We study linear control systems in infinite-dimensional Banach spaces governed by analytic semigroups. For p is an element of [1,infinity] and alpha is an element of R we introduce the notion of L(p)-admissibility of type a 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 Lp-admissibility of type a is characterized by boundedness conditions which are similar to those in the well-known Weiss conjecture. We also study L(p)-wellposedness of type a 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 not equal 2.