Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 231710, 31 pages
Research Article

On the Global Asymptotic Stability of Switched Linear Time-Varying Systems with Constant Point Delays

M. de la Sen1 and A. Ibeas2

1Department of Electricity and Electronics, Faculty of Science and Technology, University of Basque, Campus of Leioa (Bizkaia), Aptdo. 644, 48080 Bilbao, Spain
2Department of Telecommunication and Systems Engineering, Engineering School, Autonomous University of Barcelona, Cerdanyola del Vallés, 08193 Bellaterra, Barcelona, Spain

Received 22 July 2008; Accepted 25 September 2008

Academic Editor: Antonia Vecchio

Copyright © 2008 M. de la Sen and A. Ibeas. 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 asymptotic stability of switched linear time-varying systems with constant point delays under not very stringent conditions on the matrix functions of parameters. Such conditions are their boundedness, the existence of bounded time derivatives almost everywhere, and small amplitudes of the appearing Dirac impulses where such derivatives do not exist. It is also assumed that the system matrix for zero delay is stable with some prescribed stability abscissa for all time in order to obtain sufficiency-type conditions of asymptotic stability dependent on the delay sizes. Alternatively, it is assumed that the auxiliary system matrix defined for all the delayed system matrices being zero is stable with prescribed stability abscissa for all time to obtain results for global asymptotic stability independent of the delays. A particular subset of the switching instants is the so-called set of reset instants where switching leads to the parameterization to reset to a value within a prescribed set.