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Advances in Artificial Neural Systems
Volume 2013 (2013), Article ID 908602, 5 pages
http://dx.doi.org/10.1155/2013/908602
Research Article

Globally Exponential Stability of Impulsive Neural Networks with Given Convergence Rate

Department of Mathematics, Shandong Normal University, Ji'nan 250014, China

Received 29 November 2012; Accepted 12 April 2013

Academic Editor: Manwai Mak

Copyright © 2013 Chengyan Liu 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 deals with the stability problem for a class of impulsive neural networks. Some sufficient conditions which can guarantee the globally exponential stability of the addressed models with given convergence rate are derived by using Lyapunov function and impulsive analysis techniques. Finally, an example is given to show the effectiveness of the obtained results.

1. Introduction

Recently, special interest has been devoted to the dynamics analysis of neural networks due to their potential applications in different areas of science. Particularly, there has been a significant development in the theory of neural networks with impulsive effects [19], since such neural networks with impulsive effect can be used as an appropriate description of the phenomena of abrupt qualitative dynamical changes of essential continuous time systems. Based on the theory of impulsive differential equations [1017], some sufficient conditions guaranteeing the exponential stability are derived [1824]. For example, in [8], the author has obtained a criterion of exponential stability for a Hopfield neural network with periodic coefficients; in [18], by constructing the extended impulsive delayed Halanay inequality and Lyapunov functional methods, authors have got some sufficient conditions ensuring exponential stability of the unique equilibrium point of impulsive Hopfield neural networks with time delays. They all have obtained exponential stability for some kinds of neural networks through different methods. However, most of the existing results about the exponential stability of impulsive neural networks have a common feature that the exponential convergence rate cannot be derived, or derived but not the given one [8, 18, 23, 24]. The purpose of this paper is to establish some criteria which can guarantee the globally exponential stability of impulsive neural networks with the given convergence rate by using Lyapunov function and impulsive analysis techniques. This work is organized as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, the main results are presented. In Section 4, an example is discussed to illustrate the results.

2. Preliminaries

Let denote the set of real numbers, denote the set of nonnegative real numbers, denote the set of positive integers and denote the -dimensional real space equipped with the Euclidean norm .

Consider the following impulsive neural networks: where . corresponds to the number of units in a neural network; the impulse times satisfy , ; corresponds to the state of the neurons, denotes the measures of response to its incoming potentials of the unit at time ; is the input of the unit at time . for , exists for , for all but points , where is an interval, and denote the left limit and right limit of function , respectively. 0, . For given , we denote by the solution of system (1) with initial value .

In this paper, we assume that some conditions are satisfied so that the equilibrium point of system (1) does exist, see [16, 17]. Assume that is an equilibrium point of system (1). Impulsive operator is viewed as perturbation of the point of such system without impulsive effects. We assume that the following impulsive condition holds.(H0) , , , .

Furthermore, we will assume that the response function satisfies the following condition.(H1) is globally Lipschizian with Lipschitz constant , that is, , for all , .

Note that is an equilibrium point of system (1), one can derive from system (1) that the transformation , transforms such system into the following system: where , and from the condition (H1), it holds that , .

Furthermore, let , , then system (2) becomes as follows:

To prove the stability of of system (1), it is equal to prove the stability of the zero solution of system (2), and also equal to the boundedness of system (3).

In the following, the notion means the transpose of a square matrix . We will use the notation (or , , ) to denote that the matrix is a positive definite (negtive definite, positive semidefinite, an negative semidefinite) marix.

Let , , , , , , then system (3) with initial condition becomes as follows:

We introduce a definition as follows.

Definition 1. Assume is the equilibrium point of system (1), then the equilibrium point of system (1) is said to be globally exponential stable with given convergence rate . If for any initial data , there exists a constant , such that
From the transformation , , and , the globally exponential stability of the equilibrium point of system (1) can be transformed into the globally exponential stability of trivial solution of system (2), so (5) can be rewritten as follows:
Furthermore, form the transformation , , the globally exponential stability of trivial solution of system (2) can be transformed into the boundedness of the solution of system (4) and it can be rewritten as follows:

3. Main Results

Theorem 2. Given constant . The equilibrium point of the system (1) is globally exponentially stable with the given convergence rate , if the conditions (H0) and (H1) are fulfilled; moreover, suppose that (i), for all , (ii), and , , .

Proof. We only need to prove is bounded when , where is a solution of (3) through .
Consider the Lyapunov function as follows: Particularly, .
Then from conditions (H0)-(H1) and (i), we get the upper right-hand derivative of along the solutions of system (3), for , which implies the functional is nonincreasing for , . By condition (ii), it holds that For any , since is nonincreasing, it holds that ; moreover, Similarly, for any , it holds that , and Thus, it can be deduced that for ,  , Hence, we obtain that for any , which implies that where .
The proof of Theorem 2 is complete.

Remark 3. Most of the existing results about the exponential stability of impulsive neural networks cannot effectively control the convergence rate. It is interesting to see that Theorem 2 can guarantee the globally exponential stability of impulsive neural networks with the given convergence rate.

Remark 4. In particular, if in Theorem 2, where , are constant matrices, then condition can be easily checked via Matlab.

Theorem 5. Given constant . The equilibrium point of the system (1) is globally exponentially stable with the given convergence rate , if the conditions (H0)-(H1) are fulfilled; moreover, suppose that (i)there exists a constant , such that , for all , (ii), , where and , , .

Proof. We only need to prove that is bounded when , where is a solution of (3) through .
Consider the Lyapunov function as follows: In particular, .
Then from conditions (H0)-(H1) and (i), we get the upper right-hand derivative of along the solutions of system (3), for , Thus, By condition (ii), it holds that For any , it holds that , moreover Similarly, for any , it holds that , and Without loss of generality, when , , it can be deduced that Hence, we obtain that for any , which implies that where .
The proof of Theorem 5 is complete.

Remark 6. Although Theorem 5 enhances the restriction on condition (i), the impulsive restriction in (ii) is weaker; that is, is not necessary to converge to 0 as is large enough, provided that the impulsive intervals are not too small.

Theorem 7. Given constant . The equilibrium point of the system (1) is globally exponentially stable with the given exponential convergence rate , if the conditions (H0)-(H1) are fulfilled; moreover, suppose that (i)there exists a constant , such that , for all , (ii) and , where , , , .

Proof. We only need to prove that is bounded when , where is a solution of (3) through .
Consider the Lyapunov functional as follows: Particularly, .
Then from conditions (H0)-(H1) and (i), we get the upper right-hand derivative of along the solutions of system (1), for , Thus, By condition (ii), it holds that By simple induction, we can prove that for any , , which implies that where .
The proof of Theorem 7 is complete.

4. Applications

The following illustrative example will demonstrate the effectiveness of our results.

Example 8. Consider the following impulsive neural networks: where .
It is easy to see that , satisfying hypothesis (H1) with . We have , , , , , , , , , .
Let . It can be deduced that and . Hence, all the conditions of Theorem 2 are satisfied; then the equilibrium point of the above system (31) is globally exponentially stable with the given convergence rate .

Acknowledgments

This work was jointly supported by the Project of Shandong Province Higher Educational Science and Technology Program (J12LI04), Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (BS2012DX039), and National Natural Science Foundation of China (11226136, 11171192).

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