# Optimal control of multiphase free boundary problems for nonlinear parabolic equations

## TL;DR

This paper develops a method for optimal boundary heat flux control in multiphase free boundary problems modeled by nonlinear parabolic PDEs, ensuring convergence of discrete approximations to the continuous problem.

## Contribution

It introduces a finite difference approach for controlling multiphase Stefan problems and proves the existence and convergence of optimal controls.

## Key findings

- Established existence of optimal control.
- Proved convergence of discrete controls to continuous control.
- Derived uniform bounds and energy estimates for the PDE solutions.

## Abstract

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.