- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 941063, 11 pages

http://dx.doi.org/10.1155/2012/941063

## Exponential Convergence for Cellular Neural Networks with Time-Varying Delays in the Leakage Terms

^{1}School of Science, Hunan University of Technology, Hunan, Zhuzhou 412000, China^{2}College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang, Jiaxing 314001, China

Received 18 August 2012; Accepted 3 October 2012

Academic Editor: Narcisa C. Apreutesei

Copyright © 2012 Zhibin Chen and Junxia Meng. 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

We consider a class of cellular neural networks with time-varying delays in the leakage terms. By applying Lyapunov functional method and differential inequality techniques, we establish new results to ensure that all solutions of the networks converge exponentially to zero point.

#### 1. Introduction

It is well known that the delayed cellular neural networks (CNNs) have been successfully applied to signal and image processing, pattern recognition, and optimization (see [1]). Hence, they have been the object of intensive analysis by numerous authors in the past decades. In particular, extensive results on the problem of the existence and stability of the equilibrium point for CNNs are given out in many works in the literature. We refer the reader to [2–6] and the references cited therein. Recently, to consider CNNs with the incorporation of time delays in the leakage terms, Gopalsamy [7] and Wang et al. [8] investigated a class of CNNS described by in which corresponds to the number of units in a neural network, corresponds to the state vector of the th unit at the time , and represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at the time . and are the connection weights at the time , and denote the leakage delay and transmission delay, respectively, denotes the external bias on the th unit at the time , and are activation functions of signal transmission, and .

Suppose that the following conditions and are constants, where , for each , there exists a nonnegative constant such that are satisfied. Avoiding the continuously distributed delay terms, the authors of [7, 8] obtained that all solutions of system (1.1) converge to the equilibrium point or the periodic solution. However, to the best of our knowledge, few authors have considered the convergence behavior for all solutions of system (1.1) without the assumptions and . Thus, it is worthwhile to continue to investigate the convergence behavior of system (1.1) in this case.

The main purpose of this paper is to give the new criteria for the convergence behavior for all solutions of system (1.1). By applying Lyapunov functional method and differential inequality techniques, without assuming and , we derive some new sufficient conditions ensuring that all solutions of system (1.1) converge exponentially to zero point. Moreover, an example is also provided to illustrate the effectiveness of our results.

Throughout this paper, for , it will be assumed that and are continuous functions, and there exist constants and such that We also assume that the following conditions , , and hold: for each , there exist nonnegative constants and such that for all and , there exist constants and such that .

The initial conditions associated with system (1.1) are of the form where denotes real-valued-bounded continuous function defined on .

#### 2. Main Results

Theorem 2.1. *Let , , and hold. Then, for every solution of system (1.1) with any initial value , there exists a positive constant such that
*

*Proof. *
Let be a solution of system (1.1) with any initial value ,and let
In view of (1.1), we have
Let
From (1.3), , and , we can choose a positive constant such that
Then, it is easy to see that
We now claim thatIf this is not valid, then, one of the following two cases must occur:(1)there exist and such that
(2)there exist and such that
Now, we consider two cases.*Case i*. If (2.8) holds. Then, from (2.3), (2.5), and ()−(), we have
This contradiction implies that (2.8) does not hold.*Case ii*. If (2.9) holds. Then, from (2.3), (2.5), and ()−(), we get
which is a contradiction and yields that (2.9) does not hold.

Consequently, we can obtain that (2.7) is true. Thus,
This implies that the proof of Theorem 2.1 is now completed.

#### 3. An Example

*Example 3.1. *Consider the following CNNs with time-varying delays in the leakage terms:
where .

Noting that
Define a continuous function by setting
Then, we obtain
Therefore,
which, together with the continuity of , implies that we can choose positive constants and such that for all , there holds
This yields that system (3.1) satisfied , , and . Hence, from Theorem 2.1, all solutions of system (3.1) converge exponentially to the zero point .

*Remark 3.2. *Since , and CNNs (3.1) are a very simple form of CNNs with time-varying delays in the leakage terms, it is clear that the conditions and are not satisfied. Therefore, all the results in [7–9] and the references therein cannot be applicable to prove that all solutions of system (3.1) converge exponentially to the zero point.

#### Acknowledgments

The authors would like to express their sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper; this work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Hunan Provincial National Natural Science Foundation of China (12JJ3007), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grants nos. Y6110436, LY2A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).

#### References

- L. O. Chua and T. Roska, “Cellular neural networks with nonlinear and delay-type template elements,” in
*Proceedings of IEEE International Workshop on Cellular Neural Networks and Their Applications*, pp. 12–25, 1990. - H. J. Cho and J. H. Park, “Novel delay-dependent robust stability criterion of delayed cellular neural networks,”
*Chaos, Solitons and Fractals*, vol. 32, no. 3, pp. 1194–1200, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Ou, “Almost periodic solutions for shunting inhibitory cellular neural networks,”
*Nonlinear Analysis. Real World Applications*, vol. 10, no. 5, pp. 2652–2658, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Liu and L. Huang, “Global exponential stability of BAM neural networks with recent-history distributed delays and impulses,”
*Neurocomputing*, vol. 69, no. 16-18, pp. 2090–2096, 2006. View at Publisher · View at Google Scholar · View at Scopus - B. Liu, “Exponential convergence for a class of delayed cellular neural networks with time-varying coefficients,”
*Physics Letters A*, vol. 372, no. 4, pp. 424–428, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. Zhang, W. Wang, and B. Xiao, “Exponential convergence for high-order recurrent neural networks with a class of general activation functions,”
*Applied Mathematical Modelling*, vol. 35, no. 1, pp. 123–129, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Gopalsamy, “Leakage delays in BAM,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 2, pp. 1117–1132, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Wang, C. Li, and H. Xu, “Existence and global exponential stability of periodic solution of cellular neural networks with impulses and leakage delay,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 19, no. 3, pp. 831–842, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Liu, “Global exponential stability for BAM neural networks with time-varying delays in the leakage terms,”
*Nonlinear Analysis. Real World Applications*, vol. 14, no. 1, pp. 559–566, 2013. View at Publisher · View at Google Scholar