# Impulsive Pinning Markovian Switching Stochastic Complex Networks with Time-Varying Delay

Authors: Chen Xu; Jingyi Wang; Jianwen Feng; Yi Zhao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The  synchronization problem of stochastic complex networks with Markovian switching and time-varying delays is investigated by using impulsive pinning control scheme. The complex network possesses noise perturbations, Markovian switching, and internal and outer time-varying delays. Sufficient conditions for synchronization are obtained by employing the Lyapunov-Krasovskii functional method, Itö's formula, and the linear matrix inequality (LMI). Numerical examples are also given to demonstrate the validity of the theoretical results.


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## Body

## 1. Introduction and Model Description

Collective behaviors in complex networks and systems have attracted increasing attention in recent years due to their wide applications in physics, mathematics, engineering, biology, and so forth (see [1, 2] and references therein). While complex networks are ubiquitously found in nature and in the modern world, such as neural networks, socially interacting animal species, power networks, wireless sensor networks, Internet, and the World Wide Web.In the past few decades, the synchronization problems in complex networks have attracted increasing attention. Many synchronization patterns have been studied, including complete synchronization [3], cluster synchronization [4–6], phase synchronization [7], and partial synchronization [8]. There are several control methods to guide the dynamics of a complex network to a desired state, such as adaptive control [9], feedback control [10], intermittent control [11], fuzzy control [12], impulsive control [13, 14], and pinning control [5, 6, 15, 16]. Synchronization of complex networks holds particular promise for applications to many fields [17–21].Synchronization in complex dynamical networks is realized via information exchanges among the interconnect nodes [22]. The signal traveling along real physical system is usually perturbed randomly by the environmental elements, such as noises, the structures of the interconnections, time delays, and the positions of nodes [9]. One popular model is the Markovian switching model driven by continuous-time Markov chains in the sciences and industries (see [23–26] and references therein). In [23, 24], Mao et al. studied stability and controllability of stochastic differential delay equations with Markovian switching, while [25, 26] discussed the exponential stability of stochastic delayed neural networks. Liu et al. [26], on the other hand, investigated the synchronization of discrete-time stochastic complex networks with Markovian jumping and mode-dependent mixed time delays. In [16], Wang et al. investigated the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method, which is described as
(1)dxi(t)={∑j=1,i≠jN‍f(t,xi(t),xi(t-τ(t)))+∑j=1,i≠jN‍aij(r(t))Σ(xj(t)-xi(t))+∑j=1,i≠jN‍bij(r(t))Σ(xj(t-τc(t))-xi(t-τc(t)))+ui(t)∑j=1,i≠jN‍}dt+σi(x(t),x(t-τ(t)),x(t-τc(t)),r(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)ui(t)={-εi(xi(t)-s(t)),i=1,2,…,l,0,i=l+1,l+2,…,N,
and εi>0(i=1,2,…,l) are the control gains.Pinning control has been proved to be effective for the synchronization of complex dynamical networks with continuous state coupling [15, 16, 27]. In many systems, the impulsive effect is a common phenomenon due to instantaneous perturbations at certain moments [27, 28]. Impulsive control strategies have been widely used to stabilize and synchronize coupled complex dynamical systems, such as signal processing system, computer networks, automatic control systems, and telecommunications [13]. In [27], pinning impulsive strategy is proposed for the synchronization of stochastic dynamical networks with nonlinear coupling. Zhou et al. studied synchronization in complex delayed dynamical networks with impulsive effects in [28]. And Zhu et al., in [29], investigated the exponential stability of a class of stochastic neural networks with both Markovian jump parameters and mixed fixed time delays. Can the stochastic dynamical network with Markovian switching and time-varying delays be synchronized by impulsive pinning control? This paper is devoted to solving this problem.In this paper, we study the synchronization of stochastic complex networks with Markovian switching by using the impulse control method. We consider a kind of stochastic complex networks with internal time-varying delayed couplings, Markovian switching, and Wiener processes. By applying the Lyapunov-Krasovskii functional method, Itö’s formula and the linear matrix inequality (LMI), some sufficient conditions for synchronization of these networks are derived. Numerical examples are finally given to demonstrate the effectiveness of the proposed impulsive pinning strategy.Notations. Throughout this paper, ℝn will denote the n-dimensional Euclidean space and ℝn×n the set of all n×n real matrices. The superscript T will denote the transpose of a matrix or a vector. And Tr(·) stands for the trace of the corresponding matrix. 1n=(1,1,…,1)T∈ℝn, and In is the n-dimensional identity matrix. For square matrices M, the notation M>0 (<0) will mean that M is a positive-definite (negative-definite) matrix. λmax(A) and λmin(A) will denote the greatest and least eigenvalues of a symmetric matrix, respectively. pˇ=max{p1,p2,…,pn}, and p^=min{p1,p2,…,pn}.


## 2. Preliminaries

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) will denote the family of continuous functions ϕ from [-τ,0] to ℝn with the uniform norm ∥ϕ∥2=sup-τ≤s≤0ϕ(s)Tϕ(s). And Cℱ02([-τ,0];ℝn) denotes 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 Markovian switching
(3)dxi(t)={∑j=1N‍f(xi(t),xi(t-τ(t)))+∑j=1N‍aij[r(t)]Σxj(t)+∑j=1N‍bij[r(t)]Σxj(t-τc(t))}dt+σi[r(t)](x(t),x(t-τ(t)),x(t-τc(t)))dw(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(xi(t),xi(t-τ(t)))=[f1(xi(t),xi(t-τ(t))),f2(xi(t),xi(t-τ(t))),…,fn(xi(t),xi(t-τ(t)))]T is a continuous vector-valued function, Σ=diag{ϱ1,ϱ2,…,ϱn} is an inner coupling matrix of the networks that satisfies ϱj>0, j=1,2,…,n, and r(t) are the continuous-time Markov processes that describe the evolution of the nodes at time t. Here, A[r(t)]=[aij[r(t)]]∈ℝn×n and B[r(t)]=[bij[r(t)]]∈ℝn×n are the outer coupling matrices of the network at time t at nodes r(t), t-τc(t), respectively, such that aij[r(t)]≥0 for i≠j, aii[r(t)]=-∑j=i,j≠iN‍aij[r(t)], bij[r(t)]≥0 for i≠j and bii[r(t)]=-∑j=i,j≠iN‍bij[r(t)]. Figure 1 shows the topology structures of the switching networks for 5 nodes.  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[r(t)](x(t),x(t-τ(t)),x(t-τc(t)))=σi[r(t)](x1(t),…,xn(t),x1(t-τ(t)),…,xn(t-τ(t)),x1(t-τc(t)),…,xn(t-τc(t)))∈ℝn×n and w(t)=(w1(t),w2(t),…,wn(t))T∈ℝn is a bounded vector-form Weiner process, satisfying
(4)𝔼wj(t)=0,𝔼wj2(t)=1,𝔼wj(t)wj(s)=0(s≠t).
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]∈ℝM×M given by
(5)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, In this paper, A[r] is assumed to be symmetric and irreducible, and B[r] is assumed to be symmetric, for r=1,2,…,M.The topology structures of the switching networks for 5 nodes; (A1) and (A2) the topology structures of the coupling matrixA1 and A2, respectively; (B1) and (B2) the topology structures of the coupling matrix B1 and B2, respectively.
(a)
(b)The initial conditions associated with (3) are
(6)xi(s)=ξi(s),-τˇ≤s≤0,i=1,2,…,N,
where τˇ=max{τ(t),τc(t)}, ξi∈Cℱ0b([-τˇ,0],ℝn) with the norm ∥ξi∥2=sup-τˇ≤s≤0ξi(s)Tξi(s).The impulse controllers are defined by(7)Δxi(tk)=xi(tk+)-xi(tk-)=ϵikxi(tk-),t=tk,k∈Z+,i=1,2,…,N,
where ϵik are constants, and {t1,t2,t3,…} is the impulsive sequence of strictly increasing impulsive instants satisfying limk→∞tk=+∞, and tk-tk-1=T for k>1.In the case that system (3) reaches synchronization, that is, x1(t)=x2(t)=⋯=xN(t)=s(t), we have the following synchronized state equation:
(8)ds(t)=f(s(t),s(t-τ(t)))dt+σ(s(t),s(t-τ(t)))dw(t).In the paper, we would control the system (3) to the desired trajectory s(t). Define ei(t)=xi(t)-s(t)(i=1,2,…,N) as the synchronization error. Then, according to the controller (7), the error system is
(9)dei(t)={∑j=1N‍f(xi(t),xi(t-τ(t)))-f(s(t),s(t-τ(t)))+∑j=1N‍aij[r(t)]Σej(t)+∑j=1N‍bij[r(t)]Σej(t-τc(t))}dt+σi[r(t)](e(t),e(t-τ(t)),e(t-τc(t)))dw(t),t≠tk,k∈Z+,i=1,2,…,N,Δei(tk)=ϵikei(tk-),t=tk,k∈Z+,i=1,2,…,N,
where σi[r(t)](e(t),e(t-τ(t)),e(t-τc(t)))=σi[r(t)](x(t),x(t-τ(t)),x(t-τc(t)))-σ(s(t),s(t-τ(t))).Definition 1 (see [16, 27]).
The complex network (3) is said to be exponentially synchronized in mean square if the trivial solution of system (9) is such that
(10)∑i=1N‍𝔼∥ei(t,t0,ξi)∥2≤Ke-κt,
for some K>0 and κ>0 under any initial data ξi∈𝒞ℱ0b([-τˇ,0];ℝn).Definition 2 (see [9, 11, 16]).
A continuous functionf(x,y):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 constants η>0,  θ>0 such that f(·) satisfies the condition
(11)(x-y)TP((f(x,z)-f(y,w))-ΔΣ(x-y))≤-η(x-y)T(x-y)+θ(z-w)T(z-w)
for all x,y,z,w∈ℝn.The following assumptions are usual and will be used throughout this paper for establishing the synchronization conditions [9, 11, 16].(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 τ˙c(t)<τ-c<1. Let τ-ˇ=max{τ-,τ-c}.(H2)
There exist positive definite constant matricesΥi1[r], Υi2[r] and Υi3[r] for i=1,2,…,N and r=1,2,…,M such that
(12)Tr[σi[r](e(t),e(t-τ(t)),e(t-τc(t)))T×σi[r](e(t),e(t-τ(t)),e(t-τc(t)))]≤∑j=1N‍ej(t)TΥi1[r]ej(t)+∑j=1N‍ej(t-τ(t))TΥi2[r]ej(t-τ(t))+∑j=1N‍ej(t-τc(t))TΥi3[r]ej(t-τc(t)).Remark 3.
Considering Definition2 and assumption (H2), there exists a unique solution of (9) under the initial data ξi∈𝒞ℱ0b([-τ,0];ℝn) (see [23, 24]).Lemma 4 (see [23, 24]).
Consider a stochastic delayed differential equation with Markovian switching of the form(13)dx(t)=f(x(t),x(t-τ),r(t))dt+σ(x(t),x(t-τ),r(t))dω(t)
on t≥0 with initial value x0=ξ∈CF0b([-τ,0];ℝn), where
(14)f:ℝn×ℝn×S→ℝn,σ:ℝn×ℝn×S→ℝn×m.
LetC2,1(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
(15)ℒV(t,x,r)=Vt(t,x,r)+Vx(t,x,r)f(x,r)+12Tr[σ(x,r)TVxxσ(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), and Vxx(t,x,r)=(∂2V(t,x,r)/∂xixj)n×n. If   V∈C2,1(ℝ+×ℝn×S;ℝ+), and then
(16)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

In this section, we will deduce our main results.Theorem 5.
Let assumptions (H1) and (H2) be true and letf∈QUAD(P,Δ,η,θ). If there exist positive constants αr and βr such that
(17)[A[r]+δˇIN-αrINB[r]2B[r]2-βrIN]≤0,forr=1,2,…,M,τˇ≤θT,τˇ≤(1-θ)T,0≤τ-ˇ≤1-qˇ(bˇ+cˇ)1+θ,φ(τˇ+T)+2lnqˇq^|1+ϵv|-γT<0,(1b1+c1,1b2+c2,…,1bM+cM)T>Γ~-11M,
where
(18)φ=1+θ+γqˇ+bˇqˇ1-τ-eγτ+cˇqˇ1-τ-ceγτc,Γ~=diag{a1,a2,…,aM}+Γ,aˇ=maxr∈Sar,ar=λmax(-2ηIn+pˇ∑i=1NΥi1[r]+2αrPΣ)pˇ,bˇ=maxr∈Sbr,br=λmax(∑i=1NPΥi2[r]+2θIN)p^,cˇ=maxr∈Scr,cr=λmax(∑i=1NPΥi3[r]+2βrPΣ)p^,
then the solutions e1(t),e2(t),…, and eN(t) of system (9) are exponentially stable in mean square. It means that the complex dynamical network (3) can be exponentially controlled to the objective trajectory s(t) under the controllers (7).Proof.
By (44), there exists a sufficiently small constant θ>0 such that
(19)(1b1+c1,1b2+c2,…,1bM+cM)T≥(1+θ)Γ~-11.
Set (1+θ)Γ~-11=q=(q1,q2,…,qM)T. Then Γ~q=(1+θ)1M, that is,
(20)(br+cr)qr≤1,arqr+∑s=1M‍γrsqs=1+θ.
For 1≤r≤M, define the Lyapunov-Krasovskii function
(21)V(e(t),r)=qr12∑i=1N‍ei(t)TPei(t),
and let e~k(t)=(e1k(t),e2k(t),…,eNk(t))T, k=1,2,…,n. For any t∈(tk-1,tk], k=1,2,…, by Lemma 4, we have(22)ℒV(e(t),r)=qr∑i=1N‍ei(t)TP{∑j=1N‍f(xi(t),xi(t-τ(t)))-f(s(t),s(t-τ(t)))+∑j=1N‍aij[r]Σej(t)+∑j=1N‍bij[r]Σej(t-τc(t))}+12qr∑i=1N‍Tr{σi(x(t),x(t-τ(t)),x(t-τc(t)),r)T×Pσi(x(t),x(t-τ(t)),x(t-τc(t)),r)σi(x(t),x(t-τ(t)),x(t-τc(t)),r)T}+∑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=1n‍pkϱkδke~k(t)Te~k(t)+∑k=1n‍pkϱke~k(t)TA[r]e~k(t)+∑k=1n‍pkϱke~k(t)TB[r]e~k(t-τc(t))+12pˇ∑j=1N‍[∑i=1N‍ei(t)TΥj1[r]ei(t)+∑i=1N‍ei(t-τ(t))TΥj2[r]ei(t-τ(t))+∑i=1N‍ei(t-τc(t))TΥj3[r]ei(t-τc(t))]}+∑s=1M‍γrsqs12∑i=1N‍ei(t)TPei(t)=qr{∑i=1N‍ei(t)T(-ηIN+12pˇ∑j=1N‍Υj1[r]+αrPΣ)ei(t)+∑i=1N‍ei(t-τ(t))T(θ+12pˇ∑j=1N‍Υj2[r])ei(t-τ(t))+∑i=1N‍ei(t-τc(t))T(12pˇ∑j=1N‍Υj3[r]+βrPΣ)ei(t-τc(t))}+∑s=1M‍γrsqs12∑i=1N‍ei(t)TPei(t)+qr{∑k=1n‍pkϱke~k(t)T[A[r]+(δˇ-αr)IN]e~k(t)+∑k=1n‍pkϱke~k(t)TB[r]e~k(t-τc(t))-∑k=1n‍pkϱke~k(t-τc(t))Tβre~k(t-τc(t))}≤qr{∑i=1N‍ei(t)T(-ηIN+12pˇ∑j=1N‍Υj1[r]+αrPΣ)ei(t)+∑i=1N‍ei(t-τ(t))T(θIN+12pˇ∑j=1N‍Υj2[r])ei(t-τ(t))+∑i=1N‍ei(t-τc(t))T(12pˇ∑j=1N‍Υj3[r]+βrPΣ)ei(t-τc(t))}+∑s=1M‍γrsqs12∑i=1N‍ei(t)TPei(t).Let
(23)E(t)=12∑i=1N‍ei(t)TPei(t);
then we have
(24)ℒV(t)≤arqrE(t)+brqrE(t-τ(t))+crqrE(t-τc(t))+∑s=1M‍γrsqsE(t),
and by (20), we have
(25)ℒV(t)≤(1+θ)E(t)+bˇqˇE(t-τ(t))+cˇqˇE(t-τc(t)).
Define
(26)W(t)=eγtV(t).
Use (25) to compute the operator
(27)ℒW(t)=eγt[γV(t)+ℒV(t)]≤eγt[+bˇqˇE(t-τ(t))+cˇqˇE(t-τc(t))γqˇE(t)+(1+θ)E(t)+bˇqˇE(t-τ(t))+cˇqˇE(t-τc(t))].
The generalized Itô’s formula gives
(28)eγt𝔼V(t)=eγt0𝔼V(t0)+𝔼∫t0t‍ℒW(s)ds,
for any tk>t>t0>tk-1≥0. Hence we have
(29)eγt𝔼V(t)≤eγt0𝔼V(t0)+𝔼∫t0t‍eγs[γqˇE(s)+(1+θ)E(s)+bˇqˇE(s-τ(s))+cˇqˇE(s-τc(s))]ds≤qˇeγt0𝔼E(t0)+(γqˇ+1+θ)×∫t0t‍eγs𝔼E(s)ds+bˇqˇeγτ∫t0t‍eγ(s-τ(s))𝔼E(s-τ(s))ds+cˇqˇeγτc∫t0t‍eγ(s-τc(s))𝔼E(s-τc(s))ds.
By changing variable s-τ(s)=u, we have
(30)∫t0t‍eγ(s-τ(s))𝔼E(s-τ(s))ds=∫t0-τ(t0)t-τ(t)‍eγu𝔼E(u)du1-τ˙(t)≤11-τ-∫t0-τt‍eγu𝔼E(u)du.
Similarly, we have
(31)∫t0t‍eγ(s-τc(s))𝔼E(s-τ(s))ds≤11-τ-c∫t0-τct‍eγu𝔼E(u)du.
Substituting (30) and (31) into (29), we get
(32)eγt𝔼V(t)≤qreγt0𝔼V(t0)+φ∫t0-τˇt‍eγu𝔼E(u)du.
By using Gronwall inequality, we have
(33)𝔼E(t)≤qˇq^𝔼E(t0)eφ(t-t0+τˇ)+γ(t0-t).
On the other hand, from the construction of E(t), we have
(34)E(tk)≤(1+ϵk)2E(tk-),
where |1+ϵk|=maxi=1,2,…,N|1+ϵik|.
According to (33)-(34), let k=⌊(t-t0)/T⌋, for any t∈[tk-1,tk), and we get
(35)𝔼V(t)≤qˇq^𝔼E(tk)eφ(t-tk+τˇ)+γ(tk-t)≤qˇq^𝔼V(tk-1-)eφ(t-tk-1+τˇ)+γ(tk-t)+2ln|1+ϵk-1|≤⋯≤(qˇq^)k-1𝔼E(0)eφ(t+kτˇ)-γt+∑v=1k-12ln|1+ϵv|.
Let|1+ϵ|=maxv∈Z+|1+ϵv|, and we have
(36)𝔼E(t)≤𝔼E(0)eφ(t+kτˇ)-γt+2(k-1)ln(qˇ/q^)|1+ϵ|.
Using condition (43) of Theorem 5, there exist a number η such that 𝔼E(t)≤𝔼E(t0)e-ηt. Hence, 𝔼∥ei(t)∥≤(E(t0)/pˇ)1/2e-(η/2)(t-t0). The proof of Theorem 5 is completed.Remark 6.
The stochastic networks studied before are without topological switch, and the time delays are always assumed to be fixed. However, for the sake of applications in the real work, these two points above should be taken into consideration. Of course, it will enhance the difficulties of the investigations on this network. For example, if the network has Markovian switching topology, the structure of the network is fast varying and the Lyapunov function is hard to be determined. By using the Lyapunov-Krasovskii functional, Itö’s formula, and LMI, the exponential stability criterion of the pinning impulsive controlled Markovian switching stochastic dynamical network with time-varying delays was obtained. This also showed that the impulsive pinning control is a kind of cheap control strategy for guiding complex dynamical networks to the objective trajectory.To make Theorem5 more applicative, we give the following corollaries.When complex dynamic networks (3) are considered without coupled delay time (B[r(t)]=B), we can get the following corollary.Corollary 7.
Let assumptions (H1) and (H2) be true and letf∈QUAD(P,Δ,η,θ). If there exist positive constants αr and βr such that
(37)A[r]+δˇIN-αrIN≤0,forr=1,2,…,M,τ≤θT,τ≤(1-θ)T,0≤τ-≤1-qˇbˇ1+θ,φ(τˇ+T)+2lnqˇq^|1+ϵ|-γT<0,(1b1,1b2,…,1bM)T>Γ~-11M,
where
(38)φ=1+θ+γqˇ+bˇqˇ1-τ-eγτ,Γ~=diag{a1,a2,…,aM}+Γ,aˇ=maxr∈Sar,ar=λmax(-2ηIn+pˇ∑i=1NΥi1[r]+2αrPΣ)pˇ,bˇ=maxr∈Sbr,br=λmax(∑i=1NPΥi2[r]+2θIN)p^,
then the solutions e1(t),e1(t),…, and eN(t) of system (9) are exponentially stable in mean square.In another case, when we consider the system (3) without Markov switching, that is, A[r(t)]=A, B[r(t)]=B, and σi[r(t)](e(t),e(t-τ(t)),e(t-τc(t)))=σi(e(t),e(t-τ(t)),e(t-τc(t))), we can get another corollary.Corollary 8.
Let assumptions (H1) and (H2) be true and letf∈QUAD((P,Δ,η,θ). If there exist positive constants α and β such that
(39)[A+δˇIN-αINB2B2-βIN]≤0,τˇ≤θT,τˇ≤(1-θ)T,0≤τ-ˇ≤1-b+ca,φ(τˇ+T)+2ln|1+ϵv|-γT<0,b+c<a,
where
(40)φ=a+γ+b1-τ-eγτ+c1-τ-ceγτc,a=λmax(-2ηIn+pˇ∑i=1NΥi1+2αPΣ)pˇ,b=λmax(∑i=1NPΥi2+2θIN)p^,c=λmax(∑i=1NPΥi3+2βrPΣ)p^,
then the solutions e1(t),e2(t),…, and eN(t) of system (9) are exponentially stable in mean square.Furthermore, when we address the system (3) with fixed time delays, that is, τ(t)=τ, τc(t)=τc, the following corollary also can be obtained.Corollary 9.
Let assumptions (H1) and (H2) be true and letf∈QUAD(P,Δ,η,θ). If there exist positive constants αr and βr such that
(41)[A[r]+δˇIN-αrINB[r]2B[r]2-βrIN]≤0,forr=1,2,…,M,(42)τˇ≤θT,τˇ≤(1-θ)T,(43)φ(τˇ+T)+2lnqˇq^|1+ϵv|-γT<0,(44)(1b1+c1,1b2+c2,…,1bM+cM)T>Γ~-11M,
where
(45)φ=1+θ+γqˇ+bˇqˇeγτ+cˇqˇeγτc,Γ~=diag{a1,a2,…,aM}+Γ,aˇ=maxr∈Sar,ar=λmax(-2ηIn+pˇ∑i=1NΥi1[r]+2αrPΣ)pˇ,bˇ=maxr∈Sbr,br=λmax(∑i=1NPΥi2[r]+2θIN)p^,cˇ=maxr∈Scr,cr=λmax(∑i=1NPΥi3[r]+2βrPΣ)p^,
then the solutions e1(t),e2(t),…, and eN(t) of system (9) are exponentially stable in mean square.Remark 10.
In [29], the exponential stability of a class of stochastic neural networks with both Markovian jump parameters and mixed fixed time delays were investigated. Therefore, we could see our results as a further research about the stochastic dynamic network of [29].


## 4. Numerical Simulation

In this section, we present some numerical simulation results that validate the theorem in the previous section.Consider the chaotic delayed neural network(46)ds(t)={-Cs(t)+Af(s(t))+Bg(s(t-τ(t)))}dt+σ(s(t),s(t-τ(t)))dw(t),
where f(s)=g(s)=tanh(s), τ(t)=1, σ(s(t),s(t-τ(t)))=diag{s1(t),s2(t)},
(47)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 (11) is satisfied. Thus
(48)dxi(t)={f(xi(t),xi(t-τ(t)))+∑j=15‍aij[r]Σxj(t)+∑j=15‍bij[r]Σxj(t-τc(t))}dt+σi[r](x(t),x(t-τ(t)),x(t-τc(t)))dwi(t),i=1,2,…,5,r=1,2,
and Γ=[-332-2], τc(t)=0.1(et/(1+et)),
(49)σi[1](x(t),x(t-τ(t)),x(t-τc(t)))=0.1diag{xi1(t),xi2(t)},σi[2](x(t),x(t-τ(t)),x(t-τc(t)))=0.1diag{xi1(t-τ(t)),xi2(t-τ(t))}.
Computations then yield τ=1, τ-=0, τc=0.1, τ-c=0.1 and Υij=0.1I2 for i=1,2,…,N. Then the solutions of inequalities (41)–(44) are (by using the Matlab LMI toolbox) α1=2.500, β1=0.001, a1=4.305, b1=5.055, c1=0.109; α2=4.006, β2=0.009, a2=5.105, b2=5.030, and c2=0.065.The initial conditions for this simulation arexij(t0) which are constants, for i=1,2,…,5,j=1,2 and the trajectories of the impulse control gains are shown in Figure 2. Figure 3 shows the time evolution of the synchronization errors with impulse control.The trajectories of the state variables ofxi1 and xi2(i=1,2,…,5) in system (48) by impulse control.
(a)
(b)The time evolution ofei1 and ei2(i=1,2,…,5) in system (48) by impulse 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 hybrid couplings. We achieved synchronization by applying an impulse control scheme to a small fraction of the nodes and derived sufficient conditions for stability of synchronization. Finally, we considered some numerical examples that illustrate the theoretical analysis.



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*Source: 461924-2013-08-12.xml*