Mathematical Problems in Engineering

Volume 2011 (2011), Article ID 831695, 22 pages

http://dx.doi.org/10.1155/2011/831695

## Synchronization for an Array of Coupled Cohen-Grossberg Neural Networks with Time-Varying Delay

^{1}Key Laboratory of Measurement and Control of CSE, School of Automation, Southeast University, Ministry of Education, Nanjing 210096, China^{2}School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 2010016, China

Received 23 November 2010; Accepted 9 March 2011

Academic Editor: Bin Liu

Copyright © 2011 Haitao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper makes some great attempts to investigate the global exponential synchronization for arrays of coupled delayed Cohen-Grossberg neural networks with both delayed coupling and one single delayed one. By resorting to free-weighting matrix and Kronecker product techniques, two novel synchronization criteria via linear matrix inequalities (LMIs) are presented based on convex combination, in which these conditions are heavily dependent on the bounds of both the delay and its derivative. Owing to that the addressed system can include some famous neural network models as the special cases, the proposed methods can extend and improve those earlier reported ones. The efficiency and applicability of the presented conditions can be demonstrated by two numerical examples with simulations.

#### 1. Introduction

In recent years, synchronization of various chaotic systems has gained considerable attention since the pioneering works of Pecora and Carroll [1, 2]. It is widely known that many benefits of having synchronization or chaos synchronization can be existent in various engineering fields such as secure communication, image processing, and harmonic oscillation generation. Thus recently, the problem on synchronization in chaotic systems has been extensively studied owing to its potential applications in many engineering areas. Especially, since chaos synchronization in arrays of linearly coupled dynamical systems was firstly studied in [3], arrays of coupled systems including delayed chaotic ones have received much attention as they can exhibit some interesting phenomena [4, 5], and a great number of elegant results have been derived [6–28].

As a typical complex systems, delayed neural networks (DNNs) have been verified to exhibit some complex and unpredictable behaviors such as stable equilibria, periodic oscillations, bifurcation, and chaotic attractors. Thus recently, chaos synchronization for arrays of coupled DNNs have been discussed and some elegant results have been proposed [12–28]. The global synchronization of linearly coupled DNNs with delayed coupling was investigated in [12], in which the dynamical behavior of the uncoupled system could be chaotic or others. The authors in [13] have considered the robust synchronization of coupled DNNs under general impulsive control. In [14], this paper has proposed an adaptive procedure to the synchronization for coupled identical Yang-Yang fuzzy DNNs based on one simple adaptive controller. In [15], with all the parameters unknown, the authors focused on the robust synchronization between two coupled DNNs that were linearly and unidirectionally coupled, in which neither symmetry nor negative (positive) definiteness of the coupling matrix were required. However, those above-mentioned results were presented in terms of some complicated inequalities, which makes them uneasily checked and applied to real ceases by the most recently developed algorithms. By employing Kronecker product and LMI technique, the global synchronization and cluster one have been studied for DNNs with couplings, and some easy-to-test sufficient conditions have been obtained [16–25, 28]. Yet, the system forms addressed in [16–25] seemed simple and some improved techniques have not been utilized to reduce the conservatism, which make the above-mentioned results inapplicable to tackle DNNs of more general forms.

The Cohen-Grossberg neural network (CGNN) model, first proposed by Cohen and Grossberg in 1983 [29], has recently gained particular research attention, since it is quite general to include many famous network models as its special case and has promising application potentials for tasks of associative memory, parallel computation, and nonlinear optimization problems. Meanwhile, owing to complexity of CGNNs themselves, there were few works studying the global synchronization for the coupled delayed CGNNs, except for that some researchers have studied the slave-master synchronization for continuous CGNNs in [26, 27] and synchronization for coupled discrete delayed CGNNs in [28]. Thus, it is urgent and challenging to establish some easy-to-check and less conservative results ensuring the global synchronization of coupled continuous-time delayed CGNNs, which constitutes the main focus of this presented work.

In this paper, the global exponential synchronization of identical delayed CGNNs with both delayed coupling and one single delayed one is considered and two novel LMI-based conditions are derived by using Kronecker product technique, which has not been studied in the present literature. It shows that the chaos synchronization can be ensured by a suitable design of inner coupled linking matrix and the inner delayed linking ones. Moreover, some effective mathematical techniques are employed to reduce the conservatism. Finally, the efficiency of the derived criteria can be illustrated by utilizing two numerical examples.

*Notations*. denotes the -dimensional Euclidean space, and is the set of all real matrices. For the symmetric matrices (respectively, ) means that is a positive-definite (respectively, positive-semidefinite) matrix; represents the transpose of the matrix ; denote the maximum eigenvalue and minimum one of matrix , respectively; represents the identity matrix of an appropriate dimension; .

#### 2. Problem Formulations and Preliminaries

Suppose the nodes are coupled with states , , then the delayed Cohen-Grossberg neural network models can be formulated as follows: in which is the state vector of the th network at time , represents the amplification function, is the behaved function, , , ; is the external input vector; , , and are respectively the inner coupling matrices between the connected nodes and at times and .

For system (2.1), the following assumptions are introduced throughout this paper. (A1) denotes an interval time-varying delay satisfying

and we set . (A2) is the configuration matrix that is irreducible and satisfies Here if there is a connection between node and the one and otherwise, . (A3) For , each is Lipschitz continuous and there exists the positive scalars satisfying ; and there exist the positive scalars such that each function satisfies , and in which are given constants. Here we set , , , , and(A4) For any , and for , the activation function satisfies Here we denote and .

*Remark 2.1. *In (A3), the assumption on the derivative of in (2.4) is reasonable and does not result in the conservatism in many cases such as that, choosing the appropriate scalars , the function can be expressed as , , , , , , respectively. Moreover, the activation functions in system (2.1) can be of general description and those present ones in [22–26, 29] are just special cases of the system (2.1).

Based on assumption(A2), system (2.1) can be rewritten as the following forms:
To address the problem, we denote the set as the synchronization manifold for system (2.7).

*Definition 2.2 (see [16]). * Dynamical network (2.7) is said to be asymptotically synchronized, if for any initial conditions , , there exist , and sufficient large such that for all , where and are said to be the decay rate and the decay coefficient, respectively. Here denotes the Euclidean norm.

Due to the communication delay, the array of coupled nodes cannot be decoupled, and the synchronized state is always not the trajectory of an isolated node but a modified one as (2.7). Furthermore, delayed coupling matrix and the degree of the node play the important roles in the synchronized state, which has been illustrated in [21]. In the paper, we give an improved discussion for such synchronization. In the case, system (2.7) reaches the synchronization, that is, , we can deduce the synchronized state equation
where . Obviously, the synchronization is invariant for the coupled system (2.7). Therefore, to realize complete synchronization, the assumption has to be imposed on the system (2.7).

#### 3. Delay-Dependent Synchronization Criteria

Firstly, together with the Kronecker product in [16–21], we can reformulate the system (2.7) as with , , , , and .

In order to derive our results, the following lemmas are essential for obtaining the synchronization criteria.

Lemma 3.1. *Let , and be real matrices of appropriate dimensions and . Then for any vectors and with appropriate dimensions, one gets .*

Lemma 3.2 (17).. *Let , , , and with , . If and each row sum of is 0, then .*

Then by utilizing the most improved techniques for achieving the criteria in [30], we state and investigate the global exponential synchronization for the system (3.1).

Theorem 3.3. *Supposing that assumptions (A _{1})–(A_{4}) hold, then the dynamical system (3.1) is globally exponentially synchronized, if there exist matrices , , , , matrices , , making , diagonal matrices , , , , , , , , , matrices , and one scalar such that, for , the LMIs in (3.2) hold
*

*where , ,*

*with , , , , , , , , , , , , , , , , , , , , , .*

*Proof. *Firstly, we can represent the system (3.1) as the following form:
Based on assumptions (A1) and (A3), and
we construct the following Lyapunov-Krasovskii functional:
where
with two diagonal matrices , , and setting , . Based on (A3) and Lemma 3.1, one can easily verify the definite positiveness of .

Now, by directly calculating along the trajectory of the system (3.1), we can deduce
Meanwhile by (3.5), it is easy to derive that
Noting that does hold, then with Lemma 3.1 and , one can estimate as
Now combining with terms (3.12) and (3.13) yields
For any matrices , it follows from (3.5) and (3.7) that
By utilizing (A3) and (3.6), for any diagonal matrices , , the following inequality holds
Meanwhile, based on (3.14) and (3.19), it is easy to check that , and
Here we can employ the following notations to simplify the subsequent proof
Then together with in (3.14) and in (3.19), it follows from Lemma 3.2 and (3.14)–(3.22) that
For any diagonal matrices , , , , , and , , in (A3)-(A4), it can be deduced that
For any 13 matrices , it follows from Newton-Leibniz formula that
where
Now together with the terms (3.24)–(3.26), and , we can deduce that
where , and are presented in (3.2). Through using Schur-complement and convex combination, the LMIs in (3.2) can guarantee and thus, there must exist one scalar such that . Therefore, one can get
which indicates that the system (3.1) can reach the global asymptotical synchronization.

Based on (A1)–(A4), (3.9), and direct computing, there must exist three scalars such that
Letting , one can deduce that , and
By and changing the integration sequences, we have
Substituting the terms (3.32) into the relevant ones in (3.31), it is easy to have
in which . Choose an appropriate scalar such that , one has . By directly computing, there must exist a positive scalar such that
Meanwhile, . Therefore, it can be deduced that
By Definition 2.2, the system (3.1) is globally exponentially synchronized, and the proof is completed.

*Remark 3.4. *Theorem 3.3 presents a novel delay-dependent criterion guaranteeing arrays of coupled Cohen-Grossberg neural networks (2.7) to be globally synchronized. In [16–22], the authors considered global synchronization of an array of coupled neural networks of simple forms and in the paper, we derive a more general delayed neural networks and extended the case to the time variable one, which generalizes the earlier ones. Moreover, the conditions are expressed in terms of LMIs, therefore, by using LMI in Matlab Toolbox, it is straightforward and convenient to check the feasibility of the proposed results without tuning any parameters.

If there does not exist one single delayed coupling in system (2.1), that is, , which means that the restriction in is removed. Then together with the proof of Theorem 3.3, we can derive the following theorem.

Theorem 3.5. *Supposing that assumptions (A _{1})–(A_{4}) hold, then the dynamical system (3.1) is globally exponentially synchronized, if there exist matrices , , , , matrices , , making , diagonal matrices , , , , , , , , , matrices , and one scalar such that, for , the LMIs in (3.36) hold
*

*where , ,*

*with , , , , , , , , , , , , , ,*