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Mathematical Problems in Engineering
Volume 2011, Article ID 723509, 12 pages
Research Article

Common Lyapunov Function Based on Kullback–Leibler Divergence for a Switched Nonlinear System

1Graduate School of Health Sciences, The University of Tokushima, 3-18-15 Kuramoto, Tokushima 770-8509, Japan
2Institute of Health Biosciences, The University of Tokushima, 3-18-15 Kuramoto, Tokushima 770-8509, Japan

Received 21 December 2010; Revised 15 March 2011; Accepted 16 March 2011

Academic Editor: Jyh Horng Chou

Copyright © 2011 Omar M. Abou Al-Ola 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.


Many problems with control theory have led to investigations into switched systems. One of the most urgent problems related to the analysis of the dynamics of switched systems is the stability problem. The stability of a switched system can be ensured by a common Lyapunov function for all switching modes under an arbitrary switching law. Finding a common Lyapunov function is still an interesting and challenging problem. The purpose of the present paper is to prove the stability of equilibrium in a certain class of nonlinear switched systems by introducing a common Lyapunov function; the Lyapunov function is based on generalized Kullback–Leibler divergence or Csiszár's I-divergence between the state and equilibrium. The switched system is useful for finding positive solutions to linear algebraic equations, which minimize the I-divergence measure under arbitrary switching. One application of the stability of a given switched system is in developing a new approach to reconstructing tomographic images, but nonetheless, the presented results can be used in numerous other areas.