Further Regularity Properties in Quadratic Cost Problems for Parabolic Equations with Boundary Control and Non-Smoothing Final State Penalization

We consider the optimal quadratic cost problem for abstract parabolic equations over a finite horizon and with non-smoothing finite state penalization. Complementing [L-T.2], we prove regularity results for the first time derivatives of the optimal control and of the optimal trajectory, in terms of functions spaces, introduced in [D-I.1], which measure the degree of singularity at the endpoints. The setting includes all boundary control problems for parabolic, or parabolic-like, partial differential equations [L-T.3], [L-T.4].