Optimal Control of the Multiphase Stefan Problem

TL;DR

This paper develops an optimal control approach for the inverse multiphase Stefan problem, focusing on reconstructing missing boundary heat flux and free boundaries using finite difference discretization and proving convergence and well-posedness.

Contribution

It introduces a novel optimal control framework for the inverse multiphase Stefan problem, including discretization, analysis, and convergence proofs.

Findings

Proved well-posedness in Sobolev spaces.

Established convergence of discrete controls to the continuous problem.

Demonstrated convergence of finite difference solutions to the weak solution.

Abstract

We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete multiphase Stefan problem.