Weighted Admissibility and Wellposedness of Linear Systems in Banach Spaces
Résumé
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 [{\it J. London Math. Soc.} (2), 67 (2003), pp.~715--738] and Haak and Le~Merdy [{\it 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 Pruess and Simonett [{\it 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., {\it J. Dynam. Control Systems}, 8 (2002), pp.~341--370] to non-Hilbertian settings and to $p\neq 2$.