ISSN:
1432-0606
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 aC 0-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. In the unbounded case, our results are illustrated by the shift semigroup and by the heat equation on an infinite rod with distributed controls. In the bounded case, our example is an integro-differential control system.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01442887
