Differentiability of the Value Function on $H^1(\mathrm{\Omega })$ of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems under Control Constraints

An abstract framework guaranteeing the continuous differentiability of local value functions on $H^1(\mathrm{\Omega })$ associated with optimal stabilization problems subject to abstract semilinear parabolic equations in the presence of norm constraints on the control is established. It guarantees the local well-posedness of the associated Hamilton-Jacobi-Bellman equation in the classical sense. Examples illustrate that the assumptions imposed on the dynamical system are satisfied for practically relevant semilinear equations.