This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and the Neumann boundary conditions Boundary Value Problems 2012, 2012:2 doi:10.1186/1687-2770-2012-2 WeiYuan Zhang (ahzwy@163.com) JunMin Li (jmli@mail.xidian.edu.cn) ISSN 1687-2770 Article type Research Submission date 25 October 2011 Acceptance date 13 January 2012 Publication date 13 January 2012 Article URL http://www.boundaryvalueproblems.com/content/2012/1/2 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com Boundary Value Problems © 2012 Zhang and Li ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1 Global exponential synchronization of delayed BAM neural networks with reaction–diffusion terms and the Neumann boundary conditions WeiYuan Zhang *1,2 and JunMin Li 1 1 School of Science, Xidian University, Shaan Xi Xi’an 710071, P.R. China 2 Institute of Maths and Applied Mathematics, Xianyang Normal University, Xianyang, ShaanXi 712000, P.R. China * Corresponding author: ahzwy@163.com Email address: JML: jmli@mail.xidian.edu.cn Abstract In this article, a delay-differential equation modeling a bidirectional associative memory (BAM) neural networks (NNs) with reaction-diffusion terms is investigated. A feedback control law is derived to achieve the state global exponential synchronization of two identical BAM NNs with reaction-diffusion terms by constructing a suitable Lyapunov functional, using the drive-response approach and some inequality technique. A novel global exponential synchronization criterion is given in terms of inequalities, which can 2 be checked easily. A numerical example is provided to demonstrate the effectiveness of the proposed results. Keywords: neural networks; reaction–diffusion; delays; global exponential synchronization; Lyapunov functional. 1. Introduction Aihara et al. [1] firstly proposed chaotic neural network (NN) models to simulate the chaotic behavior of biological neurons. Consequently, chaotic NNs have drawn considerable attention and have successfully been applied in combinational optimization, secure communication, information science, and so on [2–4]. Since NNs related to bidirectional associative memory (BAM) have been proposed by Kosko [5], the BAM NNs have been one of the most interesting research topics and extensively studied because of its potential applications in pattern recognition, etc. Hence, the study of the stability and periodic oscillatory solution of BAM with delays has raised considerable interest in recent years, see for example [6–12] and the references cited therein. Strictly speaking, diffusion effects cannot be avoided in the NNs when electrons are moving in asymmetric electromagnetic fields. Therefore, we must consider that the activations vary in space as well as in time. In [13–27], the authors have considered various dynamical behaviors such as the stability, periodic oscillation, and synchronization of NNs with diffusion terms, which are expressed by partial differential equations. For instance, the authors of [16] discuss the impulsive control and 3 synchronization for a class of delayed reaction-diffusion NNs with the Dirichlet boundary conditions in terms of p-norm. In [25], the synchronization scheme is discussed for a class of delayed NNs with reaction-diffusion terms. In [26], an adaptive synchronization controller is derived to achieve the exponential synchronization of the drive-response structure of NNs with reaction-diffusion terms. Meanwhile, although the models of delayed feedback with discrete delays are good approximation in simple circuits consisting of a small number of cells, NNs usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, there is a distribution of conduction velocities along these pathways and a distribution of propagation delays. Therefore, the models with discrete and continuously distributed delays are more appropriate. To the best of the authors’ knowledge, global exponential synchronization is seldom reported for the class of delayed BAM NNs with reaction-diffusion terms. In the theory of partial differential equations, Poincaré integral inequality is often utilized in the deduction of diffusion operator [28]. In this article, the problem of global exponential synchronization is investigated for the class of BAM NNs with time-varying and distributed delays and reaction-diffusion terms by using Poincaré integral inequality, Young inequality technique, and Lyapunov method, which are very important in theories and applications and also are a very challenging problem. Several sufficient conditions are in the form of a few algebraic inequalities, which are very convenient to verify. 4 2. Model description and preliminaries In this article, a class of delayed BAM NNs with reaction–diffusion terms is described as follows ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 , , , , , l i i ik i i k k k n n n t ji j j ji j j ji ji ji j j i j j j u u D p u t x t x x b f v t x b f v t t x b k t s f v s x ds I t θ = −∞ = = =   ∂ ∂ ∂ = −   ∂ ∂ ∂   + + − + − + ∑ ∑ ∑ ∑ ∫ % ( ) ( ) ( ) ( ) 1 1 , , l m j j jk j j ij i i k i k k v v D q v t x d g u t x t x x ∗ = = ∂ ∂   ∂ = − +   ∂ ∂ ∂   ∑ ∑ ( ) ( ) ( ) 1 , m ij i i ij i d g u t t x τ = + − ∑ % ( ) ( ) ( ) ( ) 1 , , m t ij ij i i j i d k t s g u s x ds J t −∞ = + − + ∑ ∫ (1) where ( ) 1 2 , , , T l x x x x = , l ∈ Ω ⊂  Ω is a compact set with smooth boundary ∂Ω and 0 mes Ω > in space l  ; ( ) ( ) 1 2 1 2 , , , , , , , , T T m n m n u u u u v v v v= ∈ = ∈   ( ) , i u t x and ( ) , j v t x represent the states of the ith neurons and the jth neurons at time t and in space x , respectively. , , , ji ji ji b b b % , ij ij d d , and ij d % are known constants denoting the synaptic connection strengths between the neurons, respectively; j f and i g denote the activation functions of the neurons and the signal propagation functions, respectively; i I and j J denote the external inputs on the ith and jth neurons, respectively; i p and j q are differentiable real functions with positive derivatives defining the neuron charging time, respectively; ( ) ij t τ and ( ) ji t θ represent continuous time-varying discrete delays, respectively; 0 ik D ≥ and 0 jk D ∗ ≥ stand for the transmission diffusion coefficient along the ith and jth neurons, respectively. 1, 2, , , i m = 1, 2, , k l = and 1, 2, , . j n = 5 System (1) is supplemented with the following boundary conditions and initial values 1 2 : , , , 0, T i i i i l u u u u n x x x   ∂ ∂ ∂ ∂ = =   ∂ ∂ ∂ ∂   1 2 : , , , 0 T j j j j l v v v v n x x x ∂ ∂ ∂ ∂   = =   ∂ ∂ ∂ ∂   , 0, , t x ≥ ∈ ∂Ω (2) ( ) ( ) ( ) ( ) , , , , , , i ui j vj u s x s x v s x s x ϕ ϕ = = ( ) ( ] , ,0 . s x ∈ −∞ ×Ω (3) for any 1, 2, , i m = and 1, 2, , , j n = where n is the outer normal vector of ∂Ω , ( ) 1 1 , , , , , T u u um v vn v C ϕ ϕ ϕ ϕ ϕ ϕ ϕ   = = ∈     are bounded and continuous, where ( ] ( ] ,0 | , : . ,0 m u m n n v C ϕ ϕ ϕ ϕ ϕ +     −∞ ×     = = →         −∞ ×            It is the Banach space of continuous functions which map ( ] ( ] ,0 ,0   −∞   −∞   into m n +  with the topology of uniform converge for the norm 0 0 1 1 sup sup , 2. m n r r u ui vj s s i j v dx dx r ϕ ϕ ϕ ϕ ϕ Ω Ω −∞≤ ≤ −∞≤ ≤ = =       = = + ≥             ∑ ∑ ∫ ∫ Throughout this article, we assume that the following conditions are made. (A1) The functions ( ) ( ) , ij ji t t τ θ are piecewise-continuous of class 1 C on the closure of each continuity subinterval and satisfy ( ) ( ) ( ) ( ) 0 ,0 , 1, 1, ij ij ji ji ij ji t t t t τ θ τ τ θ θ τ µ θ µ ≤ ≤ ≤ ≤ ≤ < ≤ < & & { } { } 1 ,1 1 ,1 max , max ij ji i m j n i m j n τ τ θ θ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ = = , with some constants 0, 0, ij ji τ θ ≥ ≥ 0, 0, τ θ > > for all 0 t ≥ . 6 (A2) The functions ( ) i p ⋅ and ( ) j q ⋅ are piecewise-continuous of class 1 C on the closure of each continuity subinterval and satisfy ( ) ( ) inf 0, 0 0, i i i a p p ζ ζ ∈ ′ = > =  ( ) ( ) inf 0, 0 0. j j j c q q ζ ζ ∈ ′ = > =  (A3) The activation functions are bounded and Lipschitz continuous, i.e., there exist positive constants f j L and g i L such that for all 1 2 , η η ∈  ( ) ( ) 1 2 1 2 , f j j j f f L η η η η − ≤ − ( ) ( ) 1 2 1 2 . g i i i g g L η η η η − ≤ − (A4) The delay kernels ( ) ( ) [ ) [ ) , : 0, 0, , ji ij K s K s ∞ → ∞ ( 1, 2, , , 1, 2, , ) i m j n = = are real-valued non-negative continuous functions that satisfy the following conditions (i) ( ) ( ) 0 0 1, 1, ji ji K s ds K s ds +∞ +∞ = = ∫ ∫ (ii) ( ) ( ) 0 0 , , ji ij sK s ds sK s ds +∞ +∞ < ∞ < ∞ ∫ ∫ (iii) There exist a positive µ such that ( ) ( ) 0 0 , . s s ji ij se K s ds se K s ds µ µ +∞ +∞ < ∞ < ∞ ∫ ∫ We consider system (1) as the drive system. The response system is described by the following equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 , , , , , , , , l n i i ik i i ji j j k j k k n n t ji j j ji ji ji j j i i j j u t x u t x D p u t x b f v t x t x x b f v t t x b k t s f v s x ds I t t x θ σ = = −∞ = =   ∂ ∂ ∂ = − +   ∂ ∂ ∂   + − + − + + ∑ ∑ ∑ ∑ ∫ % % % % % % % 7 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 , , , , , , , , l m j j jk j j ij i i k i k k m m t ij i i ij ij ij i i j j i i v t x v t x D q v t x d g u t x t x x d g u t t x d k t s g u s x ds J t t x τ ϑ ∗ = = −∞ = = ∂  ∂  ∂ = − +   ∂ ∂ ∂   + − + − + + ∑ ∑ ∑ ∑ ∫ % % % % % % % (4) where ( ) , i t x σ and ( ) , j t x ϑ denote the external control inputs that will be appropriately designed for a certain control objective. We denote ( ) ( ) ( ) ( ) 1 , , , , , T m u t x u t x u t x = % % % , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , , , , , , , , , , , T T n m v t x v t x v t x t x t x t x σ σ σ = = % % % and ( ) ( ) ( ) ( ) 1 , , , , , T n t x t x t x ϑ ϑ ϑ = . The boundary and initial conditions of system (4) are 1 2 : , , , 0, T i i i i l u u u u n x x x   ∂ ∂ ∂ ∂ = =   ∂ ∂ ∂ ∂   % % % % 1 2 : , , , 0 T j j j j l v v v v n x x x ∂ ∂ ∂ ∂   = =   ∂ ∂ ∂ ∂   % % % % , 0, , t x ≥ ∈∂Ω (5) and ( ) ( ) ( ) ( ) , , , , , i ui j v j u s x s x v s x s x ψ ψ = = % % , ( ) ( ] , , 0 , s x ∈ −∞ ×Ω (6) where ( ) 1 1 , , , , , T u u u m v v n v C ψ ψ ψ ψ ψ ψ ψ   = = ∈     % % % % % % . Definition 1. Drive-response systems (1) and (4) are said to be globally exponentially synchronized, if there are control inputs ( ) , t x σ , ( ) , t x ϑ , and 2 r ≥ , further there exist constants 0 α > and 1 β ≥ such that ( ) ( ) ( ) ( ) , , , , u t x u t x v t x v t x − + − % % ( ) ( ) ( ) ( ) ( ) 2 , , , , t u u v v e s x s x s x s x α β ϕ ψ ϕ ψ − ≤ − + − % % , for all 0 t ≥ , 8 in which ( ) ( ) ( ) ( ) 1 , , , , m r i i i u t x u t x u t x u t x dx Ω = − = − ∑ ∫ % % , ( ) ( ) ( ) ( ) 1 , , , , , 2 n r j j j v t x v t x v t x v t x dx r Ω = − = − ≥ ∑ ∫ % % , and ( ) ( ) ( ) , , , u t x v t x and ( ) ( ) ( ) , , , u t x v t x % % are the solutions of drive-response systems (1) and (4) satisfying boundary conditions and initial conditions (2), (3) and (5), (6), respectively. Lemma 1. [21] (Poincaré integral inequality). Let Ω be a bounded domain of m  with a smooth boundary ∂Ω of class 2 C by Ω . ( ) u x is a real-valued function belonging to ( ) 1 0 H Ω and ( ) | 0. u x n ∂Ω ∂ = ∂ Then ( ) ( ) 2 2 1 1 , u x dx u x dx λ Ω Ω ≤ ∇ ∫ ∫ which 1 λ is the lowest positive eigenvalue of the Neumann boundary problem ( ) ( ) ( ) 1 , , | 0, . x x x u x x n ϕ λϕ ∂Ω −∆ = ∈Ω   ∂ = ∈ ∂Ω  ∂  (7) 3. Main results From the definition of synchronization, we can define the synchronization error signal ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , i i i j j j e t x u t x u t x t x v t x v t x ω = − = − % % , ( ) ( ) ( ) ( ) 1 , , , , , , T m e t x e t x e t x= and ( ) ( ) ( ) ( ) 1 , , , , , T n t x t x t x ω ω ω = . Thus, error dynamics between systems (1) and (4) can be expressed by 9 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 , , , , , , , , l n i i ik i i ji j j k j k k n n t ji j j ji ji ji j j i j j e t x e t x D p e t x b f t x t x x b f t t x b k t s f s x ds t x ω ω θ ω σ = = −∞ = =   ∂ ∂ ∂ = − +   ∂ ∂ ∂   + − + − − ∑ ∑ ∑ ∑ ∫ % % % % % ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 , , , , , , , , l m j j jk j j ij i i k i k k m m t ij i i ij ij ij i i j i i t x t x D q t x d g e t x t x x d g e t t x d k t s g e s x ds t x ω ω ω τ ϑ ∗ = = −∞ = = ∂  ∂  ∂ = − +   ∂ ∂ ∂   + − + − − ∑ ∑ ∑ ∑ ∫ % % % % % (8) where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , j j j j j j i i i i i i f t x f v t x f v t x g e t x g u t x g u t x ω = − = − % % % % , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , i i i i i i j j j j j j p e t x p u t x p u t x q t x q v t x q v t x ω = − = − % % % % . The control inputs strategy with state feedback are designed as follows: ( ) ( ) ( ) ( ) 1 1 , , , , , m n i ik k j jk k k k t x e t x t x t x σ µ ϑ ρ ω = = = = ∑ ∑ , 1,2, , , 1,2, , i m j n = = . that is, ( ) ( ) ( ) ( ) , , , , , t x e t x t x t x σ µ ϑ ρω = = , (9) where ( ) ik m m µ µ × = and ( ) jk n n ρ ρ × = are the controller gain matrices. The global exponential synchronization of systems (1) and (4) can be solved if the controller matrices µ and ρ are suitably designed. We have the following result. Theorem 1. Under the assumptions (A1)–(A4), drive-response systems (1) and (4) are in global exponential synchronization, if there exist ( ) 0 1,2, , , 2, i w i n m r > = + ≥ 0, ij γ > 0 ji β > such that the controller gain matrices µ and ρ in (9) satisfy [...]... Stability of generalized networks with reaction-diffusion terms Sci China (Series F) 44, 87–94 (2001) [24] Lou, X, Cui, B: Asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delays Comput Math Appl 52, 897– 904 (2006) [25] Wang, Y, Cao, J: Synchronization of a class of delayed neural networks with reaction-diffusion terms Phys Lett A 369, 201–211 (2007)... YM: Global exponential stability of BAM neural networks with distributed delays and reaction–diffusion terms Phys Lett A 335(2–3), 213– 225 (2005) 26 [22] Zhang, W, Li, J: Global exponential stability of reaction–diffusion neural networks with discrete and distributed time-varying delays Chin Phys B 20(3), 030701 (2011) [23] Liao, XX, Yang, SZ, Cheng, SJ, Fu, YL: Stability of generalized networks with. .. synchronization for delayed neural networks with reaction-diffusion terms IEEE Trans Neural Netw 21(1), 67– 81 (2010) [17] Wang, Z, Zhang, H: Global asymptotic stability of reaction–diffusion CohenGrossberg neural network with continuously distributed delays IEEE Trans Neural Netw 21(1), 39–49 (2010) [18] Wang, L, Zhang, R, Wang, Y: Global exponential stability of reaction–diffusion cellular neural networks with. .. 1101–1113 (2009) [19] Balasubramaniam, P, Vidhya, C: Global asymptotic stability of stochastic BAM neural networks with distributed delays and reaction-diffusion terms J Comput Appl Math 234, 3458–3466 (2010) [20] Lu, J, Lu, L: Global exponential stability and periodicity of reaction–diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions Chaos Solitons Fractals 39(4),... L: Exponential stability and periodic oscillatory solution in BAM networks with delays IEEE Trans Neural Netw 13(2), 457–463 (2002) [7] Liu, X, Martin, R, Wu, M: Global exponential stability of bidirectional associative memory neural networks with time delays IEEE Trans Neural Netw 19(2), 397– 407 (2008) [8] Lou, X, Cui, B: Stochastic exponential stability for Markovian jumping BAM neural networks with. .. Cao, J: Global exponential stability and existence of periodic solutions in BAM with delays and reaction–diffusion terms Chaos Solitons Fractals 23(2), 421– 430 (2005) [15] Cui, B, Lou, X: Global asymptotic stability of BAM neural networks with distributed delays and reaction–diffusion terms Chaos Solitons Fractals 27(5), 1347–1354 (2006) 25 [16] Hu, C, Jiang, HJ, Teng, ZD: Impulsive control and synchronization. .. Adaptive exponential synchronization of delayed neural networks with reaction–diffusion terms Chaos Solitons Fractals 40, 930–939 (2009) [27] Wang, K, Teng, Z, Jiang, H: Global exponential synchronization in delayed reaction–diffusion cellular neural networks with the Dirichlet boundary conditions Math Comput Model 52, 12–24 (2010) [28] Temam, R: Infinite Dimensional Dynamical Systems in Mechanics and Physics... the delayed BAM NNs with reaction–diffusion terms In comparison with previous literature, diffusion effects are taken into account in our models A numerical example has been given to show the effectiveness of the obtained results 22 Competing interests The authors declare that they have no competing interests Authors’ contributions WZ designed and performed all the steps of proof in this research and. .. drive-response systems (1) and (4) are globally exponentially synchronized This completes the proof of Theorem 1 Remark 1 In Theorem 1, the Poincaré integral inequality is used firstly This is a very important step Thus, the derived sufficient condition includes diffusion terms We note that, in the proof in the previous articles [24–26], a negative integral term with gradient is left out in their deduction This... article can also be extended to many other types of NNs with reaction-diffusion terms, e.g., the cellular NNs, cohen-grossberg NNs, etc Remark 3 In our result, the effects of the reaction-diffusion terms on the synchronization are considered Furthermore, we note a very interesting fact, that is, as long as diffusion coefficients in the system are large enough, then condition (10) can always satisfy This . distribution, and reproduction in any medium, provided the original work is properly cited. 1 Global exponential synchronization of delayed BAM neural networks with reaction–diffusion terms and the Neumann. class of delayed NNs with reaction-diffusion terms. In [26], an adaptive synchronization controller is derived to achieve the exponential synchronization of the drive-response structure of NNs with. reported for the class of delayed BAM NNs with reaction-diffusion terms. In the theory of partial differential equations, Poincaré integral inequality is often utilized in the deduction of diffusion