Mean-Square Exponential Synchronization of Markovian ...downloads.hindawi.com/journals/aaa/2012/298095.pdf · Stochastic perturbations and time delays are important considerations

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 298095, 18 pagesdoi:10.1155/2012/298095

Research ArticleMean-Square Exponential Synchronization ofMarkovian Switching StochasticComplex Networks with Time-Varying Delays byPinning Control

Jingyi Wang,1 Chen Xu,2 Jianwen Feng,2Man Kam Kwong,3 and Francis Austin3

1 College of Information and Engineering, Shenzhen University, Shenzhen 518060, China2 College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, China3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

Correspondence should be addressed to Chen Xu, xuchen [email protected] Jianwen Feng, [email protected]

Received 19 December 2011; Accepted 2 March 2012

Academic Editor: Marcia Federson

Copyright q 2012 Jingyi Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper investigates the mean-square exponential synchronization of stochastic complex net-works 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 thecomplex network is subject to noise perturbations, Markovian switching, and internal and outertime-varying delays. Sufficient conditions for mean-square exponential synchronization are ob-tained by using the Lyapunov-Krasovskii functional, Ito’s formula, and the linearmatrix 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). Coupledbiological systems such as neural networks and socially interacting animal species are simpleexamples 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 worldthat it is absolutely essential for us to have a thorough understanding of their dynamicalbehavior, and complex networks synchronization holds particular promise for applicationsto many fields (e.g., population dynamics, power systems and automatic control [2–6]).

2 Abstract and Applied Analysis

Chaos synchronization is a phenomenon that has been widely investigated since itwas first discovered by Pecora and Carroll in 1990, and it is a process in which two ormore dynamical systems seek to adjust a certain prescribed property of their motion to acommon behavior in the limit as time tends to infinity [7]. Many synchronization patternshave been explored (e.g., complete synchronization [8], cluster synchronization [9], phasesynchronization [10], and partial synchronization [11]), and synchronization can be achievedby the use of adaptive control [12], feedback control [13], intermittent control [14], fuzzycontrol [15], impulsive control [16], or pinning control [17–20].

Stochastic perturbations and time delays are important considerations when sim-ulating realistic complex networks because signals traveling along real physical systemsare usually randomly perturbed by the environmental elements [12] and time-delayed bychaotic 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 networkswith coupling delays ([21–27]), most of the stochastically perturbed networks that havebeen investigated were one dimensional in the sense that the same noise impacted all thetransmitted signals ([28–30]). Results on the more realistic vector-formed perturbations (inwhich 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 synchronizationof coupled neural networks. One popular model for stochastics in the sciences and industriesis the Markovian switching model driven by continuous-time Markov chains ([32–37]), andMao [32, 33] considered the stability of stochastic delayed differential equations using thismodel while others ([34–36]) discussed the exponential stability of stochastic delayed neuralnetworks. Liu et al. [37], on the other hand, investigated the synchronization of discrete-timestochastic complex networks with Markovian jumping and mode-dependent mixed timedelays.

Pinning control is a technique that applies controllers to only a small fraction of thenodes in a network, and the technique is important because it greatly reduces the number ofcontrolled 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 isrequired for synchronization, namely, for complex networks that have either a symmetric oran 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 byusing 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 Zhaoet al. [20]).

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

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

Abstract and Applied Analysis 3

2. Preliminaries

2.1. Notations

Throughout this paper, Rn shall denote the n-dimensional Euclidean space and R

n×n the setof 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 ∈ R

n andIn 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), andλmin(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 isright continuous with F0 containing all the P-null sets. C([−τ, 0];Rn) shall denote the familyof continuous functions φ from [−τ, 0] to R

n with the uniform norm ‖φ‖ = sup−τ≤s≤0|φ(s)|andC2

F0([−τ, 0];Rn) the family of allF0 measurable,C([−τ, 0];Rn)-valued stochastic variables

ξ = {ξ(θ) : −τ ≤ θ ≤ 0} such that∫0−τ E|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 switching

dxi(t) =

⎧⎨

⎩f(t, xi(t), xi(t − τ(t))) +

N∑

j=1,i /= j

aij

(raij (t)

)Σ(xj(t) − xi(t)

)

+N∑

j=1,i /= j

bij(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,

(2.1)

where xi(t) = (xi1(t), xi2(t), . . . , xin(t))T ∈ R

n 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(�1, �2, . . . , �n) is an inner couplingof the networks that satisfies �j > 0, j = 1, 2, . . . , n, and raij (t), rbij (t) and rσi(t) are thecontinuous-time Markov processes that describe the evolution of the modes at time t. Here,A(ra(t)) = [aij(raij (t))] ∈ R

n×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, suchthat aij(raij (t)) ≥ 0 for i /= j, aii(raii(t)) = −∑N

j=i,j /= i aij(raij (t)), bij(rbij (t)) ≥ 0 for i /= j, andbii(rbii(t)) = −∑N

j=i,j /= i bij(rbij (t)). τ(t) is the inner time-varying delay satisfying τ ≥ τ(t) ≥ 0and 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)) ∈ R

n×n and wi(t) = (wi1(t), wi2(t), . . . , win(t))T ∈ R

n is a bounded vector-formWeiner process, satisfying

Ewij(t) = 0, Ew2ij(t) = 1, Ewij(t)wij(s) = 0 (s /= t). (2.2)

In this paper, A(ra(t)) is assumed to be irreducible in the sense that there are no isolatednodes.


The initial conditions associated with (2.1) are

xi(s) = ξi(s), −τ ≤ s ≤ 0, i = 1, 2, . . . ,N, (2.3)

where τ = max{τ(t), τc(t)}, ξi ∈ CbF0([−τ , 0],Rn) with the norm ‖ξi‖2 = sup−τ≤s≤0ξi(s)

T ξi(s),and our objective is to control system (2.1) so that it stays in the trajectory s(t) ∈ R

n of thesystem

ds(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 lnodes be controlled. Then the network is described by

dxi(t) =

⎧⎨

⎩f(t, xi(t), xi(t − τ(t))) +

N∑

j=1,i /= j

aij

(raij (t)

)Σ(xj(t) − xi(t)

)

+N∑

j=1,i /= j

bij(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,

(2.5)

where ui(t) (i = 1, 2, . . . ,N) are the linear state feedback controllers that are defined by

ui(t) =

{−εi(xi(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} ∈R

n×n. Define ei(t) = xi(t) − s(t) (i = 1, 2, . . . ,N) as the synchronization error. Then, accordingto the controller (2.6), the error system is

dei(t) =

⎧⎨

⎩f(t, xi(t), xi(t − τ(t))) − f(t, si(t), si(t − τ(t))) +

N∑

j=1,i /= j

aij

(raij (t)

)Σ(ej(t) − ei(t)

)

+N∑

j=1,i /= j

bij(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.

(2.7)

Remark 2.1. Since the Markov chains raij (t), rbij (t), and rσi(t) are independent, we have anequivalent system as follows.


Let r(t), t > 0 be a right-continuous Markov chain on a probability space that takesvalues in a finite state space S = 1, 2, . . . ,M with a generator Γ = [γij] ∈ R

M×M given by

P{r(t + Δ) = j | r(t) = i

}=

{γijΔ + o(Δ) if i /= j,

1 + γiiΔ + o(Δ) if i = j,(2.8)

for some Δ > 0. Here γij = 0 is the transition rate from i to j if i /= j and γii = −∑i /= j γij ,

dei(t) =

⎧⎨

⎩f(t, xi(t), xi(t − τ(t))) − f(t, si(t), si(t − τ(t))) +

N∑

j=1

aij(r(t))Σej(t)

+N∑

j=1

bij(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.

(2.9)

Definition 2.2. The complex network (2.5) is said to be exponentially synchronized in mean squareif the trivial solution of system (2.9) is such that

N∑

i=1

E‖ei(t, t0, ξi)‖2 ≤ Ke−κt, (2.10)

for some K > 0 and some κ > 0 for any initial data ξi ∈ CbF0([−τ, 0];Rn).

Definition 2.3 (see [12]). A continuous function f(t, x, y) : [0,+∞] × Rn × R

n → Rn is said

to belong to the function class QUAD, denoted by f ∈ QUAD(P,Δ, η, θ) for some givenmatrix Σ = diag{�1, �2, . . . , �n} 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 > 0such 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 ∈ Rn.

Remark 2.4. The function class QUAD includes almost all the well-known chaotic systemswith or without delays such as the Lorenz system, the Rossler system, the Chen system, thedelayed Chua’s circuit, the logistic delayed differential system, the delayed Hopfield neuralnetwork, and the delayed CNNs. We shall simply write

p = max{p1, p2, . . . , pn

}, p = min

{p1, p2, . . . , pn

}, δ = max{δ1, δ2, . . . , δn}. (2.12)

The following assumptions will be used throughout this paper for establishing thesynchronization 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(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 constantmatrices Υr

i1, Υri2, and Υr

i3 for i = 1, 2, . . . ,N and r = 1, 2, . . . ,M such that

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)

]

≤N∑

j=1

ej(t)TΥri1ej(t) +

N∑

j=1

ej(t − τ(t))TΥri2ej(t − τ(t)) +

N∑

j=1

ej(t − τc(t))TΥri3ej(t − τc(t)).

(2.13)

Lemma 2.5 (see [32, 33, the generalized Ito formula]). Consider a stochastic delayed differentialequation 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 ∈ CbF0([−τ, 0];Rn), where

f : Rn × R+ × S −→ R

n, σ : Rn × R+ × S −→ R

n×m. (2.15)

Let C2,1(R+ × Rn;R+) be the family of all the nonnegative functions V (t, x, r) on R+ × R

n × S thatare twice continuously differentiable in x and once differentiable in t. Let V ∈ C2,1(R+ × R

n × S;R+).Define an operator LV from R

n × R+ × S to Rn by

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)

]+

M∑

j=1

γijV(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 V ∈ C2,1(R+ × R

n × S;R+), then

EV (t, x(t), r) = EV (t0, x(t0), r) + E

∫ t

t0

LV (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 existpositive constants αr and βr such that

⎡

⎢⎣A(r)s + δIN − Ξ + αrIN

B(r)2

B(r)T

2−βrIN

⎤

⎥⎦ ≤ 0, for r = 1, 2, . . . ,M, (3.1)

τ ≤ θT, τ ≤ (1 − θ)T, 0 ≤ τ ≤ 1−q(b + c

)

1 + θ, (3.2)

(1

b1 + c1,

1b2 + c2

, . . . ,1

bM + cM

)> Γ−11M, (3.3)

where γ > 0 is the greatest root of the equation

qγ−(1 + θ) +bq

1 − τeγτ +

cq

1 − τceγτc = 0, (3.4)

Γ = diag{a1, a2, . . . , aM} − Γ,

ar =λmin

(2ηIn − p

∑Ni=1 Υ

ri1 + 2α1PΣ

)

p, a = max

r∈Sar,

br =λmax

(∑Ni=1 PΥ

ri2 + 2ζIN

)

p, b = max

r∈Sbr ,

cr =λmax

(∑Ni=1 PΥ

ri3 + 2β2PΣ

)

p, c = max

r∈Scr .

(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

(1

b1 + c1,

1b2 + c2

, . . . ,1

bM + cM

)≥ (1 + θ)Γ−11. (3.6)

Set (1 + θ)Γ−11 = q = (q1, q2, . . . , qM)T . Then

Γq = (1 + θ)1M, (3.7)

that is,

(br + cr)qr ≤ 1, arqr −M∑

s=1

γsrqr = 1 + θ. (3.8)


For 1 ≤ r ≤ M, define the Lyapunov-Krasovskii function

V (t, e(t), r) = qr12

N∑

i=1

ei(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

LV (t, e(t), r) = qrN∑

i=1

ei(t)TP

⎧⎨

⎩f(t, xi(t), xi(t − τ(t))) − f(t, s(t), s(t − τ(t)))

+N∑

j=1

aij(r)Σej(t) +N∑

j=1

bij(r)Σej(t − τc(t)) + ui(t)

⎫⎬

⎭

+12qr

N∑

i=1

Tr{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)}+

M∑

s=1

γrsqs12

N∑

i=1

ei(t)TPei(t)

= qrN∑

i=1

ei(t)TP{f(t, xi(t), xi(t − τ(t))) − f(t, s(t), s(t − τ(t))) −ΔΣei(t)

}

+ qrN∑

i=1

ei(t)TPΔΣei(t) + qrN∑

i=1

N∑

j=1

aij(r)ei(t)TPΣej(t)

+ qrN∑

i=1

N∑

j=1

bij(r)ei(t)TPΣej(t − τc(t)) − qrl∑

i=1

εiei(t)TPΣei(t)

+12qr Tr

{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)}+

M∑

s=1

γrsqs12

N∑

i=1

ei(t)TPei(t)

≤ qr

{

−ηN∑

i=1

ei(t)Tei(t) + θN∑

i=1

ei(t − τ(t))Tei(t − τ(t)) +n∑

k=1

pk�kδkek(t)T ek(t)

+n∑

k=1

pk�kek(t)TA(r)ek(t) +

n∑

k=1

pk�kek(t)TB(r)ek(t − τc(t))

−n∑

k=1

pk�kek(t)TΞek(t)


+12p

N∑

j=1

[N∑

i=1

ei(t)TΥrj1ei(t) +

N∑

i=1

ei(t − τ(t))TΥrj2ei(t − τ(t))

+N∑

i=1

ei(t − τc(t))TΥrj3ei(t − τc(t))

]}

+M∑

s=1

γrsqs12

N∑

i=1

ei(t)TPei(t)

= qr

⎧⎨

⎩−η

N∑

i=1

ei(t)Tei(t) + θN∑

i=1

ei(t − τ(t))Tei(t − τ(t)) +12p

N∑

i=1

N∑

j=1

ei(t − τc(t))T

× Υrj3ei(t − τc(t))

⎫⎬

⎭+

M∑

s=1

γrsqs12

N∑

i=1

ei(t)TPei(t)

+ qr

⎧⎨

⎩

n∑

k=1

pk�kek(t)TA(r)ek(t)+

n∑

k=1

pk�kek(t)TB(r)ek(t − τc(t))

−n∑

k=1

pk�kek(t)TΞek(t)

+12p

⎡

⎣N∑

i=1

N∑

j=1

ei(t)TΥrj1ei(t) +

N∑

i=1

N∑

j=1

ei(t − τ(t))TΥrj2ei(t − τ(t))

⎤

⎦

+n∑

k=1

pk�kδkek(t)T ek(t)

}

= qr

⎧⎨

⎩

N∑

i=1

ei(t)T⎛

⎝−ηIN +12p

N∑

j=1

Υrj1 − αrPΣ

⎞

⎠ei(t) +N∑

i=1

ei(t − τ(t))T

×⎛

⎝θ +12p

N∑

j=1

Υrj2

⎞

⎠ei(t − τ(t))+N∑

i=1

ei(t − τc(t))T⎛

⎝12p

N∑

j=1

Υrj3 + βrPΣ

⎞

⎠

×ei(t − τc(t))

⎫⎬

⎭+

M∑

s=1

γrsqs12

N∑

i=1

ei(t)TPei(t)

+ qr

{n∑

k=1

pk�kek(t)T

[A(r) − Ξ +

(δ + αr

)IN]ek(t)

+n∑

k=1

pk�kek(t)TB(r)ek(t − τc(t)) −

n∑

k=1

pk�kek(t − τc(t))Tβr ek(t − τc(t))

}

≤ qr

⎧⎨

⎩

N∑

i=1

ei(t)T⎛

⎝−ηIN +12p

N∑

j=1

Υrj1 − αrPΣ

⎞

⎠ei(t)+N∑

i=1

ei(t − τ(t))T


×⎛

⎝βrIN +12p

N∑

j=1

Υrj2

⎞

⎠ei(t − τ(t)) +N∑

i=1

ei(t − τc(t))T⎛

⎝12p

N∑

j=1

Υrj3 + βrPΣ

⎞

⎠

×ei(t − τc(t))

⎫⎬

⎭+

M∑

s=1

γrsqs12

N∑

i=1

ei(t)TPei(t).

(3.10)

Let

E(t) =12

N∑

i=1

ei(t)TPei(t). (3.11)

Then we have

LV (t) ≤ −arqrE(t) + brqrE(t − τ(t)) + crqrE(t − τc(t)) +M∑

s=1

γrsqsE(t), (3.12)

and by (3.8), we have

LV (t) ≤ −(1 + θ)E(t) + bqE(t − τ(t)) + cqE(t − τc(t)). (3.13)

Define

W(t) = eγtV (t) (3.14)

and use (3.13) to compute the operator

LW(t) = eγt[γV (t) +LV (t)

]

≤ eγt[γpE(t) − (1 + θ)E(t) + bqE(t − τ(t)) + cqE(t − τc(t))

],

(3.15)

which, after applying the generalized Ito’s formula, gives

qreγt

EE(t) = qreγt0EE(t0) + E

∫ t

t0

LW(s)ds (3.16)


for any t > t0 ≥ 0. Hence we have

qreγt

EE(t) ≤ qEE(0) + E

∫ t

0eγs

[γE(s) − (1 − θ)E(s) + bqE(s − τ(s)) + cqE(s − τc(s))

]ds

≤ pq

2

N∑

i=1

E‖ξi‖2 +(γ − (1 + θ)

)∫ t

0eγsEE(s)ds + bqeγτ

∫ t

0eγ(s−τ(s))EE(s − τ(s))ds

+ cqeγτc∫ t

0eγ(s−τc(s))EE(s − τc(s))ds,

(3.17)

which, by using the change of variables s − τ(s) = u, gives

∫ t

0eγ(s−τ(s))EE(s − τ(s))ds =

∫ t−τ(t)

−τ(0)eγuEE(u)

du

1 − τ(t)

≤ pτ

2(1 − τ)

N∑

i=1

E‖ξi‖2 + 11 − τ

∫ t

0eγuEE(u)du,

(3.18)

and a further change of variables s − τc(s) = u gives

∫ t

0es−τc(s)EE(s − τc(s))ds =

∫ t−τc(t)

−τc(0)eγuEE(u)

du

1 − τc(t)

≤ pτc2(1 − τc)

N∑

i=1

E‖ξi‖2 + 11 − τc

∫ t

0eγuEE(u)du.

(3.19)

By Condition (3.4), we obtain

EE(t) ≤ pq

2

(

1 +bqτ

1 − τeγτ +

cqτc1 − τc

eγτc

)N∑

i=1

E‖ξi‖2e−γt (3.20)

so that

E‖e(t)‖2 ≤ pq

2p

(

1 +bqτ

1 − τeγτ +

cqτc1 − τc

eγτc

)N∑

i=1

E‖ξi‖2e−γt. (3.21)

The proof is hence complete.

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


Corollary 3.2. Let assumptions (H1) and (H2) be true and let f ∈ QUAD (P,Δ, η, θ). If there existpositive constants αr and βr such that

⎡

⎢⎣A(r)s + δIN − Ξ + αrIN

B(r)2

B(r)T

2−βrIN

⎤

⎥⎦ ≤ 0, for r = 1, 2, . . . ,M,

τ ≤ θT, τ ≤ (1 − θ)T,(

1b1 + c1

,1

b2 + c2, . . . ,

1bM + cM

)> Γ−11M ,

(3.22)

where γ > 0 is the greatest root of the equation

qγ − (1 + θ) + bqeγτ + cqeγτc = 0,

Γ = diag{a1, a2, . . . , aM} − Γ,

ar =λmin

(2ηIn − p

∑Ni=1 Υ

ri1 + 2α1PΣ

)

p, a = max

r∈Sar,

br =λmax

(∑Ni=1 PΥ

ri2 + 2ζIN

)

p, b = max

r∈Sbr,

cr =λmax

(∑Ni=1 PΥ

ri3 + 2β2PΣ

)

p, c = max

r∈Scr ,

(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(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 let f ∈ QUAD (P,Δ, η, θ). If thereexist positive constants αr and βr such that

⎡

⎢⎣As + δIN − Ξ + αrIN

B

2BT

2−βrIN

⎤

⎥⎦ ≤ 0,

τ ≤ θT, τ ≤ (1 − θ)T, 0 ≤ τ ≤ 1 −

(b + c

)

1 + θ,

(3.24)


where

ar =λmin

(2ηIn − p

∑Ni=1 Υ

ri1 + 2α1PΣ

)

p, a = max

r∈Sar,

br =λmax

(∑Ni=1 PΥ

ri2 + 2ζIN

)

p, b = max

r∈Sbr,

cr =λmax

(∑Ni=1 PΥ

ri3 + 2β2PΣ

)

p, c = max

r∈Scr ,

(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 theprevious section.

Consider the chaotic delayed neural network

ds(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 =[1 00 1

], A =

[2 −0.1−5 4.5

], B =

[−1.5 −0.1−0.2 −4

]. (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. Thus

dxi(t) =

⎧⎨

⎩f(t, xi(t), xi(t − τ(t))) +

5∑

j=1

arijΣxj(t) +

5∑

j=1

brijΣxj(t − τc(t)) − εi(xi(t) − s(t))

⎫⎬

⎭dt

+ σri (t, x(t), x(t − τ(t)), x(t − τc(t)))dwi(t), i = 1, 2, . . . , 5, r = 1, 2, 3,

(4.3)


A1 =[a1ij

]= 20

⎡

⎢⎢⎢⎢⎢⎣

−3 1 1 1 01 −4 1 1 11 1 −4 1 11 1 1 −4 10 1 1 1 −3

⎤

⎥⎥⎥⎥⎥⎦, A2 =

[a2ij

]= 15

⎡

⎢⎢⎢⎢⎢⎣

−4 1 1 1 11 −3 1 1 01 1 −4 1 11 1 1 −4 11 0 1 1 −3

⎤

⎥⎥⎥⎥⎥⎦,

A3 =[a3ij

]= 37

⎡

⎢⎢⎢⎢⎢⎣

−2 1 1 0 01 −3 1 1 01 1 −3 0 10 1 0 −1 00 0 1 0 −1

⎤

⎥⎥⎥⎥⎥⎦, B1 =

[b1ij

]= 0.1

⎡

⎢⎢⎢⎢⎢⎣

−2 1 1 0 00 −2 1 1 00 1 −2 1 00 1 1 −2 00 1 1 0 −2

⎤

⎥⎥⎥⎥⎥⎦

B2 =[b2ij

]= 0.2

⎡

⎢⎢⎢⎢⎢⎣

−1 0 0 1 00 −1 0 1 00 1 −1 0 00 1 0 −1 00 0 0 1 −1

⎤

⎥⎥⎥⎥⎥⎦, B3 =

[b3ij

]= 0.7

⎡

⎢⎢⎢⎢⎢⎣

−1 0 1 0 01 −1 0 0 00 0 −1 1 00 0 1 −1 01 0 0 0 −1

⎤

⎥⎥⎥⎥⎥⎦

Ξ = 100

⎡

⎢⎢⎢⎢⎢⎣

1 0 0 0 00 1 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎦, Γ =

⎡

⎣−3 1 22 −4 22 3 −5

⎤

⎦, τc(t) = 0.1et

1 + et

σ1i (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)},

σ2i (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))},σ3i (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))}.(4.4)

Computations then yield τ = 1, τ = 0, τc = 0.1, τc = 0.1, Υij = 0.01I2 for i = 1, 2, . . . ,N, andj = 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, and

Γ−1 =

⎡

⎣0.10526 0.014337 0.0190820.022486 0.095174 0.0187730.022176 0.025065 0.087313

⎤

⎦. (4.5)

Therefore

q = [0.1391, 0.1368, 0.1350]T , (4.6)

and, after solving (3.4), we obtain γ = 0.0349.


0 1 2 3 4 5 6 7 8 9 10

t

5

0

−5

xi1(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

t

10

5

0

−5

−10

xi2(t)

(b)

Figure 1: The trajectories of the state variables of xi1 and xi2 (i = 1, 2, . . . , 5) in system (4.3) under pinningcontrol.

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

t

−2

e i1(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

t

−2

e i2(t)

(b)

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

0 2 4 6 8 10

t

1

0.5

0

−0.5−1

−1.5−2

xi1(t)

(a)

0 2 4 6 8 10

t

xi2(t)

8

6

42

0−2−4−6−8

(b)

Figure 3: The trajectories of the state variables of xi1 and xi2 (i = 1, 2, . . . , 5) in system (4.3) under pinningcontrol.

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 periodicallyintermittent pinning control gains are shown in Figure 1. Figure 2 shows the time evolutionof the synchronization errors with periodically intermittent pinning control.

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


0 2 4 6 8 10

t

2

1.5

1

0.5

0

−0.5

e i1(t)

(a)

0 2 4 6 8 10

t

2

1.5

1

0.5

0

−0.5

e i2(t)

(b)

Figure 4: The time evolution of ei1 and ei2 (i = 1, 2, . . . , 5) in system (4.3) under pinning control.

Figure 4 shows the time evolution of the synchronization errors with periodically intermittentpinning control.

5. Conclusion

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

Acknowledgments

This work was supported by the National Science Foundation of China under Grantno. 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 KongPolytechnic University Grant G-U996 and Hong Kong Government GRF Grant B-Q21F.

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