Differentiability of the value function in control-constrained parabolic problems

TL;DR

This paper proves the differentiability of the value function in control-constrained semilinear parabolic PDE problems with respect to initial conditions and time, under certain growth assumptions, enhancing understanding of optimal control solutions.

Contribution

It establishes the differentiability of the value function in parabolic control problems with constraints, extending previous results to include time dependence and neighborhood analysis.

Findings

Differentiability with respect to initial conditions.

Differentiability with respect to time under additional assumptions.

Analysis of neighborhood differentiability using growth conditions.

Abstract

Along the optimal trajectory of an optimal control problem constrained by a semilinear parabolic partial differential equation, we prove the differentiability of the value function with respect to the initial condition and, under additional assumptions on the solution of the state equation, the differentiability of the value function with respect to the time variable. In our proof, we rely on local growth assumptions commonly associated with the study of second-order sufficient conditions. These assumptions are generally applicable to a wide range of problems, including, for instance, certain tracking-type problems. Finally, we discuss the differentiability of the value function in a neighborhood of the optimal trajectory when a growth condition for optimal controls is used.