Automatica 56 (2015) 105–112

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Passivity-based synchronization of a class of complex dynamicalnetworks with time-varying delay

Jin-Liang Wang a,b, Huai-Ning Wu b, Tingwen Huang c

a School of Computer Science & Software Engineering, Tianjin Polytechnic University, Tianjin 300387, Chinab Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191,Chinac Texas A & M University at Qatar, Doha 23874, Qatar

a r t i c l e i n f o

Article history:Received 30 June 2013Received in revised form21 January 2015Accepted 14 March 2015Available online 17 April 2015

Keywords:Complex networksReaction–diffusion neural networksTime-varying delayPassivitySynchronization

a b s t r a c t

This paper proposes a complex delayed dynamical network consisting of N linearly and diffusivelycoupled identical reaction–diffusion neural networks. By utilizing some inequality techniques, a sufficientcondition ensuring the output strict passivity is derived for the proposed networkmodel. Then, we revealthe relationship between output strict passivity and synchronization of the proposed network model.Moreover, based on the obtained passivity result and the relationship between output strict passivity andsynchronization, a criterion for synchronization is established. Finally, a numerical example is providedto illustrate the correctness and effectiveness of the proposed results.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Today various complex networks can be seen everywhere andare becoming an important part of our daily life. Some of themost well-known examples include food webs, communicationnetworks, social networks, power grids, cellular networks, WorldWide Web, metabolic systems, disease transmission networks,etc. Therefore, the topology and dynamical behavior of complexdynamical networks have been extensively studied by the re-searchers. In particular, the synchronization problem of complexdynamical networks has receivedmuchof the focus in recent years.So far, a great many important results on synchronization havebeen obtained for various complex networks (Arcak, 2011; Lü &Chen, 2005; Scardovi, Arcak, & Sontag, 2010; Slotine&Wang, 2005;

This work was supported in part by the National Natural Science Foundationof China under Grants 61403275, 61473011 and 61421063, and in part by NationalPriorities Research Program (NPRP) under Grant NPRP 4-1162-1-181 from theQatarNational Research Fund (a member of Qatar Foundation). The statements madeherein are solely the responsibility of the authors. The material in this paper wasnot presented at any conference. This paper was recommended for publication inrevised form by Associate Editor Akira Kojima under the direction of Editor Ian R.Petersen.

E-mail addresses:wangjinliang1984@163.com (J.-L. Wang), whn@buaa.edu.cn(H.-N. Wu), tingwen.huang@qatar.tamu.edu (T. Huang).

http://dx.doi.org/10.1016/j.automatica.2015.03.0270005-1098/© 2015 Elsevier Ltd. All rights reserved.

Yao, Guan, & Hill, 2009; Yao, Wang, Guan, & Xu, 2009; Zhou, Lu,& Lü, 2006, 2008). In Slotine and Wang (2005), the authors stud-ied the synchronization of a network with fixed and switchingtopologies by using partial contraction theory. In Arcak (2011), asufficient conditionwas obtained to guarantee the synchronizationof a compartmental ODE model by utilizing the properties of theLaplacian matrix and the Mean-Value theorem. However, in theseexisting works, the node state is only dependent on the time. But,in many circumstances, the node state is not only dependent onthe time, but also intensively dependent on space variable.

As a special class of complex networks, arrays of coupledneural networks have attracted much attention in recent years.Especially, the synchronization problem of arrays of coupled neu-ral networks has stirred much research interest due to its fruit-ful applications in various fields. In Hoppensteadt and Izhikevich(2000), the authors proposed an architecture of coupled neuralnetworks to store and retrieve complex oscillatory patterns as syn-chronization states. In Zhang and He (1997), a secure commu-nication system based on coupled cellular neural networks waspresented. In addition, the study of synchronization of coupledneural networks is an important step for understanding brain sci-ence (Gray, 1994; Ukhtomsky, 1978). Therefore, it is interestingto investigate the synchronization of coupled neural networks. InTang and Fang (2009), the authors considered a general modelof an array of N linearly coupled delayed neural networks with

106 J.-L. Wang et al. / Automatica 56 (2015) 105–112

Markovian jumping hybrid coupling, which is composed of con-stant coupling, discrete and distributed time-varying delay cou-pling. In Li, Song, and Fei (2010), an array of coupled discrete-timeCohen–Grossberg neural networks with time-varying delay wasdiscussed. However, in these existing works (Li et al., 2010; Tang &Fang, 2009), the diffusion effects have not been considered. Strictlyspeaking, diffusion effects cannot be avoided in neural networkswhen electrons are moving in asymmetric electromagnetic fields,thus we must consider the diffusion effects in neural networks. Toour knowledge, very few researchers have investigated the syn-chronization in coupled reaction–diffusion neural networks (Liu,2010; Wang, Teng, & Jiang, 2012). In Liu (2010), the author inves-tigated a class of linearly coupled reaction–diffusion neural net-workswith unbounded timedelays. InWang, Teng et al. (2012), theadaptive synchronization in an array of linearly coupledneural net-works with reaction–diffusion terms and time delays was studied.

Recently, passivity theory has also received a great deal of atten-tion, andmany results on this topic have been reported. The passiv-ity theory was firstly proposed in the circuit analysis (Bevelevich,1968), and since then has found successful applications in diverseareas such as stability, complexity, signal processing, chaos con-trol and synchronization, fuzzy control, group coordination, powercontrol, flow control, energymanagement, and so on (Arcak, 2007;Hill & Moylan, 1977; Wu, 2001; Xie, Fu, & Li, 1998). Although re-search on passivity has attracted so much attention, little of thathad been devoted to the passivity properties of the spatially andtemporally complex dynamical networks until Wang, Wu and Guo(Wang, Wu, & Guo, 2011) obtained the conditions for passivityof reaction–diffusion neural networks. To the best of our knowl-edge, the passivity of arrays of coupled reaction–diffusion neuralnetworks has not yet been considered. Therefore, it is importantand interesting to study the passivity of coupled reaction–diffusionneural networks. On the other hand, the passivity theory has longbeen a nice tool for analyzing the synchronization of the complexnetworks. But in most existing works, it is assumed that the nodestate is only dependent on the time. Therefore, it is essential to in-vestigate the relationship between passivity and synchronizationof the coupled reaction–diffusion neural networks.

The objective of this paper is to study the synchronization prob-lem of arrays of coupled reaction–diffusion neural networks by us-ing the passivity theory. Themain contributions of this paper are asfollows. First, we establish a criterion for the output strict passiv-ity by utilizing some inequality techniques. Second, we reveal therelationship between output strict passivity and synchronizationof the proposed network model. Third, by employing the obtainedpassivity result and the relationship between output strict passiv-ity and synchronization, a sufficient condition for synchronizationof the complex dynamical network is derived.

The rest of this paper is organized as follows. In Section 2,our mathematical model of complex network is presented andsome preliminaries are given. The main results of this paper aregiven in Section 3. In Section 4, a numerical example is providedto illustrate the effectiveness of the theoretical results. Finally,Section 5 concludes the investigation.

2. Network model and preliminaries

Let R = (−∞, +∞), R+= [0, +∞), Rn be the n-dimensional

Euclidean space and Rn×m be the space of n×m real matrices. P ∈

Rn×n > 0(P ∈ Rn×n 6 0) means that matrix P is symmetric andsemi-positive (semi-negative) definite. P ∈ Rn×n > 0(P ∈ Rn×n <0) means that matrix P is symmetric and positive (negative)definite. In denotes the n × n real identity matrix. BT denotes thetranspose of matrix B. ⊗ denotes the Kronecker product of twomatrices. λm(·) and λM(·) denote the minimum and the maximumeigenvalue of the corresponding matrix, respectively. Ω = x =

(x1, x2, . . . , xq)T | |xk| < lk, k = 1, 2, . . . , q is an openbounded domain in Rq with smooth boundary ∂Ω , Ω = Ω ∪

∂Ω , and mesΩ denotes the measure of Ω . For any e(x, t) =

(e1(x, t), e2(x, t), . . . , en(x, t))T ∈ Rn, (x, t) ∈ Ω × R, ∥e(·, t)∥2denotes

∥e(·, t)∥2 =

Ω

ni=1

e2i (x, t)dx

12

.

In addition, we define ∥e(·, t)∥τ = sup−τ6θ60 ∥e(·, t + θ)∥2.In this paper, we consider a complex dynamical network

consisting of N identical reaction–diffusion neural networks. Tofacilitate the readers, the complex network model is presented ina step-by-step format.

A single reaction–diffusion neural network with Dirichletboundary conditions is described by the following partial differ-ential equations (PDEs):

∂wi(x, t)∂t

= diwi(x, t) − aiwi(x, t) + Ji

+

nj=1

bijfj(wj(x, t)) (1)

where i = 1, 2, . . . , n, n is the number of neurons in the network;x = (x1, x2, . . . , xq)T ∈ Ω ⊂ Rq

; wi(x, t) ∈ R is the state of theith neuron at time t and in space x; =

qk=1

∂2

∂x2kis the Laplace

diffusion operator on Ω; di > 0 represents the transmission dif-fusion coefficient along the ith neuron; fj(·) denotes the activationfunction of the jth neuron; ai > 0 represents the rate with whichthe ith neuron will reset its potential to the resting state when dis-connected from the network and external input; bij denotes thestrength of the jth neuron on the ith neuron; Ji is a constant ex-ternal input.

The initial value and boundary value conditions associatedwithsystem (1) are given in the form

wi(x, 0) = φi(x), x ∈ Ω, (2)wi(x, t) = 0, (x, t) ∈ ∂Ω × [0, +∞) (3)

where φi(x)(i = 1, 2, . . . , n) is bounded and continuous on Ω .Throughout this paper, the function fj(·)(j = 1, 2, . . . , n) sat-

isfies the Lipschitz condition, that is, there exists positive constantρj such that

|fj(ξ1) − fj(ξ2)| 6 ρj|ξ1 − ξ2|

for any ξ1, ξ2 ∈ R, where | · | is the Euclidean norm.We can rewrite system (1) in a compact form as follows:

∂w(x, t)∂t

= Dw(x, t) − Aw(x, t) + J + Bf (w(x, t)) (4)

where D = diag(d1, d2, . . . , dn), B = (bij)n×n, J = (J1, J2,. . . , Jn)T , A = diag(a1, a2, . . . , an), f (w(x, t)) = (f1(w1(x, t)),f2(w2(x, t)), . . . , fn(wn(x, t)))T , w(x, t) = (w1(x, t), w2(x, t), . . . ,wn(x, t))T .

N mutually coupled reaction–diffusion neural networks (4) canresult in a complex network, which is described by

∂zi(x, t)∂t

= Dzi(x, t) − Azi(x, t) + Bf (zi(x, t))

+ c1Nj=1

G1ijΓ1zj(x, t) + J

+ c2Nj=1

G2ijΓ2zj(x, t − τ(t)) + ui(x, t) (5)

J.-L. Wang et al. / Automatica 56 (2015) 105–112 107

Fig. 1. Complex dynamical network (5) (i, j ∈ 1, 2, . . . ,N, p, q ∈ 1, 2, . . . , n).

where i = 1, 2, . . . ,N , N is the number of nodes in the network;τ(t) is the time-varying delay with 0 6 τ(t) 6 τ ; zi(x, t) =

(zi1(x, t), zi2(x, t), . . . , zin(x, t))T ∈ Rn is the state vector of nodei; ui(x, t) ∈ Rn is the control input; c1 and c2 are positive realnumbers, which represent the coupling strength for non-delayedconfiguration and delayed one, respectively; Γ1 = diag(γ 1

1 , γ 12 ,

. . . , γ 1n ) ∈ Rn×n and Γ2 = diag(γ 2

1 , γ 22 , . . . , γ 2

n ) ∈ Rn×n are pos-itive definite matrices, which describe the individual coupling be-tween two nodes for non-delayed configuration and delayed one,respectively; G1

= (G1ij)N×N and G2

= (G2ij)N×N represent the

topological structure of network and coupling strength betweennodes for non-delayed configuration anddelayedone, respectively,whereG1

ij (G2ij can be defined similarly) is defined as follows: if there

exists a connection between node i and node j, then G1ij = G1

ji > 0;otherwise, G1

ij = G1ji = 0(i = j), and the diagonal elements of ma-

trix G1 are defined by

G1ii = −

Nj=1j=i

G1ij, i = 1, 2, . . . ,N.

In this paper, the topological structure of complex network (5)is undirected and weighted. The initial value and boundary valueconditions associated with network (5) are given in the form

zi(x, t) = Φi(x, t) ∈ Rn, (x, t) ∈ Ω × [−τ , 0], (6)zi(x, t) = 0, (x, t) ∈ ∂Ω × [−τ , +∞) (7)

where Φi(x, t)(i = 1, 2, . . . ,N) is a bounded and continuousfunction on Ω × [−τ , 0].

Remark 1. Many systems in nature and society can be modeled ascomplex networks, which consist of nodes connected by edges. In

a complex network, each node represents a fundamental unit withspecific activity, while edges represent the relationship of thesefundamental units. In this paper, we consider a complex dynamicalnetwork consisting of N identical nodes, in which each node isan n-dimensional reaction–diffusion neural network (RDNN) (seeFig. 1). In complex network (5), G1

ij and G2ij denote the impact

strength of the jth node on the ith node for non-delayed case anddelayed case, respectively.

In what follows, we introduce some useful definitions.

Definition 2.1 (See Willems, 1972). A system with supply rate ϑis said to be dissipative if there exists a nonnegative function S :

R+→ R+, called the storage function, such that tp

t0ϑ(u, y)dt > S(tp) − S(t0)

for any tp, t0 ∈ R+ and tp > t0, where u(x, t) ∈ Rn and y(x, t) ∈ Rn

are the input and output of the system at time t and in space x,respectively, (x, t) ∈ Ω × R+.

Definition 2.2 (See Hill & Moylan, 1976; Wang et al., 2011). Asystem is said to be strictly passive if it is dissipative with respectto

ϑ(u, y) =

Ω

yT (x, t)u(x, t)dx − γ1

Ω

uT (x, t)u(x, t)dx

− γ2

Ω

yT (x, t)y(x, t)dx

for γ1 > 0, γ2 > 0, γ1 + γ2 > 0, where u(x, t) ∈ Rn andy(x, t) ∈ Rn are the input and output of the system at time t andin space x, respectively, (x, t) ∈ Ω × R+.

The system is said to be input-strictly passive if γ1 > 0 andoutput-strictly passive if γ2 > 0.

3. Passivity-based synchronization of the complex delayeddynamical network

Suppose that w∗(x) = (w∗

1(x), w∗

2(x), . . . , w∗n(x))

T is an equi-librium solution of the reaction–diffusion neural network (1), thenit satisfies (3) and

0 = Dw∗(x) − Aw∗(x) + J + Bf (w∗(x)). (8)

Remark 2. In this section, we assume that w∗(x) is an equilibriumsolution of the reaction–diffusion neural network (1), but we donot require that the equilibrium solution be unique. Obviously,the uniqueness of equilibrium solution is also an important andinteresting problem. Practically, in Lu (2007), a class of intervalreaction–diffusionHopfield neural networkswithDirichlet bound-ary conditions was considered, and reaction–diffusion neuralnetwork (1) is a special case of this model. The author obtained asufficient condition for the uniqueness of the equilibrium solutionin Lu (2007).

Next, we give the definition of synchronization for the complexdynamical network (5).

Definition 3.1. The complex network (5) is said to achievesynchronization if

limt→+∞

∥zi(·, t) − w∗(·)∥2 = 0 for all i = 1, 2, . . . ,N

under the condition that ui(x, t) = 0, i = 1, 2, . . . ,N .

108 J.-L. Wang et al. / Automatica 56 (2015) 105–112

Defining ei(x, t) = zi(x, t) − w∗(x), then the dynamics of theerror vector ei(x, t) is governed by the following equation:∂ei(x, t)

∂t= Dei(x, t) − Aei(x, t) + Bf (zi(x, t))

− Bf (w∗(x)) + c1Nj=1

G1ijΓ1ej(x, t)

+ c2Nj=1

G2ijΓ2ej(x, t − τ(t)) + ui(x, t) (9)

where i = 1, 2, . . . ,N .In what follows, we establish the passivity of system (9). The

output vector yi(x, t) of system (9) is defined as

yi(x, t) = Fei(x, t) + Hui(x, t) (10)

for the passivity scheme, where F ,H ∈ Rn×n are known realmatrices, i = 1, 2, . . . ,N .

For the convenience, we denote

D = diag(D,D, . . . ,D), A = diag(A, A, . . . , A),

B = diag(B, B, . . . , B), F = diag(F , F , . . . , F),

H = diag(H,H, . . . ,H), Θ = diag(ρ21 , ρ

22 , . . . , ρ

2n ).

By using Kronecker product, we can rewrite systems (9) and (10)in a compact form as follows:∂e(x, t)

∂t= De(x, t) − Ae(x, t) + Bf (z(x, t))

− Bf (w∗(x)) + c1(G1⊗ Γ1)e(x, t)

+ c2(G2⊗ Γ2)e(x, t − τ(t)) + u(x, t)

y(x, t) = F e(x, t) + Hu(x, t) (11)

where e(x, t) = (eT1(x, t), eT2(x, t), . . . , e

TN(x, t))T , u(x, t) = (uT

1(x, t), uT

2(x, t), . . . , uTN(x, t))T , y(x, t) = (yT1(x, t), y

T2(x, t),

. . . , yTN(x, t))T , f (z(x, t)) = (f T (z1(x, t)), f T (z2(x, t)), . . . , f T (zN(x, t)))T , f (w∗(x)) = (f T (w∗(x)), f T (w∗(x)), . . . , f T (w∗(x)))T .

Lemma 3.1. Let τ (t) 6 σ < 1. If there exist matrices P =

diag(P1, P2, . . . , PN) ∈ RnN×nN > 0,Q ∈ RnN×nN > 0 and a scalarγ > 0 such that

PD + DP > 0 (12)χ P − F T

+ γ F T HP − F + γ HT F −H − HT

+ γ HT H

6 0 (13)

where Θ = diag(Θ, Θ, . . . , Θ), χ = −q

k=11l2k(P D+ DP)− P A−

AP + Θ + P BBT P + c1P(G1⊗Γ1)+ c1(G1

⊗Γ1)P +Q

1−σ+ c22 P(G2

⊗

Γ2)Q−1(G2⊗Γ2)P+γ F T F , then system (11) is output-strictly passive

in the sense of Definition 2.2.

Proof. See Appendix B.

Remark 3. To our knowledge, very few researchers have discussedthe passivity of reaction–diffusion neural networks (Wang et al.,2011), in which the input and output variables are varied withthe time and space variables. Especially, the passivity of coupledreaction–diffusion neural networks has not yet been investigated.In this paper, we consider a complex network consisting of Nlinearly and diffusively coupled identical reaction–diffusion neuralnetworks, which is totally different from the network modelconsidered in Wang et al. (2011). In Lemma 3.1, a sufficientcondition ensuring output strict passivity is established, which canbe applied to analyze linearly coupled reaction–diffusion neuralnetworks (5) with output vector defined by (10).

Lemma 3.2. Suppose that V : R+× C → R+ is continuously

differentiable and satisfies the following condition

v1(∥e(·, t)∥2) 6 V (t, et(x, θ)) 6 v2(∥e(·, t)∥τ ) (14)

where et(x, θ) , e(x, t + θ), −τ 6 θ 6 0, v1, v2 : R+→ R+ are

continuous and strictly monotonically nondecreasing functions, v1(s)and v2(s) are positive for s > 0with v1(0) = v2(0) = 0. The complexnetwork (5) is locally synchronized in the sense of Definition 3.1 ifsystem (11) is output-strictly passive with respect to storage functionV and matrix F is nonsingular.

Proof. System (11) is output-strictly passive with respect tostorage function V , then there obviously exists a positive constantγ such that

V (t + ε) − V (t) 6

t+ε

t

Ω

yT (x, s)u(x, s)dxds

− γ

t+ε

t

Ω

yT (x, s)y(x, s)dxds (15)

for any t ∈ R+ and ε > 0. Obviously, we can derive from (15) that

V (t + ε) − V (t)ε

6

t+ε

t

ΩyT (x, s)u(x, s)dxds

ε

− γ

t+ε

t

ΩyT (x, s)y(x, s)dxds

ε. (16)

By taking limit ϵ → 0 in (16), we have

V (t) 6

Ω

yT (x, t)u(x, t)dx − γ

Ω

yT (x, t)y(x, t)dx. (17)

Letting u(x, t) = 0, we can get from (17) that

V (t) 6 −γ λm(F T F)∥e(·, t)∥22.

From Lemma A.2, we can conclude that system (11) is internallylocally asymptotically stable. Therefore, the complex network (5)is locally synchronized in the sense of Definition 3.1. The proof iscompleted.

From Lemmas 3.1 and 3.2, we can obtain the followingconclusion.

Theorem 3.1. Let τ (t) 6 σ < 1. If there exist matrices F ,H ∈ Rn×n,P = diag(P1, P2, . . . , PN) ∈ RnN×nN > 0,Q ∈ RnN×nN > 0 and ascalar γ > 0 such that

PD + DP > 0 (18)χ P − F T

+ γ F T HP − F + γ HT F −H − HT

+ γ HT H

6 0 (19)

where F is a nonsingular matrix, Θ = diag(Θ, Θ, . . . , Θ), F =

diag(F , F , . . . , F), H = diag(H,H, . . . ,H), χ = −q

k=11l2k

(P D + DP) − P A − AP + Θ + P BBT P + c1P(G1⊗ Γ1) + c1(G1

⊗

Γ1)P +Q

1−σ+ c22 P(G2

⊗ Γ2)Q−1(G2⊗ Γ2)P + γ F T F , then complex

network (5) is locally synchronized in the sense of Definition 3.1.

Remark 4. In Chopra (2012) and Chopra and Spong (2006),the authors studied the output synchronization of multi-agentsystems, in which the agents’ dynamics are input–output passiveand described by affine nonlinear ODE systems. In this paper,we propose a complex delayed dynamical network consistingof N identical reaction–diffusion neural networks. Firstly, weanalyze the output strict passivity of the proposed networkmodel, which cannot be dealt with by those techniques used in

J.-L. Wang et al. / Automatica 56 (2015) 105–112 109

Chopra (2012) and Chopra and Spong (2006). Then, we reveal therelationship between output strict passivity and synchronizationof the complex delayed dynamical network (5). Moreover, by usingthe obtained passivity result and the relationship between outputstrict passivity and synchronization, a sufficient condition ensuringsynchronization is obtained.

Remark 5. To our knowledge, there are very few works on thesynchronization of coupled reaction–diffusion neural networks(Liu, 2010; Wang, Teng et al., 2012), and the asymptotic stabilityof synchronization error system has not yet been investigated. InTheorem3.1, a sufficient condition is established to ensure that thesynchronization error system is locally asymptotically stable.

4. Numerical example

In this section, an illustrative example is provided to verify theeffectiveness of the proposed theoretical results.

Consider a complex network consisting of 5 identical nodeswith diffusive coupling, in which each node is a 3-dimensionalreaction–diffusion neural network described by∂wi(x, t)

∂t= diwi(x, t) − aiwi(x, t) + Ji

+

3j=1

bijfj(wj(x, t)) (20)

where i = 1, 2, 3, Ω = x | −0.5 < x < 0.5, fj(ξ) =|ξ+1|−|ξ−1|

4 , d1 = 0.3, d2 = 0.2, d3 = 0.4, a1 = 0.4, a2 =

0.3, a3 = 0.3, J1 = J2 = J3 = 0, and the matrix B = (bij)3×3 ischosen as

B =

0.2 0.2 0.30.1 0.3 0.30.3 0.1 0.2

.

Obviously, w∗(x) = (0, 0, 0)T ∈ R3 is an equilibrium solutionof the reaction–diffusion neural network (20), and fj(·)(j =

1, 2, 3) satisfies the Lipschitz condition with ρj =12 . In the

following, we analyze the asymptotic stability of synchronizedstate [(w∗(x))T , (w∗(x))T , . . . , (w∗(x))T ]T .

We take c1 = 0.2, c2 = 0.1, Γ1 = diag(0.4, 0.2, 0.3), Γ2 =

diag(0.3, 0.1, 0.2) and τ(t) = 0.1 − 0.1e−t . The matrices G1 andG2 are chosen as, respectively,

G1=

−0.5 0.2 0 0.3 00.2 −0.8 0.2 0.2 0.20 0.2 −0.5 0 0.30.3 0.2 0 −0.5 00 0.2 0.3 0 −0.5

,

G2=

−0.3 0.1 0 0.2 00.1 −0.6 0.2 0.1 0.20 0.2 −0.4 0 0.20.2 0.1 0 −0.3 00 0.2 0.2 0 −0.4

.

By using the YALMIP Toolbox of MATLAB, we can find thefollowing matrices:

F =

0.8605 −0.0401 −0.0185−0.0427 1.0158 −0.0226−0.0178 −0.0204 0.7640

,

H =

0.6442 0.0048 0.00230.0048 0.6406 0.00250.0023 0.0025 0.6471

,

P = I5 ⊗

1.1401 −0.0413 −0.0153−0.0413 1.3780 −0.0203−0.0153 −0.0203 1.0000

> 0,

Fig. 2. The change processes of ∥zi(·, t)∥2, i = 1, 2, . . . , 5.

Q = I5 ⊗

1.2851 −0.1148 −0.0773−0.1148 0.8580 −0.0708−0.0773 −0.0708 1.4187

> 0

satisfying (18) and (19) with γ = 0.4127. According to Theo-rem 3.1, the complex network (5) with above given parameters islocally synchronized in the sense of Definition 3.1. The simulationresults are shown in Fig. 2.

From Fig. 2, we clearly see that ∥zi(·, t)∥2, i = 1, 2, . . . , 5are very close to 0 when the time t increases gradually to 2.5s,and this state is maintained along with the increasing of the time.These results show that zi(x, t)(i = 1, 2, . . . , 5) asymptoticallyconverges to w∗(x) = (0, 0, 0)T , thus the complex network(5) is synchronized. Fig. 2 demonstrates the correctness andeffectiveness of the obtained result in Theorem 3.1.

Remark 6. It is obvious that the delay has a strong influence onthe rate of change of the node state (see (5)). The change processesof ∥zi(·, t)∥2, i = 1, 2, . . . , 5, may be different if we take othertime-varying delays in Fig. 2, although we omit this comparisonfor brevity. On the other hand, since

Nj=1 G

1ij = 0 and

Nj=1 G

2ij =

0(i = 1, 2, . . . ,N),N

j=1 G1ijΓ1zj(x, t) and

Nj=1 G

2ijΓ2zj(x, t −τ(t))

will tend to zero asymptotically if the complex network (5) issynchronized.

Remark 7. YALMIP Toolbox is a very convenient tool to solvelinear matrix inequality (LMI), which has been widely usedin analyzing the stability of various systems. Using the Schurcomplement, we can obtain that (19) is equivalent to

W1 +Q

1 − σP − F T F T P B W2

P − F −H − HT HT 0 0

F H −1γInN 0 0

BT P 0 0 −InN 0WT

2 0 0 0 −Q

6 0,

whereW1 = −q

k=11l2k(P D+DP)− P A− AP+Θ +c1P(G1

⊗Γ1)+

c1(G1⊗ Γ1)P, W2 = c2P(G2

⊗ Γ2). Then, by utilizing the YALMIPToolbox of MATLAB, we can find the F ,H, P,Q , γ satisfying (18)and the above LMI.

110 J.-L. Wang et al. / Automatica 56 (2015) 105–112

5. Conclusion

In this paper, a general array model of coupled reaction–diffusion neural networks with delay coupling has been intro-duced. The output strict passivity of the proposed network modelhas been taken into consideration, and a sufficient conditionhas been established. Furthermore, the relationship between out-put strict passivity and synchronization of the proposed networkmodel has been revealed. In addition, a sufficient condition en-suring synchronization has been derived by utilizing the obtainedpassivity result and the relationship between output strict passiv-ity and synchronization. A numerical example has been providedto verify the correctness and effectiveness of the theoretical re-sults. In future work, we shall study the passivity of coupled re-action–diffusion neural networks with impulsive effects.

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir valuable comments and suggestions.

Appendix A. Some useful lemmas

Lemma A.1 (See Lu, 2008). Let Ω be a cube |xk| < lk(k =

1, 2, . . . , q) and let h(x) be a real-valued function belonging to C1(Ω)which vanishes on the boundary ∂Ω of Ω , i.e., h(x)|∂Ω = 0. Then

Ω

h2(x)dx 6 l2k

Ω

∂h∂xk

2

dx

where x = (x1, x2, . . . , xq)T .

Consider the following delay partial differential equation(DPDE) system

∂ϕ(x, t)∂t

= Dϕ(x, t) + g(ϕ(x, t − τ(t))) + f (ϕ(x, t))

ϕ(x, t) = φ(x, t), (x, t) ∈ Ω × [−τ , 0]ϕ(x, t) = 0, (x, t) ∈ ∂Ω × [−τ , +∞)

(21)

where x = (x1, x2, . . . , xq)T ∈ Ω ⊂ Rq; ϕ(x, t) = (ϕ1(x, t),ϕ2(x, t), . . . , ϕn(x, t))T ∈ Rn is the state vector of system attime t and in space x; =

qk=1

∂2

∂x2kis the Laplace operator on

Ω; τ(t) is the time-varying delay with 0 6 τ(t) 6 τ ; D =

diag(d1, d2, . . . , dn) > 0; φ(x, t) ∈ Rn is bounded and continuouson Ω × [−τ , 0]; the functions f (·) ∈ Rn and g(·) ∈ Rn satisfyf (0) = 0, g(0) = 0, and the following assumption:

(A1) The functions f (·) and g(·) are continuous, and there existtwo positive constants α1 and α2 such that

|f (ξ1) − f (ξ2)| 6 α1|ξ1 − ξ2|,

|g(ξ1) − g(ξ2)| 6 α2|ξ1 − ξ2|

for any ξ1, ξ2 ∈ Rn, where | · | is the Euclidean norm.

Definition A.1. Let ϕ(x, t, φ) be the state trajectory of system(21) with initial condition φ(x, t). The equilibrium solution ϕ∗

=

0 of system (21) is said to be asymptotically stable if for anyϵ > 0, there exists δ(ϵ) such that ∥φ(·, 0)∥τ < δ(ϵ) implies∥ϕ(·, t, φ)∥2 < ϵ for t > 0, and there is a b0 > 0 such that∥φ(·, 0)∥τ < b0 implies ∥ϕ(·, t, φ)∥2 → 0 as t → +∞.

To our knowledge, there are very few works on the asymptoticstability (in the sense of Definition A.1) of spatially and temporallycomplex dynamical networks. It is well known that Lyapunov

functional is a very convenient tool to analyze the asymptoticstability of DODE systems. In the following, we give a sufficientcondition for the asymptotic stability of the equilibrium solutionϕ∗

= 0 of system (21).

Lemma A.2 (See Wang, Wu, & Guo, 2012). Suppose that v1, v2, v3 :

R+→ R+ are continuous and strictly monotonically nondecreasing

functions, v1(s), v2(s), v3(s) are positive for s > 0 with v1(0) =

v2(0) = 0. If there is a continuous functional V : R+×C → R+ such

that

v1(∥ϕ(·, t)∥2) 6 V (t, ϕt(x, θ)) 6 v2(∥ϕ(·, t)∥τ ),

V (t, ϕt(x, θ)) 6 −v3(∥ϕ(·, t)∥2)

where ϕt(x, θ) , ϕ(x, t + θ), −τ 6 θ 6 0, V is the derivativeof V along the solution of system (21), then the equilibrium solutionϕ∗

= 0 of system (21) is asymptotically stable.

Appendix B. Proof of Lemma 3.1

First, construct a Lyapunov functional for system (11) asfollows:

V (t) =

Ω

eT (x, t)Pe(x, t)dx

+1

1 − σ

t

t−τ(t)

Ω

eT (x, s)Qe(x, s)dxds. (22)

Calculating the time derivative of V (t) along the trajectory ofsystem (11), we can get

V (t) = 2

Ω

eT (x, t)P∂e(x, t)

∂tdx +

11 − σ

Ω

eT (x, t)Qe(x, t)dx

−1 − τ (t)1 − σ

Ω

eT (x, t − τ(t))Qe(x, t − τ(t))dx

6

Ω

[eT (x, t)P De(x, t) + (e(x, t))T DPe(x, t)

+ eT (x, t)(−P A − AP + c1P(G1⊗ Γ1) +

Q1 − σ

+ c1(G1⊗ Γ1)P)e(x, t) + 2eT (x, t)P B(f (z(x, t))

− f (w∗(x))) + 2eT (x, t)Pu(x, t) + 2c2eT (x, t)P× (G2

⊗ Γ2)e(x, t − τ(t))]dx

−

Ω

eT (x, t − τ(t))Qe(x, t − τ(t))dx.

From Green’s formula and the boundary condition, we haveΩ

eil(x, t)eij(x, t)dx = −

qk=1

Ω

∂eil(x, t)∂xk

∂eij(x, t)∂xk

dx

where l, j ∈ 1, 2, . . . , n, i = 1, 2, . . . ,N . Letting P i= (pijl)n×n,

we can obtainΩ

eT (x, t)P De(x, t)dx

=

Ni=1

Ω

eTi (x, t)PiDei(x, t)dx

=

Ni=1

nj=1

nl=1

pijldl

Ω

eij(x, t)eil(x, t)dx

= −

qk=1

Ni=1

nj=1

nl=1

pijldl

Ω

∂eij(x, t)∂xk

∂eil(x, t)∂xk

dx

J.-L. Wang et al. / Automatica 56 (2015) 105–112 111

= −

qk=1

Ω

∂e(x, t)

∂xk

T

P D∂e(x, t)

∂xkdx,

Ω

(e(x, t))T DPe(x, t)dx

=

Ni=1

Ω

(ei(x, t))TDP iei(x, t)dx

=

Ni=1

nj=1

nl=1

djpijl

Ω

eil(x, t)eij(x, t)dx

= −

qk=1

Ω

∂e(x, t)

∂xk

T

DP∂e(x, t)

∂xkdx.

Then, we can getΩ

(eT (x, t)P De(x, t) + (e(x, t))T DPe(x, t))dx

= −

qk=1

Ω

∂e(x, t)

∂xk

T

(P D + DP)∂e(x, t)

∂xkdx.

On the other hand, there obviously exists a real matrix C ∈ RnN×nN

such that

P D + DP = CTC,

then∂e(x, t)

∂xk

T

(P D + DP)∂e(x, t)

∂xk=

∂e(x, t)

∂xk

T

CTC∂e(x, t)

∂xk

=

∂(Ce(x, t))

∂xk

T∂(Ce(x, t))

∂xk.

Let ϑ(x, t) = Ce(x, t), for (x, t) ∈ ∂Ω × [−τ , +∞) from theboundary condition (7), we have ϑ(x, t) = Ce(x, t) = 0. In view ofLemma A.1, one has

qk=1

Ω

∂ϑ(x, t)

∂xk

T∂ϑ(x, t)

∂xkdx

>

qk=1

1l2k

Ω

ϑT (x, t)ϑ(x, t)dx

=

qk=1

1l2k

Ω

eT (x, t)(P D + DP)e(x, t)dx. (23)

Furthermore, we can easily derive

2eT (x, t)P B(f (z(x, t)) − f (w∗(x)))

= 2Ni=1

eTi (x, t)PiB(f (zi(x, t)) − f (w∗(x)))

6

Ni=1

eTi (x, t)PiBBTP iei(x, t) +

Ni=1

eTi (x, t)Θei(x, t)

= eT (x, t)(P BBT P + Θ)e(x, t), (24)

2eT (x, t)P(G2⊗ Γ2)e(x, t − τ(t))

6 c2eT (x, t)P(G2⊗ Γ2)Q−1(G2

⊗ Γ2)Pe(x, t)

+1c2

eT (x, t − τ(t))Qe(x, t − τ(t)). (25)

It follows from (23)–(25) that

V (t) 6

Ω

eT (x, t)

−

qk=1

1l2k

(P D + DP) − P A − AP

+ Θ + P BBT P + c1P(G1⊗ Γ1) + c1(G1

⊗ Γ1)P

+Q

1 − σ+ c22 P(G2

⊗ Γ2)Q−1(G2⊗ Γ2)P

e(x, t)dx

+ 2

Ω

eT (x, t)Pu(x, t)dx.

Therefore,

V (t) − 2

Ω

yT (x, t)u(x, t)dx + γ

Ω

yT (x, t)y(x, t)dx

6

Ω

eT (x, t)

−

qk=1

1l2k

(P D + DP) − P A − AP

+ Θ + P BBT P + c1P(G1⊗ Γ1) + c1(G1

⊗ Γ1)P

+Q

1 − σ+ c22 P(G2

⊗ Γ2)Q−1(G2⊗ Γ2)P + γ F T F

× e(x, t)dx + 2

Ω

eT (x, t)(P − F T+ γ F T H)u(x, t)dx

+

Ω

uT (x, t)(−H − HT+ γ HT H)u(x, t)dx

=

Ω

ξ T (x, t)

χ χ1

χ T1 χ2

ξ(x, t)dx,

where ξ(x, t) = (eT (x, t), uT (x, t))T , χ1 = P − F T+ γ F T H, χ2 =

−H − HT+ γ HT H . From (13), we can get

V (t) + γ

Ω

yT (x, t)y(x, t)dx 6 2

Ω

yT (x, t)u(x, t)dx. (26)

By integrating Eq. (26) with respect to t over the time period t0 totp, we can obtain

2 tp

t0

Ω

yT (x, t)u(x, t)dxdt

> V (tp) − V (t0) + γ

tp

t0

Ω

yT (x, t)y(x, t)dxdt.

Namely, tp

t0

Ω

yT (x, t)u(x, t)dxdt

>12(V (tp) − V (t0)) +

γ

2

tp

t0

Ω

yT (x, t)y(x, t)dxdt

for any tp, t0 ∈ R+ and tp > t0. The proof is completed.

References

Arcak, M. (2007). Passivity as a design tool for group coordination. IEEE Transactionson Automatic Control, 52, 1380–1390.

Arcak, M. (2011). Certifying spatially uniform behavior in reaction–diffusion PDEand compartmental ODE systems. Automatica, 47, 1219–1229.

Bevelevich, V. (1968). Classical network synthesis. New York: Van Nostrand.Chopra, N. (2012). Output synchronization on strongly connected graphs. IEEE

Transactions on Automatic Control, 57, 2896–2901.Chopra, N., & Spong, M. W. (2006). Passivity-based control of multi-agent systems.

In S. Kawamura, & M. Svinin (Eds.), Advances in robot control: from everydayphysics to human-like movements (pp. 107–134). Berlin: Springer-Verlag.

Gray, C. M. (1994). Synchronous oscillations in neuronal systems: mechanisms andfunctions. Journal of Computational Neuroscience, 1, 11–38.

Hill, D., & Moylan, P. (1976). The stability of nonlinear dissipative systems. IEEETransactions on Automatic Control, AC-21, 708–711.

112 J.-L. Wang et al. / Automatica 56 (2015) 105–112

Hill, D. J., & Moylan, P. J. (1977). Stability results for nonlinear feedback systems.Automatica, 13, 377–382.

Hoppensteadt, F. C., & Izhikevich, E. M. (2000). Pattern recognition via synchro-nization in phase-locked loop neural networks. IEEE Transactions on Neural Net-works, 11, 734–738.

Li, T., Song, A., & Fei, S. (2010). Synchronization control for arrays of coupleddiscrete-time delayed Cohen–Grossberg neural networks. Neurocomputing , 74,197–204.

Liu, X. (2010). Synchronization of linearly coupled neural networks withreaction–diffusion terms and unbounded time delays. Neurocomputing , 73,2681–2688.

Lu, J. G. (2007). Robust global exponential stability for interval reaction–diffusionhopfield neural networks with distributed delays. IEEE Transactions on Circuitsand Systems II: Express Briefs, 54, 1115–1119.

Lu, J. G. (2008). Global exponential stability and periodicity of reaction–diffusiondelayed recurrent neural networks with Dirichlet boundary conditions. Chaos,Solitons and Fractals, 35, 116–125.

Lü, J., & Chen, G. (2005). A time-varying complex dynamical network model and itscontrolled synchronization criteria. IEEE Transactions on Automatic Control, 50,841–846.

Scardovi, L., Arcak, M., & Sontag, E. D. (2010). Synchronization of interconnectedsystemswith applications to biochemical networks: an input–output approach.IEEE Transactions on Automatic Control, 55, 1367–1379.

Slotine, J. J. E., &Wang,W. (2005). A study of synchronization and group cooperationusing partial contraction theory. In Lecture notes in control and informationscience: Vol. 309. Cooperative control (pp. 207–228). Springer.

Tang, Y., & Fang, J. A. (2009). Adaptive synchronization in an array of chaotic neuralnetworks with mixed delays and jumping stochastically hybrid coupling.Communications in Nonlinear Science and Numerical Simulation, 14, 3615–3628.

Ukhtomsky, A. A. (1978). Collected works. (pp. 107–237). Leningrad, Russia: Nauka.Wang, K., Teng, Z., & Jiang, H. (2012). Adaptive synchronization in an array of

linearly coupled neural networks with reaction–diffusion terms and timedelays. Communications in Nonlinear Science and Numerical Simulation, 17,3866–3875.

Wang, J.-L., Wu, H.-N., & Guo, L. (2011). Passivity and stability analysis ofreaction–diffusion neural networks with Dirichlet boundary conditions. IEEETransactions on Neural Networks, 22, 2105–2116.

Wang, J.-L., Wu, H.-N., & Guo, L. (2012). Pinning control of spatially and temporallycomplex dynamical networks with time-varying delays. Nonlinear Dynamics,70, 1657–1674.

Willems, J. C. (1972). Dissipative dynamical systems part I: general theory. Archivefor Rational Mechanics and Analysis, 45, 321–351.

Wu, C. W. (2001). Synchronization in arrays of coupled nonlinear systems:passivity, circle criterion, and observer design. IEEE Transactions on Circuits andSystems—I: Fundamental Theory and Applications, 48, 1257–1261.

Xie, L., Fu,M., & Li, H. (1998). Passivity analysis and passification for uncertain signalprocessing systems. IEEE Transactions on Signal Processing , 46, 2394–2403.

Yao, J., Guan, Z. H., & Hill, D. J. (2009). Passivity-based control and synchronizationof general complex dynamical networks. Automatica, 45, 2107–2113.

Yao, J., Wang, H. O., Guan, Z. H., & Xu, W. (2009). Passive stability andsynchronization of complex spatio-temporal switching networks with timedelays. Automatica, 45, 1721–1728.

Zhang, Y., & He, Z. (1997). A secure communication scheme based on cellularneural network. In Proceedings of the IEEE international conference on intelligentprocessing systems (pp. 521–524).

Zhou, J., Lu, J. A., & Lü, J. (2006). Adaptive synchronization of an uncertain complexdynamical network. IEEE Transactions on Automatic Control, 51, 652–656.

Zhou, J., Lu, J. A., & Lü, J. (2008). Pinning adaptive synchronization of a generalcomplex dynamical network. Automatica, 44, 996–1003.

Jin-Liang Wang was born in Hebei, China, in 1984. Hereceived the Ph.D. degree in Control Theory and ControlEngineering from the School of Automation Science andElectrical Engineering, Beihang University, Beijing, China,in 2014.

In January 2014, he joined the School of Computer Sci-ence & Software Engineering, Tianjin Polytechnic Univer-sity, Tianjin, China. Hewas a ProgramAidwith Texas A&MUniversity at Qatar, Doha, Qatar, in 2014 for two months.His current research interests include complex networks,multi-agent systems, and distributed parameter systems.

He was the recipient of the 2011 Excellent Master Degree Thesis Award ofChongqing city and the Scholarship Award for Excellent Doctoral Student grantedby Ministry of Education (2012).

Huai-Ning Wu was born in Anhui, China, in 1972. He re-ceived the B.E. degree in Automation from Shandong Insti-tute of Building Materials Industry, Jinan, China, and thePh.D. degree in Control Theory and Control Engineeringfrom Xi’an Jiaotong University, Xi’an, China, in 1992 and1997, respectively.

From August 1997 to July 1999, he was a Postdoc-toral Researcher at the Department of Electronic Engi-neering, Beijing Institute of Technology, Beijing, China. InAugust 1999, he joined the School of Automation Scienceand Electrical Engineering, Beihang University (formerly

Beijing University of Aeronautics and Astronautics), Beijing. From December 2005to May 2006, he was a Senior Research Associate at the Department of Manufac-turing Engineering and Engineering Management (MEEM), City University of HongKong, Kowloon, HongKong. FromOctober toDecember during 2006–2008 and fromJuly to August in 2010, hewas a Research Fellowwith theDepartment ofMEEM, CityUniversity of Hong Kong. From July to August in 2011, he was a Research Fellowwith the Department of Systems Engineering and Engineering Management, CityUniversity of Hong Kong. He is currently a Professor with Beihang University. Hiscurrent research interests include robust control, fault-tolerant control, distributedparameter systems, and fuzzy/neural modeling and control.

He serves as an Associate Editor of the IEEE Transactions on Systems, Man & Cy-bernetics: Systems. He is a member of the Committee of Technical Process FailureDiagnosis and Safety, Chinese Association of Automation.

Tingwen Huang received his B.S. degree in Mathematicsfrom Southwest Normal University, Chongqing, China, in1990, M.S. degree in Applied Mathematics from SichuanUniversity, Chengdu, China, in 1993 and Ph.D. degree inMathematics from Texas A&MUniversity, College Station,Texas, USA, in 2002.

He was a Lecturer with Jiangsu University, Zhenjiang,China, from 1994 to 1998, and a Visiting AssistantProfessor with Texas A & M University, College Station,TX, USA, in 2003. From 2003 to 2009, he was an AssistantProfessor, from 2009 to 2013, an Associate Professor, and,

since 2013, a Professor with Texas A & M University at Qatar, Doha, Qatar. Hiscurrent research interests include neural networks, complex networks, chaos anddynamics of systems, and operator semi-groups and their applications.

He serves as an editorial board member for three international journals, suchas the IEEE Transactions on Neural Networks and Learning Systems, CognitiveComputation, and Advances in Artificial Neural Systems.