We prove L 2 -maximal regularity of linear non-autonomous evolutionary Cauchy probleṁwhere the operator A(t) arises from a time depending sesquilinear form a(t, ., .) on a Hilbert space H with constant domain V. We prove the maximal regularity in H when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed in [9], [10] and [13]. As a consequence, we obtain an invariance criterion for convex and closed sets of H.

This paper is devoted to the study of L p -maximal regularity for non-autonomous linear evolution equations of the forṁwhere {A(t), t ∈ [0, T ]} is a family of linear unbounded operators whereas the operators {B(t), t ∈ [0, T ]} are bounded and invertible. In the Hilbert space situation we consider operators A(t), t ∈ [0, T ], which arise from sesquilinear forms. The obtained results are applied to parabolic linear differential equations in one spatial dimension.

We consider a linear non-autonomous evolutionary Cauchy probleṁ1) where the operator A(t) arises from a time depending sesquilinear form a(t, ., .) on a Hilbert space H with constant domain V . Recently, a result on L 2 -maximal regularity in H, i.e. for each given f ∈ L 2 (0, T, H) and u0 ∈ V the problem (0.1) has a unique solution u ∈ L 2 (0, T, V ) ∩ H 1 (0, T, H), is proved in Dier (J. Math. Anal. Appl. 425:33-54, 2015) under the assumption that a is symmetric and of bounded variation. The aim of this paper is to prove that the solutions of an approximate nonautonomous Cauchy problem in which a is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form. In particular, this provides an alternative proof of the result in Dier (J. Math. Anal. Appl. 425:33-54, 2015) on L 2 -maximal regularity in H. Mathematics Subject Classification. 35K90, 35K50, 35K45, 47D06.

We study a class of non-autonomous linear boundary control and observation systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.

This paper gives further regularity properties of the evolution family associated with a non-autonomous evolution equatioṅarise from non-autonomous sesquilinear forms a(t, ·, ·) on a Hilbert space H with constant domain V ⊂ H. Results on norm continuity, compactness and results on the Gibbs character of the evolution family are established. The abstract results are applied to the Laplacian operator with time dependent Robin boundary conditions.