Feedback stabilizability in Hilbert Spaces

This paper is divided in four sections. In the first one, we recall the main notions related to the infinite dimensional control system such as the controllability and stability notions. The second section is devoted to a brief survey of the literature relevant to this paper. In the third section, we concentrate on the weak stabilizability of semigroups which are similar to contraction semigroups. The result presented here, which is believed to be new is that: IfA generates a semigroup which is similar to aC₀ contraction semigroup, the system is weakly stabilizable if and only if the weakly unstable states of the system are approximately controllable. Finally, in the last section, we present two counterexamples showing the limitations of the weak stabilizability approach and discuss its possible extensions in view of some other fundamental problems.This research was supported in part by the United States Air Force Office of Scientific Research, Applied Mathematics Division, under Grant 73-2492