  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 $p\neq 2$.
