Local and global exponential synchronization of complex delayed dynamical networks with general topology

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2011.16.393DYNAMICAL SYSTEMS SERIES BVolume 16, Number 1, July 2011 pp. 393–408

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF

COMPLEX DELAYED DYNAMICAL NETWORKS WITH

GENERAL TOPOLOGY

Jin-Liang Wang

National Key Laboratory of Science and Technology on Holistic ControlSchool of Automation Science and Electrical Engineering

Beihang University, Beijing 100191, China

Zhi-Chun Yang

College of Mathematics Science, Chongqing Normal UniversityChongqing 400047, China

Tingwen Huang

Department of Mathematics and ScienceTexas A&M University at QatarPO Box 23874, Doha, Qatar

Mingqing Xiao

Department of Mathematics, Southern Illinois UniversityCarbondale, IL 62901, USA

(Communicated by David Yang Gao)

Abstract. In this paper, we consider a generalized complex network possess-ing general topology, in which the coupling may be nonlinear, time-varying,nonsymmetric and the elements of each node have different time-varying de-lays. Some criteria on local and global exponential synchronization are derivedin form of linear matrix inequalities (LMIs) for the complex network by con-structing suitable Lyapunov functionals. Our results show that the obtainedsufficient conditions are less conservative than ones in previous publications.Finally, two numerical examples and their simulation results are given to illus-trate the effectiveness of the derived results.

1. Introduction. In the past ten years, there have been many researchers study-ing the topology and dynamical behavior of complex networks across many fieldsof science and engineering, such as power grids, communication networks, Internet,World Wide Web, metabolic systems, food webs and so on [1, 2]. In particular,the synchronization is one of the most significant and interesting dynamical prop-erties of the complex networks. Many interesting results on synchronization havebeen derived, e.g., see also [3]-[25]. In [3], a general complex delayed dynamical

2000 Mathematics Subject Classification. Primary: 37N99, 93D05; Secondary: 93A30.Key words and phrases. Complex networks, time-varying delays, exponential synchronization.This work was supported in part by National Natural Science Foundation of China under

Grants 10971240, 6100404 & 61074057, in part by Natural Science Foundation of Chongqing underGrant CSTC2008BB2364, in part by Foundation of Science and Technology project of ChongqingEducation Commission under Grant KJ080806, in part by Fundamental Research Funds for theCentral Universities of China under Grant YWF-10-01-A19, in part by NSF 1021203 of the U.S..

393

394 J.-L. WANG, Z.-C. YANG, T. HUANG AND M. XIAO

network model with nonsymmetric coupling was introduced, and several synchro-nization criteria on delay-independent and delay-dependent synchronization wereprovided by employing the matrix Jordan canonical formalization method. In work[4], complex delayed dynamical networks possessing general topology were inves-tigated and some global exponential synchronization criteria were established. Liand Yi [5] considered a general complex network model in which the coupling con-figurations are not restricted to the symmetric and irreducible connections or thenon-negative off-diagonal links. Several stability criteria were obtained by usingLyapunov-Krasovskii functional method and subspace projection method in [5].

However, in most existing works (see also the above mentioned references), thereare some simplified assumptions that the coupling among the nodes of the complexnetworks are linear, time invariant, symmetric and so on. In fact, such simplificationdoes not match the peculiarities of real networks in many circumstances. Firstly,the interplay of two different nodes in a network can not be described accuratelyby linear functions of their states because they are naturally nonlinear functions ofstates. Secondly, many complex networks in reality, are more likely to have differentcoupling strengths for different connections, and coupling strength are frequentlyvaried with time. Moreover, the coupling topology is likely to be directed andweighted in many real-world networks such as the food web, metabolic networks,World-Wide-Web, epidemic networks, document citation networks and so on. Inthis case, the coupling matrices are nonsymmetric and may not be diagonalizable.In addition, one should note that the phenomena of time delays are common incomplex networks, in which delays even may be frequently varied with time and theelements of each node have different time-varying delays. Therefore, It is interestingto further study such a general complex network model with nonlinear, time-varying,nonsymmetric and delayed coupling.

Motivated by the above discussions, we formulate a model of complex dynamicalnetworks with general topology, in which the coupling may be nonlinear, time-varying and nonsymmetric, and the elements of each node have different time-varying delays. Some sufficient conditions ensuring exponential synchronization areobtained by LMIs and Lyapunov functional method for the complex networks.

The rest of this paper is organized as follows. A complex delayed dynamicalnetwork model is introduced and some useful preliminaries are given in Section 2.Several criteria for both local and global exponential synchronization stability areobtained in Section 3. In Section 4, two numerical examples are given to illustratethe theoretical results. Finally, conclusions are presented in Section 5.

2. Network model and preliminaries. In this paper, we consider a complexnetwork consisting of N identical nodes with diffusive and delay coupling, in whicheach node is an n-dimensional dynamical system. The mathematical model of thecoupled network can be described by the following functional differential equations:

xi(t) = f(xi(t)) +N∑

j=1

Gij(t)h(xj(t)) +N∑

j=1

Gij(t)h(xj(t− τ(t))) (1)

where i = 1, 2, · · · , N, f(·) is continuously differentiable function, xi(t) = (xi1(t),

xi2(t), · · · , xin(t))T ∈ Rn is the state variable of node i, and xi(t− τ(t)) = [xi1(t−

τ1(t)), xi2(t − τ2(t)), · · · , xin(t − τn(t))]T , τl(t)(l = 1, 2, · · · , n) is the time-varying

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF NETWORKS 395

delay with 0 ≤ τl(t) ≤ τl ≤ τ . In this model, h(·), h(·) are continuously functionsdescribe the coupling relations between two nodes for non-delayed configuration and

delayed one, respectively, G(t) = (Gij(t))N×N and G(t) = (Gij(t))N×N representthe topological structure of the network and coupling strength between nodes fornon-delayed configuration and delayed one at time t, respectively. The diagonalelements of G(t) and G(t) are defined as follows:

Gii(t) = −N∑

j=1j 6=i

Gij(t), Gii(t) = −N∑

j=1j 6=i

Gij(t), i = 1, 2, · · · , N.

One should note that, in this model, the individual couplings between two connectednodes may be nonlinear, and the coupling configurations are not restricted to thesymmetric and irreducible connections or the non-negative off-diagonal links.

In the following, we give several useful denotations, definitions and lemmas.Let Rn be the n-dimensional Euclidean space and Rn×m be the space of n×m

real matrices. P > 0(P < 0) means matrix P is symmetrical and positive (neg-ative) definite. P ≥ 0(P ≤ 0) means matrix P is symmetrical and semi-positive(semi-negative) definite. ‖ · ‖ is the Euclidean norm. In denotes the n × n realidentity matrix. C([−τ, 0], Rn) is a Banach space of continuous functions mappingthe interval [−τ, 0] into Rn with the norm ‖φ‖τ = sup−τ≤θ≤0 ‖φ(θ)‖.

Definition 2.1. Let X(t, t0, φ) = (xT1 (t, t0, φ), x

T2 (t, t0, φ), · · · , x

TN (t, t0, φ))

T be asolution of (1) with initial conditions

X(t0 + s) = φ(s), s ∈ [−τ, 0], φ = (φT1 , φ

T2 , · · · , φ

TN )T , φi ∈ C([−τ, 0], Rn).

Then the network (1) is local exponential synchronization if there exist constants

α > 0, δ > 0 and M > 0 such that for any initial function φ with ‖φ(θ)−S0‖τ < δ ,we have

‖X(t, t0, φ)− S(t, t0, S0)‖ ≤ M−α(t−t0), t ≥ t0

where S0 = (sT0 , sT0 , · · · , s

T0 )

T , S(t, t0, S0) = (sT (t, t0, s0), sT (t, t0, s0), · · · , s

T (t, t0,s0))

T and s(t, t0, s0) is a solution of an isolated node, namely,

s(t) = f(

s(t))

. (2)

Here s(t) can be an equilibrium, a periodic trajectory or a chaotic attractor of thesystem (2).

Definition 2.2. Let X(t, t0, φ) = (xT1 (t, t0, φ), x

T2 (t, t0, φ), · · · , x

TN (t, t0, φ))

T be asolution of (1) with initial conditions

X(t0 + s) = φ(s), s ∈ [−τ, 0], φ = (φT1 , φ

T2 , · · · , φ

TN )T , φi ∈ C([−τ, 0], Rn).

Then the network (1) is global exponential synchronization, if there exist constantsα > 0, λ > 0 such that

‖X(t, t0, φ)− S(t, t0, s0)‖ ≤ αe−λ(t−t0)‖φ(θ) − S0‖τ

where S0 = (sT0 , sT0 , · · · , s

T0 )

T , S(t, t0, S0) = (sT (t, t0, s0), sT (t, t0, s0), · · · , s

T (t, t0,s0))

T . Here s(t, t0, s0) denotes the same meaning as that in Definition 2.1.

396 J.-L. WANG, Z.-C. YANG, T. HUANG AND M. XIAO

Definition 2.3. For A = (aij)m×n ∈ Rm×n and B = (bij)p×q ∈ Rp×q, the Kro-necker product between two matrices is defined by

A⊗B =

a11B a12B · · · a1nB

a21B a22B · · · a2nB...

... · · ·...

am1B am2B · · · amnB

∈ Rmp×nq.

Lemma 2.4. (see [11]) For any vectors x, y ∈ Rn and n× n square matrix P > 0,the following LMI holds:

xT y + yTx ≤ xTPx+ yTP−1y.

3. Synchronization criteria. In order to obtain our main results, two usefulassumptions are introduced.

(A1) Suppose that there exist two constants L1 < 0, L2 > 0 such that

[x(t) − y(t)]T [f(x(t))− f(y(t))] ≤ L1[x(t) − y(t)]T [x(t) − y(t)],

[f(x(t)) − f(y(t))]T [f(x(t))− f(y(t))] ≤ L2[x(t) − y(t)]T [x(t) − y(t)]

hold for any t. Here x(t), y(t) ∈ Rn are time-varying vectors.(A2) Suppose that there exist two positive constants L3 > 0, L4 > 0 such that

‖h(x(t))− h(y(t))‖ ≤ L3‖x(t)− y(t)‖,

‖h(x(t)) − h(y(t))‖ ≤ L4‖x(t)− y(t)‖

hold for any t. Here x(t), y(t) ∈ Rn are time-varying vectors.In the following, we first give the local exponential synchronization stability

criteria.

Theorem 3.1. Let (A2) hold, and τl(t) ≤ σ < 1, l = 1, 2, · · · , n. If there exist amatrix P ∈ RNn×Nn > 0, and a positive constant γ > 1 such that

P [IN ⊗Df(s(t))] + [IN ⊗Df(s(t))]TP + (L23 + γ)INn +

L24

1− σP (G(t)⊗ In)×

(G(t)⊗ In)TP + P (G(t)⊗ In)(G(t) ⊗ In)

TP ≤ 0,

(3)

then the complex network (1) achieves local exponential synchronization.

Proof. Firstly, according to the LMI (3), there obviously exists a small constantε > 0, such that the following LMI hold

P [IN ⊗Df(s(t))] + [IN ⊗Df(s(t))]TP + εP + E−1ε + L2

3INn +L24

1− σP (G(t)⊗ In)

× (G(t)⊗ In)TP + P (G(t) ⊗ In)(G(t)⊗ In)

TP ≤ 0

where Eε = diag{E,E, · · · , E} ∈ RNn×Nn, E = (Eij)n×n = diag{e−ετ1, e−ετ2 , · · · ,e−ετn}. In the following, we can rewrite system (1) in a compact form as follows:

x(t) = F (x(t)) + (G(t) ⊗ In)H(x(t)) + (G(t)⊗ In)H(x(t− τ(t))) (4)

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF NETWORKS 397

where

x(t) = (xT1 (t), x

T2 (t), . . . , x

TN (t))T ,

F (x(t)) = [fT (x1(t)), fT (x2(t)), . . . , f

T (xN (t))]T ,

H(x(t)) = [hT (x1(t)), hT (x2(t)), . . . , h

T (xN (t))]T ,

x(t− τ(t)) = [x1(t− τ(t))T, x2(t− τ(t))

T, . . . , xN (t− τ(t))

T]T ,

H(x(t − τ(t))) = [hT (x1(t− τ(t))), hT (x2(t− τ(t))), . . . , hT (xN (t− τ(t)))]T .

Set xi(t) = s(t) + ηi(t), i = 1, 2, · · · , N. We have the following matrix equation:

η(t) = [F (x(t)) − F (s(t))] + (G(t) ⊗ In)H1(η(t)) + (G(t)⊗ In)H1(η(t− τ(t))) (5)

where

η(t) = (ηT1 (t), ηT2 (t), . . . , η

TN (t))T ,

F (s(t)) = [fT (s(t)), fT (s(t)), . . . , fT (s(t))]T ,

H(s(t)) = [hT (s(t)), hT (s(t)), . . . , hT (s(t))]T ,

H1(η(t)) = H(x(t)) −H(s(t)),

H1(η(t− τ(t))) = H(x(t − τ(t)))− H(s(t− τ(t))),

η(t− τ(t)) = [η1(t− τ(t))T, η2(t− τ(t))

T, . . . , ηN (t− τ(t))

T]T ,

H(s(t− τ(t))) = [hT (s(t− τ(t))), hT (s(t− τ(t))), . . . , hT (s(t− τ(t)))]T .

It is easy to know that system (5) is locally exponentially stable if the followingsystem is globally exponentially stable:

η(t) = (IN ⊗Df(s(t)))η(t) + (G(t)⊗ In)H1(η(t)) + (G(t)⊗ In)H1(η(t− τ(t))) (6)

where Df(s(t)) := f ′(s(t)) ∈ Rn×n is the Jacobian of f(x(t)) at s(t). Next, con-struct Lyapunov functional for system (6) as follows

V (η(t)) = ηT (t)Pη(t)eεt +

N∑

i=1

n∑

l=1

∫ t

t−τl(t)

1

Ell

η2il(s)eεsds. (7)

The derivative of V (η(t)) satisfies

V (η(t)) ≤ ηT (t)Pη(t)eεt + ηT (t)P η(t)eεt + εηT (t)Pη(t)eεt + ηT (t)E−1ε η(t)eεt

−(1− σ)η(t − τ(t))Tη(t− τ(t))eεt

= ηT (t)[P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε ]η(t)eεt

+ηT (t)P (G(t) ⊗ In)H1(η(t))eεt + ηT (t)P (G(t)⊗ In)H1(η(t− τ(t)))

×eεt +HT1 (η(t))(G(t) ⊗ In)

TPη(t)eεt

−(1− σ)η(t − τ(t))Tη(t− τ(t))eεt

+HT1 (η(t− τ(t)))(G(t)⊗ In)

TPη(t)eεt.

Applying Lemma 2.4, then we have

HT1 (η(t))(G(t) ⊗ In)

TPη(t) + ηT (t)P (G(t) ⊗ In)H1(η(t)) ≤ HT1 (η(t))H1(η(t))

+ηT (t)P (G(t) ⊗ In)(G(t) ⊗ In)TPη(t), (8)

398 J.-L. WANG, Z.-C. YANG, T. HUANG AND M. XIAO

HT1 (η(t− τ(t)))(G(t)⊗ In)

TPη(t) + ηT (t)P (G(t)⊗ In)H1(η(t− τ(t)))

≤1− σ

L24

HT1 (η(t− τ(t)))H1(η(t− τ(t))) +

L24

1− σηT (t)P (G(t)⊗ In)(G(t)⊗ In)

T

×Pη(t). (9)

Hence, we can easily obtain

V (η(t)) ≤ ηT (t)[P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε ]η(t)eεt

+HT1 (η(t))H1(η(t))e

εt + ηT (t)P (G(t) ⊗ In)(G(t) ⊗ In)TPη(t)eεt

+1− σ

L24

HT1 (η(t− τ(t)))H1(η(t− τ(t)))eεt

+L24

1− σηT (t)P (G(t)⊗ In)(G(t)⊗ In)

TPη(t)eεt

−(1− σ)η(t− τ(t))Tη(t− τ(t))eεt.

Using (A2), we have

HT1 (η(t))H1(η(t)) =

N∑

i=1

[h(xi(t))− h(s(t))]T [h(xi(t))− h(s(t))]

≤

N∑

i=1

L23[xi(t)) − s(t)]T [xi(t)− s(t)]

= L23η

T (t)η(t), (10)

and

HT1 (η(t− τ(t)))H1(η(t− τ(t)))

=

N∑

i=1

[h(xi(t− τ(t)))− h(s(t− τ(t)))]T [h(xi(t− τ(t)))− h(s(t− τ(t)))]

≤N∑

i=1

L24[xi(t− τ(t)) − s(t− τ(t))]T [xi(t− τ(t)) − s(t− τ(t))]

= L24η(t− τ(t))

Tη(t− τ(t)). (11)

Then the derivative of V (η(t)) satisfies

V (η(t))

≤ ηT (t)[P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε + L2

3INn +

L24

1− σP (G(t)⊗ In)(G(t)⊗ In)

TP + P (G(t)⊗ In)(G(t) ⊗ In)TP ]η(t)eεt

≤ 0.

Hence, we have V (η(t)) ≤ V (η(t0)), for all t ≥ t0. According to definition of V (η(t)),we have

V (η(t)) ≥ ηT (t)Pη(t)eεt ≥ λmin‖η(t)‖2eεt,

and

V (η(t0)) ≤ B‖φ(θ)‖2τeεt0

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF NETWORKS 399

where λmin is the smallest eigenvalue of matrix P,B > 0 is a positive number,φ(θ) = (φT

1 , φT2 , · · · , φ

TN )T , φi = φi(θ) ∈ C([−τ, 0], Rn). Thus, we can draw

‖η(t)‖ ≤ B‖φ(θ)‖τe−ε(t−t0)

2 for t ≥ t0

where B ≥ 1 is a positive number. Then, systems (6) is global exponentially stableand we have local exponential synchronization of complex network (1). The proofis complete.

One should note that we do not assume that the coupling matrix is symmetric andirreducible and its off-diagonal elements are nonnegative. Moreover, the individualcouplings between two connected nodes of may be nonlinear in this paper. Thetheorem generalizes the related results in the literature [6, 21].

In fact, when τl(t) = τ(t), l = 1, 2, · · · , n, h(xj(t)) = Axj(t) (A is a constant

matrix), j = 1, 2, · · · , N , G(t) = 0 and G(t) = G is a constant symmetric irreduciblematrix with non-negative off-diagonal elements, then by the Schur decompositionwe get φTGφ = Λ, where φ = (φ1, φ2, · · · , φN ) ∈ RN×N ,Λ = diag(λ1, λ2, · · · , λN ).Especially, we suppose that 0 = λ1 > λ2 ≥ λ3 ≥ · · · ≥ λN . Take a nonsingulartransform

(φT ⊗ In)η(t) = ν(t) =(

νT1 (t), νT2 (t), · · · , ν

TN (t)

)T∈ RNn.

From system (6), we have the following matrix equation:

ν(t) = (IN ⊗Df(s(t)))ν(t) + (Λ⊗A)ν(t − τ(t)), (12)

where ν(t− τ(t)) = [νT1 (t− τ(t)), νT2 (t− τ(t)), · · · , νTN (t− τ(t))]T .Choose the Lyapunov functional for the system (12) as follows:

V (ν(t)) = νT (t)Pν(t)eεt +

N∑

i=1

n∑

l=1

∫ t

t−τ(t)

1

Ell

ν2il(s)eεsds, (13)

where Eε = diag{E,E, · · · , E} ∈ RNn×Nn, E = (Eij)n×n = diag{e−ετ , e−ετ , · · · ,e−ετ}. Then we can easily obtain the following Corollary

Corollary 1. Suppose that 0 ≤ τ(t) ≤ τ and τ (t) ≤ σ < 1. If there exist a matrixP = diag(P1, P2, · · · , PN ), Pi ∈ Rn×n > 0(i = 1, 2, · · · , N), and a positive constantγ > 1 such that

P [IN ⊗Df(s(t))] + [IN ⊗Df(s(t))]TP + γINn +P (Λ⊗A)(Λ ⊗A)TP

1− σ≤ 0,

then the system (12) is globally exponentially stable.

Remark 1. It is obvious the condition we derive in Corollary 1 is consistent withthe result in [6, Theorem 1] and [21, Theorem 1, Corollary 1]. This means that thesynchronization stability condition derived in this paper is less conservative thanthe result in [6] and [21].

Theorem 3.2. Let (A2) hold and τl(t) ≤ σ < 1. If there exist matrices P ∈

RNn×Nn > 0, Z = diag(Z1, Z2, · · · , ZN ), Zi = diag(zi1, zi2, · · · , zin), zil > 0, andtwo positive constants γ > 0, ξ > 1, such that

P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + [γ(L23 + 1) + ξ]INn ≤ 0, (14)

400 J.-L. WANG, Z.-C. YANG, T. HUANG AND M. XIAO

W3 W1 W2

WT1 (G(t)⊗ In)

TM(G(t) ⊗ In) (G(t)⊗ In)TM(G(t)⊗ In)

WT2 (G(t)⊗ In)

TM(G(t) ⊗ In) (G(t)⊗ In)TM(G(t)⊗ In)

− γ

INn 0 00 INn 00 0 INn

≤ 0,

(15)

1− σ − γL24 ≥ 0 (16)

where

W1 = (IN ⊗Df(s(t)))TM(G(t)⊗ In) + P (G(t)⊗ In),

W2 = (IN ⊗Df(s(t)))TM(G(t)⊗ In) + P (G(t)⊗ In),

W3 = (IN ⊗Df(s(t)))TM(IN ⊗Df(s(t))),

Mi = diag(τ1zi1, τ2zi2, · · · , τnzin),

M = diag(M1,M2, · · · ,MN)

where i = 1, 2, 3, · · · , N, l = 1, 2, · · · , n, then the complex network (1) achieves localexponential synchronization.

Proof. Firstly, according to the LMI (14), there obviously exists a small constantε > 0 such that the following LMI hold

P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε + γ(L2

3 + 1)INn ≤ 0

where E = (Eij)n×n = diag{e−ετ1, e−ετ2 , · · · , e−ετn}, Eε = diag{E,E, · · · , E} ∈RNn×Nn. Define Lyapunov functional for the system (6) as follows:

V (η(t)) = ηT (t)Pη(t)eεt +

N∑

i=1

n∑

l=1

∫ 0

−τl(t)

∫ t

t+β

zilη2il(s)e

εsdsdβ

+

N∑

i=1

n∑

l=1

∫ t

t−τl(t)

1

Ell

η2il(s)eεsds.

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF NETWORKS 401

Set ζ(t) = [ηT (t), HT1 (η(t)), H

T1 (η(t− τ(t)))]T . Then the derivative of V (η(t)) sat-

isfies

V (η(t))

≤ ηT (t)Pη(t)eεt + ηT (t)P η(t)eεt + εηT (t)Pη(t)eεt + ηT (t)E−1ε η(t)eεt

−(1− σ)η(t− τ(t))Tη(t− τ(t))eεt +

N∑

i=1

n∑

l=1

τl(t)zilη2il(t)e

εt

+

N∑

i=1

n∑

l=1

τl(t)

∫ t

t−τl(t)

zilη2il(s)e

εsds−

N∑

i=1

n∑

l=1

∫ t

t−τl(t)

zilη2il(s)e

εsds

≤ ηT (t)[P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε ]η(t)eεt

+ηT (t)P (G(t) ⊗ In)H1(η(t))eεt + ηT (t)P (G(t)⊗ In)H1(η(t− τ(t)))eεt

+HT1 (η(t))(G(t) ⊗ In)

TPη(t)eεt − (1− σ)η(t − τ(t))Tη(t− τ(t))eεt

+HT1 (η(t − τ(t)))(G(t)⊗ In)

TPη(t)eεt + [(IN ⊗Df(s(t)))η(t)

+(G(t)⊗ In)H1(η(t)) + (G(t)⊗ In)H1(η(t− τ(t)))]TM ×

[(IN ⊗Df(s(t)))η(t) + (G(t) ⊗ In)H1(η(t))

+(G(t)⊗ In)H1(η(t− τ(t)))]eεt

= ηT (t)[P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε ]η(t)eεt

−(1− σ)η(t− τ(t))Tη(t− τ(t))eεt + ζT (t)×

W3 W1 W2

WT1 (G(t) ⊗ In)

TM(G(t)⊗ In) (G(t) ⊗ In)TM(G(t)⊗ In)

WT2 (G(t)⊗ In)

TM(G(t)⊗ In) (G(t)⊗ In)TM(G(t)⊗ In)

×

ζ(t)eεt.

Using LMI (15), then we have

V (η(t)) ≤ ηT (t)[P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε ]η(t)eεt

−(1− σ)η(t− τ(t))Tη(t− τ(t))eεt + γHT

1 (η(t))H1(η(t))eεt

+γηT (t)η(t)eεt + γHT1 (η(t− τ(t)))H1(η(t − τ(t)))eεt.

According to (10), (11) and (16), we can obtain

V (η(t)) ≤ ηT (t)[P (IN ⊗Df(s(t))) + (IN ⊗Df(s(t)))TP + εP + E−1ε

+γ(L23 + 1)INn]η(t)e

εt

≤ 0.

Similarly, we can conclude that system (6) is globally exponentially stable. Hence,we have the local exponential synchronization of complex network (1). The proofis completed.

In the above, we give two sufficient conditions on local exponential synchroniza-tion stability criteria. In the following, we discuss global exponential synchroniza-tion.

402 J.-L. WANG, Z.-C. YANG, T. HUANG AND M. XIAO

Theorem 3.3. Let (A1) and (A2) hold, and τl(t) ≤ σ < 1, l = 1, 2, · · · , n. If thereexists a positive constant γ > 1 such that

(L23+2L1+γ)INn+

L24(G(t)⊗ In)(G(t)⊗ In)

T

1− σ+(G(t)⊗In)(G(t)⊗In)

T ≤ 0, (17)

then the complex network (1) achieves global exponential synchronization.

Proof. Firstly, according to the LMI (17), there obviously exists a small constantε > 0 such that the following LMI hold

(L23+2L1+ε)INn+E−1

ε +L24(G(t)⊗ In)(G(t)⊗ In)

T

1− σ+(G(t)⊗In)(G(t)⊗In)

T ≤ 0,

where Eε = diag{E,E, · · · , E} ∈ RNn×Nn, E = (Eij)n×n = diag{e−ετ1, e−ετ2 , · · · ,e−ετn}. Construct Lyapunov functional as follows

V (η(t)) = ηT (t)η(t)eεt +

N∑

i=1

n∑

l=1

∫ t

t−τl(t)

1

Ell

η2il(s)eεsds.

We calculate the derivative of V (η(t)) along the solution of the system (5)

V (η(t))

≤ ηT (t)η(t)eεt + ηT (t)η(t)eεt + εηT (t)η(t)eεt + ηT (t)E−1ε η(t)eεt

−(1− σ)η(t − τ(t))Tη(t− τ(t))eεt

= ηT (t)[F (x(t)) − F (s(t))]eεt + [F (x(t)) − F (s(t))]T η(t)eεt + ηT (t)(εINn +

E−1ε )η(t)eεt + ηT (t)(G(t) ⊗ In)H1(η(t))e

εt

+ηT (t)(G(t)⊗ In)H1(η(t− τ(t)))eεt +HT1 (η(t))(G(t) ⊗ In)

T η(t)eεt +

HT1 (η(t− τ(t)))(G(t)⊗ In)

T η(t)eεt − (1− σ)η(t− τ(t))Tη(t− τ(t))eεt.

Applying Lemma 2.4, then we have

HT1 (η(t))(G(t) ⊗ In)

T η(t) + ηT (t)(G(t) ⊗ In)H1(η(t)) ≤ HT1 (η(t))H1(η(t))

+ηT (t)(G(t) ⊗ In)(G(t) ⊗ In)T η(t),

HT1 (η(t − τ(t)))(G(t)⊗ In)

T η(t) + ηT (t)(G(t)⊗ In)H1(η(t − τ(t)))

≤1− σ

L24

HT1 (η(t− τ(t)))H1(η(t− τ(t)))

+L24

1− σηT (t)(G(t)⊗ In)(G(t)⊗ In)

T η(t).

Hence, we can easily obtain

V (η(t)) ≤ ηT (t)[F (x(t)) − F (s(t))]eεt + [F (x(t)) − F (s(t))]T η(t)eεt

+ηT (t)(εINn + E−1ε )η(t)eεt +HT

1 (η(t))H1(η(t))eεt

+ηT (t)(G(t) ⊗ In)(G(t) ⊗ In)T η(t)eεt +

1− σ

L24

HT1 (η(t− τ(t)))

×H1(η(t− τ(t)))eεt +L24

1− σηT (t)(G(t)⊗ In)(G(t)⊗ In)

T η(t)eεt

−(1− σ)η(t− τ(t))Tη(t− τ(t))eεt.

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF NETWORKS 403

Using (A1), we have

ηT (t)[F (x(t)) − F (s(t))] =

N∑

i=1

[xi(t)) − s(t)]T [f(xi(t))− f(s(t))]

≤N∑

i=1

L1[xi(t))− s(t)]T [xi(t)− s(t)]

= L1ηT (t)η(t). (18)

According to (10), (11) and (18), then we have the derivative of V (η(t)) satisfies

V (η(t)) ≤ ηT (t)[(L23 + 2L1 + ε)INn + E−1

ε +L24

1− σ(G(t)⊗ In)(G(t)⊗ In)

T

+(G(t)⊗ In)(G(t) ⊗ In)T ]η(t)eεt

≤ 0.

Hence, we can easily obtain V (η(t)) ≤ V (η(t0)), for all t ≥ t0. This implies that

‖η(t)‖ ≤ B‖φ(θ)‖τe−ε(t−t0)

2 for t ≥ t0

where B ≥ 1 is a positive number, φ(θ) = (φT1 , φ

T2 , · · · , φ

TN )T , φi = φi(θ) ∈

C([−τ, 0], Rn). Then, we can conclude that system (5) is globally exponentiallystable. Therefore, we have the global exponential synchronization of complex net-work (1). The proof is completed.

Theorem 3.4. Let (A1) and (A2) hold, and τl(t) ≤ σ < 1, l = 1, 2, · · · , n. If thereexist a matrix Z = diag(Z1, Z2, · · · , ZN), Zi = diag(zi1, zi2, · · · , zin), zil > 0, andtwo positive constants γ > 0, ξ > 1 such that

1− σ − γL24 > 0, (19)

2L1 + ξ + γ(L23 + 1 + L2) ≤ 0, (20)

0 0 (G(t) ⊗ In) (G(t)⊗ In)

0 M M(G(t)⊗ In) M(G(t)⊗ In)(G(t) ⊗ In)

T (G(t) ⊗ In)TM W4 W5

(G(t)⊗ In)T (G(t)⊗ In)

TM WT5 W6

− γ

INn 0 0 00 INn 0 00 0 INn 00 0 0 INn

≤ 0

(21)

where

W4 = (G(t) ⊗ In)TM(G(t)⊗ In),

W5 = (G(t) ⊗ In)TM(G(t)⊗ In),

W6 = (G(t)⊗ In)TM(G(t)⊗ In),

Mi = diag(τ1zi1, τ2zi2, · · · , τnzin),

M = diag(M1,M2, · · · ,MN ),

i = 1, 2, 3, · · · , N, l = 1, 2, · · · , n, then the complex network (1) achieves globalexponential synchronization.

404 J.-L. WANG, Z.-C. YANG, T. HUANG AND M. XIAO

Proof. By (20), there obviously exists a small constant ε > 0 such that the followingLMI hold

2L1INn + εINn + E−1ε + γ(L2

3 + 1 + L2)INn ≤ 0

where Eε = diag{E,E, · · · , E} ∈ RNn×Nn, E = (Eij)n×n = diag{e−ετ1, e−ετ2 , · · · ,e−ετn}. Define the following Lyapunov functional for the system (5)

V (η(t)) = ηT (t)η(t)eεt +N∑

i=1

n∑

l=1

∫ 0

−τl(t)

∫ t

t+β

zilη2il(s)e

εsdsdβ

+

N∑

i=1

n∑

l=1

∫ t

t−τl(t)

1

Ell

η2il(s)eεsds.

Set ξ(t) = [ηT (t), FT (x(t)) − FT (s(t)), HT1 (η(t)), HT

1 (η(t − τ(t)))]T . Then the de-rivative of V (η(t)) satisfies

V (η(t))

≤ ηT (t)η(t)eεt + ηT (t)η(t)eεt + εηT (t)η(t)eεt + ηT (t)E−1ε η(t)eεt

−(1− σ)η(t− τ(t))Tη(t− τ(t))eεt +

N∑

i=1

n∑

l=1

τl(t)zilη2il(t)e

εt

+

N∑

i=1

n∑

l=1

τl(t)

∫ t

t−τl(t)

zilη2il(s)e

εsds−

N∑

i=1

n∑

l=1

∫ t

t−τl(t)

zilη2il(s)e

εsds

≤ ηT (t)[F (x(t)) − F (s(t))]eεt + [F (x(t)) − F (s(t))]T η(t)eεt

+ηT (t)(εINn + E−1ε )η(t)eεt + ηT (t)(G(t) ⊗ In)H1(η(t))e

εt

+ηT (t)(G(t)⊗ In)H1(η(t − τ(t)))eεt +HT1 (η(t))(G(t) ⊗ In)

T η(t)eεt

+HT1 (η(t− τ(t)))(G(t)⊗ In)

T η(t)eεt − (1− σ)η(t − τ(t))Tη(t− τ(t))eεt

+{[F (x(t))− F (s(t))] + (G(t)⊗ In)H1(η(t)) + (G(t)⊗ In)×

H1(η(t− τ(t)))}TM{[F (x(t))− F (s(t))] + (G(t)⊗ In)H1(η(t))

+(G(t)⊗ In)H1(η(t − τ(t)))}eεt

= ηT (t)[F (x(t)) − F (s(t))]eεt + [F (x(t)) − F (s(t))]T η(t)eεt

+ηT (t)(εINn + E−1ε )η(t)eεt − (1− σ)η(t− τ(t))

Tη(t− τ(t))eεt + ξT (t)×

0 0 (G(t) ⊗ In) (G(t)⊗ In)

0 M M(G(t)⊗ In) M(G(t)⊗ In)(G(t) ⊗ In)

T (G(t) ⊗ In)TM W4 W5

(G(t)⊗ In)T (G(t)⊗ In)

TM WT5 W6

ξ(t)eεt.

According to LMI (21), then we have

V (η(t)) ≤ ηT (t)[F (x(t)) − F (s(t))]eεt + [F (x(t)) − F (s(t))]T η(t)eεt

+ηT (t)(εINn + E−1ε )η(t)eεt − (1− σ)η(t − τ(t))

Tη(t− τ(t))eεt

+γηT (t)η(t)eεt + γ[F (x(t))− F (s(t))]T [F (x(t)) − F (s(t))]eεt

+γHT1 (η(t))H1(η(t))e

εt + γHT1 (η(t− τ(t)))H1(η(t− τ(t)))eεt.

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF NETWORKS 405

Using (A1), we can easily obtain

[F (x(t)) − F (s(t))]T [F (x(t)) − F (s(t))]

≤

N∑

i=1

[f(xi(t))− f(s(t))]T [f(xi(t))− f(s(t))]

≤

N∑

i=1

L2[xi(t)) − s(t)]T [xi(t)− s(t)]

= L2ηT (t)η(t). (22)

It follows from inequalities (10), (11), (18), (19) and (22) that

V (η(t)) ≤ ηT (t)[2L1INn + εINn + E−1ε + γ(L2

3 + 1 + L2)INn]η(t)eεt

≤ 0.

Similarly, we can conclude that system (5) is globally exponentially stable. There-fore, we have the global exponential synchronization of complex network (1). Theproof is completed.

Remark 2. The conditions in the above theorems do not restrict the networkconfigurations to be symmetric and irreducible and non-negative off-diagonal, whilethese restrictive conditions are imposed in [3, 4, 7, 8, 9, 18, 20]. Also, our theoremsdon’t require the restrictive assumption that the derivative of the time-varyingdelays is non-negative or non-positive in some previous results (see, e.g.,[9, 10]).Therefore, our criteria may be more general and verifiable.

4. Examples. In this section, we give two examples and their simulation resultsto show the effectiveness of the above obtained theoretical criteria.

0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

8

t

x il,i,l=

1,2,

3

Figure 1. Local exponential synchronization of complex networkwith time-varying delays

406 J.-L. WANG, Z.-C. YANG, T. HUANG AND M. XIAO

Example 1 Consider a three-order nonlinear system as the dynamical node ofthe complex network, which is described by

x1 = −7x1 + x22

x2 = −8x2

x3 = −9x3 + x2x3.

Clearly, the state s(t) = (0, 0, 0)T is an equilibrium of the isolated node, and theJacobin matrix is Df(s(t)) = diag(−7,−8,−9).

For simplicity, we assume that

h(x) =x

2, x ∈ R3, h(x) =

x

4, x ∈ R3,

G(t) =

−1 0.5 0.50.2 0.3 −0.50.2 0 −0.2

, G(t) =

−1 0.4 0.6−0.3 0.3 00.5 0.5 −1

.

It is obvious that we can take L3 = 12 and L4 = 1

4 , and the coupling configurationsare not restricted to the symmetric and non-negative off-diagonal.

In the following, we analyze the synchronization of complex network with differ-ent time-varying delays.

Set τl(t) = 1− 13+l

e−t, then we have τl(t) ≤ τl = τ = 1, τl(t) =1

3+le−t ≤ 1

4 < 1,for t ≥ 0, l = 1, 2, 3.

We can find the following positive-definite matrix P satisfying the LMI (3) withγ = 1.1,

P = diag(7.5648, 8.6235, 9.2189, 8.2314, 9.4976, 8.6785, 8.4568, 8.9456, 8.6784).

According to the Theorem 3.1, the network achieves local exponential synchroniza-tion. The simulation results are shown in Figure 1.

0 2 4 6 8 10−5

−4

−3

−2

−1

0

1

2

3

4

5

t

x il,i,l=

1,2,

3

Figure 2. Global exponential synchronization of complex networkwith time-varying delays

LOCAL AND GLOBAL EXPONENTIAL SYNCHRONIZATION OF NETWORKS 407

Example 2 Consider a three-order linear system as the dynamical node of thecomplex network, which is described by

x1 = −6x1

x2 = −7x2

x3 = −8x3.

Clearly, the state s(t) = (0, 0, 0)T is an equilibrium of the isolated node, and we cantake L1 = −6.

Assume that

h(x) =x

3, x ∈ R3, h(x) =

x

4, x ∈ R3,

G(t) =

−2 1 10.1 0.3 −0.40.2 0.1 −0.3

, G(t) =

−0.5 0.4 0.10.3 −0.2 −0.10.5 0 −0.5

.

It is obvious that we can take L3 =13 and L4 = 1

4 , and the coupling configurationsare not restricted to the symmetric and non-negative off-diagonal.

Set τl(t) =5

lt+5 , then we have τl(t) ≤ τl = τ = 1, τl(t) = − 5l(lt+5)2 ≤ 0 < 1, for

t ≥ 0, l = 1, 2, 3.We can find a γ = 3.4 satisfying the LMI (17). According to the Theorem 3.3,

the network achieves global exponential synchronization. The simulation results areshown in Figure 2.

5. Conclusion. We have studied the stability of the synchronized state of complexnetworks with different time-varying delays and nonlinear coupling. We not onlyconsidered the case that the coupling strength and topology structure are frequentlyvaried with time, but also took into account the case that the couplings relation andthe coupling configurations are related to the current states and the delayed states.Several theorems and corollaries of exponential synchronization of complex networkwere established. Two illustrative examples were presented to show the effectivenessof the derived results. Our conditions are less conservative and verifiable.

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Received June 2010; revised January 2011.

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