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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 325752, 4 pages
http://dx.doi.org/10.1155/2013/325752
Research Article

Global Exponential Stability of Discrete-Time Neural Networks with Time-Varying Delays

1Department of Mathematics, Kasetsart University, Kamphaeng Saen Campus, Nakhonpathom 73140, Thailand
2Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand
3Center of Excellence in Mathematics CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 13 August 2013; Revised 20 November 2013; Accepted 21 November 2013

Academic Editor: Lifeng Ma

Copyright © 2013 S. Udpin and P. Niamsup. 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 presents some global stability criteria of discrete-time neural networks with time-varying delays. Based on a discrete-type inequality, a new global stability condition for nonlinear difference equation is derived. We consider nonlinear discrete systems with time-varying delays and independence of delay time. Numerical examples are given to illustrate the effectiveness of our theoretical results.

1. Introduction

In recent years, neural networks (NNs) have been investigated extensively due to their broad applications in information processing problems, associative memory, parallel computation, pattern recognition, signal processing, and optimization problems. It is well known that delays are often the sources of instability and oscillation in system. In practical studies, discrete-time systems have been used for a variety of phenomena in electrical networks, genetics, ecological systems, and so forth. Therefore, the stability analysis of discrete-time neural networks (DNNs) with delays has become an important topic of theoretical studies in neural networks; for example, asymptotic stability and exponential stability of neural networks have been studied by many researchers. In [1], the authors have studied robust stability of discrete-time linear-parameter-dependent (LPD) neural networks with time-varying delay. In order to derive stability criteria of discrete-time, one common approach is the use of appropriate inequalities for difference equations. Another approach is the use of Lyapunov stability theory. In [2], the authors have studied global exponential stability of impulsive discrete-time neural networks with time-varying delays, based on some inequality analysis techniques. In [3], the authors have studied new discrete-type inequalities and global stability of nonlinear difference equation. In [4], the authors have studied global exponential stability of discrete-time Hopfield neural networks with variable by using the difference inequality. In [5], the authors have considered the problem of robust stability analysis of generalized neural networks with multiple discrete delays and multiple distributed delays by using the Lyapunov-Krasovskii functional method and the linear matrix inequality technique. In [6], the authors have studied delay-dependent exponential stability criteria for discrete-time nonlinear system with multiple time-varying delays. In this paper, we propose to study global exponential stability of discrete-time neural networks with time-varying delays. In Section 2, we have introduced discrete-time neural networks with time-varying delays and presented some preliminaries. In Section 3, we have derived new discrete-type inequalities; global exponential stability criteria are derived by using new discrete-type inequalities. Finally, numerical examples are given to illustrate the effectiveness of our theoretical results.

2. Notations, Definitions, and Preliminaries Results

In this section, we give some notations definitions and preliminaries results which will be used throughout this paper.

Let denote the set of all real numbers;the set of positive real numbers;the set of nonnegative real numbers; the set of integers; andthe set of positive integers;. We consider the following discrete-time neural networks with time-varying delays: whereis the neuron state vector,is constant input vector,, are positive integers denoting the time-varying delays satisfying whereare known positive integers,,,andare the interconnection matrices, andis the activation function where,satisfy The initial conditions of system (1) are of the form whereis an initial functions,, andis the maximum value of. Letbe an equilibrium point of system (1). We shift the equilibrium pointto the origin by using the transformation. Then, we obtain the new system whereis the state vector of the transformed system,, where,and the transformed activation functions satisfy the condition

Definition 1 (see [4]). The equilibrium point at the origin of system (5) is globally exponentially stable if for any solutionwith the initial condition, there exist constantandsuch that where.

3. Main Result

Throughout this section, we denotebyIn this section, we provide global exponential stability criteria for system (5). First, we introduce new discrete-type inequalities which will be used to derive global exponential stability condition.

Lemma 2. Let , , ; , where and , and let be a sequence of real numbers satisfying the inequality where . Then, there existssuch that Moreover,might be chosen as the smallest positive root of the polynomial which lies in the interval.

Proof. Letbe a solution of the difference equation Fromand, one may follow the proof of Theoremin [3] to show that ifsatisfies (8) andfor, thenfor all. For givenand, the sequencedefined byis a solution of (11) if and only ifis a root of the polynomial (10). Sinceand, it follows from continuity ofthat there exists a smallest real numbersuch that. Thus, for any, the sequence is a solution of (11). Let. Therefore,is a solution of (11) and obviously, we have, for. Hence, by using the first part of the proof, we conclude that,.

Now, we will provide a global exponential stability condition for system (5).

Theorem 3. The equilibrium point at the origin of system (5) is globally exponentially stable if where,, andis maximum of vector.

Proof. LetThen, from (5), the difference of system is given by where From Lemma 2, it follows assumptions of theorem that there existsuch that Thus, we obtain By Definition 1, we conclude that (5) is globally exponentially stable. The proof is complete.

Remark 4. In our main result, we derived global exponential stability criteria for discrete-time neural networks with multiple time-varying delays by using discrete-type inequality. In [4], the global exponential stability criteria of discrete-time Hopfield neural networks are given. Nevertheless, the stability criteria in [4] cannot be applied to discrete-time system with multiple time-varying delays.

4. Numerical Example

Example 1. Consider the NNs (1) withwhere Then, ,,,,,,,, and .
The equilibrium point of system in Example 1 is From which it follows that Therefore, from Theorem 3, it follows that the equilibrium point at the origin of system (1) is globally exponentially stable.
The trajectory of solutions of Example 1 with the initial state for is shown in Figure 1.

325752.fig.001
Figure 1: The trajectory of solutions of system (1) in Example 1.

Example 2. Consider the NNs (1) withwhere Then,,,,,, and. The equilibrium point of system in Example 2 is From which it follows that Therefore, from Theorem 3, it follows that the equilibrium point at the origin of system (1) is globally exponentially stable.
The trajectory of solutions of Example 2 with the initial statefor is shown in Figure 2.

325752.fig.002
Figure 2: The trajectory of solutions of system (1) in Example 2.

5. Conclusion

In this paper, we obtained sufficient condition for global exponential stability of discrete-time neural networks with multiple time-varying delays. The stability criteria are derived by using a discrete-type inequality. Numerical examples are given to show the effectiveness of our theoretical results.

Acknowledgments

This research is supported by Research Promotion and Technology Transfer Center (RPTTC), Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng Saen Campus, Thailand, and the Center of Excellence in Mathematics, Thailand, and Commission for Higher Education, Thailand. The authors would like to thank anonymous reviewers for valuable comments and suggestions to improve the exposition of the paper.

References

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