Fixed Point Theory and Applications 
Volume 2006 (2006), Article ID 41480, 14 pages
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
Received 9 December 2004; Revised 28 June 2005; Accepted 6 July 2005

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.