Weighted Admissibility and Wellposedness of linear systems in Banach spaces

We study linear control systems in infinite--dimensional Banach spaces governed by analytic semigroups. For $p\in[1,\infty]$ and $\alpha\in\RR$ we introduce the notion of $L^p$--admissibility of type $\alpha$ for unbounded observation and control operators. Generalising earlier work by Le Merdy and the first named author and Le Merdy we give conditions under which $L^p$--admissibility of type $\alpha$ is characterised 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. 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 from Byrnes, Gilliam, Shubov and Weiss to non--Hilbertian settings and to $p\neq 2$.