Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 480187, 20 pages
About Robust Stability of Dynamic Systems with Time Delays through Fixed Point Theory
Faculty of Science and Technology, University of the Basque Country, Leioa (Bizkaia), Aptdo. 644 de Bilbao, 48080 Bilbao, Spain
Received 22 September 2008; Revised 10 November 2008; Accepted 19 November 2008
Academic Editor: Andrzej Szulkin
Copyright © 2008 M. De la Sen. 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.
This paper investigates the global asymptotic stability independent
of the sizes of the delays of linear time-varying systems with internal point delays which possess a limiting equation via fixed point
theory. The error equation between the solutions of the limiting
equation and that of the current one is considered as a perturbation equation
in the fixed- point and stability analyses. The existence of a unique
fixed point which is later proved to be an asymptotically stable equilibrium point is investigated. The stability conditions are basically concerned with the matrix measure of the delay-free matrix of dynamics to be negative and to have a modulus larger than the contribution of the error dynamics with respect to the
limiting one. Alternative conditions are obtained concerned with the matrix dynamics for zero delay to be negative and to have a modulus larger than an appropriate contributions of the error dynamics of the current dynamics with respect to the limiting one. Since global stability is guaranteed under some
deviation of the current solution related to the limiting one, which is considered as nominal, the stability is robust against such errors for certain tolerance margins.