²=sup_{−̌𝜏≤𝑠≤0}𝜉_𝑖(𝑠)^𝑇𝜉_𝑖(𝑠), and our objective is to control system (2.1) so that it stays in the trajectory 𝑠(𝑡)∈ℝ^𝑛 of the system𝑑𝑠(𝑡)=𝑓(𝑡,𝑠(𝑡),𝑠(𝑡−𝜏(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 by𝑑𝑥_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=_{1,𝑖≠𝑗}+𝑎_𝑖𝑗𝑟^𝑎_𝑖𝑗(𝑡)Σ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)_𝑁_𝑗=_{1,𝑖≠𝑗}+𝜎_𝑖𝑏_𝑖𝑗𝑟^𝑏_𝑖𝑗(𝑡)Σ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)−𝑥_𝑖𝑡−𝜏_𝑐(𝑡)+𝑢_𝑖(𝑡)𝑑𝑡𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟_𝜎𝑖(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,𝑁,(2.5) where 𝑢_𝑖(𝑡)  (𝑖=1,2,…,𝑁) are the linear state feedback controllers that are defined by𝑢_𝑖(𝑡)=−𝜀^𝑖𝑥_𝑖0,(𝑡)−𝑠(𝑡),𝑖𝑖==1𝑙,+2,1…,𝑙+,𝑙,2,…,𝑁,(2.6) where 𝜀_𝑖>0 (𝑖=1,2,…,𝑙) are the control gains, denoted by Ξ=diag{𝜀₁,𝜀₂,…,𝜀_𝑙,0,…,0}∈ℝ^𝑛×𝑛. Define 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡) (𝑖=1,2,…,𝑁) as the synchronization error. Then, according to the controller (2.6), the error system is𝑑𝑒_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓𝑡,𝑠_𝑖(𝑡),𝑠_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=_{1,𝑖≠𝑗}+𝑎_𝑖𝑗𝑟^𝑎_𝑖𝑗(𝑡)Σ𝑒_𝑗(𝑡)−𝑒_𝑖(𝑡)_𝑁_𝑗=_{1,𝑖≠𝑗}+𝜎_𝑖𝑏_𝑖𝑗𝑟^𝑏_𝑖𝑗(𝑡)Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)−𝑒_𝑖𝑡−𝜏_𝑐(𝑡)+𝑢_𝑖(𝑡)𝑑𝑡𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡),𝑟_𝜎𝑖(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,𝑁.(2.7)

Remark 2.1. Since the Markov chains 𝑟_𝑎𝑖𝑗(𝑡), 𝑟_𝑏𝑖𝑗(𝑡), and 𝑟_𝜎𝑖(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=[𝛾_𝑖𝑗]∈ℝ^𝑀×𝑀 given by 𝑃{𝑟(𝑡+Δ)=𝑗∣𝑟(𝑡)=𝑖}=𝛾^𝑖𝑗1+Δ𝛾+_𝑖𝑖𝑜(Δ)if𝑖≠𝑗,Δ+𝑜(Δ)if𝑖=𝑗,(2.8) for some Δ>0. Here 𝛾_𝑖𝑗=0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 and 𝛾_𝑖𝑖=−∑_𝑖≠𝑗𝛾_𝑖𝑗, 𝑑𝑒_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓𝑡,𝑠_𝑖(𝑡),𝑠_𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑗=1+𝑎_𝑖𝑗(𝑟(𝑡))Σ𝑒_𝑗(𝑡)_𝑁_𝑗=1+𝜎_𝑖𝑏_𝑖𝑗(𝑟(𝑡))Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)+𝑢_𝑖(𝑡)𝑑𝑡𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡),𝑟(𝑡)𝑑𝑤_𝑖(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 ^𝑁_𝑖=1𝔼‖‖𝑒_𝑖𝑡,𝑡₀,𝜉_𝑖‖‖²≤𝐾𝑒^−𝜅𝑡,(2.10) for some 𝐾>0 and some 𝜅>0 for any initial data 𝜉_𝑖∈𝒞^𝑏ℱ₀([−𝜏,0];ℝ^𝑛).

Definition 2.3 (see [12]). A continuous function 𝑓(𝑡,𝑥,𝑦)∶[0,+∞]×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛 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,…,𝜚_𝑛} if there exists a positive definite diagonal matrix 𝑃=diag{𝑝₁,𝑝₂,…,𝑝_𝑛}, a diagonal matrix Δ=diag{𝛿₁,𝛿₂,…,𝛿_𝑛}, and a constant 𝜂>0,𝜃>0 such that 𝑓(⋅) satisfies the condition (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)(2.11) for all 𝑥,𝑦,𝑧,𝑤∈ℝ^𝑛.

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 𝑝̌=max𝑝₁,𝑝₂,…,𝑝_𝑛,𝑝̂=min𝑝₁,𝑝₂,…,𝑝_𝑛,𝛿̌=max𝛿₁,𝛿₂,…,𝛿_𝑛.(2.12)

The following assumptions will be used throughout this paper for establishing the synchronization conditions. (H1)𝜏(𝑡), and 𝜏_𝑐(𝑡) are bounded and continuously differentiable functions such that 0<𝜏(𝑡)≤𝜏, 𝜏.(𝑡)<𝜏<1, 0<𝜏_𝑐(𝑡)≤𝜏_𝑐 and 𝜏._𝑐(𝑡)<𝜏_𝑐<1. Let 𝜏̌‌=max{𝜏,𝜏_𝑐}.(H2) Let 𝜎(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡)),𝑟) = 𝜎(𝑡,𝑒₁(𝑡),…,𝑒_𝑁(𝑡),𝑒₁(𝑡−𝜏(𝑡)),…,𝑒_𝑁(𝑡−𝜏(𝑡)),𝑒₁(𝑡−𝜏_𝑐(𝑡)),…,𝑒_𝑁(𝑡−𝜏_𝑐(𝑡)),𝑟). Then there exist positive definite constant matrices Υ^𝑟𝑖1, Υ^𝑟𝑖2, and Υ^𝑟𝑖3 for 𝑖=1,2,…,𝑁 and 𝑟=1,2,…,𝑀 such thatTr≤𝜎_𝑖𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡),𝑟^𝑇𝜎_𝑖𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡),𝑟_𝑁_𝑗=1𝑒_𝑗(𝑡)^𝑇Υ^𝑟𝑖1𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑒_𝑗(𝑡−𝜏(𝑡))^𝑇Υ^𝑟𝑖2𝑒_𝑗(𝑡−𝜏(𝑡))+^𝑁_𝑗=1𝑒_𝑗𝑡−𝜏_𝑐(𝑡)^𝑇Υ^𝑟𝑖3𝑒_𝑗𝑡−𝜏_𝑐(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(𝑡−𝜏),𝑟(𝑡))𝑑𝑡+𝜎(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏),𝑟(𝑡))𝑑𝜔(𝑡)(2.14) on 𝑡≥0 with initial value 𝑥₀=𝜉∈𝐶^𝑏𝐹₀([−𝜏,0];ℝ^𝑛), where 𝑓∶ℝ^𝑛×ℝ₊×𝑆⟶ℝ^𝑛,𝜎∶ℝ^𝑛×ℝ₊×𝑆⟶ℝ^𝑛×𝑚.(2.15) Let 𝐶^2,1(ℝ₊×ℝ^𝑛;ℝ₊) be the family of all the nonnegative functions 𝑉(𝑡,𝑥,𝑟) on ℝ₊×ℝ^𝑛×𝑆 that are twice continuously differentiable in 𝑥 and once differentiable in 𝑡. Let 𝑉∈𝐶^2,1(ℝ₊×ℝ^𝑛×𝑆;ℝ₊). Define an operator ℒ𝑉 from ℝ^𝑛×ℝ₊×𝑆 to ℝ^𝑛 by ℒ𝑉(𝑡,𝑥,𝑟)=𝑉_𝑡(𝑡,𝑥,𝑟)+𝑉_𝑥(𝑡,𝑥,𝑟)𝑓(𝑡,𝑥,𝑟)+12Tr𝜎(𝑡,𝑥,𝑟)^𝑇𝑉_𝑥𝑥𝜎(𝑡,𝑥,𝑟)+^𝑀_𝑗=1𝛾_𝑖𝑗𝑉(𝑡,𝑥,𝑗),(2.16) where 𝑉_𝑡(𝑡,𝑥,𝑟)=𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑡, 𝑉_𝑥(𝑡,𝑥,𝑟)=(𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑥₁,…,𝜕𝑉(𝑡,𝑥,𝑟)/𝜕𝑥_𝑛), 𝑉_𝑥𝑥(𝑡,𝑥,𝑟)=(𝜕²𝑉(𝑡,𝑥,𝑟)/𝜕𝑥_𝑖𝑥_𝑗)_𝑛×𝑛. If 𝑉∈𝐶^2,1(ℝ₊×ℝ^𝑛×𝑆;ℝ₊), then 𝔼𝑉(𝑡,𝑥(𝑡),𝑟)=𝔼𝑉𝑡₀,𝑥𝑡₀,𝑟+𝔼^𝑡𝑡₀ℒ𝑉(𝑠,𝑥(𝑠),𝑟)𝑑𝑠,(2.17) for all ∞>𝑡>𝑡₀≥0, as long as the expectations of the integrals exist.

3. Main Result

Theorem 3.1. Let assumptions (H1) and (H2) be true and let 𝑓∈QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼_𝑟 and 𝛽_𝑟 such that ⎡⎢⎢⎢⎣𝐴(𝑟)^𝑠+𝛿̌𝐼_𝑁−Ξ+𝛼_𝑟𝐼_𝑁𝐵(𝑟)𝐵(𝑟)_𝑇22−𝛽^𝑟𝐼_𝑁𝜏̌≤𝜃𝑇,𝜏̌≤(1−𝜃)𝑇,0≤̌⎤⎥⎥⎥⎦≤0,for𝑟=1,2,…,𝑀,(3.1)𝜏≤1−𝑞̌𝑏̌+𝑐̌11+𝜃,(3.2)𝑏₁+𝑐₁,1𝑏₂+𝑐₂,…,1𝑏_𝑀+𝑐_𝑀>Γ⁻¹𝟏_𝑀,(3.3) where 𝛾>0 is the greatest root of the equation 𝑞̌𝛾−(1+𝜃)+𝑏̌𝑞̌1−𝜏𝑒^𝛾𝜏+𝑐̌𝑞̌1−𝜏_𝑐Γ=diag𝑎₁𝑒^𝛾𝜏𝑐=0,(3.4)𝑎,𝑎₂,…,𝑎_𝑀−Γ,_𝑟=𝜆^min2𝜂𝐼^𝑛−𝑝̌∑^𝑁𝑖=1Υ^𝑟𝑖1+2𝛼₁𝑃Σ𝑝̌,𝑎̌=max^𝑟∈𝑆𝑏_𝑟𝑎_𝑟,=𝜆^max∑^𝑁𝑖=1𝑃Υ^𝑟𝑖2+2𝜁𝐼_𝑁𝑝̂,𝑏̌=max^𝑟∈𝑆𝑐_𝑟𝑏_𝑟,=𝜆^max∑^𝑁𝑖=1𝑃Υ^𝑟𝑖3+2𝛽₂𝑃Σ𝑝̂,𝑐̌=max^𝑟∈𝑆𝑐_𝑟.(3.5) Then the solutions 𝑥₁(𝑡), 𝑥₁(𝑡), … and 𝑥_𝑁(𝑡) of system (2.9) are globally and exponentially stable.

Proof. By (3.3), there exists a sufficiently small constant 𝜃>0 such that 𝑏₁+1𝑐₁,1𝑏₂+𝑐₂,…,1𝑏_𝑀+𝑐_𝑀≥(1+𝜃)Γ⁻¹𝟏.(3.6) Set (1+𝜃)Γ⁻¹𝟏=𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑀)^𝑇. Then Γ𝑞=(1+𝜃)𝟏_𝑀,(3.7) that is, 𝑏_𝑟+𝑐_𝑟𝑞_𝑟≤1,𝑎_𝑟𝑞_𝑟−^𝑀_𝑠=1𝛾_𝑠𝑟𝑞_𝑟=1+𝜃.(3.8) For 1≤𝑟≤𝑀, define the Lyapunov-Krasovskii function 𝑉(𝑡,𝑒(𝑡),𝑟)=𝑞_𝑟12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)(3.9) and let 𝑒^𝑘̃(𝑡)=(𝑒_1𝑘(𝑡),𝑒_2𝑘(𝑡),…,𝑒_𝑁𝑘(𝑡))^𝑇, 𝑘=1,2,…,𝑛. By Lemma 2.5, we have ℒ𝑉(𝑡,𝑒(𝑡),𝑟)=𝑞_𝑟𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃+𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓(𝑡,𝑠(𝑡),𝑠(𝑡−𝜏(𝑡)))_𝑁_𝑗=1𝑎_𝑖𝑗(𝑟)Σ𝑒_𝑗(𝑡)+^𝑁_𝑗=1+1𝑏_𝑖𝑗(𝑟)Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)+𝑢_𝑖(𝑡)2𝑞^𝑟𝑁_𝑖=1Tr𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟^𝑇𝑃𝜎_𝑖×𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1=𝑞_𝑟𝑁𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)_𝑖=1+𝑞_𝑟𝑁𝑒_𝑖(𝑡)^𝑇𝑃𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓(𝑡,𝑠(𝑡),𝑠(𝑡−𝜏(𝑡)))−ΔΣ𝑒_𝑖(𝑡)_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃ΔΣ𝑒_𝑖(𝑡)+𝑞_𝑟𝑁_𝑁𝑖=1_𝑗=1+𝑞_𝑟𝑁𝑎_𝑖𝑗(𝑟)𝑒_𝑖(𝑡)^𝑇𝑃Σ𝑒_𝑗(𝑡)_𝑁𝑖=1_𝑗=1𝑏_𝑖𝑗(𝑟)𝑒_𝑖(𝑡)^𝑇𝑃Σ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)−𝑞_𝑟𝑙_𝑖=1+1𝜀_𝑖𝑒_𝑖(𝑡)^𝑇𝑃Σ𝑒_𝑖(𝑡)2𝑞^𝑟Tr𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟^𝑇𝑃𝜎_𝑖×𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡),𝑟+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1≤𝑞_𝑟𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+𝜃^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇𝑒_𝑖(𝑡−𝜏(𝑡))+^𝑛_𝑘=1+𝑝_𝑘𝜚_𝑘𝛿_𝑘𝑒^𝑘̃(𝑡)^𝑇𝑒^𝑘̃(𝑡)_𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴(𝑟)𝑒^𝑘̃(𝑡)+^𝑛_𝑘=1−𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐵(𝑟)𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)_𝑛_𝑘=1+1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇Ξ𝑒^𝑘̃(𝑡)2𝑝̌^𝑁_𝑗=1^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇Υ^𝑟𝑗1𝑒_𝑖(𝑡)+^𝑁_𝑖=1+𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇Υ^𝑟𝑗2𝑒_𝑖(𝑡−𝜏(𝑡))_𝑁_𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇Υ^𝑟𝑗3𝑒_𝑖𝑡−𝜏_𝑐(𝑡)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1=𝑞_𝑟𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+𝜃^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇𝑒_𝑖(𝑡−𝜏(𝑡))+12𝑝̌^𝑁_𝑁𝑖=1_𝑗=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇×Υ^𝑟𝑗3𝑒_𝑖𝑡−𝜏_𝑐(𝑡)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1+𝑞_𝑟𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)^𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴(𝑟)𝑒^𝑘̃(𝑡)+^𝑛_𝑘=1−𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐵(𝑟)𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)_𝑛_𝑘=1+1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇Ξ𝑒^𝑘̃(𝑡)2𝑝̌^𝑁_𝑁𝑖=1_𝑗=1𝑒_𝑖(𝑡)^𝑇Υ^𝑟𝑗1𝑒_𝑖(𝑡)+^𝑁_𝑁𝑖=1_𝑗=1+𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇Υ^𝑟𝑗2𝑒_𝑖(𝑡−𝜏(𝑡))_𝑛_𝑘=1=𝑞_𝑟𝑝_𝑘𝜚_𝑘𝛿_𝑘𝑒^𝑘̃(𝑡)^𝑇𝑒^𝑘̃(𝑡)^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇−𝜂𝐼^𝑁+12𝑝̌^𝑁_𝑗=1Υ^𝑟𝑗1−𝛼_𝑟𝑃Σ𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇×𝜃+12𝑝̌^𝑁_𝑗=1Υ^𝑟𝑗2𝑒^𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇21𝑝̌^𝑁_𝑗=1×𝑒_𝑖Υ^𝑟𝑗3+𝛽_𝑟𝑃Σ𝑡−𝜏_𝑐(𝑡)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1+𝑞_𝑟𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)^𝑛_𝑘=1+𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴(𝑟)−Ξ+𝛿̌+𝛼_𝑟𝐼_𝑁𝑒^𝑘̃(𝑡)_𝑛_𝑘=1𝑝_𝑘𝜚_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐵(𝑟)𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)−^𝑛_𝑘=1≤𝑞_𝑟𝑝_𝑘𝜚_𝑘𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)^𝑇𝛽_𝑟𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇−𝜂𝐼^𝑁+12𝑝̌^𝑁_𝑗=1Υ^𝑟𝑗1−𝛼_𝑟𝑃Σ𝑒_𝑖(𝑡)+^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇×𝛽_𝑟𝐼_𝑁+12𝑝̌^𝑁_𝑗=1Υ^𝑟𝑗2𝑒^𝑖(𝑡−𝜏(𝑡))+^𝑁_𝑖=1𝑒_𝑖𝑡−𝜏_𝑐(𝑡)^𝑇12𝑝̌^𝑁_𝑗=1×𝑒_𝑖Υ^𝑟𝑗3+𝛽_𝑟𝑃Σ𝑡−𝜏_𝑐(𝑡)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡).(3.10) Let 𝐸(𝑡)=12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡).(3.11) Then we have ℒ𝑉(𝑡)≤−𝑎_𝑟𝑞_𝑟𝐸(𝑡)+𝑏_𝑟𝑞_𝑟𝐸(𝑡−𝜏(𝑡))+𝑐_𝑟𝑞_𝑟𝐸𝑡−𝜏_𝑐(𝑡)+^𝑀_𝑠=1𝛾_𝑟𝑠𝑞_𝑠𝐸(𝑡),(3.12) and by (3.8), we have ℒ𝑉(𝑡)≤−(1+𝜃)𝐸(𝑡)+𝑏̌𝑞̌𝐸(𝑡−𝜏(𝑡))+𝑐̌𝑞̌𝐸𝑡−𝜏_𝑐(𝑡).(3.13) Define 𝑊(𝑡)=𝑒^𝛾𝑡𝑉(𝑡)(3.14) and use (3.13) to compute the operator ℒ𝑊(𝑡)=≤𝑒𝑒^𝛾𝑡_𝛾𝑡[𝛾𝑉(𝑡)+ℒ𝑉(𝑡)]𝛾𝑝𝐸̌(𝑡)−(1+𝜃)𝐸(𝑡)+𝑏𝑞̌̌𝐸(𝑡−𝜏(𝑡))+𝑐̌𝑞̌𝐸𝑡−𝜏_𝑐(𝑡),(3.15) which, after applying the generalized Itô's formula, gives 𝑞_𝑟𝑒^𝛾𝑡𝔼𝐸(𝑡)=𝑞_𝑟𝑒^𝛾𝑡0𝔼𝐸𝑡₀+𝔼^𝑡𝑡₀ℒ𝑊(𝑠)𝑑𝑠(3.16) for any 𝑡>𝑡₀≥0. Hence we have 𝑞_𝑟𝑒^𝛾𝑡𝔼𝐸(𝑡)≤≤𝑞𝑝̌𝔼𝑞̌̌𝐸(0)+𝔼^𝑡0𝑒^𝛾𝑠𝛾𝐸(𝑠)−(1−𝜃)𝐸(𝑠)+𝑏̌𝑞̌𝐸(𝑠−𝜏(𝑠))+𝑐̌𝑞̌𝐸𝑠−𝜏_𝑐(𝑠)𝑑𝑠2^𝑁_𝑖=1𝔼‖‖𝜉_𝑖‖‖²+(𝛾−(1+𝜃))^𝑡0𝑒^𝛾𝑠𝔼𝐸(𝑠)𝑑𝑠+𝑏𝑞̌̌𝑒^𝛾𝜏^𝑡0+𝑐̌𝑞̌𝑒_𝛾𝜏𝑐𝑒^{𝛾(𝑠−𝜏(𝑠))}𝔼𝐸(𝑠−𝜏(𝑠))𝑑𝑠^𝑡0𝑒^{𝛾(𝑠−𝜏𝑐(𝑠))}𝔼𝐸𝑠−𝜏_𝑐(𝑠)𝑑𝑠,(3.17) which, by using the change of variables 𝑠−𝜏(𝑠)=𝑢, gives ^𝑡0𝑒^{𝛾(𝑠−𝜏(𝑠))}𝔼𝐸(𝑠−𝜏(𝑠))𝑑𝑠=^{𝑡−𝜏(𝑡)−𝜏(0)}𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢≤𝑝𝜏̌1−𝜏.(𝑡)21−𝜏^𝑁_𝑖=1𝔼‖‖𝜉_𝑖‖‖²+11−𝜏^𝑡0𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢,(3.18) and a further change of variables 𝑠−𝜏_𝑐(𝑠)=𝑢 gives ^𝑡0𝑒^{𝑠−𝜏𝑐(𝑠)}𝔼𝐸𝑠−𝜏_𝑐(𝑠)𝑑𝑠=^{𝑡−𝜏𝑐(𝑡)−𝜏}_𝑐(0)𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢≤𝑝𝜏̌_𝑐1−𝜏._𝑐(𝑡)21−𝜏_𝑐^𝑁_𝑖=1𝔼‖‖𝜉_𝑖‖‖²+11−𝜏_𝑐^𝑡0𝑒^𝛾𝑢𝔼𝐸(𝑢)𝑑𝑢.(3.19) By Condition (3.4), we obtain 𝔼𝐸(𝑡)≤𝑝𝑞̌̌21+1𝑏−̌𝑞̌𝜏𝜏𝑒^𝛾𝜏+𝑐̌𝑞̌𝜏^𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐^𝑁_𝑖=1𝔼‖‖𝜉_𝑖‖‖²𝑒^−𝛾𝑡(3.20) so that 𝔼‖𝑒(𝑡)‖²≤𝑝𝑞̌̌2𝑝̂1+1𝑏−̌𝑞̌𝜏𝜏𝑒^𝛾𝜏+𝑐̌𝑞̌𝜏^𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐^𝑁_𝑖=1𝔼‖‖𝜉_𝑖‖‖²𝑒^−𝛾𝑡.(3.21) The proof is hence complete.

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

Corollary 3.2. Let assumptions (H1) and (H2) be true and let 𝑓∈QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼_𝑟 and 𝛽_𝑟 such that ⎡⎢⎢⎢⎣𝐴(𝑟)^𝑠+𝛿̌𝐼_𝑁−Ξ+𝛼_𝑟𝐼_𝑁𝐵(𝑟)𝐵(𝑟)_𝑇22−𝛽^𝑟𝐼_𝑁1𝜏̌≤𝜃𝑇,𝜏̌≤⎤⎥⎥⎥⎦(≤1−0,𝜃)f𝑇o,r𝑟=1,2,…,𝑀,𝑏₁+𝑐₁,1𝑏₂+𝑐₂,…,1𝑏_𝑀+𝑐_𝑀>Γ⁻¹𝟏_𝑀,(3.22) where 𝛾>0 is the greatest root of the equation 𝑞̌𝛾Γ−=(d1ia+g𝜃)𝑎+₁𝑏̌𝑞̌𝑒^𝛾𝜏+𝑐̌𝑞̌𝑒^𝛾𝜏𝑐=0,𝑎,𝑎₂,…,𝑎_𝑀−Γ,_𝑟=𝜆^min2𝜂𝐼^𝑛−𝑝̌∑^𝑁𝑖=1Υ^𝑟𝑖1+2𝛼₁𝑃Σ𝑝̌,𝑎̌=max^𝑟∈𝑆𝑏_𝑟𝑎_𝑟,=𝜆^max∑^𝑁𝑖=1𝑃Υ^𝑟𝑖2+2𝜁𝐼_𝑁𝑝̂,𝑏̌=max^𝑟∈𝑆𝑐_𝑟𝑏_𝑟,=𝜆^max∑^𝑁𝑖=1𝑃Υ^𝑟𝑖3+2𝛽₂𝑃Σ𝑝̂,𝑐̌=max^𝑟∈𝑆𝑐_𝑟,(3.23) then the solutions 𝑥₁(𝑡), 𝑥₁(𝑡),… and 𝑥_𝑁(𝑡) of system (2.9) are globally and exponentially stable.

When 𝐴(𝑟(𝑡))=𝐴, 𝐵(𝑟(𝑡))=𝐵, and 𝜎_𝑖(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡)),𝑟(𝑡))=𝜎_𝑖(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡))), we can get the following corollary.

Corollary 3.3. Let assumptions (H1) and (H2) be true, and let 𝑓∈QUAD(𝑃,Δ,𝜂,𝜃). If there exist positive constants 𝛼_𝑟 and 𝛽_𝑟 such that ⎡⎢⎢⎣𝐴^𝑠+𝛿̌𝐼_𝑁−Ξ+𝛼_𝑟𝐼_𝑁𝐵𝐵_𝑇22−𝛽^𝑟𝐼_𝑁𝜏̌≤𝜃𝑇,𝜏̌≤(1−𝜃)𝑇,0≤̌⎤⎥⎥⎦≤0,𝜏≤1−𝑏̌+𝑐̌1+𝜃,(3.24) where 𝑎_𝑟=𝜆^min2𝜂𝐼^𝑛−𝑝̌∑^𝑁𝑖=1Υ^𝑟𝑖1+2𝛼₁𝑃Σ𝑝̌,𝑎̌=max^𝑟∈𝑆𝑏_𝑟𝑎_𝑟,=𝜆^max∑^𝑁𝑖=1𝑃Υ^𝑟𝑖2+2𝜁𝐼_𝑁𝑝̂,𝑏̌=max^𝑟∈𝑆𝑐_𝑟𝑏_𝑟,=𝜆^max∑^𝑁𝑖=1𝑃Υ^𝑟𝑖3+2𝛽₂𝑃Σ𝑝̂,𝑐̌=max^𝑟∈𝑆𝑐_𝑟,(3.25) then the solutions 𝑥₁(𝑡), 𝑥₁(𝑡), … and 𝑥_𝑁(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 network𝑑𝑠(𝑡)=[−𝐶𝑠(𝑡)+𝐴𝑓(𝑠(𝑡))+𝐵𝑔(𝑠(𝑡−𝜏(𝑡)))]𝑑𝑡,(4.1) where 𝑓(𝑠)=𝑔(𝑠)=tanh(𝑠), 𝜏(𝑡)=1,𝐶=⎡⎢⎢⎣1001⎤⎥⎥⎦,𝐴=⎡⎢⎢⎣−25−40.5.1⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣−−10..52−−04.1⎤⎥⎥⎦.(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. Thus𝑑𝑥_𝑖(𝑡)=𝑓𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))+⁵_𝑗=1𝑎^𝑟𝑖𝑗Σ𝑥_𝑗(𝑡)+⁵_𝑗=1+𝜎_𝑟𝑖𝑏^𝑟𝑖𝑗Σ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)−𝜀_𝑖𝑥_𝑖(𝑡)−𝑠(𝑡)𝑑𝑡𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,5,𝑟=1,2,3,(4.3)𝐴𝐴₁=𝑎^1𝑖𝑗=20⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−11103−11114−11114−11114−01113⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐴₂=𝑎^2𝑖𝑗=15⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−11114−11103−11114−11114−10113₃⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐵=𝑎^3𝑖𝑗=37⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−11002−11103−10113−10001−00101⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐵¹=𝑏^1𝑖𝑗=0.1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−00002−11112−11112−01102−00002₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=𝑏^2𝑖𝑗=0.2Ξ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−0000=11−0011001⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−100000000101000−101100000100000−0000100000⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,,Γ𝐵₃==⎡⎢⎢⎢⎢⎣−22𝑏3^3𝑖𝑗−13=40−.2275⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−,10011−0000𝜏1_𝑐−10101−00101−00001⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(𝑡)=0.1𝑒^𝑡1+𝑒^𝑡𝜎𝜎^1𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)=0.1diag𝑥_𝑖1(𝑡)−𝑥_𝑖+1,1(𝑡),𝑥_𝑖2(𝑡)−𝑥_𝑖+1,2(𝑡)_2𝑖,𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)=0.1diag−𝑥𝑥_𝑖1(_𝑖+1,2𝑡−𝜏(𝑡))−𝑥_𝑖+1,1(𝑡−𝜏(𝑡)),𝑥_𝑖2(𝑡−𝜏(𝑡))𝜎(𝑡−𝜏(𝑡))_3𝑖,(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏(𝑡)))=0.1diag−𝑥𝑥_𝑖1_𝑖+1,2𝑡−𝜏_𝑐(𝑡)−𝑥_𝑖+1,1𝑡−𝜏_𝑐(𝑡),𝑥_𝑖2𝑡−𝜏_𝑐(𝑡)𝑡−𝜏_𝑐(𝑡).(4.4) Computations then yield 𝜏=1, 𝜏=0, 𝜏_𝑐=0.1, 𝜏_𝑐=0.1, Υ_𝑖𝑗=0.01𝐼₂ for 𝑖=1,2,…,𝑁, and 𝑗=1,2. Let 𝐽={1,2} and the control strength 𝜀_𝑖=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, andΓ⁻¹=⎡⎢⎢⎢⎢⎣000..00.122022541287666000...000192455103763457000...000181879730718332⎤⎥⎥⎥⎥⎦.(4.5) Therefore𝑞=[0.1391,0.1368,0.1350]^𝑇,(4.6) and, after solving (3.4), we obtain 𝛾=0.0349.

The initial conditions for this simulation are 𝑥_𝑖𝑗(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.

(a)

(b)

Figure 1 

The trajectories of the state variables of 𝑥𝑖1 and 𝑥𝑖2 (𝑖=1,2,…,5) in system (4.3) under pinning control.

(a)

(b)

Figure 2 

The time evolution of 𝑒𝑖1 and 𝑒𝑖2 (𝑖=1,2,…,5) in system (4.3) under pinning control.

Next, let 𝑡_𝑐(𝑡)=0.1, we get the 𝛾=0.0402. The initial conditions for this simulation are 𝑥_𝑖𝑗(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.

(a)

(b)

Figure 3 

The trajectories of the state variables of 𝑥𝑖1 and 𝑥𝑖2 (𝑖=1,2,…,5) in system (4.3) under pinning control.

(a)

(b)

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.