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Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and the Neumann boundary conditions
Boundary Value Problems volume 2012, Article number: 2 (2012)
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 be checked easily. A numerical example is provided to demonstrate the effectiveness of the proposed results.
Aihara et al.  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 , 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  discuss the impulsive control and synchronization for a class of delayed reaction-diffusion NNs with the Dirichlet boundary conditions in terms of p-norm. In , the synchronization scheme is discussed for a class of delayed NNs with reaction-diffusion terms. In , 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 . 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.
2. Model description and preliminaries
In this article, a class of delayed BAM NNs with reaction-diffusion terms is described as follows
where x = (x1, x2 ,..,x l ) T ∈ Ω ⊂ ℝ l , Ω is a compact set with smooth boundary ∂Ω and mes Ω > 0 in space ℝ l ; u = (u1,u2,...,u m ) T ∈ ℝ m , (v1,v2,...,v n ) T ∈ ℝ n, u i (t,x) and v j (t,x) and represent the states of the i th neurons and the j th neurons at time t and in space x, respectively. , and are known constants denoting the synaptic connection strengths between the neurons, respectively; f i and g i denote the activation functions of the neurons and the signal propagation functions, respectively; I i and J i denote the external inputs on the i th and j th neurons, respectively; p i and q j are differentiable real functions with positive derivatives defining the neuron charging time, respectively; τ ij (t) and θ ji (t) represent continuous time-varying discrete delays, respectively; D ik ≥ 0 and stand for the transmission diffusion coefficient along the i th and j th neurons, respectively. i = 1, 2, ..., m, k = 1, 2, l and j = 1, 2,..., n.
System (1) is supplemented with the following boundary conditions and initial values
for any i = 1,2,..., m and j = 1,2,..., n where is the outer normal vector of ∂Ω, are bounded and continuous, where It is the Banach space of continuous functions which map into ℝm+n with the topology of uniform converge for the norm
Throughout this article, we assume that the following conditions are made.
(A1) The functions τ ij (t), θ ji (t) are piecewise-continuous of class C1 on the closure of each continuity subinterval and satisfy
with some constants τ ij ≥ 0, θ ji ≥ 0, τ > 0, θ > 0, for all t ≥ 0.
(A2) The functions p i (·)and q j (·) are piecewise-continuous of class C1 on the closure of each continuity subinterval and satisfy
(A3) The activation functions are bounded and Lipschitz continuous, i.e., there exist positive constants and such that for all η1, η2 ∈ ℝ
(A4) The delay kernels (i = 1, 2,...,m, j = 1, 2,...,n) are real-valued non-negative continuous functions that satisfy the following conditions
(iii)There exist a positive μ such that
We consider system (1) as the drive system. The response system is described by the following equations
where σ i (t,x) and ϑ j (t,x) denote the external control inputs that will be appropriately designed for a certain control objective. We denote , and ϑ(t,x) = (ϑ1(t,x),..., ϑ n (t,x)) T .
The boundary and initial conditions of system (4) are
Definition 1. Drive-response systems (1) and (4) are said to be globally exponentially synchronized, if there are control inputs σ(t,x), ϑ(t,x), and r ≥ 2, further there exist constants α > 0 and β ≥ 1 such that
, for all t ≥ 0,
in which , , and (u(t,x), v(t,x)) and are the solutions of drive-response systems (1) and (4) satisfying boundary conditions and initial conditions (2), (3) and (5), (6), respectively.
Lemma 1.  (Poincaré integral inequality). Let Ω be a bounded domain of ℝ m with a smooth boundary ∂Ω of class C2 by Ω. u(x) is a real-valued function belonging to and Then
which λ1 is the lowest positive eigenvalue of the Neumann boundary problem
3. Main results
From the definition of synchronization, we can define the synchronization error signal , e(t,x) = (e1(t,x),...,e m (t,x)) T , and ω(t,x) = (ω1(t,x),..., ω n (t,x)) T . Thus, error dynamics between systems (1) and (4) can be expressed by
where , .
The control inputs strategy with state feedback are designed as follows:
where and 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 w i > 0(i = 1,2,..., n+m), r ≥ 2, γ ij > 0, β ji > 0 such that the controller gain matrices μ and ρ in (9) satisfy
in which i = 1, 2, ..., m, j = 1, 2,..., n, and are Lipschitz constants, λ1 is the lowest positive eigenvalue of problem (7).
Proof. If (10) holds, we can always choose a positive number δ > 0 (may be very small) such that
where i = 1, 2,..., m, j = 1, 2,..., n.
Let us consider functions
where i = 1, 2, ..., m, j = 1, 2, ..., n.
From (12) and (A4), we derive
F i (0) < -δ < 0, G j (0) < -δ < 0; and are continuous for Moreover, as and as , thus there exist constants ε i ,ν j ∈ [0, +∞) such that
By using obviously, we get
Multiplying both sides of the first equation of (8) by e i (t,x) and integrating over Ω yields
It is easy to calculate by the Neumann boundary conditions (2) that
Moreover, from Lemma 1, we can derive
From (13)-(17), (A2), and (A3), we obtain that
Multiplying both sides of the second equation of (8) by ω j (t,x), similarly, we also have
Consider the following Lyapunov functional
Its upper Dini-derivative along the solution to system (8) can be calculated as
From (21) and Young inequality, we can conclude
From (10), we can conclude