Module structure of constant linear systems and its applications to controllability

Abstract

We shall introduce a new module structure to a large class of continuous-time constant linear systems. This is done as a natural extension of the classical k[z]-module structure of finite-dimensional constant linear systems. This module action is used to investigate the relationship between reachability and controllability of linear systems. After introducing the notion of K-controllability due to Kamen [12], we give the following result in Section 5: If a constant linear system is described by a functional differential equation x ̇ = Fx + Gu, where x and G belong to a Banach space X, and if G is K-controllable to zero, then every reachable state is reachable and controllable in bounded time. (The result given in Section 5 is a little more general than this.) We also give a simple example in Section 6 to illustrate this result.