Abstract and Applied Analysis
Volume 2008 (2008), Article ID 381791, 29 pages
doi:10.1155/2008/381791
Research Article
Robust Stability and Stability Radius for Variational Control Systems
Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, Blvd. V. Pârvan 4, 300223 Timişoara, Romania
Received 26 September 2007; Revised 15 January 2008; Accepted 27 February 2008
Academic Editor: Stephen Clark
Copyright © 2008 Bogdan Sasu. 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
We consider an integral variational control system on a Banach space X and we study the connections between its uniform exponential stability and the (I(ℝ+,X),O(ℝ+,X)) stability, where I and O are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input space. We analyze the robustness of exponential stability in the presence of structured perturbations. We deduce general estimations for the lower bound of the stability radius of a variational control system in terms of input-output operators acting on translation-invariant spaces. We apply the main results at the study of the exponential stability of nonautonomous systems and analyze in the nonautonomous case the robustness of this asymptotic property.