On exact controllability and complete stabilizability for linear systems

Abstract

In this paper we consider linear systems with control described by the equation $\dot x = \mathcal A x +\mathcal B u$ where functions u and x take values in U and X respectively. For such object, a short review of results concerning relations between exact controllability and complete stabilizability (stabilizability with arbitrary decay rate) is given. The analysis is done in various situations: bounded or unbounded state and control operators $\mathcal A$ and $\mathcal B$, Banach or Hilbert spaces U and X. The well known equivalence between complete controllability and pole assignment in the situation of finite dimensional spaces is no longer true in general in infinite dimensional spaces. Exact controllability is not sufficient for complete stabilizability if U and X are Banach spaces. In Hilbert space setting this implication holds true. The converse also is not so simple: in some situations, complete stabilizability implies exact controllability (Banach space setting with bounded operators), in other situation, it is not true. The corresponding results are given with some ideas for the proofs. Complete technical development are indicated in the cited literature. Several examples are given. Special attention is paid to the case of infinite dimensional systems generated by delay systems of neutral type in some general form (distributed delays). The question of the relation between exact null controllability and complete stabilizability is more precisely investigated. In general there is no equivalence between the two notions. However for some classes of neutral type equations there is an equivalence. The question how the equivalence occurs for more general systems is still open. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces. Our works in this area were initiated by V. I. Korobov during the 70th of the past century in Kharkov State University.