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

Authors: Jingyi Wang; Chen Xu; Jianwen Feng; Man Kam Kwong; Francis Austin
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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 Section2, 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,ℝn shall denote the n-dimensional Euclidean space and ℝn×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)T∈ℝn 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,ℱ,{ℱt}t≥0,𝒫) be a complete probability space with a filtration {ℱt}t≥0 that is right continuous with ℱ0 containing all the 𝒫-null sets. C([-τ,0];ℝn) shall denote the family of continuous functions ϕ from [-τ,0] to ℝn with the uniform norm ∥ϕ∥=sup-τ≤s≤0|ϕ(s)| and Cℱ02([-τ,0];ℝn) the family of all ℱ0 measurable, C([-τ,0];ℝn)-valued stochastic variables ξ={ξ(θ):-τ≤θ≤0} such that ∫-τ0𝔼|ξ(s)|2ds≤∞, where 𝔼 stands for the correspondent expectation operator with respect to the given probability measure 𝒫.Consider a complex network consisting ofN identical nodes with nondelayed and time-varying delayed linear coupling and Markovian switching(2.1)dxi(t)={f(t,xi(t),xi(t-τ(t)))+∑j=1,i≠jNaij(raij(t))Σ(xj(t)-xi(t))+∑j=1,i≠jNbij(rbij(t))Σ(xj(t-τc(t))-xi(t-τc(t)))}dt+σi(t,x(t),x(t-τ(t)),x(t-τc(t)),rσi(t))dwi(t),i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈ℝn is the state vector of the ith node of the network, f(t,xi(t),xi(t-τ(t)))=[f1(t,xi(t),xi(t-τ(t))),f2(t,xi(t),xi(t-τ(t))),…,fn(t,xi(t),xi(t-τ(t)))]T is a continuous vector-valued function, Σ=diag(ϱ1,ϱ2,…,ϱn) is an inner coupling of the networks that satisfies ϱj>0, j=1,2,…,n, and 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))]∈ℝn×n and B(rb(t))=[bij(rbij(t))]∈ℝn×n are the outer coupling matrices of the network at time t at nodes raij(t), t-τc(t) and rbij(t), respectively, such that aij(raij(t))≥0 for i≠j, aii(raii(t))=-∑j=i,j≠iNaij(raij(t)), bij(rbij(t))≥0 for i≠j, and bii(rbii(t))=-∑j=i,j≠iNbij(rbij(t)). τ(t) is the inner time-varying delay satisfying τ≥τ(t)≥0 and τc(t) is the coupling time-varying delay satisfying τc≥τc(t)≥0. Finally, σi(t,x(t),x(t-τ(t)),x(t-τc(t)),rσi(t))=σi(t,x1(t),…,xn(t),x1(t-τ(t)),…,xn(t-τ(t)),x1(t-τc(t)),…,xn(t-τc(t)),rσi(t))∈ℝn×n and wi(t)=(wi1(t),wi2(t),…,win(t))T∈ℝn is a bounded vector-form Weiner process, satisfying(2.2)Ewij(t)=0,Ewij2(t)=1,Ewij(t)wij(s)=0(s≠t).
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) are(2.3)xi(s)=ξi(s),-τ̌≤s≤0,i=1,2,…,N,
where τ̌=max{τ(t),τc(t)}, ξi∈Cℱ0b([-τ̌,0],ℝn) with the norm ∥ξi∥2=sup-τ̌≤s≤0ξi(s)Tξi(s), and our objective is to control system (2.1) so that it stays in the trajectory s(t)∈ℝn of the system(2.4)ds(t)=f(t,s(t),s(t-τ(t)))dt
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 by(2.5)dxi(t)={f(t,xi(t),xi(t-τ(t)))+∑j=1,i≠jNaij(raij(t))Σ(xj(t)-xi(t))+∑j=1,i≠jNbij(rbij(t))Σ(xj(t-τc(t))-xi(t-τc(t)))+ui(t)}dt+σi(t,x(t),x(t-τ(t)),x(t-τc(t)),rσi(t))dwi(t),i=1,2,…,N,
where ui(t)  (i=1,2,…,N) are the linear state feedback controllers that are defined by(2.6)ui(t)={-ɛi(xi(t)-s(t)),i=1,2,…,l,0,i=l+1,l+2,…,N,
where ɛi>0 (i=1,2,…,l) are the control gains, denoted by Ξ=diag{ɛ1,ɛ2,…,ɛl,0,…,0}∈ℝn×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 is(2.7)dei(t)={f(t,xi(t),xi(t-τ(t)))-f(t,si(t),si(t-τ(t)))+∑j=1,i≠jNaij(raij(t))Σ(ej(t)-ei(t))+∑j=1,i≠jNbij(rbij(t))Σ(ej(t-τc(t))-ei(t-τc(t)))+ui(t)}dt+σi(t,e(t),e(t-τ(t)),e(t-τc(t)),rσi(t))dwi(t),i=1,2,…,N.Remark 2.1.
Since the Markov chainsraij(t), rbij(t), and rσi(t) are independent, we have an equivalent system as follows.
Letr(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]∈ℝM×M given by
(2.8)P{r(t+Δ)=j∣r(t)=i}={γijΔ+o(Δ)ifi≠j,1+γiiΔ+o(Δ)ifi=j,
for some Δ>0. Here γij=0 is the transition rate from i to j if i≠j and γii=-∑i≠jγij,
(2.9)dei(t)={f(t,xi(t),xi(t-τ(t)))-f(t,si(t),si(t-τ(t)))+∑j=1Naij(r(t))Σej(t)+∑j=1Nbij(r(t))Σej(t-τc(t))+ui(t)}dt+σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r(t))dwi(t),i=1,2,…,N.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
(2.10)∑i=1NE‖ei(t,t0,ξi)‖2≤Ke-κt,
for some K>0 and some κ>0 for any initial data ξi∈𝒞ℱ0b([-τ,0];ℝn).Definition 2.3 (see [12]).
A continuous functionf(t,x,y):[0,+∞]×ℝn×ℝn→ℝn is said to belong to the function class QUAD, denoted by f∈QUAD(P,Δ,η,θ) for some given matrix Σ=diag{ϱ1,ϱ2,…,ϱn} if there 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
(2.11)(x-y)TP((f(t,x,z)-f(t,y,w))-ΔΣ(x-y))≤-η(x-y)T(x-y)+θ(z-w)T(z-w)
for all x,y,z,w∈ℝn.Remark 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(2.12)p̌=max⁡{p1,p2,…,pn},p̂=min⁡{p1,p2,…,pn},δ̌=max⁡{δ1,δ2,…,δn}.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, τ̇(t)<τ¯<1, 0<τc(t)≤τc and τ̇c(t)<τ¯c<1. Let τ¯̌=max{τ¯,τ¯c}.(H2)
Letσ(t,e(t),e(t-τ(t)),e(t-τc(t)),r) = σ(t,e1(t),…,eN(t),e1(t-τ(t)),…,eN(t-τ(t)),e1(t-τc(t)),…,eN(t-τc(t)),r). Then there exist positive definite constant matrices Υi1r, Υi2r, and Υi3r for i=1,2,…,N and r=1,2,…,M such that(2.13)Tr⁡[σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r)Tσi(t,e(t),e(t-τ(t)),e(t-τc(t)),r)]≤∑j=1Nej(t)TΥi1rej(t)+∑j=1Nej(t-τ(t))TΥi2rej(t-τ(t))+∑j=1Nej(t-τc(t))TΥi3rej(t-τc(t)).Lemma 2.5 (see[32, 33, the generalized Itô formula]).
Consider a stochastic delayed differential equation with Markovian switching of the form(2.14)dx(t)=f(t,x(t),x(t-τ),r(t))dt+σ(t,x(t),x(t-τ),r(t))dω(t)
on t≥0 with initial value x0=ξ∈CF0b([-τ,0];ℝn), where
(2.15)f:Rn×R+×S⟶Rn,σ:Rn×R+×S⟶Rn×m.
Let C2,1(ℝ+×ℝn;ℝ+) be the family of all the nonnegative functions V(t,x,r) on ℝ+×ℝn×S that are twice continuously differentiable in x and once differentiable in t. Let V∈C2,1(ℝ+×ℝn×S;ℝ+). Define an operator ℒV from ℝn×ℝ+×S to ℝn by
(2.16)LV(t,x,r)=Vt(t,x,r)+Vx(t,x,r)f(t,x,r)+12Tr⁡[σ(t,x,r)TVxxσ(t,x,r)]+∑j=1MγijV(t,x,j),
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 V∈C2,1(ℝ+×ℝn×S;ℝ+), then
(2.17)EV(t,x(t),r)=EV(t0,x(t0),r)+E∫t0tLV(s,x(s),r)ds,
for all ∞>t>t0≥0, as long as the expectations of the integrals exist.


## 2.1. Notations

Throughout this paper,ℝn shall denote the n-dimensional Euclidean space and ℝn×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)T∈ℝn 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,ℱ,{ℱt}t≥0,𝒫) be a complete probability space with a filtration {ℱt}t≥0 that is right continuous with ℱ0 containing all the 𝒫-null sets. C([-τ,0];ℝn) shall denote the family of continuous functions ϕ from [-τ,0] to ℝn with the uniform norm ∥ϕ∥=sup-τ≤s≤0|ϕ(s)| and Cℱ02([-τ,0];ℝn) the family of all ℱ0 measurable, C([-τ,0];ℝn)-valued stochastic variables ξ={ξ(θ):-τ≤θ≤0} such that ∫-τ0𝔼|ξ(s)|2ds≤∞, where 𝔼 stands for the correspondent expectation operator with respect to the given probability measure 𝒫.Consider a complex network consisting ofN identical nodes with nondelayed and time-varying delayed linear coupling and Markovian switching(2.1)dxi(t)={f(t,xi(t),xi(t-τ(t)))+∑j=1,i≠jNaij(raij(t))Σ(xj(t)-xi(t))+∑j=1,i≠jNbij(rbij(t))Σ(xj(t-τc(t))-xi(t-τc(t)))}dt+σi(t,x(t),x(t-τ(t)),x(t-τc(t)),rσi(t))dwi(t),i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈ℝn is the state vector of the ith node of the network, f(t,xi(t),xi(t-τ(t)))=[f1(t,xi(t),xi(t-τ(t))),f2(t,xi(t),xi(t-τ(t))),…,fn(t,xi(t),xi(t-τ(t)))]T is a continuous vector-valued function, Σ=diag(ϱ1,ϱ2,…,ϱn) is an inner coupling of the networks that satisfies ϱj>0, j=1,2,…,n, and 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))]∈ℝn×n and B(rb(t))=[bij(rbij(t))]∈ℝn×n are the outer coupling matrices of the network at time t at nodes raij(t), t-τc(t) and rbij(t), respectively, such that aij(raij(t))≥0 for i≠j, aii(raii(t))=-∑j=i,j≠iNaij(raij(t)), bij(rbij(t))≥0 for i≠j, and bii(rbii(t))=-∑j=i,j≠iNbij(rbij(t)). τ(t) is the inner time-varying delay satisfying τ≥τ(t)≥0 and τc(t) is the coupling time-varying delay satisfying τc≥τc(t)≥0. Finally, σi(t,x(t),x(t-τ(t)),x(t-τc(t)),rσi(t))=σi(t,x1(t),…,xn(t),x1(t-τ(t)),…,xn(t-τ(t)),x1(t-τc(t)),…,xn(t-τc(t)),rσi(t))∈ℝn×n and wi(t)=(wi1(t),wi2(t),…,win(t))T∈ℝn is a bounded vector-form Weiner process, satisfying(2.2)Ewij(t)=0,Ewij2(t)=1,Ewij(t)wij(s)=0(s≠t).
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) are(2.3)xi(s)=ξi(s),-τ̌≤s≤0,i=1,2,…,N,
where τ̌=max{τ(t),τc(t)}, ξi∈Cℱ0b([-τ̌,0],ℝn) with the norm ∥ξi∥2=sup-τ̌≤s≤0ξi(s)Tξi(s), and our objective is to control system (2.1) so that it stays in the trajectory s(t)∈ℝn of the system(2.4)ds(t)=f(t,s(t),s(t-τ(t)))dt
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 by(2.5)dxi(t)={f(t,xi(t),xi(t-τ(t)))+∑j=1,i≠jNaij(raij(t))Σ(xj(t)-xi(t))+∑j=1,i≠jNbij(rbij(t))Σ(xj(t-τc(t))-xi(t-τc(t)))+ui(t)}dt+σi(t,x(t),x(t-τ(t)),x(t-τc(t)),rσi(t))dwi(t),i=1,2,…,N,
where ui(t)  (i=1,2,…,N) are the linear state feedback controllers that are defined by(2.6)ui(t)={-ɛi(xi(t)-s(t)),i=1,2,…,l,0,i=l+1,l+2,…,N,
where ɛi>0 (i=1,2,…,l) are the control gains, denoted by Ξ=diag{ɛ1,ɛ2,…,ɛl,0,…,0}∈ℝn×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 is(2.7)dei(t)={f(t,xi(t),xi(t-τ(t)))-f(t,si(t),si(t-τ(t)))+∑j=1,i≠jNaij(raij(t))Σ(ej(t)-ei(t))+∑j=1,i≠jNbij(rbij(t))Σ(ej(t-τc(t))-ei(t-τc(t)))+ui(t)}dt+σi(t,e(t),e(t-τ(t)),e(t-τc(t)),rσi(t))dwi(t),i=1,2,…,N.Remark 2.1.
Since the Markov chainsraij(t), rbij(t), and rσi(t) are independent, we have an equivalent system as follows.
Letr(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]∈ℝM×M given by
(2.8)P{r(t+Δ)=j∣r(t)=i}={γijΔ+o(Δ)ifi≠j,1+γiiΔ+o(Δ)ifi=j,
for some Δ>0. Here γij=0 is the transition rate from i to j if i≠j and γii=-∑i≠jγij,
(2.9)dei(t)={f(t,xi(t),xi(t-τ(t)))-f(t,si(t),si(t-τ(t)))+∑j=1Naij(r(t))Σej(t)+∑j=1Nbij(r(t))Σej(t-τc(t))+ui(t)}dt+σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r(t))dwi(t),i=1,2,…,N.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
(2.10)∑i=1NE‖ei(t,t0,ξi)‖2≤Ke-κt,
for some K>0 and some κ>0 for any initial data ξi∈𝒞ℱ0b([-τ,0];ℝn).Definition 2.3 (see [12]).
A continuous functionf(t,x,y):[0,+∞]×ℝn×ℝn→ℝn is said to belong to the function class QUAD, denoted by f∈QUAD(P,Δ,η,θ) for some given matrix Σ=diag{ϱ1,ϱ2,…,ϱn} if there 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
(2.11)(x-y)TP((f(t,x,z)-f(t,y,w))-ΔΣ(x-y))≤-η(x-y)T(x-y)+θ(z-w)T(z-w)
for all x,y,z,w∈ℝn.Remark 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(2.12)p̌=max⁡{p1,p2,…,pn},p̂=min⁡{p1,p2,…,pn},δ̌=max⁡{δ1,δ2,…,δn}.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, τ̇(t)<τ¯<1, 0<τc(t)≤τc and τ̇c(t)<τ¯c<1. Let τ¯̌=max{τ¯,τ¯c}.(H2)
Letσ(t,e(t),e(t-τ(t)),e(t-τc(t)),r) = σ(t,e1(t),…,eN(t),e1(t-τ(t)),…,eN(t-τ(t)),e1(t-τc(t)),…,eN(t-τc(t)),r). Then there exist positive definite constant matrices Υi1r, Υi2r, and Υi3r for i=1,2,…,N and r=1,2,…,M such that(2.13)Tr⁡[σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r)Tσi(t,e(t),e(t-τ(t)),e(t-τc(t)),r)]≤∑j=1Nej(t)TΥi1rej(t)+∑j=1Nej(t-τ(t))TΥi2rej(t-τ(t))+∑j=1Nej(t-τc(t))TΥi3rej(t-τc(t)).Lemma 2.5 (see[32, 33, the generalized Itô formula]).
Consider a stochastic delayed differential equation with Markovian switching of the form(2.14)dx(t)=f(t,x(t),x(t-τ),r(t))dt+σ(t,x(t),x(t-τ),r(t))dω(t)
on t≥0 with initial value x0=ξ∈CF0b([-τ,0];ℝn), where
(2.15)f:Rn×R+×S⟶Rn,σ:Rn×R+×S⟶Rn×m.
Let C2,1(ℝ+×ℝn;ℝ+) be the family of all the nonnegative functions V(t,x,r) on ℝ+×ℝn×S that are twice continuously differentiable in x and once differentiable in t. Let V∈C2,1(ℝ+×ℝn×S;ℝ+). Define an operator ℒV from ℝn×ℝ+×S to ℝn by
(2.16)LV(t,x,r)=Vt(t,x,r)+Vx(t,x,r)f(t,x,r)+12Tr⁡[σ(t,x,r)TVxxσ(t,x,r)]+∑j=1MγijV(t,x,j),
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 V∈C2,1(ℝ+×ℝn×S;ℝ+), then
(2.17)EV(t,x(t),r)=EV(t0,x(t0),r)+E∫t0tLV(s,x(s),r)ds,
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 letf∈QUAD(P,Δ,η,θ). If there exist positive constants αr and βr such that
(3.1)[A(r)s+δ̌IN-Ξ+αrINB(r)2B(r)T2-βrIN]≤0,forr=1,2,…,M,(3.2)τ̌≤θT,τ̌≤(1-θ)T,0≤τ¯̌≤1-q̌(b̌+č)1+θ,(3.3)(1b1+c1,1b2+c2,…,1bM+cM)>Γ̃-11M,
where γ>0 is the greatest root of the equation
(3.4)q̌γ-(1+θ)+b̌q̌1-τ¯eγτ+čq̌1-τ¯ceγτc=0,(3.5)Γ̃=diag⁡{a1,a2,…,aM}-Γ,ar=λmin⁡(2ηIn-p̌∑i=1NΥi1r+2α1PΣ)p̌,ǎ=max⁡r∈Sar,br=λmax⁡(∑i=1NPΥi2r+2ζIN)p̂,b̌=max⁡r∈Sbr,cr=λmax⁡(∑i=1NPΥi3r+2β2PΣ)p̂,č=max⁡r∈Scr.
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
(3.6)(1b1+c1,1b2+c2,…,1bM+cM)≥(1+θ)Γ̃-11.
Set (1+θ)Γ̃-11=q=(q1,q2,…,qM)T. Then
(3.7)Γ̃q=(1+θ)1M,
that is,
(3.8)(br+cr)qr≤1,arqr-∑s=1Mγsrqr=1+θ.
For 1≤r≤M, define the Lyapunov-Krasovskii function
(3.9)V(t,e(t),r)=qr12∑i=1Nei(t)TPei(t)
and let ẽk(t)=(e1k(t),e2k(t),…,eNk(t))T, k=1,2,…,n. By Lemma 2.5, we have
(3.10)LV(t,e(t),r)=qr∑i=1Nei(t)TP{∑j=1Nf(t,xi(t),xi(t-τ(t)))-f(t,s(t),s(t-τ(t)))+∑j=1Naij(r)Σej(t)+∑j=1Nbij(r)Σej(t-τc(t))+ui(t)}+12qr∑i=1NTr⁡{σi(t,x(t),x(t-τ(t)),x(t-τc(t)),r)TPσi×(t,x(t),x(t-τ(t)),x(t-τc(t)),r)(t,x(t),x(t-τ(t)),x(t-τc(t)),r)T}+∑s=1Mγrsqs12∑i=1Nei(t)TPei(t)=qr∑i=1Nei(t)TP{f(t,xi(t),xi(t-τ(t)))-f(t,s(t),s(t-τ(t)))-ΔΣei(t)}+qr∑i=1Nei(t)TPΔΣei(t)+qr∑i=1N∑j=1Naij(r)ei(t)TPΣej(t)+qr∑i=1N∑j=1Nbij(r)ei(t)TPΣej(t-τc(t))-qr∑i=1lɛiei(t)TPΣei(t)+12qrTr⁡{σi(t,x(t),x(t-τ(t)),x(t-τc(t)),r)TPσi×(t,x(t),x(t-τ(t)),x(t-τc(t)),r)T(t,x(t),x(t-τ(t)),x(t-τc(t)),r)}+∑s=1Mγrsqs12∑i=1Nei(t)TPei(t)≤qr{-η∑i=1Nei(t)Tei(t)+θ∑i=1Nei(t-τ(t))Tei(t-τ(t))+∑k=1npkϱkδkẽk(t)Tẽk(t)+∑k=1npkϱkẽk(t)TA(r)ẽk(t)+∑k=1npkϱkẽk(t)TB(r)ẽk(t-τc(t))-∑k=1npkϱkẽk(t)TΞẽk(t)+12p̌∑j=1N[∑i=1Nei(t)TΥj1rei(t)+∑i=1Nei(t-τ(t))TΥj2rei(t-τ(t))+∑i=1Nei(t-τc(t))TΥj3rei(t-τc(t))∑i=1Nei]}+∑s=1Mγrsqs12∑i=1Nei(t)TPei(t)=qr{-η∑i=1Nei(t)Tei(t)+θ∑i=1Nei(t-τ(t))Tei(t-τ(t))+12p̌∑i=1N∑j=1Nei(t-τc(t))T∑i=1N×Υj3rei(t-τc(t))∑i=1N∑j=1Nei(t-τc(t))T∑i=1Nei(t-τ(t))T}+∑s=1Mγrsqs12∑i=1Nei(t)TPei(t)+qr{∑j=1N∑k=1npkϱkẽk(t)TA(r)ẽk(t)+∑k=1npkϱkẽk(t)TB(r)ẽk(t-τc(t))-∑k=1npkϱkẽk(t)TΞẽk(t)+12p̌[∑i=1N∑j=1Nei(t)TΥj1rei(t)+∑i=1N∑j=1Nei(t-τ(t))TΥj2rei(t-τ(t))]+∑k=1npkϱkδkẽk(t)Tẽk(t)∑k=1npkϱkẽk(t)TΞ}=qr{∑i=1Nei(t)T(-ηIN+12p̌∑j=1NΥj1r-αrPΣ)ei(t)+∑i=1Nei(t-τ(t))T×(θ+12p̌∑j=1NΥj2r)ei(t-τ(t))+∑i=1Nei(t-τc(t))T(12p̌∑j=1NΥj3r+βrPΣ)×ei(t-τc(t))(12p̌∑j=1NΥj3r+βrPΣ)}+∑s=1Mγrsqs12∑i=1Nei(t)TPei(t)+qr{∑k=1npkϱkẽk(t)T[A(r)-Ξ+(δ̌+αr)IN]ẽk(t)+∑k=1npkϱkẽk(t)TB(r)ẽk(t-τc(t))-∑k=1npkϱkẽk(t-τc(t))Tβrẽk(t-τc(t))}≤qr{∑i=1Nei(t)T(-ηIN+12p̌∑j=1NΥj1r-αrPΣ)ei(t)+∑i=1Nei(t-τ(t))T×(βrIN+12p̌∑j=1NΥj2r)ei(t-τ(t))+∑i=1Nei(t-τc(t))T(12p̌∑j=1NΥj3r+βrPΣ)×ei(t-τc(t))(12p̆∑j=1NΥj3r+βrPΣ)}+∑s=1Mγrsqs12∑i=1Nei(t)TPei(t).
Let
(3.11)E(t)=12∑i=1Nei(t)TPei(t).
Then we have
(3.12)LV(t)≤-arqrE(t)+brqrE(t-τ(t))+crqrE(t-τc(t))+∑s=1MγrsqsE(t),
and by (3.8), we have
(3.13)LV(t)≤-(1+θ)E(t)+b̌q̌E(t-τ(t))+čq̌E(t-τc(t)).
Define
(3.14)W(t)=eγtV(t)
and use (3.13) to compute the operator
(3.15)LW(t)=eγt[γV(t)+LV(t)]≤eγt[γp̌E(t)-(1+θ)E(t)+b̌q̌E(t-τ(t))+čq̌E(t-τc(t))],
which, after applying the generalized Itô's formula, gives
(3.16)qreγtEE(t)=qreγt0EE(t0)+E∫t0tLW(s)ds
for any t>t0≥0. Hence we have
(3.17)qreγtEE(t)≤q̌EE(0)+E∫0teγs[γE(s)-(1-θ)E(s)+b̌q̌E(s-τ(s))+čq̌E(s-τc(s))]ds≤p̌q̌2∑i=1NE‖ξi‖2+(γ-(1+θ))∫0teγsEE(s)ds+b̌q̌eγτ∫0teγ(s-τ(s))EE(s-τ(s))ds+čq̌eγτc∫0teγ(s-τc(s))EE(s-τc(s))ds,
which, by using the change of variables s-τ(s)=u, gives
(3.18)∫0teγ(s-τ(s))EE(s-τ(s))ds=∫-τ(0)t-τ(t)eγuEE(u)du1-τ̇(t)≤p̌τ2(1-τ¯)∑i=1NE‖ξi‖2+11-τ¯∫0teγuEE(u)du,
and a further change of variables s-τc(s)=u gives
(3.19)∫0tes-τc(s)EE(s-τc(s))ds=∫-τc(0)t-τc(t)eγuEE(u)du1-τ̇c(t)≤p̌τc2(1-τ¯c)∑i=1NE‖ξi‖2+11-τ¯c∫0teγuEE(u)du.
By Condition (3.4), we obtain
(3.20)EE(t)≤p̌q̌2(1+b̌q̌τ1-τ¯eγτ+čq̌τc1-τ¯ceγτc)∑i=1NE‖ξi‖2e-γt
so that
(3.21)E‖e(t)‖2≤p̌q̌2p̂(1+b̌q̌τ1-τ¯eγτ+čq̌τc1-τ¯ceγτc)∑i=1NE‖ξi‖2e-γt.
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 letf∈QUAD(P,Δ,η,θ). If there exist positive constants αr and βr such that
(3.22)[A(r)s+δ̌IN-Ξ+αrINB(r)2B(r)T2-βrIN]≤0,forr=1,2,…,M,τ̌≤θT,τ̌≤(1-θ)T,(1b1+c1,1b2+c2,…,1bM+cM)>Γ̃-11M,
where γ>0 is the greatest root of the equation
(3.23)q̌γ-(1+θ)+b̌q̌eγτ+čq̌eγτc=0,Γ̃=diag⁡{a1,a2,…,aM}-Γ,ar=λmin⁡(2ηIn-p̌∑i=1NΥi1r+2α1PΣ)p̌,ǎ=max⁡r∈Sar,br=λmax⁡(∑i=1NPΥi2r+2ζIN)p̂,b̌=max⁡r∈Sbr,cr=λmax⁡(∑i=1NPΥi3r+2β2PΣ)p̂,č=max⁡r∈Scr,
then the solutions x1(t), x1(t),… and xN(t) of system (2.9) are globally and exponentially stable.WhenA(r(t))=A, B(r(t))=B, and σi(t,e(t),e(t-τ(t)),e(t-τc(t)),r(t))=σi(t,e(t),e(t-τ(t)),e(t-τc(t))), we can get the following corollary.Corollary 3.3.
Let assumptions (H1) and (H2) be true, and letf∈QUAD(P,Δ,η,θ). If there exist positive constants αr and βr such that
(3.24)[As+δ̌IN-Ξ+αrINB2BT2-βrIN]≤0,τ̌≤θT,τ̌≤(1-θ)T,0≤τ¯̌≤1-(b̌+č)1+θ,
where
(3.25)ar=λmin⁡(2ηIn-p̌∑i=1NΥi1r+2α1PΣ)p̌,ǎ=max⁡r∈Sar,br=λmax⁡(∑i=1NPΥi2r+2ζIN)p̂,b̌=max⁡r∈Sbr,cr=λmax⁡(∑i=1NPΥi3r+2β2PΣ)p̂,č=max⁡r∈Scr,
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(4.1)ds(t)=[-Cs(t)+Af(s(t))+Bg(s(t-τ(t)))]dt,
where f(s)=g(s)=tanh(s), τ(t)=1,(4.2)C=[1001],A=[2-0.1-54.5],B=[-1.5-0.1-0.2-4].
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. Thus(4.3)dxi(t)={f(t,xi(t),xi(t-τ(t)))+∑j=15aijrΣxj(t)+∑j=15bijrΣxj(t-τc(t))-ɛi(xi(t)-s(t))}dt+σir(t,x(t),x(t-τ(t)),x(t-τc(t)))dwi(t),i=1,2,…,5,r=1,2,3,(4.4)A1=[aij1]=20[-311101-411111-411111-410111-3],A2=[aij2]=15[-411111-311011-411111-411011-3],A3=[aij3]=37[-211001-311011-301010-100010-1],B1=[bij1]=0.1[-211000-211001-210011-200110-2]B2=[bij2]=0.2[-100100-101001-100010-100001-1],B3=[bij3]=0.7[-101001-100000-110001-101000-1]Ξ=100[1000001000000000000000000],Γ=[-3122-4223-5],τc(t)=0.1et1+etσi1(t,x(t),x(t-τ(t)),x(t-τc(t)))=0.1diag⁡{xi1(t)-xi+1,1(t),xi2(t)-xi+1,2(t)},σi2(t,x(t),x(t-τ(t)),x(t-τc(t)))=0.1diag⁡{xi1(t-τ(t))-xi+1,1(t-τ(t)),xi2(t-τ(t))-xi+1,2(t-τ(t))},σi3(t,x(t),x(t-τ(t)),x(t-τ(t)))=0.1diag⁡{xi1(t-τc(t))-xi+1,1(t-τc(t)),xi2(t-τc(t))-xi+1,2(t-τc(t))}.
Computations then yield τ=1, τ¯=0, τc=0.1, τ¯c=0.1, Υij=0.01I2 for i=1,2,…,N, and j=1,2. Let J={1,2} and the control strength ɛi=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, and(4.5)Γ̃-1=[0.105260.0143370.0190820.0224860.0951740.0187730.0221760.0250650.087313].
Therefore(4.6)q=[0.1391,0.1368,0.1350]T,
and, after solving (3.4), we obtain γ=0.0349.The initial conditions for this simulation arexij(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.The trajectories of the state variables ofxi1 and xi2 (i=1,2,…,5) in system (4.3) under pinning control.
(a)(b)The time evolution ofei1 and ei2 (i=1,2,…,5) in system (4.3) under pinning control.
(a)(b)Next, lettc(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.The trajectories of the state variables ofxi1 and xi2 (i=1,2,…,5) in system (4.3) under pinning control.
(a)(b)The time evolution ofei1 and ei2 (i=1,2,…,5) in system (4.3) under pinning control.
(a)(b)


## 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.



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*Source: 298095-2012-05-15.xml*