Invariance theory for infinite dimensional linear control systems

Abstract

In this paper we characterize the situation wherein a subspaceS of a separable Hilbert state space is holdable under the abstract linear autonomous control system\(\dot x = Ax + Bu\), whereA is the infinitesimal generator of aC0-semigroup of operators and whereB is a bounded linear operator mapping a Hilbert space Ω intoX. WhenS⊥∩D(A*) is dense inS⊥, it is shown that a necessary (but insufficient) condition for holdability is (1):\(A[S \cap D\left( A \right)] \subset \bar S + B\Omega\). A stronger condition than (1) is shown to be sufficient for a type of approximate holdability. In the finite dimensional setting, (1) reduces to (A, B)-invariance, which is known to be equivalent to the existence of a (bounded) linear feedback control law which achieves holdability inS. We prove that this equivalence holds in infinite dimensions as well, whenA is bounded and the linear spacesS, BΩ andS+ BΩ are closed.