Robustness of closed-loop stability for infinite dimensional systems under sample and hold-counterexamples

We consider continuous-time, linear control systems for which a static state feedback stabilizes the system. If we construct a sampled-data controller by applying an idealized sample-and-hold process to the continuous-time stabilizing feedback, it is known that if the state and control spaces are finite dimensional, then this sampled-data controller stabilizes the system for all sufficiently small sampling times. In this paper we show that this robustness with respect to sampling times is not true in general for infinite dimensional systems. We consider systems where the state space X and the control space U are Hilbert spaces, the system is of the form x˙(t)=Ax(t)+Bu(t), and A is the generator of a strongly continuous semigroup on X. Suppose that the continuous time feedback is u(t)=Fx(t), where F is compact. Then it is known that if either B is bounded, or if A generates an analytic semigroup on X (in which case B is allowed to be unbounded in a general sense), then the sampled-data controller stabilizes the system for all sufficiently small sampling times. In this paper we show that the first condition is sharp in the following sense: we give a counterexample to show that the result is not true if B is barely unbounded, that is, B is unbounded but A^-δB is bounded for all δ>0. We also give an easy counterexample if F is not compact