Optimal Control of Singular Parabolic PDEs Modeling Multiphase Stefan-type Free Boundary Problems

TL;DR

This paper develops a finite difference approach to optimally control singular nonlinear parabolic PDEs modeling complex multiphase Stefan-type free boundary problems, proving convergence and stability under minimal assumptions.

Contribution

It introduces a novel finite difference approximation scheme for singular PDEs in multiphase free boundary problems, establishing existence, convergence, and stability of optimal controls.

Findings

Finite difference method converges for the singular PDE control problem.

Existence and uniqueness of optimal control are proved.

Stability estimates are established under minimal regularity.

Abstract

Optimal control of the singular nonlinear parabolic PDE which is a distributional formulation of multidimensional and multiphase Stefan-type free boundary problem is analyzed. Approximating sequence of finite-dimensional optimal control problems is introduced via finite differences. Existence of the optimal control and the convergence of the sequence of discrete optimal control problems both with respect to functional and control is proved. In particular, convergence of the method of finite differences, and existence, uniqueness and stability estimations are established for the singular PDE problem under minimal regularity assumptions on the coefficients.