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Global exponential synchronization of delayed BAM neural networks with reaction-diffusion terms and the Neumann boundary conditions

WeiYuan Zhang12* and JunMin Li1

Author affiliations

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

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Citation and License

Boundary Value Problems 2012, 2012:2  doi:10.1186/1687-2770-2012-2


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/2


Received:25 October 2011
Accepted:13 January 2012
Published:13 January 2012

© 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.

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 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 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.

2. Model description and preliminaries

In this article, a class of delayed BAM NNs with reaction-diffusion terms is described as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M1">View MathML</a>

(1)

where x = (x1, x2 ,..,xl)T ∈ Ω ⊂ ℝl, Ω is a compact set with smooth boundary ∂Ω and mesΩ > 0 in space ℝl; u = (u1,u2,...,um)T ∈ ℝm, (v1,v2,...,vn)T ∈ ℝn, ui(t,x) and vj(t,x) and represent the states of the ith neurons and the jth neurons at time t and in space x, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M3">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M4">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M5">View MathML</a> are known constants denoting the synaptic connection strengths between the neurons, respectively; fi and gi denote the activation functions of the neurons and the signal propagation functions, respectively; Ii and Ji denote the external inputs on the ith and jth neurons, respectively; pi and qj are differentiable real functions with positive derivatives defining the neuron charging time, respectively; τij(t) and θji(t) represent continuous time-varying discrete delays, respectively; Dik ≥ 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M6">View MathML</a> stand for the transmission diffusion coefficient along the ith and jth 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M7">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M8">View MathML</a>

(3)

for any i = 1,2,..., m and j = 1,2,..., n where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M9">View MathML</a> is the outer normal vector of ∂Ω,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M10">View MathML</a> are bounded and continuous, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M11">View MathML</a> It is the Banach space of continuous functions which map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M12">View MathML</a> into ℝm+n with the topology of uniform converge for the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M13">View MathML</a>

Throughout this article, we assume that the following conditions are made.

(A1) The functions τij(t), θji(t) are piecewise-continuous of class C1 on the closure of each continuity subinterval and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M14">View MathML</a>

with some constants τij ≥ 0, θji ≥ 0, τ > 0, θ > 0, for all t ≥ 0.

(A2) The functions pi (·)and qj(·) are piecewise-continuous of class C1 on the closure of each continuity subinterval and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M15">View MathML</a>

(A3) The activation functions are bounded and Lipschitz continuous, i.e., there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M17">View MathML</a> such that for all η1, η2 ∈ ℝ

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M18">View MathML</a>

(A4) The delay kernels <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M19">View MathML</a>(i = 1, 2,...,m, j = 1, 2,...,n) are real-valued non-negative continuous functions that satisfy the following conditions

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M20">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M21">View MathML</a>

(iii)There exist a positive μ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M22">View MathML</a>

We consider system (1) as the drive system. The response system is described by the following equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M23">View MathML</a>

(4)

where σi (t,x) and ϑj(t,x) denote the external control inputs that will be appropriately designed for a certain control objective. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M26">View MathML</a> and ϑ(t,x) = (ϑ1(t,x),..., ϑn(t,x))T.

The boundary and initial conditions of system (4) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M27">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M28">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M29">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M30">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M31">View MathML</a>, for all t ≥ 0,

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M33">View MathML</a>, and (u(t,x), v(t,x)) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M34">View MathML</a> 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 ℝm with a smooth boundary ∂Ω of class C2 by Ω. u(x) is a real-valued function belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M36">View MathML</a> Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M37">View MathML</a>

which λ1 is the lowest positive eigenvalue of the Neumann boundary problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M38">View MathML</a>

(7)

3. Main results

From the definition of synchronization, we can define the synchronization error signal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M39">View MathML</a>, e(t,x) = (e1(t,x),...,em(t,x))T, and ω(t,x) = (ω1(t,x),..., ωn(t,x))T . Thus, error dynamics between systems (1) and (4) can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M40">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M43">View MathML</a>.

The control inputs strategy with state feedback are designed as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M44">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M45">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M47">View MathML</a> 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 wi > 0(i = 1,2,..., n+m), r ≥ 2, γij > 0, βji > 0 such that the controller gain matrices μ and ρ in (9) satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M48">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M49">View MathML</a>

(10)

in which i = 1, 2, ..., m, j = 1, 2,..., n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M51">View MathML</a> are Lipschitz constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M52">View MathML</a>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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M53">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M54">View MathML</a>

(11)

where i = 1, 2,..., m, j = 1, 2,..., n.

Let us consider functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M55">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M56">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M57">View MathML</a>i = 1, 2, ..., m, j = 1, 2, ..., n.

From (12) and (A4), we derive

Fi(0) < -δ < 0, Gj(0) < -δ < 0; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M59">View MathML</a> are continuous for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M60">View MathML</a> Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M61">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M63">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M64">View MathML</a>, thus there exist constants εij ∈ [0, +∞) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M65">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M66">View MathML</a>

(13)

By using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M67">View MathML</a> obviously, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M68">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M69">View MathML</a>

(14)

Multiplying both sides of the first equation of (8) by ei (t,x) and integrating over Ω yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M70">View MathML</a>

(15)

It is easy to calculate by the Neumann boundary conditions (2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M71">View MathML</a>

(16)

Moreover, from Lemma 1, we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M72">View MathML</a>

(17)

From (13)-(17), (A2), and (A3), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M73">View MathML</a>

(18)

Multiplying both sides of the second equation of (8) by ωj (t,x), similarly, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M74">View MathML</a>

(19)

Consider the following Lyapunov functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M75">View MathML</a>

(20)

Its upper Dini-derivative along the solution to system (8) can be calculated as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M76">View MathML</a>

(21)

From (21) and Young inequality, we can conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M77">View MathML</a>

(22)

From (10), we can conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M78">View MathML</a>

(23)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M79">View MathML</a>

(24)

Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M80">View MathML</a>

(25)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M81">View MathML</a>

Clearly, β ≥ 1.

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M82">View MathML</a>

(26)

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.

Some famous NN models are a special case of model (1). In system (1), ignoring the role of reaction-diffusion, then system (1) will degenerate into the following delayed BAM NNs

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M83">View MathML</a>

(27)

and the corresponding response system (4) will become the following form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M85">View MathML</a>

(28)

Define the synchronization error signal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M87">View MathML</a>, then the error dynamics between systems (27) and (28) can be expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M88">View MathML</a>

(29)

We consider the following control inputs strategy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M90">View MathML</a>

(30)

As a consequence of Theorem 1, we have the following result:

Corollary 1. Under the assumptions (A1)-(A4), drive-response systems (27) and (28) are in global exponential synchronization, if there exist wi > 0 (i = 1, 2,...,n+m), r ≥ 2, γij > 0, βji > 0 such that the controller gain matrices μ and ρ in (9) satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M91">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M92">View MathML</a>

(31)

in which i = 1, 2,..., m, j = 1, 2,..., n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M94">View MathML</a> are Lipschitz constants.

4. Illustration example

To illustrate the effectiveness of our criterion, we give the following example.

Example 1. Consider the following system on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M95">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M96">View MathML</a>

(32)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M98">View MathML</a>

(33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M100">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M101">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M102">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M103">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M104">View MathML</a>

By simple calculation with w1 = w2 = w3 = w4 = 1, β11 = β12 = β21 = β22 = 1,

and γ11 = γ12 = γ21 = γ22 = 1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M105">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/2/mathml/M106">View MathML</a>

Hence, it follows from Theorem 1 that (32) and (33) are globally exponentially synchronized.

5. Conclusions

In this article, global exponential synchronization has been considered for a class of BAM NNs with time-varying and distributed delays and reaction-diffusion terms. We have established a new sufficient condition which includes the diffusion coefficients by constructing the suitable Lyapunov functional, introducing many real parameters and applying inequality techniques. From condition (10) in Theorem 1, we see that diffusion coefficients directly affect the synchronization behavior of 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.

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 also wrote the paper. JL participated in the design of the study and suggest many good ideas that made this paper possible and helped to draft the first manuscript. All authors read and approved the final manuscript.

Acknowledgements

This study was partially supported by the National Natural Science Foundation of China under Grant No. 60974139 and partially supported by the Fundamental Research Funds for the Central Universities under Grant No. 72103676, the Natural Science Foundation of Shannxi Province, China under Grant No. 2010JQ1013, and the Special research projects in Shannxi Province Department of Education under Grant No. 2010JK896.

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