Fixed Point Theory and Applications
Volume 2006 (2006), Issue 1, Pages Article 41480, 14 p.
doi:10.1155/FPTA/2006/41480
Fixed points and controllability in delay systems
Hang Gao1
and Bo Zhang2
1Department of Mathematics, Northeast Normal University, Changchun, Jilin 130024, China
2Department of Mathematics and Computer Science, Fayetteville State University, NC 28301-4298, Fayetteville, USA
Abstract
Schaefer's fixed point theorem is used to study the controllability in an infinite delay system x′(t)=G(t,xt)+(Bu)(t). A compact map or homotopy is constructed enabling us to show that if there is an a priori bound on all possible solutions of the companion control system x′(t)=λ[G(t,xt)+(Bu)(t)],0<λ<1, then there exists a solution for λ=1. The a priori bound is established by means of a Liapunov functional or applying an integral inequality. Applications to integral control systems are given to illustrate the approach.