Generalized Sampled-Data Stabilization of Well-Posed Linear Infinite-Dimensional Systems

We consider well-posed linear infinite-dimensional systems, the outputs of which are sampled in a generalized sense using a suitable weighting function. Under certain natural assumptions on the system, the weighting function, and the sampling period, we show that there exists a generalized hold function such that unity sampled-data feedback renders the closed-loop system exponentially stable (in the state-space sense) as well as L 2-stable (in the input-output sense). To illustrate our main result, we describe an application to a structurally damped Euler–Bernoulli beam. 1. Introduction. The design of sampled-data controllers is important both for applications, because of digital implementation issues, and for theoretical development. Sampled-data control for infinite-dimensional systems has been considered in In this paper we develop generalized sampled-data control for well-posed linear continuous-time infinite-dimensional systems. Generalized sampled-data control has been frequently studied for finite-dimensional systems (see, for instance, [2, 10]) and for infinite-dimensional systems in Tarn et al. [28] and Tarn, Zavgren, and Zeng [29]. A well-posed system Σ has generating operators (A, B, C), where A is the generator of a strongly continuous semigroup T = (T t) t≥0 governing the state evolution of the uncontrolled system, B is the control operator, and C is the observation operator; see, for example, [5, 23, 25, 27, 31]. Denote by u and y the input and output of Σ. For a given sampling period τ > 0, a generalized sampled-data feedback control will have the form