On non-exact controllable systems

Abstract

In this paper we are concerned with linear control systems of first or second order, defined on Banach spaces. We established some negative facts for these systems, fundamentally related with linear compact operators. In particular, we show several sufficient conditions, of a general character, for the non-exact controllability or these systems. We also show that it is not possible to uniformly stabilize a first-order control system, with states in a Banach space, using a compact control feedback. This generalizes Gibson's (1980) result for systems on Hilbert spaces.