Weighted Admissibility and Wellposedness of Linear Systems in Banach Spaces

Abstract

We study linear control systems in infinite‐dimensional Banach spaces governed by analytic semigroups. For $p\in[1,\infty]$ and $\alpha\in\mathbb{R}$ we introduce the notion of $L^p$‐admissibility of type $\alpha$ for unbounded observation and control operators. Generalizing earlier work by Le Merdy [J. London Math. Soc. (2), 67 (2003), pp. 715–738] and Haak and Le Merdy [Houston J. Math., 31 (2005), pp. 1153–1167], we give conditions under which $L^p$‐admissibility of type $\alpha$ is characterized by boundedness conditions which are similar to those in the well‐known Weiss conjecture. We also study $L^p$‐wellposedness of type $\alpha$ for the full system. Here we use recent ideas due to Pruss and Simonett [Arch. Math. (Basel), 82 (2004), pp. 415–431]. Our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take Lebesgue and Besov spaces as state space. This extends the considerations in [C. I. Byrnes et al., J. Dynam. Control Systems, 8 (2002), pp. 341–370] to non‐Hilbertian settings and to $p\neq2$.