Value function and optimality condition for semilinear control problems. II: Parabolic case

In this paper we continue to study properties of the value function and of optimal solutions of a semilinear Mayer problem in infinite dimensions. Applications concern systems governed by a state equation of parabolic type. In particular, the issues of the joint Lipschitz continuity and semiconcavity of the value function are treated in order to investigate the differentiability of the value function along optimal trajectories.The research of P. Cannarsa was partially supported by the Italian National Project MURST Equazioni di Evoluzione e Applicazioni Fisico-Matematiche.