Cluster synchronization in an array of coupled stochastic delayed neural networks via pinning control

Neurocomputing 74 (2011) 846–856

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Cluster synchronization in an array of coupled stochastic delayed neural networks via pinning control Lulu Li, Jinde Cao  Department of Mathematics, Southeast University, Nanjing 210096, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 16 November 2009 Received in revised form 5 September 2010 Accepted 29 November 2010 Communicated by S. Arik Available online 15 December 2010

In this paper, cluster synchronization problem is studied for an array of coupled stochastic delayed neural networks by using pinning control strategy. Based on the free matrix approach and stochastic analysis techniques, some sufficient criteria are derived to ensure cluster synchronization of the network model if a single linear or adaptive feedback controller is added to each cluster. Furthermore, two specific methods are given to achieve desired cluster synchronization pattern. Finally, a numerical example is provided to demonstrate the effectiveness of the obtained theoretical results. & 2010 Elsevier B.V. All rights reserved.

Keywords: Delayed neural networks Lyapunov function Cluster synchronization Stochastic disturbances Pinning control

1. Introduction Complex dynamical networks are everywhere in nature and our daily life, such as biological neural networks, ecosystems, the Internet, the WWW, and so on [1,3]. Over the past two decades, they have been widely investigated in various fields of physics, mathematics, engineering, biology and sociology [2,3,5–12,23]. Most previous works have concentrated on the structure properties of various dynamical networks [1–4]. However, research of collective behavior in arrays of coupled nonlinear systems arrests increasing attention since they can exhibit many interesting phenomena such as spatiotemporal chaos, autowaves, spiral waves, etc. [5,6]. Over them, as a typical collective behavior of coupled nonlinear systems, synchronization has been a hot topic due to its potential application in engineering [7–12]. Different kinds of synchronization have been found and studied, such as complete synchronization [7–10], cluster synchronization [19–23], mSynchronization [11], phase synchronization [15], lag synchronization [16,31] and generalized synchronization [14]. Cluster synchronization means that synchronization occurs in one group with another, but there is no synchronization among the different groups. It has been shown that cluster synchronization is significant in biological science and communication engineering [17]. In [19], an approach to constructing different coupling schemes to realize cluster synchronization for coupled Josephson equations was proposed. Ma and Liu [20] showed that the arbitrarily selected cluster synchronization manifolds could be

 Corresponding author.

E-mail address: [email protected] (J. Cao). 0925-2312/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2010.11.006

stabilized by constructing a special coupled matrix for connected chaotic networks. Cluster synchronization in two-dimensional and three-dimensional lattices of diffusively coupled chaotic oscillators was studied in [21,22]. In [23], the relation between the cluster synchronization and global synchronization was discussed. Synchronization of delayed neural networks have been extensively studied owing to its importance on the fundamental science (e.g., the self-organization behavior in the brain). Self-organization means the spontaneous formation of patterns and pattern change in open, nonequilibrium systems. The human brain is the typical selforganized system governed by potentially discoverable, nonlinear dynamical laws [24]. Synchronization among neurons, as an important self-organized activity of brain, play the key role in the communication and even the basis of communication [25]. In 1992, it was founded that the effective coupling among neurons varied temporally in a rather short time scale and the degree of synchronization among pairs of neurons changed both temporally and by the choice of pairs [18]. Therefore, the study of cluster synchronization phenomena of coupled neural networks is an important step toward both basic theory (e.g., brain science) and technological practice. However, most existing works have focused on the complete synchronization, i.e., the states of the coupled systems are asymptotically equal [8,27]. In [34], a cluster synchronization scheme is presented for the following coupled neural networks with delays: N X dxi ðtÞ Gð1Þ ¼ Cxi ðtÞ þ Af ðxi ðtÞÞ þ Bf ðxi ðttÞÞ þ IðtÞ þ ij D1 xj ðtÞ dt j¼1

þ

N X j¼1

Gð2Þ ij D2 xj ðttÞ þ

N X j¼1

Gð3Þ ij D3

Z

t tt

xj ðsÞ ds,

i ¼ 1,2, . . . ,N,

ð1Þ

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

where xi ðtÞ ¼ ðxi1 ðtÞ,xi2 ðtÞ, . . . ,xin ðtÞÞT A Rn is the state vector of the ith network at time t. C¼diag(c1,c2,y,cn), with ci 40 denotes the rate with which the cell i resets its potential to the resting when being isolated from other cells and inputs, A ¼ ðarj Þnn , B ¼ ðbrj Þnn represent the connection weight matrix and the delayed connection weight matrix respectively, f is the activation function with f(xi(t))¼(f1(xi1(t)), f2 ðxi2 ðtÞÞ, . . . ,fn ðxin ðtÞÞÞT , t 40 is the transmission delay, I ¼ ðI1 ðtÞ,I2 ðtÞ, . . . ,In ðtÞÞT A Rn is an external input vector. In [34], by constructing a special coupling matrix, several sufficient criteria for cluster synchronization are derived based on Lyapunov stability theory and linear matrix inequality (LMI) technique. However, due to the existence of embedding of invariant synchronization manifolds, it may occur that the system can reach different clustering patterns from different initial values by means of the cluster synchronization scheme proposed in [34]. Therefore, a suitable method to achieve the desired cluster synchronization pattern independent of initial values was needed. In [28], cluster synchronization under pinning control of the complex network with community structure was studied. In [29], Wu et al. propose a pinning control strategy to steer a dynamical network to an expected cluster synchronization pattern. In this paper, pinning control strategy will be used to realize the desired cluster synchronization pattern for coupled delayed neural networks. On the other hand, in real nervous networks, it has been found that synaptic transmission, as a noisy process, is brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Therefore, to investigate and simulate more realistic dynamical process for synchronization of coupled neural networks, the effect of noise should be taken into account. Recent research works on complete synchronization of stochastic delayed neural networks can be seen in [30–33]. Unfortunately, to the best of the authors’ knowledge, the cluster synchronization problem for coupled stochastic neural networks with time-varying delays has not been studied yet, and it is, therefore, the purpose of this paper to tackle shorten such a gap. Motivated by the aforementioned discussions, this paper aims to analyze the cluster synchronization problem for an array of delayed neural systems with stochastic disturbances via pinning control. By employing stochastic analysis techniques and linear matrix inequality (LMI) technique, several cluster synchronization criteria are derived for addressed coupled neural networks. Furthermore, two methods are given to guarantee the cluster synchronization of the proposed model. The organization of the remaining part is as follows. In Section 2, the model of an array of coupled stochastic delayed neural networks is presented and some preliminaries are introduced. The pinning control of the proposed model is discussed and sufficient conditions of cluster synchronization are developed in Section 3. In Section 4, a numerical simulation is given to show the effectiveness of the theoretical results. Finally, concluding remarks are drawn in Section 5. Notations: Throughout this paper, Rn and Rmn represent ndimensional Euclidean space and the set of m  n real matrices, respectively. The notation X 4 0 ðX o0Þ means the symmetric matrix X is positive (negative) definite; In represents the ndimensional identity matrix; lmax ðÞ or lmin ðÞ denotes the largest or smallest eigenvalue of a symmetric matrix, respectively; Efg stands for the mathematical expectation operator.

2. Problem formulation and preliminaries

847

follows: " dxi ðtÞ ¼ Cxi ðtÞ þ Af ðxi ðtÞÞ þBgðxi ðttðtÞÞÞ þD

Z

t ttðtÞ

þ

N X

hðxi ðsÞÞ ds þIðtÞ

cð1Þ Gð1Þ Dð1Þ xj ðtÞ þ ij

j¼1

þ

N X

N X

cð2Þ Gð2Þ Dð2Þ xj ðttðtÞÞ ij

j¼1

cð3Þ Gð3Þ Dð3Þ ij

Z

j¼1

ttðtÞ

xj ðsÞ dsþ ui ðtÞ dt

þ si ðt,x1 ðtÞ, . . . ,xN ðtÞ, x1 ðttðtÞÞ, . . . ,xN ðttðtÞÞÞdoi ðtÞ, i ¼ 1,2, . . . ,N,

ð2Þ

ð2Þ ð3Þ where Gð1Þ ¼ ðGð1Þ ¼ ðGð2Þ ¼ ðGð3Þ ij ÞNN , G ij ÞNN and G ij ÞNN , which

may not be identical, represent the coupling configuration of the array and satisfy the diffusive coupling connections: GijðlÞ Z0, GiiðlÞ ¼ 

ia j, l ¼ 1,2,3, N X j ¼ 1,j a i

GðlÞ ij ,

i,j ¼ 1,2, . . . ,N,

ð3Þ

D(1), D(2) and Dð3Þ A Rnn represent the inner coupling matrix, the discrete-delay inner coupling matrix, and the distributed-delay inner coupling matrix of the neural networks, respectively, and for (1) (1) (2) simplicity, we assume that D(1) ¼diag(d(1) ¼ 1 , d2 ,y,dn ), D (2) (2) (2) (3) (3) (3) (3) diag(d1 , d2 ,y,dn ) and D ¼diag(d1 , d2 ,y,dn ) with d(1) i , and dð3Þ i ¼ 1,2, . . . ,n; f(xi(t)) ¼(f1(xi1(t)),f2(xi2(t)),y, d(2) i i Z 0, fn(xin(t)))T, gðxi ðttðtÞÞÞ ¼ ðg1 ðxi1 ðttðtÞÞÞ,g2 ðxi2 ðttðtÞÞÞ, . . . , gn ðxin ðttðtÞÞÞÞT , h(xi(t)) ¼(h1(xi1(t)), h2(xi2(t)),y,hn(xin(t)))T are activation functions; cð1Þ 4 0, cð2Þ 4 0 and cð3Þ 4 0 represent the coupling strength; tðtÞ is the transmission delay satisfying 0 o tðtÞ r t. The notations xi(t), C, A, and B are the same as in model (1); ui(t) is the control input; oi ¼ ðoi1 , oi2 , . . . , oin ÞT is an ndimensional Brown motion defined on a complete probability space ðO,F ,PÞ with a nature filtration fF t gt Z 0 ; oi1 , oi2 , . . . , oin are independent scalar Wiener process satisfying

Efoij ðtÞg ¼ 0, Efo2ij ðtÞg ¼ 1, Efoij ðtÞoij ðsÞg ¼ 0 ðsa tÞ, i,j ¼ 1,2, . . . ,N,

si A Rnn is the noise intensity matrix. The initial conditions associated with (2) are given as xi ðsÞ ¼ xi ðsÞ,

t r s r0, i ¼ 1,2, . . . ,N,

for any xi A L2F 0 ð½t,0, Rn Þ, where L2F 0 ð½t,0; Rn Þ is the family of all F 0 measurable Cð½-t,0; Rn Þvalued random variables satisfying supt r s r 0 EJxi ðsÞJ2 o 1, and Cð½t,0; Rn Þ denotes the family of all continuous Rn -valued functions xi ðsÞ on ½t,0 with the norm Jxi ðsÞJ ¼ supt r s r 0 Jxi ðsÞJ. Suppose we want to control network (1) onto some desired inhomogeneous state defined by xm0 þ 1 ðtÞ, . . . ,xm1 ðtÞ-s1 ðtÞ, xm1 þ 1 ðtÞ, . . . ,xm2 ðtÞ-s2 ðtÞ, . . . xmk1 þ 1 ðtÞ, . . . ,xmk ðtÞ-sk ðtÞ, m0 ¼ 0, mk ¼ N, mj1 omj , j ¼ 1,2, . . . ,k, i.e., M ¼ ðs1 ðtÞ, . . . , s1 ðtÞ, . . . ,sk ðtÞ, . . . sk ðtÞÞ  RnN is the desired cluster synchronization pattern under pinning control, where xi ðtÞ-sl ðtÞ means that limt- þ 1 EJxi ðtÞsl ðtÞJ2 ¼ 0, for i¼1,2,y,N; l ¼1,2,y,k. sl(t) is defined as

2.1. Problem formulation In this paper, we consider a general model of an array of coupled delayed neural networks with stochastic disturbances as

#

t

dsl ðtÞ ¼ ½Csl ðtÞ þ Af ðsl ðtÞÞ þ Bgðsl ðttðtÞÞÞþ D

Z

t ttðtÞ

hðsl ðuÞÞ du þ IðtÞ dt

ð4Þ

848

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

for l¼1,y,k. Moreover, sl(t) can be an equilibrium point, periodic orbit or chaotic attractor in the phase space. Without loss of generality, to achieve the goal of cluster synchronization, we apply the pinning control strategy on the nodes set J¼{m1,m2,y,mk}, and adopt pinning controller of the following form: ( ei ðxi ðtÞsl ðtÞÞ, i ¼ ml , l ¼ 1,2, . . . ,k, ui ðtÞ ¼ ð5Þ 0, ia ml :

2.2. Preliminaries Before stating the main results of this paper, some useful basic definitions and lemmas need to be recalled. Definition 1. A stochastic network with N nodes is said to realize cluster synchronization, if the N nodes are splitted into several clusters G1, G2,y,Gk, such as {G1 ¼ (1,2,y,m1), G2 ¼(m1 + 1, m1 + 2,y,m2),y,Gk ¼(mk  1 + 1, mk  1 + 2,y,mk), m0 ¼ 0, mk ¼N, mj1 omj , j ¼ 1,2, . . . ,kg such that the nodes in the same cluster synchronize with one another, i.e., for the states xi(t) and xj(t) of arbitrary nodes i and j in the same cluster, limt- þ 1 EJxi ðtÞ xj ðtÞJ2 ¼ 0 holds.

PN

j ¼ 1,j a i

aij ¼ 

PN

j ¼ 1,j a i

Lemma 5 (Chen et al. [26]). If A ¼ ðaij Þ A Rmm is an irreducible P matrix satisfying aij ¼ aji Z 0, if ia j, and N j ¼ 1 aij ¼ 0, for i ¼1, 2,y,m. Then, all eigenvalues of the matrix 0 1 a11 e a12    a1m B C a22    a2m C B a21 C ð10Þ A ¼B B ^ ^ & ^ C @ A am2    amm am1 are negative. Lemma 6 (Wu et al. [29]). For a matrix B A Rpq , denote aðBÞ ¼ then,

1 2 max½p,qmaxi,j jbij j,

holds for all x A Rp , y A Rq .

aji ,

Throughout this paper, the following assumptions are imposed. Assumption 1. The activation functions f ðÞ,gðÞ,hðÞ satisfy the Lipschitz condition, i.e., there exist positive definite matrices L, F and C such that

Then we say A A A1 . Definition 0 A11 B B A21 A¼B B ^ @ Ak1

Lemma 4 (Zhou et al. [27]). Assume that M is a diagonal matrix whose ikth ð1 r ik r N,1 rk r l,1 rl r NÞ diagonal elements are m and the others are 0, where m 4 0 is a constant. Then for a symmetric matrix G which has the same dimension with M, GM o0 is equivalent to Gl o 0 when m is large enough, where Gl denotes the minor matrix of a matrix G by removing all the ikth ð1 r ik r N, 1r k rl,1 r l rNÞ row–column pairs of G.

xT By r aðBÞðxT x þ yT yÞ

Definition 2. Suppose A ¼ ðaij Þ A RNN . If (i) aij Z0, for ia j, and aii ¼  i ¼ 1,2, . . . ,N, (ii) A is irreducible.

Vt ðt,xÞ ¼ @Vðt,xÞ=@t, Vx ðt,xÞ ¼ ð@Vðt,xÞ=@x1 , . . . ,@Vðt,xÞ=@xn Þ, Vxx ðt,xÞ ¼ ð@2 Vðt,xÞ=@xi xj Þnn .

Jf ðxÞf ðyÞJ r JLðxyÞJ,

3. For an N  N matrix 1 A12    A1k C A22    A2k C C, ^ & ^ C A Ak2    Akk ðmi mi1 Þðmi mi1 Þ

JgðxÞgðyÞJ r JFðxyÞJ, ð6Þ

JhðxÞhðyÞJ rJCðxyÞJ, for all x,y A Rn .

ðmi mi1 Þðmj mj1 Þ

, Aij A R , i,j ¼ 1,2, . . . ,k, if with Aii A R each block Aij is a zero-row-sum matrix, then we say A A M1 ðkÞ. Furthermore, if Aii A A1 , i ¼ 1,2, . . . ,k, then we say AA M2 ðkÞ.

Assumption 2. f(0) ¼g(0) ¼h(0) ¼0 and sðt,s1 ðtÞ, . . . ,s1 ðtÞ, . . . ,sk ðtÞ, . . . ,sk ðtÞ,s1 ðttðtÞÞ, . . ., s1 ðttðtÞÞ, . . . ,sk ðttðtÞÞ, . . . ,sk ðt tðtÞÞÞ ¼ 0, where sl(t) are defined as (4) for l¼1,y,k.

Lemma 1 (Xu [36]). Let O1 , O2 , O3 be real matrices of appropriate dimensions with O3 4 0. Given any vector x, y of appropriate dimensions, then the following inequality holds:

Assumption 3. tðtÞ is a bounded and continuously differentiable function, satisfying 0 o tðtÞ r t and 0 r t_ ðtÞ r m o 1.

2xT OT1 O2 yr xT OT1 O3 O1 x þ yT OT2 O1 3 O2 y: Lemma 2 (Gu [35]). Assume that the vector function o : ½0,r! Rn is well defined for the following integrations. For any symmetric matrix W A Rnn and scalar r 40, one has Z r T Z r  Z r oðsÞT W oðsÞ dsZ oðsÞ ds W oðsÞ ds : r 0

0

0

si ðt,e1 ðtÞ, . . . ,eN ðtÞ, e1 ðttðtÞÞ, . . . , Assumption 4. Denote eN ðttðtÞÞÞ ¼ si ðt,eðtÞ, eðttðtÞÞÞ. There exist constant matrices of appropriate dimensions Sj1 and Sj2 such that tr½sTi ðt,eðtÞ,eðttðtÞÞÞsi ðt,eðtÞ,eðttðtÞÞÞ N N X X r JSj1 ej ðtÞJ2 þ JSj2 ej ðttðtÞÞJ2 , j¼1

ð11Þ

j¼1

Lemma 3 (Khasminskii [37]). Consider an n-dimensional stochastic differential equation dxðtÞ ¼ f ðt,xðtÞ,xðttÞÞ dt þ sðt,xðtÞ,xðttðtÞÞÞ doðtÞ:

ð7Þ

Let C2,1 ðR þ  Rn ; R þ Þ denote the family of all nonnegative functions V(t,x) on R þ  Rn , which are twice continuously differentiable in x and once differentiable in t. For each V A C2,1 ðR þ  Rn ; R þ Þ, the stochastic derivative of V along trajectories of (7) can be expressed as follows: dVðt,xÞ ¼ LVðt,xðtÞ,xðttðtÞÞÞ dt þVx ðt,xðtÞÞsðt,xðtÞ,xðttðtÞÞÞ doðtÞ, ð8Þ where LV is an operator, defined by LVðt,x,yÞ ¼ Vt ðt,xÞ dt þ Vx ðt,xÞf ðt,x,yÞ þ 12tr½sðt,x,yÞT Vxx sðt,x,yÞ,

ð9Þ

where si ðt,eðtÞ,eðttðtÞÞÞ ¼ si ðt,x1 ðtÞ, . . . ,xN ðtÞ,x1 ðttðtÞÞ, . . . , xN ðt tðtÞÞÞsðt,s1 ðtÞ, . . . ,s1 ðtÞ, . . . ,sk ðtÞ, . . . ,sk ðtÞ,s1 ðttðtÞÞ, . . . ,s1 ðttðtÞÞ, . . . , sk ðttðtÞÞ, . . . ,sk ðttðtÞÞÞ. Remark 1. In this paper, a nontrivial coupling scheme with cooperative and competitive couplings is used to achieve the desired cluster synchronization pattern. Specifically, if GijðmÞ o 0, m ¼ 1,2,3, ia j, we will say neural networks i and j are competitive, which can be viewed as a mechanism to desynchronize i and j. Contrarily, neural networks i and j are said to be cooperative. It is obvious that network structure with cooperative and competitive couplings can characterize the real world better.

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

0

3. Cluster synchronization criteria

(c)

In this section, we will investigate the cluster synchronization of the considered network (2) under pinning control. Without loss of generality, we let the controlled neural network set J¼{m1,m2,y,mk} in theorem 2. Define the error variables as ei(t) ¼xi sl(t), for l¼1,2,y,k; i¼ml  1 + 1,y,ml. The target of pin-

B Gj ¼ B B @

849

pj dð1Þ G~ ð1Þ X þmj IN j

ð2Þ ~ ð2Þ 1 2 pj dj G

ð3Þ ~ ð3Þ 1 2 pj dj G

ð2Þ ~ ð2Þ 1 2 pj dj G

ð1oÞqj IN

0

ð3Þ ~ ð3Þ 1 2 pj dj G

0

dIN

1 C C C o0, A

j ¼ 1,2, . . . ,n,

ð16Þ

ning control is limt- þ 1 EJei ðtÞJ2 ¼ 0.

where P ¼diag{p1,p2,y,pn}, Q¼diag{q1,q2,y,qn}, R¼diag ðmÞ {r1,r2,y,rn}, M¼diag{m1, m2,y,mn}, G~ ¼ ðGðmÞ þðGðmÞ ÞT Þ=2

Theorem 1. Suppose Assumptions 1–4 hold and GðmÞ A M1 ðkÞ, m ¼ 1,2,3. If there exist diagonal positive definite matrices P, Q, R, diagonal matrix M and positive constants d, o such that the following conditions hold:

Ui ¼ PC þ 12 PAAT P þ 12 LT L þ 12 PBBT P þ rN STi1 ,m ¼ 1,2,3, Si1 þ Q þ tRM, X ¼ diagf0, . . . , em1 , . . . ,0, . . . , emk g and r is the maximum eigenvalues of the symmetric matrix P. Then, the controlled network (14) will achieve cluster synchronization.

(a)

t 2

CT C þ dtIn ð1mÞR r0,

(b) P ¼ i (c)

0 B Gj ¼ B B @

0

Ui 1 T 2 F Fþ

0

! o 0,

rN STi2 Si2 oQ

i ¼ 1,2, . . . ,N,

ð12Þ

cð1Þ pj dð1Þ G~ ð1Þ X þ mj IN j

ð2Þ ~ ð2Þ 1 ð2Þ 2 c pj dj G

ð3Þ ~ ð3Þ 1 ð3Þ 2 c pj dj G

ð2Þ ~ ð2Þ 1 ð2Þ 2 c pj dj G

ð1moÞqj IN

0

ð3Þ ~ ð3Þ 1 ð3Þ 2 c pj dj G

0

 dI N

o 0,

j ¼ 1,2, . . . ,n,

1 C C C A

ð13Þ

Q¼diag{q1,q2,y,qn}, R¼ diag ðmÞ {r1,r2,y,rn}, M¼diag{m1, m2,y,mn}, G~ ¼ ðGðmÞ þ ðGðmÞ ÞT Þ=2, where

P¼diag{p1,p2,y,pn},

m ¼ 1,2,3,

¼ PC þ 12 PAAT P þ 12 LT L þ 12 PBBT P þ 12 PDDT P þ

r

tRM, X ¼ diagf0, . . . , em1 , . . . ,0, . . . , emk g and r

Ui NSTi1 Si1 þ Q þ

is the maximum eigenvalues of the diagonal matrix P. Then, the controlled network (2) will achieve the desired cluster synchronization pattern. Proof. See Appendix A. Remark 2. In Theorem 1, the cluster synchronization problem for an array of coupled stochastic delayed neural networks has been studied. The model (2) analyzed here is quite general and stochastic disturbances is considered in the cluster synchronization problem for the first time. Moreover, it should be emphasized that the coupling matrix is not assumed to be symmetric. However, most of the former works on network cluster synchronization are based on this assumption [19,20,28,29]. When the matrix D ¼0, tðtÞ ¼ t, c(1) ¼c(2) ¼ c(3) ¼1 and f ðÞ ¼ gðÞ, we can get 2 N X Gð1Þ Dð1Þ xj ðtÞ dxi ðtÞ ¼ 4Cxi ðtÞ þ Af ðxi ðtÞÞ þ Bf ðxi ðttÞÞ þ IðtÞ þ ij j¼1

þ

N X

Gð2Þ Dð2Þ xj ðt ij

tÞ þ

j¼1

N X

Gð3Þ Dð3Þ ij

Z

j¼1

t tt

3

xj ðsÞ dsþ ui ðtÞ5dt

þ si ðt,x1 ðtÞ, . . . ,xN ðtÞ,x1 ðttðtÞÞ, . . . ,xN ðttðtÞÞÞ doi ðtÞ, i ¼ 1,2, . . . ,N: ð14Þ

Remark 3. In the conclusions of [34], two questions have been proposed: (Q1) Due to the existence of embedding of invariant synchronization manifolds, which method can be used to achieve the desired cluster synchronization pattern? (Q2) How to extend the obtained result to uncertain stochastic delayed neural networks? Corollary 1 gives possible answers for these two questions. Let P¼ In, Q ¼ aIn , R ¼ gIn and M ¼ bIn , we can easily get the following corollary by Lemma 4 and Schur complement [13]. Corollary 2. Suppose Assumptions 1–4 hold and GðmÞ A M1 ðkÞ, m ¼ 1,2,3. If there exist positive constants a, b, g, d, o such that the following conditions hold: (a) ðt=2ÞCT C þ dtIn ð1mÞgIn r0, (b) C þ 12 AAT þ 12 LT L þ 12 BBT þ 12 DDT þ rNSTi1 Si1 þ aIn þ tgIn bIn o 0, i ¼ 1,2, . . . ,N, (c) 12 FT F þ rN STi2 Si2 oaIn o 0, i ¼ 1,2, . . . ,N, (d) lmin ðDkðjÞ Þ o0, j ¼ 1,2, . . . ,n, ðmÞ ¼ ðGðmÞ þ ðGðmÞ ÞT Þ=2, m ¼ 1,2,3, X ¼ diagf0, . . . , where o o 1m, G~ ð1Þ ð3Þ em , . . . ,0, . . . , em g, D ¼ cð1Þ dð1Þ G~ X þ bIN þ ð1=4dÞðcð3Þ dð3Þ G~ Þ2 þ 1

k

ð1=4ð1moÞaÞðcð2Þ dð2Þ G~ j

Corollary 1. Suppose Assumptions 1–4 hold and GðmÞ A M1 ðkÞ, m ¼ 1,2,3. If there exist diagonal positive definite matrices P, Q, R, diagonal matrix M and positive constants d, o such that the following conditions hold: (a) dtIn Rr 0, (b) P ¼ i

0

Ui 0

1 T 2 L Lþ

rN STi2 Si2 oQ

! o 0,

i ¼ 1,2, . . . ,N,

ð15Þ

j

Þ , DkðjÞ denotes the minor matrix of DðjÞ

by removing all the ml th ð1 rl r kÞ row–column pairs of DðjÞ and r is the maximum eigenvalues of the symmetric matrix P. Then the controlled system (2) is synchronized to (s1(t),y,s1(t),y,sk(t),ysk(t)). Remark 4. If Assumptions 1–4 hold and GðmÞ A M1 ðkÞ, m ¼ 1,2,3. According to Corollary 2, we can realize the cluster synchronization of network (2) by adding controllers in each cluster and adjusting control gains. Specifically, the method can be divided into three steps: (i) Choose suitable a, b, g, d, and o, such that the following conditions hold: 8 o o1m, > > t  > > > < ð1mÞgdt Z lmax CT C , 2 ð17Þ > > b atg 4 lmax ðOðiÞ Þ, i ¼ 1,2, . . . ,N, > > > : oa 4 l ð1 FT F þ rN ST S Þ, i ¼ 1,2, . . . ,N, max 2

Then, we have the following corollary:

j ð2Þ 2

i2

i2

where OðiÞ ¼ C þ 12 AAT þ 12 LT L þ 12 BBT þ 12 DDT þ rNSTi1 Si1 . (ii) Fix a, b, g, d, and o based on step (i) and calculate the principle minor of DðjÞ in sequence. If DðjÞ contain an (N l)-order negative definite principle minor and DðjÞ does not contain an (N  l+ 1)-order negative definite principle minor, then we should add controllers in each cluster to the nodes which are not included in the (N  l)-order negative definite principle minor. (iii) Adjust control gains to each pinning controller.

850

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

Generally, only a small fraction of nodes are pinned in pinningcontrolled networks. However, the number of controller required in the previous method may be too large. To solve this problem and realize cluster synchronization of network (2), we can construct coupling matrix G(1) and select suitable c(1) and eml such that Corollary 2 holds. Theorem 2. Suppose Assumptions 1–4 hold and GðmÞ A M1 ðkÞ, m ¼ 2, 3. e0l 4 0 is a constant, and 0 1 H11 H12    H1k B C B H21 H22    H2k C C A M2 ðkÞ, H¼B ð18Þ B ^ ^ & ^ C @ A Hk1 Hk2    Hkk with Hii A Rðmi mi1 Þðmi mi1 Þ , Hij A Rðmi mi1 Þðmj mj1 Þ , i,j ¼ 1,2, . . . ,k. Denoting X0l ¼ diagf0, . . . ,0, e0l g A Rml ml , l ¼ 1,2, . . . ,k, we can choose eml ¼ cð1Þ e0l and coupling matrix 0 1 1 H11 H    c1ð1Þ H1k cð1Þ 12 B 1 C B cð1Þ H21 H22    c1ð1Þ H2k C C ð19Þ Gð1Þ ¼ B B ^ ^ & ^ C @ A 1 1 H H  Hkk cð1Þ k1 cð1Þ k2

þD

Z

t ttðtÞ

þ

N X

hðxi ðsÞÞ ds þIðtÞ þ

N X

cð1Þ Gð1Þ Dð1Þ xj ðtÞ ij

j¼1

ð2Þ cð2Þ Gð2Þ ij D xj ðttðtÞÞ

j¼1

þ

N X

ð3Þ cð3Þ Gð3Þ ij D

j¼1

Z

#

t ttðtÞ

xj ðsÞ ds þui ðtÞ dt

þ si ðt,x1 ðtÞ, . . . ,xN ðtÞ,x1 ðttðtÞÞ, . . . ,xN ðttðtÞdoi ðtÞ, i ¼ 1,2, . . . ,20,

ð21Þ

where xi ðtÞ ¼ ðxi1 ðtÞ,xi2 ðtÞÞT , tðtÞ ¼ 1, IðtÞ ¼ 0, Dð1Þ ¼ Dð2Þ ¼ Dð3Þ ¼ I2 , f ðxi ðtÞÞ ¼ gðxi ðtÞÞ ¼ hðxi ðtÞÞ ¼ ðtanhðxi1 ðtÞÞ,tanhðxi2 ðtÞÞÞT ,     1:8 0:15 1 0 C¼ , A¼ , 5:2 3:5 0 1     1:7 0:12 0:6 0:15 B¼ , D¼ : 0:26 2:5 2 0:1

si ðt,x1 ðtÞ, . . . ,xN ðtÞ, x1 ðttðtÞÞ, . . . ,xN ðttðtÞÞÞ doi ðtÞ ¼ Denote si ðt,xðtÞ, xðttðtÞÞÞ doi ðtÞ, si ðt,xðtÞ,xðttðtÞÞÞ

and ~ 2ðk1Þdð1Þ j maxu a v aðH lv Þ þ x , cð1Þ 4 max ð1Þ ~ lmax ðd H ll X0 Þ 1rjrn 1rlrk

j

ð20Þ

¼

l

~ lv ¼ Hlv þHT , l, v ¼ 1,2, . . . ,k, X ¼ cð1Þ diagfX0 , . . . , X0 g, where H 1 k vl 1 x ¼ b þ 4 þ1=4ð1moÞ. Then, system (2) will realize cluster synchronization. Proof. See Appendix B. In Theorem 2, we have proved that cluster synchronization can be realized if the coupling strength c(1) is large enough. However, it is known that the theoretical value of coupling strength are usually much larger than needed in practice. In order to find relatively smaller control strength, we will introduce adaptive control strategy to system (2) as follows. Theorem 3. Suppose e0l 4 0, l ¼ 1,2, . . . ,k, are constants. Let P eml ðtÞ ¼ cð1Þ ðtÞe0l and c_ ð1Þ ðtÞ ¼ ðW=2Þ Ni¼ 1 ei ðtÞT ei ðtÞ in controlled sysð1Þ tem (2), where c ð0Þ 4 0 and W 40. If we choose coupling matrix G(1) according to (19) and (20), then, system (2) can achieve the desired cluster synchronization pattern.

81 1 > 6 diagðxi1 ðtÞxi þ 1,1 ðtÞ,xi2 ðtÞxi þ 1,2 ðtÞÞ þ 6 diagðxi1 ðttðtÞÞ > > > < xi þ 1,1 ðttðtÞÞ,xi2 ðttðtÞÞxi þ 1,2 ðttðtÞÞÞ,

ia ml ,

1 1 > > 6 diagðxi1 ðtÞxi1,1 ðtÞ,xi2 ðtÞxi1,2 ðtÞÞþ 6 diagðxi1 ðttðtÞÞ > > : x ðttðtÞÞ,x ðttðtÞÞx ðttðtÞÞÞ,

i ¼ ml , l ¼ 1, . . . ,k:

i1,1

i2

i1,2

ð22Þ Assume the desired cluster synchronization states of system (21) are u1(t) and u2(t), which satisfy dui ðtÞ ¼ ½Cui ðtÞ þ Af ðui ðtÞÞ þBgðui ðttðtÞÞÞ Z t hðui ðsÞÞ ds þIðtÞ, i ¼ 1,2, þD

with initial values u1(0)¼ [0.2, 0.5]T, u2(0)¼[ 0.8, 0.1]T (Fig. 1). In this simulation, we want to realize two-cluster synchronization with the controlled neural network sets {10, 20}. Obviously, Assumptions 1–4 hold and L ¼ F ¼ C ¼ I2 and Sj1 ¼ Sj2 ¼ 13 I2 , j ¼ 1,2, . . . ,20. In the following, we will use two methods discussed in this paper to achieve cluster synchronization of system (21). (i) Let cð1Þ ¼ 2, cð2Þ ¼ cð3Þ ¼ 12, Gð1Þ ¼ Gð2Þ ¼ Gð3Þ , 0

 92 B B 0 B B ^ Gð1Þ ¼ 11 B B @ 0

Proof. See Appendix B. Remark 5. Remark 4 and Theorem 3 provide two detailed methods which ensure cluster synchronization of network model (2). They suggest that: (i) the network structure and coupling strength are key factors for cluster synchronization of neural networks and, perhaps, the realization of cluster synchronization was by cooperative interaction among neurons to change network structure and coupling strength. (ii) It is possible to control the cluster synchronization of neural networks. (iii) In practice engineering application, for achieving the cluster synchronization of the network, we should make the tradeoff between the design of network structure and the design of controller. 4. Numerical examples In this section, an example is given to illustrate the effectiveness of the theoretical results. Consider the following coupled delayed neural networks with stochastic disturbances as follows: " dxi ðtÞ ¼ Cxi ðtÞ þ Af ðxi ðtÞÞ þ Bgðxi ðttðtÞÞÞ

ð23Þ

ttðtÞ

0 ð1Þ

G

¼@

Gð1Þ 11

Gð1Þ 12

Gð1Þ 21

Gð1Þ 22

1 A,

0 0

 92

B 1 B 2 ð1Þ B Gð1Þ 12 ¼ G22 ¼ B ^ @ 1 2

1 2



 92 ^

 &

1 2



1 2 1 2

1 2



4



1 2 1 2

^

&

^

0



 12

0



0

1 2 1 2

1

C C C C C 1C A 2 0

,

1010

1

C C C ^ C A  92

,

Gð1Þ 21 ¼ 0:

1010

Choose a ¼ 4, b ¼ 30, g ¼ 1, d ¼ 12 and o ¼ 34, such that (17) is satisfied. By computation, we find that all neural networks should be pinned in (21) to achieve cluster synchronization according to Remark 3. Choose control gains ei ¼ 33, i ¼ 1,2, . . . ,20, then all conditions of Corollary 2 are satisfied. The corresponding state trajectories under controlling all neural networks are shown in Fig. 2. Fig. 3 shows the cluster synchronization errors ei1 and ei2 (i¼ 1,2,y,20) between an array of coupled stochastic DNNs (21) and the isolated DNNs (23), respectively.

6

6

4

4

2

2

0

0

s2

s2

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

–2

–2

–4

–4

–6 –1

–0.5

0 s1

0.5

–6 –0.8

1

–0.6

851

–0.4

–0.2

0 s1

0.2

0.4

0.6

0.8

Fig. 1. The desired cluster synchronization states u1 and u2 .

5 1

4 3 xi2(t),i=1,...20

xi1(t),i=1,...20

0.5 0 –0.5 –1

2 1 0 –1 –2 –3 –4

–1.5 0

1

2

3

4

5

6

7

8

–5

9

0

2

4

6

8

t

10

12

14

16

18

t

Fig. 2. Time evolution of xi ðtÞ ð1 r ir 20Þ in system (21) under controlling all neural networks.

0.8

0.6

0.6 0.4

0.2

ei2(t),i=1,...20

ei1(t),i=1,...20

0.4

0 –0.2

0.2 0 –0.2

–0.4

–0.4

–0.6

–0.6

–0.8

–0.8 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t

t Fig. 3. Time evolution of ei ðtÞ ð1 r i r 20Þ in system (21) under controlling all neural networks.

(ii) Next, for reducing the number of controller and realizing cluster synchronization of network (21), we will construct coupling matrix G(1) and adjust the coupling strength c(1) based on Theorem 3. Let H¼

H11

H12

H21

H22

0

 92 B 1 B 2 H11 ¼ H12 ¼ H22 ¼ B B ^ @ 1 2

! ,

c

ð2Þ

¼c

ð3Þ

¼

1 2,

ð2Þ

G

ð3Þ

¼G

1 2



 92



^

&

1 2



¼ H, and G

1 2 1 2

1

C C C ^ C A  92 (1)

,

H21 ¼ 0,

1010

is constructed by (19). Choose

e01 ¼ e02 ¼ 10, and the controlled neural networks set is {10,20}. Fig. 4

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

3

6

2

4

1

2

xi2(t),i=1,...20

xi1(t),i=1,...20

852

0 –1

–2 –4

–2 –3

0

0

5

10

15

20 t

25

30

35

–6

40

0

5

10

15

20 t

25

30

35

40

25

30

35

40

Fig. 4. Time evolution of xi ðtÞ ð1r ir 20Þ in system (21) under pinning control.

3

4 3

2 ei2(t),i=1,...20

ei1(t),i=1,...20

2 1 0 –1

1 0 –1 –2

–2 –3

–3 0

5

10

15

20 t

25

30

35

40

–4

0

5

10

15

20 t

Fig. 5. Time evolution of ei ðtÞ ð1 r i r 20Þ in system (21) under pinning control.

5. Conclusions

4 3.5

c(1)(t)

3 2.5 2 1.5 1

0

5

10

15

20 t

25

30

35

40

Fig. 6. Time evolution of coupling strength cð1Þ ðtÞ in the pinning controlled networks.

In this paper, stochastic disturbances have been considered in the model of coupled stochastic delayed neural networks, which can be more realistic to characterize the dynamical behaviors of coupled DNNs. Cluster synchronization problem of this model has been studied in detail. Two efficient schemes to realize cluster synchronization have been proposed: adding the number of controller or adjusting the coupling strength and control gains. The simulations verify the effectiveness of the theoretical results. The following two issues about cluster synchronization should be paid attention in future. Firstly, for realizing expected cluster synchronization pattern, we required the coupling matrix should be specified in this paper. Can we obtain the results on cluster synchronization under the generalized coupling matrix? Secondly, are there other methods to achieve the desired cluster synchronization pattern independent of initial values besides the control strategy?

Appendix A. Proof of Theorem 1 Since GðmÞ A M1 ðkÞ, we have

indicates how xi ðtÞ ð1r i r30Þ evolute in the pinning-controlled network (21). Fig. 5 depicts the evolution of the cluster synchronization errors ei (t) (i¼1,2,y,20). The evolution of c(1)(t) with c(1)(0)¼1 in the system (21) is illustrated by Fig. 6. From these simulations, one can concluded that using the proposed two methods in this paper, cluster synchronization of network (21) can be achieved.

N X

GðmÞ DðmÞ xj ðtÞ ij

j¼1

¼

k X

ml X

l ¼ 1 j ¼ ml1 þ 1

GðmÞ DðmÞ ½ej ðtÞ þ sl ðtÞ ij

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

¼

ml X

k X

GijðmÞ DðmÞ ej ðtÞ þ

l ¼ 1 j ¼ ml1 þ 1

¼

N X

ml X

k X

GðmÞ DðmÞ sl ðtÞ ij

N h X

¼

GðmÞ DðmÞ ej ðtÞ, ij

eTi ðtÞPCei ðtÞ þ eTi ðtÞPAf~ ðei ðtÞÞ þ eTi ðtÞPBg~ ðei ðttðtÞÞÞ

i¼1

l ¼ 1 j ¼ ml1 þ 1

þeTi ðtÞPD

m ¼ 1,2,3:

j¼1

Therefore, we can easily get the error system as follows: " de ðtÞ ¼ Ce ðtÞ þ Af~ ðe ðtÞÞ þ Bg~ ðe ðttðtÞÞÞ i

i

i

N X

þ

N X

þ Z

t

þD ttðtÞ N X

þ

c

ð2Þ

N X

~ ðsÞÞ dsþ hðe i

j¼1

tðtÞÞ

N X

ð3Þ cð3Þ Gð3Þ ij D

Z

j¼1

t ttðtÞ

0 eTi ðtÞP@

i¼1

c

ð2Þ

þ

j¼1

Gð2Þ Dð2Þ ej ðt ij

tðtÞÞ

N X

c

ð3Þ

Gð3Þ Dð3Þ ij

Z

j¼1

1

t ttðtÞ

ej ðsÞ dsA

N X

e~ i eTi ðtÞPei ðtÞ

i¼1

½eTi ðtÞQei ðtÞð1t_ ðtÞÞei ðttðtÞÞT Qei ðttðtÞÞ

i¼1 N X

ej ðsÞ dse~ i ei ðtÞ dt

þ si ðt,e1 ðtÞ, . . . ,eN ðtÞ,e1 ðttðtÞÞ, . . . ,eN ðttðtÞÞÞ doi ðtÞ, i ¼ 1,2, . . . ,N,

N X

tðtÞeTi ðtÞRei ðtÞð1t_ ðtÞÞ

þ

#

ttðtÞ

 X N N X ~ ðsÞÞ ds þ hðe eTi ðtÞP cð1Þ Gð1Þ Dð1Þ ej ðtÞ i ij

i¼1 N  X

ð2Þ Gð2Þ ij D ej ðt

j¼1

þ

N X

þ

t

j¼1

i¼1 ð1Þ cð1Þ Gð1Þ ij D ej ðtÞ

Z

eTi ðtÞP

i¼1

i

853

Z

t

ttðtÞ

 eTi ðsÞRei ðsÞ ds

trace½sTi ðt,eðtÞ,eðttðtÞÞÞP si ðt,eðtÞ,eðttðtÞÞÞ:

ð26Þ

i¼1

ð24Þ It follows from Lemmas 1, 2 and Assumption 1 that

where (

e~ i ¼

T eTi ðtÞPAf~ ðei ðtÞÞ r 12 eTi ðtÞPAAT Pei ðtÞ þ 12f~ ðei ðtÞÞf~ ðei ðtÞÞ

ei , i ¼ ml ,l ¼ 1,2, . . . ,k,

r 12 eTi ðtÞPAAT Pei ðtÞ þ 12eTi ðtÞLT Lei ðtÞ,

i a ml ,

0,

eTi ðtÞPBg~ ðei ðttðtÞÞÞ r 12 eTi ðtÞPBBT Pei ðtÞ þ 12g~ T ðei ðttðtÞÞÞg~ ðei ðttðtÞÞÞ

f~ ðei ðtÞÞ ¼ f ðei ðtÞ þsl ðtÞÞf ðsl ðtÞÞ,

r 12 eTi ðtÞPBBT Pei ðtÞ þ 12eTi ðttðtÞÞFT Fei ðttðtÞÞ

g~ ðei ðttðtÞÞÞ ¼ gðei ðttðtÞÞ þ sl ðttðtÞÞÞgðsl ðttðtÞÞÞ,

ð28Þ

~ ðsÞÞ ¼ hðe ðsÞ þ s ðsÞÞhðs ðsÞÞ, hðe i i l l

and

and si ðt,e1 ðtÞ, . . . ,eN ðtÞ,e1 ðttðtÞÞ, . . . ,eN ðttðtÞÞÞ is defined in (11). Define the Lyapunov functional as

eTi ðtÞPD

N N Z t X 1X eTi ðtÞPei ðtÞ þ eTi ðsÞQei ðsÞ ds VðtÞ ¼ 2i¼1 i ¼ 1 ttðtÞ Z t N Z t X eTi ðyÞRei ðyÞ dy ds: þ i¼1

ttðtÞ

s

si ðt,e1 ðtÞ, . . . ,eN ðtÞ,e1 ðttðtÞÞ, . . . ,eN ðttðtÞÞÞ ¼ si ðt,eðtÞ, Denote eðttðtÞÞÞ. According to Lemma 3, the time derivative of V(t) along the trajectories of (2), we have dVðt,eðtÞÞ ¼ LVðt,eðtÞÞ dt þ

N X

ð27Þ

eTi ðtÞP

si ðt,eðtÞ,eðttðtÞÞÞ doi ðtÞ,

i¼1

where LVðt,eðtÞÞ " N X T ei ðtÞP Cei ðtÞ þ Af~ ðei ðtÞÞ þ Bg~ ðei ðttðtÞÞÞ ¼

ð25Þ

Z

t ttðtÞ

~ ðsÞÞ ds hðe i Z

t

r

1 T 1 e ðtÞPDDT Pei ðtÞ þ 2 i 2

r

1 T tðtÞ e ðtÞPDDT Pei ðtÞ þ 2 i 2

ttðtÞ Z t

1 tðtÞ r eTi ðtÞPDDT Pei ðtÞ þ 2 2

Z

T h~ ðei ðsÞÞ ds

ttðtÞ t ttðtÞ

Z

t ttðtÞ

~ ðsÞÞ ds hðe i

T ~ ðsÞÞ ds h~ ðei ðsÞÞhðe i

eTi ðsÞCT Cei ðsÞ ds:

ð29Þ

By Assumption 4, we can get tr½sTi ðt,eðtÞ,eðttðtÞÞÞPsi ðt,eðtÞ,eðttðtÞÞÞ r lmax ðPÞtr½sTi ðt,eðtÞ,eðttðtÞÞÞsi ðt,eðtÞ,eðttðtÞÞÞ N X rr ½eTj ðtÞSTj1 Sj1 ej ðtÞ þeTj ðttðtÞÞSTj2 Sj2 ej ðttðtÞÞ:

ð30Þ

j¼1

i¼1

þD

Z

t

ttðtÞ

þ

N X

~ ðsÞÞ ds þ hðe i

N X

N X

ð1t_ ðtÞÞ rð1mÞ:

j¼1

ð2Þ cð2Þ Gð2Þ ij D ej ðttðtÞÞ

cð3Þ Gð3Þ Dð3Þ ij

j¼1

Z

#

t ttðtÞ

ej ðsÞ dse~ i ei ðtÞ

N h i X eTi ðtÞQei ðtÞð1t_ ðtÞÞei ðttðtÞÞT Qei ðttðtÞÞ þ

þ þ

i¼1 N  X

tðtÞeTi ðtÞRei ðtÞð1t_ ðtÞÞ

i¼1 N X i¼1

ð31Þ

Substituting inequalities (27)–(31) into Eq. (26), it can be derived that

j¼1

þ

From Assumption 3, we also have

ð1Þ cð1Þ Gð1Þ ij D ej ðtÞ

Z

t ttðtÞ

 ei ðsÞT Rei ðsÞ ds #

trace½sTi ðt,eðtÞ,eðttðtÞÞÞP si ðt,eðtÞ,eðttðtÞÞÞ

LVðt,eðtÞÞ N  X 1 1 eTi ðtÞPCei ðtÞ þ eTi ðtÞPAAT Pei ðtÞ þ eTi ðtÞLT Lei ðtÞ r 2 2 i¼1 1 T 1 e ðtÞPBBT Pei ðtÞ þ eTi ðttðtÞÞFT Fei ðttðtÞÞ 2 i 2 Z  1 T t t T eTi ðsÞCT Cei ðsÞ ds þ ei ðtÞPDD Pei ðtÞ þ 2 2 ttðtÞ N N X X  e~ i eTi ðtÞPei ðtÞ þ ½eTi ðtÞQei ðtÞð1mÞei ðttðtÞÞT Qei ðttðtÞÞ þ

i¼1

i¼1

854

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

þ

N  X

Z

teTi ðtÞRei ðtÞð1mÞ

ttðtÞ

i¼1

þ cð1Þ þ cð3Þ

n X

ð3Þ ~T pj dð3Þ j e j ðtÞG

Z

j¼1

þr

N X

þ

n X

pj dð2Þ e~ Tj ðtÞGð2Þ e~ j ðttðtÞÞ j

ttðtÞ

T

By Lemma 3, we have Vðt,eðtÞÞVð0,eð0ÞÞ Z Z t LVðs,eðsÞÞ dsþ ¼

Z

e~ j ðsÞ ds:

According to Lemma 2, we have T Z t n Z t n Z X X d e~ j ðsÞ ds e~ j ðsÞ dsþ dt ¼ d

ttðtÞ

n Z X

ttðtÞ

j¼1

þ dt

t

N Z X

t

ttðtÞ

i¼1

j¼1

T Z e~ j ðsÞ ds

ttðtÞ

e~ Tj ðsÞe~ j ðsÞ ds

e~ j ðsÞ ds ð33Þ

Using conditions (a)–(c), it yields that  N X 1 1 eTi ðtÞ PC þ PAAT P þ LT L LVðt,eðtÞÞ r 2 2 i¼1

 1 1 þ PBBT P þ PDDT P þ rN STi1 Si1 þQ þ tRM ei ðtÞ 2 2   N X 1 T eTi ðttðtÞÞ F F þ rN STi2 Si2 oQ ei ðttðtÞÞ þ 2 i¼1 Z N h i t X t þ eTi ðsÞ CT C þ dtIn ð1mÞR ei ðsÞ ds 2 i ¼ 1 ttðtÞ n X

qj e~ Tj ðttðtÞÞe~ j ðttðtÞÞ

j¼1

þ þ þ

n X j¼1 n X j¼1 n X

e~ Tj ðtÞðcð1Þ pj dð1Þ G~ j

ð1Þ

e~ Tj ðtÞðcð2Þ pj dð2Þ G~ j

ð2Þ

e~ Tj ðtÞðcð3Þ pj dð3Þ G~ j

ð3Þ

X þ mj IN Þe~ j ðtÞ Þe~ j ðttðtÞÞ Z

j¼1

d

n Z X

ttðtÞ

j¼1

¼

N Z X i¼1

t

t ttðtÞ

T Z e~ j ðsÞ ds

eTi ðsÞ

ht 2

t

Þ ttðtÞ

e~ j ðsÞ ds e~ j ðsÞ ds

i C C þ dtIn ð1mÞR ei ðsÞ ds T

0

Furthermore, N Z t X 1 E Jei ðsÞJ2 dsr EVð0,eð0ÞÞ, i¼1

l0

0

PN

i¼1

EJei ðtÞJ2 ¼ 0. This completes the

Appendix B. Proof of Theorem 2 Firstly, choose a, b, g, d, o, such that b, o small enough and the following conditions hold: 8 ~ ð2Þ 2 ~ ð3Þ 2 o o1m, a 4 lmax ððcð2Þ dð2Þ d 4 14 lmax ððcð3Þ dð3Þ > > j G Þ Þ, j G Þ Þ, > >   > t > > < ð1mÞgdt Z lmax CT C , 2 > > batg 4 lmax ðOðiÞ Þ, i ¼ 1,2, . . . ,N, > >   > > > 1 T : oa 4 l i ¼ 1,2, . . . ,N, F F þ rN ST S , max 2

i2

i2

ð36Þ where OðiÞ ¼ C þ 12 AAT þ 12 LT L þ 12 BBT þ 12 DDT þ rNSTi1 Si1 . From (36), we can easily get that (a)–(c) hold in Corollary 2 and ð1Þ

1 ð3Þ ð3Þ ~ ð3Þ 2 ðc dj G Þ 4d ð2Þ ð2Þ ðcð2Þ d G~ Þ2

~ DðjÞ ¼ cð1Þ dð1Þ j G X þ bIN þ þ

1 4ð1moÞa

r cð1Þ dð1Þ G~ j

ð1Þ

~ ¼ cð1Þ dð1Þ j G

ð1Þ

j

X þ bIN þ

1 1 IN þ IN 4 4ð1moÞ

X þ xIN ,

j ¼ 1,2, . . . ,n:

ð37Þ

Since H A M2 ðkÞ, we know Hll A A1 , l ¼ 1,2, . . . ,k. It is easy to prove ~ ll A A1 from the definition of H ~ ll and Definition 2. that H For any vector x ¼ ½x1 , . . . ,xN T A RN , let x 1 ¼ ½x1 , . . . ,xm1 T , x 2 ¼ ½xm1 þ 1 , . . . ,xm2 T , . . . ,x k ¼ ½xmk1 þ 1 , . . . ,xmk T . Then, for all j¼1,2,y,n, from Lemma 6, we can get xT ½cð1Þ dð1Þ G~ j

t ttðtÞ

i¼1

which imply that limt- þ 1 proof of Theorem 1. t

eTi ðsÞPsi ðs,eðsÞ,eðstðsÞÞÞ doi ðsÞ:

0

ð32Þ

eTi ðsÞei ðsÞ dsZ 0:

ð1moÞ

0 i¼1

Let l0 ¼ min1 r i r N lmin ðPi Þ, taking the expectation on both side of (35) yields Z t N Z t X EVðt,eðtÞÞEVð0,eð0ÞÞ ¼ E LVðs,eðsÞÞ ds rl0 E Jei ðsÞJ2 ds:

t ttðtÞ

N X

t

ð35Þ

t ttðtÞ

j¼1

j¼1

0

ð2Þ ~ e~ Tj ðtÞðcð2Þ pj dð2Þ j G Þe j ðttðtÞÞ

ttðtÞ

ð34Þ

where Pi , Gj are defined in (12) and (13), xi ¼ ½eTi eTi ðttðtÞÞ and Rt T ~ ZTj ðtÞ ¼ ½e~ Tj ðtÞ e~ Tj ðttðtÞÞ ð t tðtÞ e j ðsÞ dsÞ  for i¼1,2,y,N; j ¼1,2,y,n.

½eTj ðtÞSTj1 Sj1 ej ðtÞ þeTj ðttðtÞÞSTj2 Sj2 ej ðttðtÞÞ

Gð3Þ Þ e~ Tj ðtÞðcð3Þ pj dð3Þ j

xTi ðtÞPi xi ðtÞ,

e~ j ðsÞ ds

1 þ PDDT P þ rNSTi1 Si1 þ Q þ tRÞei ðtÞ 2   N X 1 T eTi ðttðtÞÞ F F þ rNSTi2 Si2 ð1mÞQ ei ðttðtÞÞ þ 2 i¼1 Z t ht i þ eTi ðsÞ CT Cð1mÞR ei ðsÞ ds 2 ttðtÞ n X e~ Tj ðtÞðcð1Þ pj dð1Þ Gð1Þ XÞe~ j ðtÞ þ j

þ

N X

ZTj ðtÞGj Zj ðtÞ

j¼1

t

 1 1 1 eTi ðtÞ PC þ PAAT P þ LT L þ PBBT P ¼ 2 2 2 i¼1

j¼1 n X

n X

xTi ðtÞPi xi ðtÞ þ

i¼1

N X

þ

r

j¼1

i¼1j¼1

j¼1 n X

N X i¼1

pj dð1Þ e~ Tj ðtÞGð1Þ e~ j ðtÞ þ cð2Þ j

j¼1 n X

N X

 eTi ðsÞRei ðsÞ ds

t

¼

k X

ð1Þ

X þ xIN x

k X ð1Þ T X ~ lv x v þ cð1Þ dð1Þ ~ ll x l dj x l H x Tl H j

l ¼ 1 val

c

ð1Þ

k X l¼1

l¼1

x Tl X0l x l þ x

k X l¼1

x Tl x l

L. Li, J. Cao / Neurocomputing 74 (2011) 846–856

r

k X X dð1Þ aðH~ lv Þðx Tl x l þx Tv x v Þ j

k X k k X X X ~ lv y þ cð1Þ ðtÞdð1Þ ~ ll y cð1Þ ðtÞ dð1Þ y Tl H y Tl H y Tl X0l y l v l j j

¼

l ¼ 1 val

þ

k X

l ¼ 1 val

ð1Þ 0 ~ x Tl ðcð1Þ dð1Þ j H ll c Xl þ xIml ml1 Þx l

l¼1

þ½xzðccð1Þ ðtÞÞ

l¼1

uav

þ

k X

k X

x Tl x l

r

x Tl

uav

xl

þ r

l¼1 1 "0 ~ lv Þ þ x k X 2ðk1Þdð1Þ maxu a v aðH j ð1Þ @ A r c  ~ ll X0 Þ H lmax ðdð1Þ l¼1 l j # 0 ~ lmax ðpj g H ll X Þ x T x l : l

l¼1 0 ð1Þ ð1Þ ~ y Tl ½cð1Þ ðtÞdð1Þ j H ll c ðtÞXl þ zc ðtÞIml ml1 y l

 k  X ~ uv Þ þ xzc y T y 2ðk1Þdð1Þ  max a ð H l l j uav

l¼1

~ ll X0 Þx T x l lmax ðdð1Þ H l l j

j

k X l¼1

uav

k X

l

y Tl y l

 X k ~ uv Þ  2ðk1Þdð1Þ þ xzc r maxaðH y Tl y l j

k  X ~ uv Þ þ xx T x l r 2ðk1Þdð1Þ  maxaðH l j l¼1

l¼1 k X l¼1

uav

ð1Þ 0 ~ þ cð1Þ dð1Þ j H ll c Xl

k X k X X ~ ll cð1Þ ðtÞX0 Þy H dð1Þ aðH~ lv Þðy Tl y l þ y Tv y v Þ þ y Tl ðcð1Þ ðtÞdð1Þ l l j j

þ½xzðccð1Þ ðtÞÞ

  ^ 2ðk1Þdð1Þ j  maxaðH uv Þ þ x Iml ml1

l¼1

þ cð1Þ

l¼1

y Tl y l

l ¼ 1 val

l¼1

~ ll cð1Þ X0 þ xIm m Þx l x Tl ðcð1Þ dð1Þ H l l l1 j

l¼1

¼

k X l¼1

~ uv Þ  2ðk1Þdð1Þ r maxaðH j k X

855

þcð1Þ ðtÞ

k X

T 0 ~ ½lmax ðdð1Þ j H ll Xl Þ þ zy l y l :

ð40Þ

l¼1

~ Similar as in Theorem 2, since cð1Þ ðtÞ Z0 and lmax ðdð1Þ j H ll  ð38Þ

X0l Þ o 0,

~ ll X0 Þ we can choose z and c, such that z o lmax ðdð1Þ H l j

~ and 2ðk1Þdð1Þ j  maxu a v aðH uv Þ þ xzc o0 for all l ¼1,2,y,k; j ¼ ~ ll X0 Þ o 0. So, according to (20) and (38), we By Lemma 5, lmax ðdð1Þ H l j have that xT ½cð1Þ dð1Þ G~ j

ð1Þ

X þ xIN x o 0

References

holds for all x A RN \f0g and j¼ 1,2,y,n, i.e., DðjÞ o0. From Lemma 4, it follows that DkðjÞ o 0. Based on Corollary 2, we can say system (2) will realize cluster synchronization.

Appendix C. Proof of Theorem 3 Define the Lyapunov functional as VðtÞ ¼

N N Z t X 1X eTi ðtÞPei ðtÞ þ eTi ðsÞQei ðsÞ ds 2i¼1 i ¼ 1 ttðtÞ Z t N Z t X z þ eTi ðyÞRei ðyÞ dy ds þ ðccð1Þ ðtÞÞ2 , i¼1

ttðtÞ

s

W

where positive constants z and c will be decided later. Similar to the proof of Theorem 1, we can get the same conditions (a)–(c) with Corollary 2 and condition (d): ð1Þ

DðjÞ ðtÞ ¼ cð1Þ ðtÞdð1Þ G~ ðtÞXðtÞ þ bIn þ j þ

1 ð3Þ ð3Þ ~ ð3Þ 2 ðc dj G Þ 4d

ð2Þ 1 ðcð2Þ dð2Þ G~ Þ2 zðccð1Þ ðtÞÞ j 4ð1moÞa

o 0,

ð39Þ

which ensure network (2) achieve the desired cluster synchronization pattern. Adopting the same procedure as in Theorem 2, we can choose a, b, g, d, o, such that conditions (a)–(c) hold and ð1Þ

ð1Þ ~ DðjÞ ðtÞ r cð1Þ ðtÞdð1Þ j G ðtÞXðtÞ þ ½xzðcc ðtÞÞIN :

For any vector y ¼ ½y1 , . . . ,yN T A RN , let y 1 ¼ ½y1 , . . . ,ym1 T , y 2 ¼ ½ym1 þ 1 , . . . ,ym2 T , . . . ,y k ¼ ½ymk1 þ 1 , . . . ,ymk T . Then, we have G~ yT fcð1Þ ðtÞdð1Þ j

ð1Þ

1,2,y,n. It yields DðjÞ ðtÞ o0 for j ¼1,2,y,n. This completes the proof of Theorem 3.

ðtÞXðtÞ þ ½xzðccð1Þ ðtÞÞIN gy

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Lulu Li was born in Anhui Province, China. He received his B.S. degree in mathematics and applied mathematics from the Department of Mathematics and Computer Science of Anhui Normal University, Wuhu, China, in 2007 and the M.S. degree in mathematics from Southeast University, Nanjing, China, in 2010. Now he is working toward the Ph.D. degree at the City University of Hong Kong, Hong Kong. His research interests include nonlinear systems, stability theory, neural networks and chaos synchronization.

Jinde Cao received the B.S. degree in mathematics/applied mathematics from Anhui Normal University, Wuhu, China, in 1986, the M.S. degree in mathematics/applied mathematics from Yunnan University, Kunming, China, in 1989, and the Ph.D. degree in mathematics/applied mathematics from Sichuan University, Chengdu, China, in 1998. From March 1989 to May 2000, he was with Yunnan University, where he was a Professor from 1996 to 2000. Since May 2000, he has been with the Department of Mathematics, Southeast University, Nanjing, China, where he is currently a TePin Professor and a Doctoral Advisor. From July 2001 to June 2002, he was a Postdoctoral Research Fellow with the Department of Automation and Computer-aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. In 2006, 2007, and 2008, he was a Visiting Research Fellow and a Visiting Professor with the School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, UK. He is an Associate Editor for the Journal of the Franklin Institute, Mathematics and Computers in Simulation, Neurocomputing, IEEE Transactions on Neural Networks, International Journal of Differential Equations, and Differential Equations and Dynamical Systems. He is the author or coauthor of more than 160 journal papers and five edited books, and is a reviewer of Mathematical Reviews and Zentralblatt-Math. His research interests include nonlinear systems, neural networks, complex systems, complex networks, stability theory, and applied mathematics.