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An abstract framework guaranteeing the continuous differentiability of localvalue functions on $H^1(\Omega)$ associated with optimal stabilization problemssubject to abstract semilinear parabolic equations in the presence of normconstraints on the control is established. It guarantees the localwell-posedness of the associated Hamilton-Jacobi-Bellman equation in theclassical sense. Examples illustrate that the assumptions imposed on thedynamical system are satisfied for practically relevant semilinear equations.

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