Fixed points and controllability in delay systems

Schaefer's fixed point theorem is used to study the controllability in an infinite delay system [inline-graphic not available: see fulltext]. 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 [inline-graphic not available: see fulltext], then there exists a solution for [inline-graphic not available: see fulltext]. 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.