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 and the control space are Hilbert spaces, the system is of the form , where is the generator of a strongly continuous semigroup on , and the continuous time feedback is . The answer to the above question is known to be ``yes' if and are finite-dimensional spaces. In the infinite-dimensional case, if 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 is a bounded operator from into . Moreover, if is unbounded, we show that the answer ``yes' remains correct, provided that the semigroup generated by is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.