Differentiability of the Value Function of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems on $L^2(\Omega)$ under Control Constraints

TL;DR

This paper establishes a framework ensuring the local differentiability of the value function in infinite-horizon optimal control problems for semilinear parabolic equations, enabling classical Hamilton-Jacobi-Bellman equation solutions.

Contribution

It introduces an abstract framework that guarantees the local continuous differentiability of the value function under control constraints for semilinear parabolic equations.

Findings

Framework guarantees differentiability of the value function.

Value function satisfies Hamilton-Jacobi-Bellman equation classically.

Applicable to specific semilinear parabolic equations.

Abstract

An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton-Jacobi-Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations.