Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 231710, 31 pages
doi:10.1155/2008/231710
Research Article
On the Global Asymptotic Stability of Switched Linear Time-Varying Systems with Constant Point Delays
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.
Abstract
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.