Extensions of Rank Conditions for Controllability and Observability to Banach Spaces and Unbounded Operators

Abstract

Generalizations of the familiar rank conditions for controllability and observability of linear autonomous finite-dimensional systems to the general case when both the state space and the control space are infinite-dimensional Banach spaces and the operator A acting on the state is only assumed to generate a strongly continuous semigroup (group) are sought. It is shown that a suitable version of the rank condition, although generally only sufficient for approximate controllability (observability), is however “essentially” necessary and sufficient in two important cases: (i) when A generates an analytic semigroup, (ii) when A generates a group. Such generalization of the rank condition is then used to derive, in turn, easy-to-check tests for approximate controllability (observability) for the important class of normal operators with compact resolvent. In the case of finite number of scalar controls (observations), the tests are expressed by a sequence of rank conditions, using the complete set of eigenvect...