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We suppose that a continuous-time feedback is input-output stabilizing for an infinite-dimensional system. We address the question of whether the sampled-data controller obtained by applying idealized sample-and-hold to this continuous-time feedback is also input-output stabilizing if the sampling time is small enough. This question has been previously addressed for fairly general systems under various conditions. In this note, we restrict our attention to Riesz spectral systems, for which we generalize the existing results. Specifically, we give two relatively simple conditions which, combined, are sufficient for the sampled-data controller to be stabilizing. The first condition is a spectrum decomposition for the open-loop system generator, which by itself is necessary, but not sufficient, for the system to be stabilizable by sampled-data control. The second is a summability condition relating the real part of the spectrum of the generator and the expansion coefficients for the input and feedback operators

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 paper, 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.

In this paper we consider the stabilization of continuous-time distributed parameter systems by piecewise polynomial controls. We obtain necessary and sufficient conditions for stabilization by piecewise polynomial controls.