We consider the maximal regularity problem for non-autonomous evolution equationsEach operator A(t) is associated with a sesquilinear form a(t; ·, ·) on a Hilbert space H. We assume that these forms all have the same domain and satisfy some regularity assumption with respect to t (e.g., piecewise α-Hölder continuous for some α > 1 /2). We prove maximal Lpregularity for all u 0 in the real-interpolation space (H, D(A(0))) 1−1/p,p . The particular case where p = 2 improves previously known results and gives a positive answer to a question of J.L. Lions [16] on the set of allowed initial data u 0 .We then set A(t)u := h. We mention that equality of the form domains, i.e., D(a(t; ·, ·)) = V for t ∈ [0, τ ] does not imply equality of the domains D(A(t)) of the corresponding operators. For each fixed u ∈ V , φ := a(t; u, ·) defines a continuous (anti-)linear functional on V , i.e. φ ∈ V ′ , then it induces a linear operator1991 Mathematics Subject Classification. 35K90, 35K50, 35K45, 47D06.

We study linear systems, described by operators A, B, C for which the state space X is a Banach space. We suppose that −A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss' conjecture. The crucial assumptions on A are boundedness of an H ∞ -calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L p (Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω. Mathematics Subject Classification (2000). Primary 93C05, 93C20, 47D06, 47A10; Secondary 47A60.

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 p -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 p -admissibility of type α is characterised by boundedness conditions which are similar to those in the well-known Weiss conjecture. We also study L p -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 = 2.

We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier–Stokes–Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an Lp‐Lq setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the scriptR‐sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.