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 ∈ [1,∞] and α ∈ R we introduce the notion of L–admissibility of type α for unbounded observation and control operators. Generalising earlier work by Le Merdy [20] and the first named author and Le Merdy [12] we give conditions under which L–admissibility of type α is characterised by boundedness conditions which are similar to those in the well–known Weiss conjecture. We also study L–wellposedness of type α 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 in [4] to non–Hilbertian settings and to p 6= 2.