On the State Constrained Optimal Control of the Stefan Type Free Boundary Problems

TL;DR

This paper formulates and analyzes a state constrained inverse Stefan free boundary problem as an optimal control problem within Sobolev-Besov spaces, establishing existence, differentiability, and convergence of discretized solutions.

Contribution

It introduces a novel framework for controlling Stefan problems with state constraints, proving existence, differentiability, and convergence of solutions in Besov spaces.

Findings

Existence of optimal control established

Frechet differentiability proved under minimal regularity

Discretized solutions converge to continuous problem

Abstract

We analyze the state constrained inverse Stefan type parabolic free boundary problem as an optimal control problem in the Sobolev-Besov spaces framework. Boundary heat flux, density of heat sources, and free boundary are components of the control vector. Cost functional is the sum of the $L_2$-norm declinations of the temperature measurement at the final moment, the phase transition temperature, the final position of the free boundary, and the penalty term, taking into account the state constraint on the temperature. We prove the existence of optimal control, Frechet differentiability, and optimality condition in the Besov spaces under minimal regularity assumptions on the data. We pursue space-time discretization through finite differences and prove that the sequence of discrete optimal control problems converges to the original problem both with respect to functional and control.