Impulsive Stabilization for a Class of Neural Networks with Both Time-Varying and Distributed Delays
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Research Article Impulsive Stabilization for a Class of Neural Networks with Both Time-Varying and Distributed Delays Lizi Yin1 and Xiaodi Li2 1 2
School of Science, University of Jinan, Jinan 250022, China Department of Mathematics, Xiamen University, Xiamen 361005, China
Correspondence should be addressed to Lizi Yin, ss [email protected] Received 16 January 2009; Accepted 4 March 2009 Recommended by Paul Eloe The impulsive control method is developed to stabilize a class of neural networks with both timevarying and distributed delays. Some exponential stability criteria are obtained by using Lyapunov functionals, stability theory, and control by impulses. A numerical example is also provided to show the effectiveness and feasibility of the impulsive control method. Copyright q 2009 L. Yin and X. Li. 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.
1. Introduction During the last decades, neural networks such as Hopfield neural networks, cellular neural networks, Cohen-Grossberg neural networks, and bidirectional associative memory neural networks have been extensively studied. There have appeared a number of important results; see 1–13 and references therein. It is well known that the properties of stability and convergence are important in design and application of neural networks, for example, when designing a neural network to solve linear programming problems and pattern recognition problems, we foremost guarantee that the models of neural network are stable. However, it may become unstable or even divergent because the model of a system is highly uncertain or the nature of the problem itself. So it is necessary to investigate stability and convergence of neural networks from the control point of view. It is known that impulses can make unstable systems stable or, otherwise, stable systems can become unstable after impulse effects; see 14–18. The problem of stabilizing the solutions by imposing proper impulse controls has been used in many fields such as neural network, engineering, pharmacokinetics, biotechnology, and population dynamics 19–25. Recently, several good impulsive control
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Advances in Difference Equations
approaches for real world systems have been proposed; see 22–32. In 26, Yang and Xu investigate the global exponential stability of Cohen-Grossberg neural networks with variable delays by establishing some impulsive differential inequalities. The criteria not only present an approach to stabilize the unstable neural networks by utilizing impulsive effects but also show that the stability still remains under certain impulsive perturbations for some continuous stable neural networks. In 27, Li et al. consider the impulsive control of Lotka-Volterra predator-prey system by employing the method of Lyapunov functions. In 28, Wang and Liu investigate the impulsive stabilization of delay differential systems via the Lyapunov
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