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

It is a well-known principle, for finite-dimensional systems, that applying sampled-and-hold in the feedback loop around a stabilizing state feedback (or dynamic) controller results in a stable sampled-data feedback control system if the sampling period is small enough. The principle extends to infinite-dimensional systems with compact state feedback if either the input operator is bounded or the given state-space system is analytic. In this note, we give an example for which this principle fails but which nevertheless satisfies certain necessary conditions arising in sampled-data control of infinite-dimensional systems.