RESEARC H Open Access Synchronization of nonidentical chaotic neural networks with leakage delay and mixed time- varying delays Qiankun Song 1 and Jinde Cao 2* * Correspondence: jdcao@seu.edu. cn 2 Department of Mathematics, Southeast University, Nanjing 210096, China Full list of author information is available at the end of the article Abstract In this paper, an integral sliding mode control approach is presented to investigate synchronization of nonidentical chaotic neural networks with discrete and distributed time-varying delays as well as leakage delay. By considering a proper sliding surface and constructing Lyapunov-Krasovskii functional, as well as employing a combination of the free-weighting matrix method, Newton-Leibniz formulation and inequality technique, a sliding mode controller is designed to achieve the asymptotical synchronization of the addressed nonidentical neural networks. Moreover, a sliding mode control law is also synthesized to guarantee the reachability of the specified sliding sur face. The provided conditions are expressed in terms of linear matrix inequalities, and are dependent on the discrete and distributed time delays as well as leakage delay. A simulation example is given to verify the theoretical results. Keywords: Synchronization, Chaotic neural network, Leakage delay, Discrete time- varying delays, Distributed time-varying delays 05.45.Xt 05.45.Gg Introduction In the past few years, neural networks have attracted much attention due to the bac k- ground of a wide range applications such as associative memory, pattern recognition, image processing and model identification [1]. In such applications, the qualitative ana- lysis of the dynamical behaviors is a necessary step for the practical design of neural networks [2]. In hardware implementation, time delays occur due to finite switching speed of the amplifiers and communication time. The existence of time delay may lead to some complex dynamic behaviors such as oscillation, divergence, chao s, instability, or other poor performance of the neural networks [3]. Therefore, the study of dynamical beha- viors with consideration of time delays becomes extremely important to m anufacture high-quality neural networks [4]. Many results on dynamical behaviors have been reported for delayed neural networks, for example, see [1-10] and references therein. On the other hand, it was found that some delayed neural networks can exhibit chaotic behavior [11-13]. These kinds of chaotic neural networks have been utilized to solve optimization problems [14]. Since the drive-response concept for considering synchronization of coupled chaotic systems was proposed in 1990 [15], the synchroni- zation of chaotic systems has attracted considerable attention due to its benefits of Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 © 2011 Song and Cao; licensee Spr inger. This is an Open Access article dis trib uted under the terms of the Creative Commons Attribution License (http://creativecommons.org/lice nses/by/2.0), which permits unrestricted use, distr ibution, and reproduction in any medium, provided the original work is properly cited. having chaos synchronization in some engineering applications such as secure commu- nication, chemical reactions, information processing and harmonic oscillation genera- tion [16]. Therefore, some chaotic neural networks with delays could be treated as models when we study the synchronization. Recently, some works dealing w ith synchronization phenomena in de layed neural networks have also appeared, for example, see [17-29] and references therein. In [17-20], the coupled connected neural networks with delays were c onsidered, several sufficient conditions for synchronization of such neural networks were obtained by Lyapunov stability theory and the li near matrix inequality (LMI) technique. In [21-29], the authors investigated the synchronization problem of some chaotic neural networks with delays. Using the drive-response concept, the control laws were derived to achieve the synchronization of two identical chaotic neural networks. It is worth pointing out that, the reported works in [17-29] focused on synchronizing of two identical chao tic neural networks with different initial conditi ons. In practice, the chaotic systems are inevitably subject to some environmental changes, which may render their parameters to be variant. Furthermore, from the point of view of engineer- ing, it is very difficult to keep the two chaotic systems to be identical all the time. Therefore, it is important to study the synchronization problem of nonidentical chaotic neural networks. Obviously, when the considered drive a nd response neural networks are distinct and with time delay, it becomes more complex and challenging. On the study for synchronization problem of two nonidentical chaotic systems, one usually adopts adaptive control approach to establish synchronization conditions, for example, see [30-32], and refere nces therein. Recently, the integral sliding mode control approach is also employed to investigate synchronization of nonidentical chaotic delayed neura l networks [33-38]. In [33], an integral sliding mode control appr oach is proposed to address synchronization for two nonidentical chaotic neural networks with constant delay. Based on the drive-response concept and Lyapunov stability t heory, both delay-independent and delay-dependent conditions in LMIs are derived under which the resulting error system is globally asymptotically stable in the specified switc hing surface, and a sliding mode controller is synthesized to guarantee the reach- ability of the specified sliding surface. In [34], the authors investigated synchronization for two chaotic neural networks with discrete and distributed constant delays. By using Lyapunov functional method and LMI technique, a delay-dependent condition was obtained to ensure that the drive system synchronizes with the identical response sys- tem. When the parameters and activation functions of two chaotic neural networks mismatched, the synchronization criterion is also derived by sliding mode control approach. In [35], the projective synchronization for two nonidentical chaotic neural networks with constant delay was investigated, a delay-dependent sufficient condition was derived by sliding mode control approach, LMI technique and Lyapunov stability theory. However, to the best of the authors’ knowledge, there are no results on the problem of synchronization for chaotic neural networks with leakag e delay. As pointed out in [39], neural networks with leakage delay is a class of important neural networks; time delay in the leakage term also has great impact on the dynamics of neural net- works because time delay in the stabilizing negative feedback term has a tendency to destabilize a system [39-43]. Therefore, it is necessary to further investigate the syn- chronization problem for two chaotic neural networks with leakage delay. Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 2 of 17 Motivated by the above discussions, the objective of this paper is to present a sys- tematic design procedure for synchronization of two nonidentical chaotic neural net- works with discrete and distributed time-varying delays as well as leakage delay. By constructing a proper sliding surface and Lyapunov-Krasovskii functional, and employ- ing a combination of the free-weighting matrix method, Newton-Leibniz formulation and inequality technique, a sliding mode controller is designed to achieve the asympto- tical synchroni zation of the addressed nonidentical neural networks. Moreo ver, a sli d- ing mode control law is a lso synthesized to guarantee the reachability of the specified sliding surface. The provided conditions are expressed in terms of LMI, and are depen- dent on the discrete and distributed time delays as well as leakage delay. Differing from the results in [33-35], the main contributions of this study are to investi gate the effect of the leakage delay on the synchronization of two nonidentical chaotic neural net- works with discrete and distributed time-varying delays as well as leakage delay and to propose an integral sliding mode control approach to solving it. Problem formulation and preliminaries In this paper, we consider the following neural network model ˙ y(t )=− D 1 y(t − δ)+A 1 f (y(t)) + B 1 f (y(t − τ(t )) ) + C 1 t t−σ ( t ) f (y(s))ds + I 1 (t ), t ≥ 0, (1) where y(t)=(y 1 (t), y 2 (t), , y n (t)) T Î R n is the state vector of the network at time t, n corresponds to the number of neurons; D 1 Î R n × n is a positive diagonal matrix, A 1 , B 1 , C 1 Î R n × n are, respectively, the connection weight matrix, the discretely delayed connection weight matrix and distributively delayed connection weight matrix; f(y(t)) = (f 1 (y 1 (t)), f 2 (y 2 (t)), , f n (y n (t))) T Î R n denotes the neuron activation at time t; I 1 (t) Î R n is an external input vector; δ ≥ 0, τ(t) ≥ 0ands(t) ≥ 0 denote the leakage delay, the discrete time-varying delay and the distributed time-varying dela y, respectively, and sati sfy 0 ≤ τ(t) ≤ τ,0≤ s(t) ≤ s,whereδ, τ and s are constants. It is assumed that the measured output of system (1) is dependent on the state and the delayed states with the following form: w ( t ) = K 1 y ( t ) + K 2 y ( t − δ ) + K 3 y ( t − τ ( t )) + K 4 y ( t − σ ( t )), (2) where w(t) Î R m , K i Î R m × n (i = 1, 2, 3, 4) are known constant matrices. The initial condition associated with model (1) is given by y ( s ) = φ ( s ) , s ∈ [−ρ,0] , where j(s) is bounded and continuously differential on [-r, 0], r = max {δ, τ, s}. We consider the system (1) as the drive system. The response system is as follows: ˙ z (t )=− D 2 z(t − δ)+A 2 g(z(t)) + B 2 g(z(t − τ(t)) ) + C 2 t t−σ ( t ) g(z(s))ds + I 2 (t )+u(t), t ≥ 0, (3) with initial condition z(s)=(s), s Î [-r, 0], where (s) is bounded and continuously differential on [-r,0],u(t) is the appropriate control input that will be designed in order to obtain a certain control objective. Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 3 of 17 Let x(t)=y(t)-z(t) be the error state, then the error system can be obtained from (1) and (3) as follows: ˙ x(t )=− D 1 x(t − δ)+A 1 h(x(t)) + B 1 h(x(t − τ (t))) + C 1 t t−σ (t) h(x(s))d s +(D 2 − D 1 )z(t − δ) − A 2 g(z(t)) − B 2 g(z(t − τ(t))) − C 2 t t−σ (t) g(z(s))ds + A 1 f (z(t)) + B 1 f (z(t − τ (t))) + C 1 t t−σ ( t ) f (z(s))ds − u(t)+I 1 (t ) − I 2 (t ), (4) where h(x(t)) = f(y(t)-f(z(t)), and x(s)=j(s)-(s), s Î [-r, 0]. Definition 1 Thedrivesystem(1)andtheresponsesystem(3)issaidtobeglobally asymptotically synchronized, if system (4) is globally asymptotically stable. The aim of the paper is to design a controller u(t) to let the response system (3) syn- chronize with the drive system (1). Since dynamic behavior of error system (4) relies on both error state x(t) and chaotic state z(t) of response system (3), complete synchronization between two nonidentical chaotic neural networks ( 1) and (3) cannot b e achieved only by utilizing output feed- back control. To overcome the difficulty, an integral sliding mode c ontrol approach will be proposed to investigate the synchronization problem of two nonidentical chao- tic neural networks (1) and (3). In other words, an integral slid ing mode controller is designed such that the sliding motion is globally asymptotically stable, and the state trajectory of the error system (4) is globally driven onto the specified sliding surface and maintained there for all subsequent time. To utilize the information of the measured output w(t), a suitable sliding surface is constructed as S(t )=x(t)+ t 0 D 1 x(ξ − δ) − A 1 h(x(ξ)) − B 1 h(x(ξ − τ (ξ))) − C 1 ξ ξ−σ (ξ ) h(x(s))ds + K w(ξ) − K 1 z(ξ) − K 2 z(ξ − δ ) − K 3 z(ξ − τ (ξ)) − K 4 z(ξ − σ (ξ )) dξ, (5) where K Î R n × m is a gain matrix to be determined. It follows from (2), (4) and (5) that S(t )=x(0) + t 0 (D 2 − D 1 )z(ξ − δ) − A 2 g(z(ξ )) − B 2 g(z(ξ − τ(ξ)) ) − C 2 ξ ξ−σ (ξ ) g(z(s))ds + A 1 f (z(ξ)) + B 1 f (z(ξ − τ (ξ))) + C 1 ξ ξ−σ (ξ ) f (z(s))ds − u(ξ)+I 1 (ξ) − I 2 (ξ)+KK 1 x(ξ) + KK 2 x(ξ − δ)+KK 3 x(ξ − τ(ξ )) + KK 4 x(ξ − σ(ξ )) dξ. (6) According to the sliding mode control theory [44], it is true that S(t)=0and ˙ S ( t ) = 0 as the state trajectories of the error system (4) enter into the sliding mode. It Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 4 of 17 thus follows from (6) and ˙ S ( t ) = 0 that an equivalent control law can be designed as u (t )=(D 2 − D 1 )z(t − δ) − A 2 g(z(t)) − B 2 g(z(t − τ(t))) − C 2 t t−σ (t) g(z(s))ds + A 1 f (z(t)) + B 1 f (z(t − τ (t))) + C 1 t t−σ (t) f (z(s))ds + I 1 (t ) − I 2 (t ) + KK 1 x ( t ) + KK 2 x ( t − δ ) + KK 3 x ( t − τ ( t )) + KK 4 x ( t − σ ( t )). (7) Substituting (7) into (4), the sliding mode dynamics can be obtained and described by ˙ x(t )=− KK 1 x(t ) − (D 1 + KK 2 )x(t − δ) − KK 3 x(t − τ (t)) − KK 4 x(t − σ (t) ) + A 1 h(x(t)) + B 1 h(x(t − τ (t))) + C 1 t t−σ ( t ) h(x(s))ds. (8) Throughout this paper, we make the following assumption: (H). For any j Î {1, 2, , n}, there exist constants F − j , F + j , G − j and G + j such that F − j ≤ f j (α 1 ) − f j (α 2 ) α 1 − α 2 ≤ F + j , G − j ≤ g j (α 1 ) − g j (α 2 ) α 1 − α 2 ≤ G + j for all a 1 ≠ a 2 . To prove our result, the following lemma that can be found in [41] is necessary. Lemma 1 For any constant matrix W Î R m × m , W >0,scalar 0 <h(t) <h,vector function ω : [0, h] ® R m such that the integrations concerned are well defined, then h(t) 0 ω(s )ds T W h(t) 0 ω(s )ds ≤ h( t) h(t) 0 ω T (s)Wω(s)ds . Main results For presentation convenience, in the following, we denote F 1 = diag(F − 1 , F − 2 , , F − n ), F 2 = diag(F + 1 , F + 2 , , F + n ) , F 3 = diag(F − 1 F + 1 , F − 2 F + 2 , , F − n F + n ), F 4 = diag( F − 1 + F + 1 2 , F − 2 + F + 2 2 , , F − n + F + n 2 ). Theorem 1 Assume that the condition (H) holds and the measured output of drive neural network (1) is condition (2). If there exist five symmetric positive definite matrices P i (i =1,2,3,4,5), four positive diagonal matr ices R i (i =1,2,3,4), and ten matrices M, N, L, Y, X ij (i, j =1,2,3,i ≤ j) such that the following two LMIs hold: X = ⎡ ⎣ X 11 X 12 X 13 X T 12 X 22 X 23 X T 13 X T 23 X 33 ⎤ ⎦ > 0 , (9) Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 5 of 17 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 11 12 13 14 15 16 17 18 19 1,10 ∗−P 3 0 24 25 26 27 28 29 0 ∗∗ 33 34 −YK 3 −YK 4 37 P 1 B 1 P 1 C 1 0 ∗∗∗−P 2 000000 ∗∗∗∗ 55 00F 4 R 4 00 ∗∗∗∗ ∗ 66 00 0 6,10 ∗∗∗∗ ∗ ∗ 77 00 0 ∗∗∗∗ ∗ ∗ ∗−R 4 00 ∗∗∗∗ ∗ ∗ ∗∗−P 4 0 ∗∗∗∗ ∗ ∗ ∗∗ ∗ 10 , 10 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0 , (10) in which 11 = −P 1 D 1 − D 1 P 1 − YK 1 − K T 1 Y T + P 2 + δ 2 P 3 + τ X 11 + X 13 + X T 1 3 − F 3 R 3 + M + M T , 13 = −F 1 R 1 + F 2 R 2 − K T 1 Y T , 13 = −F 1 R 1 + F 2 R 2 − K T 1 Y T , Ω 14 =-YK 2 , 15 = −YK 3 + τ X 12 − X 13 + X T 2 3 , Ω 16 =-YK 4 -M T + N, Ω 17 = P 1 A 1 + F 4 R 3 , Ω 18 = P 1 B 1 , Ω 19 = P 1 C 1 , Ω 1,10 = L-M T , Ω 24 = D 1 YK 2 , Ω 25 = D 1 YK 3 , Ω 26 = D 1 YK 4 , Ω 27 = -D 1 P 1 A 1 , Ω 28 = D 1 P 1 B 1 , Ω 29 = D 1 P 1 C 1 , Ω 33 = τ X 33 + s 2 P 5 -2P 1 , Ω 34 =-P 1 D 1 - Y K 2 , Ω 37 = R 1 - R 2 + P 1 A 1 , 55 = τX 22 − X 23 − X T 2 3 − F 3 R 4 , Ω 66 =-N - N T , Ω 6,10 =-L - N T , Ω 77 = s 2 P 4 - R 3 , Ω 10,10 =-P 5 - L - L T , then the response neural network (3) can globally asymptotically synchronize the drive neural network (1), and the gain matrix K can be designed as K = P − 1 1 Y . (11) Proof 1 Let R i = diag(r (i) 1 , r (i) 2 , , r (i) n )(i =1,2 ) , ν ( ξ, s ) = ( x T ( ξ ) , x T ( ξ − τ ( ξ )) , ˙ x T ( s )) T , and consider the following Lyapunov-Krasovskii functional as V( t ) = V 1 ( t ) + V 2 ( t ) + V 3 ( t ) + V 4 ( t ) + V 5 ( t ) + V 6 ( t ) + V 7 ( t ), (12) where V 1 (t)= x(t) − D 1 t t−δ x(s)ds T P 1 x(t) − D 1 t t−δ x(s)ds , (13) V 2 (t)=2 n i=1 r (1) i x i ( t ) 0 (h i (s) − F − i s)ds +2 n i=1 r (2) i x i ( t ) 0 (F + i s − h i (s))ds , (14) V 3 (t)= t t−δ x T (s)P 2 x(s)ds + δ 0 −δ t t+ ξ x T (s)P 3 x(s)ds dξ , (15) V 4 (t)= 0 −τ t t+ ξ ˙ x T (s)X 33 ˙ x(s)ds dξ , (16) V 5 (t)=σ 0 −σ t t+ ξ h T (x(s))P 4 h(x(s))ds dξ , (17) V 6 (t)=σ 0 −σ t t+ ξ ˙ x T (s)P 5 ˙ x(s)ds dξ , (18) V 7 (t)= t 0 ξ ξ−τ ( ξ ) ν T (ξ, s)Xν(ξ , s)ds dξ . (19) Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 6 of 17 Calculating the time derivative of V 1 (t) along the trajectories of model (8), we obtain ˙ V 1 (t)=2 x(t) − D 1 t t−δ x(s)ds T P 1 −(D 1 + KK 1 )x(t) − KK 2 x(t − δ ) − KK 3 x(t − τ (t)) − KK 4 x(t − σ (t)) + A 1 h(x(t)) +B 1 h(x(t − τ (t))) + C 1 t t−σ (t) h(x(s))ds = x T (t)(−2P 1 D 1 − 2P 1 KK 1 )x(t) +2x T (t)(D 1 P 1 D 1 + K T 1 K T P 1 D 1 ) t t−δ x(s)ds − 2x T (t)P 1 KK 2 x(t − δ) − 2x T (t)P 1 KK 3 x(t − τ (t)) − 2x T (t)P 1 KK 4 x(t − σ (t)) + 2x T (t)P 1 A 1 h(x(t)) +2x T (t)P 1 B 1 h(x(t − τ (t))) + 2x T (t)P 1 C 1 t t−σ (t) h(x(s))ds +2 t t−δ x(s)ds T D 1 P 1 KK 2 x(t − δ) +2 t t−δ x(s)ds T D 1 P 1 KK 3 x(t − τ (t)) +2 t t−δ x(s)ds T D 1 P 1 KK 4 x(t − σ (t)) − 2 t t−δ x(s)ds T D 1 P 1 A 1 h(x(t)) − 2 t t−δ x(s)ds T D 1 P 1 B 1 h(x(t − τ (t))) − 2 t t−δ x(s)ds T D 1 P 1 C 1 t t−σ ( t ) h(x(s))ds. (20) Calculating the time derivatives of V i (t) (i =2,3,4,5,6,7), we have ˙ V 2 (t )=2 ˙ x T (t )R 1 (h(x(t)) − F 1 x(t )) + 2 ˙ x T (t )R 2 (F 2 x(t ) − h(x( t)) ) =2x T ( t )( −F 1 R 1 + F 2 R 2 ) ˙ x ( t ) +2 ˙ x T ( t )( R 1 − R 2 ) h ( x ( t )) , (21) ˙ V 3 (t )=x T (t )(P 2 + δ 2 P 3 )x(t) − x T (t − δ)P 2 x(t − δ) − δ t t−δ x T (s)P 3 x(s)d s ≤ x T (t )(P 2 + δ 2 P 3 )x(t) − x T (t − δ)P 2 x(t − δ) − t t−δ x(s)ds T P 3 t t−δ x(s)ds , (22) ˙ V 4 (t )=τ ˙ x T (t )X 33 ˙ x(t ) − t t −τ ˙ x T (s)X 33 ˙ x(s)ds, (23) ˙ V 5 (t )=σ 2 h T (x(t))P 4 h(x(t)) − σ t t−σ h T (x(s))P 4 h(x(s))ds ≤ σ 2 h T (x(t))P 4 h(x(t)) − σ (t) t t−σ (t) h T (x(s))P 4 h(x(s))d s ≤ σ 2 h T (x(t))P 4 h(x(t)) − t t−σ ( t ) h(x(s))ds T P 4 t t−σ ( t ) h(x(s))ds , (24) Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 7 of 17 ˙ V 6 (t)=σ 2 ˙ x T (t)P 5 ˙ x(t) − σ t t−σ ˙ x T (s)P 5 ˙ x(s)ds ≤ σ 2 ˙ x T (t)P 5 ˙ x(t) − σ (t) t t−σ (t) ˙ x T (s)P 5 ˙ x(s)ds ≤ σ 2 ˙ x T (t)P 5 ˙ x(t) − t t−σ ( t ) ˙ x(s)ds T P 5 t t−σ ( t ) ˙ x(s)ds , (25) ˙ V 7 (t)= t t−τ (t) ν T (t, s)Xν(t, s)ds = τ (t) x(t) x(t − τ (t)) T X 11 X 12 X T 12 X 22 x(t) x(t − τ (t)) +2x T (t)X 13 x(t ) − 2x T (t)X 13 x(t − τ(t)) + 2x T (t − τ(t))X 23 x(t) − 2x T (t − τ (t))X 23 x(t − τ (t)) + t t−τ (t) ˙ x T (s)X 33 ˙ x(s)ds ≤ x T (t)(τ X 11 +2X 13 )x(t)+2x T (t)(τ X 12 − X 13 + X T 23 )x(t − τ (t)) + x T (t − τ (t))(τ X 22 − 2X 23 )x(t − τ(t)) + t t − τ ˙ x T (s)X 33 ˙ x(s)ds. (26) In deriving inequalities (22), (24) and (25), we have made use of 0 ≤ s (t) ≤ s,0≤ τ(t) ≤ τ and Lemma 1. It follows from inequalities (20)-(26) that ˙ V (t) ≤ x T (t)(−2P 1 D 1 − 2P 1 KK 1 + P 2 + δ 2 P 3 + τ X 11 +2X 13 )x(t) +2x T (t)(D 1 P 1 D 1 + K T 1 K T P 1 D 1 ) t t−δ x(s)ds +2x T (t)(−F 1 R 1 + F 2 R 2 ) ˙ x(t) − 2x T (t)P 1 KK 2 x(t − δ) +2x T (t)(−P 1 KK 3 + τ X 12 − X 13 + X T 23 )x(t − τ (t)) − 2x T (t)P 1 KK 4 x(t − σ (t)) + 2x T (t)P 1 A 1 h(x(t)) +2x T (t)P 1 B 1 h(x(t − τ (t))) + 2x T (t)P 1 C 1 t t−σ (t) h(x(s))ds − t t−δ x(s)ds T P 3 t t−δ x(s)ds +2 t t−δ x(s)ds T D 1 P 1 KK 2 x(t − δ) +2 t t−δ x(s)ds T D 1 P 1 KK 3 x(t − τ (t)) +2 t t−δ x(s)ds T D 1 P 1 KK 4 x(t − σ (t)) − 2 t t−δ x(s)ds T D 1 P 1 A 1 h(x(t)) − 2 t t−δ x(s)ds T D 1 P 1 B 1 h(x(t − τ (t))) − 2 t t−δ x(s)ds T D 1 P 1 C 1 t t−σ (t) h(x(s))ds + ˙ x T (t)(τ X 33 + σ 2 P 5 ) ˙ x(t)+2 ˙ x T (t)(R 1 − R 2 )h(x(t)) − x T (t − δ)P 2 x(t − δ)+x T (t − τ (t))(τ X 22 − 2X 23 )x(t − τ (t)) + σ 2 h T (x(t))P 4 h(x(t)) − t t−σ (t) h(x(s))ds T P 4 t t−σ (t) h(x(s))ds − t t−σ (t) ˙ x(s)ds T P 5 t t−σ (t) ˙ x(s)ds = α T ( t ) α ( t ) , (27) Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 8 of 17 where α (t )= x T (t ), t t−δ x(s)ds, ˙ x T (t ), x T (t − δ), x T (t − τ (t)), x T (t − σ (t)), h T (x(t)), h T (x(t − τ (t))), t t−σ (t) h T (x(s))ds, t t−σ (t) ˙ x T (s)ds T , = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 11 12 13 14 15 16 P 1 A 1 P 1 B 1 P 1 C 1 0 ∗−P 3 0 24 25 26 27 28 29 0 ∗∗ 33 000R 1 − R 2 000 ∗∗∗−P 2 00 0 0 0 0 ∗∗∗∗ 55 00 000 ∗∗∗∗∗00 000 ∗∗∗∗∗∗σ 2 P 4 000 ∗∗∗∗∗∗ ∗ 000 ∗∗∗∗∗∗ ∗ ∗−P 4 0 ∗∗∗∗∗∗ ∗ ∗ ∗−P 5 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ with 11 = −P 1 D 1 − D 1 P 1 − P 1 KK 1 − K T 1 K T P 1 + P 2 + δ 2 P 3 + τ X 11 + X 13 + X T 1 3 , 12 = D 1 P 1 D 1 + K T 1 K T P 1 D 1 , Π 13 =-F 1 R 1 +F 2 R 2 , Π 14 =-P 1 KK 2 , 15 = −P 1 KK 3 + τ X 12 − X 13 + X T 2 3 , Π 16 =-P 1 KK 4 , Π 24 = D 1 P 1 KK 2 , Π 25 = D 1 P 1 KK 3 , Π 26 = D 1 P 1 KK 4 , Π 27 =-D 1 P 1 A 1 , Π 28 =-D 1 P 1 B 1 , Π 29 =-D 1 P 1 C 1 , Π 33 = τX 33 + s 2 P 5 , 55 = τ X 22 − X 23 − X T 2 3 . In addition, for any n × ndiagonalmatricesR 3 >0 and R 4 >0, we can get from assumption (H) that [45] x(t ) h(x(t)) T F 3 R 3 −F 4 R 3 −F 4 R 3 R 3 x(t ) h(x(t)) ≤ 0 (28) x(t − τ (t)) h(x(t − τ (t))) T F 3 R 4 −F 4 R 4 −F 4 R 4 R 4 x(t − τ (t)) h(x(t − τ (t))) ≤ 0 . (29) From Newton-Leibniz formulation x(t ) − x(t − σ (t)) − t t−σ ( t ) ˙ x(s)ds = 0 , we have 0=2 x(t ) − x(t − σ (t)) − t t−σ (t) ˙ x(s)ds T × Mx(t )+Nx(t − σ (t)) + L t t−σ ( t ) ˙ x(s)ds . (30) Noting this fact 0=2 ˙ x T (t )P 1 − ˙ x(t) − KK 1 x(t ) − (D 1 + KK 2 )x(t − δ ) − KK 3 x(t − τ (t)) − KK 4 x(t − σ (t )) + A 1 h(x(t)) + B 1 h(x(t − τ (t))) + C 1 t t−σ ( t ) h(x(s))ds . (31) Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 9 of 17 It follows from (27)-(31) that ˙ V (t ) ≤ α T (t )α(t) − x(t ) h(x(t)) T F 3 R 3 −F 4 R 3 −F 4 R 3 R 3 x(t ) h(x(t)) − x(t − τ (t)) h(x(t − τ (t))) T F 3 R 4 −F 4 R 4 −F 4 R 4 R 4 x(t − τ (t)) h(x(t − τ (t))) +2 x(t ) − x(t − σ (t)) − t t−σ (t) ˙ x(s)ds T × Mx(t )+Nx(t − σ (t)) + L t t−σ (t) ˙ x(s)ds +2 ˙ x T (t )P 1 − ˙ x(t ) − KK 1 x(t ) − (D 1 + KK 2 )x(t − δ) − KK 3 x(t − τ (t)) − KK 4 x(t − σ (t )) + A 1 h(x(t)) +B 1 h(x(t − τ (t))) + C 1 t t−σ (t) h(x(s))ds = α T ( t ) α ( t ) , (32) where = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 11 12 13 14 15 16 17 P 1 B 1 P 1 C 1 1,10 ∗−P 3 0 24 25 26 27 28 29 0 ∗∗ 33 34 −P 1 KK 3 −P 1 KK 4 37 P 1 B 1 P 1 C 1 0 ∗∗∗−P 2 000000 ∗∗∗∗ 55 00F 4 R 4 00 ∗∗∗∗ ∗ 66 00 0 6,10 ∗∗∗∗ ∗ ∗ 77 00 0 ∗∗∗∗ ∗ ∗ ∗−R 4 00 ∗∗∗∗ ∗ ∗ ∗ ∗−P 4 0 ∗∗∗∗ ∗ ∗ ∗ ∗ ∗ 10 , 10 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ with Γ 11 = Γ 11 - F 3 R 3 +M +M T , 13 = 13 − K T 1 K T P 1 , Γ 16 =-P 1 KK 4 M T + N, Γ 17 = P 1 A 1 + F 4 R 3 , Γ 1,10 = L-M T , Γ 33 = Π 33 -2P 1 , Γ 34 =-P 1 D 1 - P 1 KK 2 , Γ 37 = R 1 - R 2 +P 1 A 1 , Γ 55 = Π 55 -F 3 R 4 , Γ 66 =-N-N T , Γ 6,10 =-L-N T , Γ 77 = s 2 P 4 - R 3 , Γ 10,10 =-P5 - L-L T . From (10) and (11), we get that Γ = Ω <0. There must exist a small scalar r >0 such that +dia g {ρI,0,0,0,0,0,0,0,0,0}≤0 . (33) It follows from (32) and (33), we get that ˙ V( t ) ≤−ρα T ( t ) α ( t ) ≤−ρx T ( t ) x ( t ) , t ≥ 0 , which implies that the error dynamical system (8) is globally asymptotically stable by the Lyapunov stability theory. Accordingly, the response neural network (3) can globally asymptotically synchronize the drive neural network (1). The proof is completed. When there is no leakage delay, the drive neural network (1) and the response neural network (3) become, respectively, the following models ˙ y(t)=− D 1 y(t)+A 1 f (y(t)) + B 1 f (y(t − τ (t)) ) + C 1 t t−σ ( t ) f (y(s))ds + I 1 (t ), t ≥ 0, (34) Song and Cao Advances in Difference Equations 2011, 2011:16 http://www.advancesindifferenceequations.com/content/2011/1/16 Page 10 of 17 [...]... been investigated for nonidentical chaotic neural networks with discrete and distributed time-varying delays as well as leakage delay, which is more difficult and challenging than the ones for identical chaotic neural networks and nonidentical chaotic neural networks with constant delay but without leakage delay An integral sliding mode control approach has been presented to deal with this problem By... 2010, 60:479-487 33 Huang H, Feng G: Synchronization of nonidentical chaotic neural networks with time delays Neural Netw 2009, 22:869-874 34 Gan QT, Xu R, Kang XB: Synchronization of chaotic neural networks with mixed time delays Commun Nonlinear Sci Numer Simul 2011, 16:966-974 35 Zhang D, Xu J: Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode... asymptotical synchronization of chaotic neural networks by output feedback impulsive control: an LMI approach Chaos Soliton Fract 2009, 41:2293-2300 26 Gao XZ, Zhong SM, Gao FY: Exponential synchronization of neural networks with time-varying delay Nonlinear Anal 2009, 71:2003-2011 27 Tang Y, Fang JA, Miao QY: On the exponential synchronization of stochastic jumping chaotic neural networks with mixed delays and. .. an array of nonlinearly coupled chaotic neural networks with delay coupling Int J Bifurc Chaos 2008, 18:3101-3111 20 Cao JD, Li LL: Cluster synchronization in an array of hybrid coupled neural networks with delay Neural Netw 2009, 22:335-342 21 Yu WW, Cao JD: Synchronization control of stochastic delayed neural networks Physica A 2007, 373:252-260 22 Yan JJ, Lin JS, Hung ML, Liao TL: On the synchronization. .. TL: On the synchronization of neural networks containing time-varying delays and sector nonlinearity Phys Lett A 2007, 361:70-77 23 Park JH: Synchronization of cellular neural networks of neutral type via dynamic feedback controller Chaos Soliton Fract 2009, 42:1299-1304 24 Karimi HR, Maass P: Delay- range-dependent exponential H∞ synchronization of a class of delayed neural networks Chaos Soliton Fract... required in this paper Remark 2 In [33-35], the synchronization of two nonidentical chaotic neural networks with constant delay is investigated It is worth pointing out that the presented methods cannot be applied to analyze the synchronization of two nonidentical chaotic neural networks with time-varying delays Example Example 1 Consider a two-dimensional drive neural network (1), where f (y) = (tanh(y1... drive neural network (1) Figure 5 depicts the synchronization errors of state variables between drive and response systems The numerical simulations clearly verify the effectiveness of the developed sliding mode control approach to the synchronization of nonidentical two chaotic neural networks with discrete and distributed time-varying delays as well as leakage delay Conclusions In this paper, the synchronization. .. recurrent neural networks with time delay in the leakage term under impulsive perturbations Nonlinear Anal Real World Appl 2010, 11:4092-4108 42 Peng SG: Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms Nonlinear Anal Real World Appl 2010, 11:2141-2151 43 Li XD, Cao JD: Delay- dependent stability of neural networks of neutral type with. .. Neural Netw 2006, 19:76-83 7 Wang ZD, Liu YR, Liu XH: State estimation for jumping recurrent neural networks with discrete and distributed delays Neural Netw 2009, 22:41-48 8 Xu DY, Yang ZC: Impulsive delay differential inequality and stability of neural networks J Math Anal Appl 2005, 305:107-120 9 Zeng ZG, Wang J: Improved conditions for global exponential stability of recurrent neural networks with. .. mode synchronization controller design with neural network for uncertain chaotic systems Chaos Soliton Fract 2009, 39:1856-1863 37 Zhen R, Wu XL, Zhang JH: Sliding model synchronization controller design for chaotic neural network with timevarying delay Proceedings of the 8th World Congress on Intelligent Control and Automation 2010, 3914-3919 38 Mei R, Wu QX, Jiang CS: Lag synchronization of delayed chaotic . Synchronization of nonidentical chaotic neural networks with time delays. Neural Netw 2009, 22:869-874. 34. Gan QT, Xu R, Kang XB: Synchronization of chaotic neural networks with mixed time delays gate the effect of the leakage delay on the synchronization of two nonidentical chaotic neural net- works with discrete and distributed time-varying delays as well as leakage delay and to propose. identical chaotic neural networks and nonidentical chaotic neural networks with constant delay but without leakage delay. An integral s liding mode control approach has been presented to deal with