Frechet Differentiability in Besov Spaces in the Optimal Control of Parabolic Free Boundary Problems

TL;DR

This paper establishes Frechet differentiability in Besov spaces for an inverse Stefan free boundary problem, enabling the use of gradient methods for optimal control in complex heat transfer scenarios.

Contribution

It proves Frechet differentiability in Besov spaces for the inverse Stefan problem, providing a foundation for numerical optimization methods.

Findings

Proved Frechet differentiability in Besov spaces under minimal regularity.

Derived explicit formula for the Frechet differential.

Enabled application of gradient-based numerical methods.

Abstract

We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.