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 $X$ and the control space $U$ are Hilbert spaces, the system is of the form $\dot x(t) = Ax(t) + Bu(t)$, where $A$ is the generator of a strongly continuous semigroup on $X$, and the continuous time feedback is $u(t) = Fx(t)$. The answer to the above question is known to be ``yes'' if $X$ and $U$ are finite-dimensional spaces. In the infinite-dimensional case, if $F$ 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 $B$ is a bounded operator from $U$ into $X$. Moreover, if $B$ is unbounded, we show that the answer ``yes'' remains correct, provided that the semigroup generated by $A$ is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.