Stability of infinite-dimensional sampled-data systems

Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space X X and the control space U U are Hilbert spaces, the system is of the form x ˙ ( t ) = A x ( t ) + B u ( t ) x(t) = Ax(t) + Bu(t) , where A A is the generator of a strongly continuous semigroup on X X , and the continuous time feedback is u ( t ) = F x ( t ) u(t) = Fx(t) . The answer to the above question is known to be “yes” if X X and U U are finite-dimensional spaces. In the infinite-dimensional case, if F F is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is “yes”, if B B is a bounded operator from U U into X X . Moreover, if B B is unbounded, we show that the answer “yes” remains correct, provided that the semigroup generated by A A is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.