Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and the Neumann boundary conditions
© Zhang and Li; licensee Springer. 2012
Received: 25 October 2011
Accepted: 13 January 2012
Published: 13 January 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
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.
Throughout this article, we assume that the following conditions are made.
with some constants τ ij ≥ 0, θ ji ≥ 0, τ > 0, θ > 0, for all t ≥ 0.
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 .
, 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.
3. Main results
where , .
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.
in which i = 1, 2, ..., m, j = 1, 2,..., n, and are Lipschitz constants, λ1 is the lowest positive eigenvalue of problem (7).
where i = 1, 2,..., m, j = 1, 2,..., n.
where i = 1, 2, ..., m, j = 1, 2, ..., n.
From (12) and (A4), we derive
Clearly, β ≥ 1.
for any t ≥ 0 where β ≥ 1 is a constant. This implies that 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 leads to those criteria that are irrelevant to the diffusion term. Therefore, Theorem 1 is essentially new and more effectiveness than those obtained.
Remark 2. It is noted that we construct a novel Lyapunov functional here as defined in (20) since the considered model contains time-varying and distributed delays and reaction-diffusion terms. We can see that the results and research method obtained in 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 shows that a large enough diffusion coefficient may always make the system globally exponentially synchronous.