Boundary controllability of phase-transition region of a two-phase Stefan problem

TL;DR

This paper demonstrates the boundary controllability of the phase-transition region in a two-phase Stefan problem, showing that the mushy region can be manipulated to include any measurable set at the final time.

Contribution

It introduces a novel boundary control method for the Stefan problem's phase-transition region using an optimal control approach, Carleman's inequality, and fixed point theorem.

Findings

Controllability of the mushy region at final time.

Existence of boundary control for arbitrary measurable sets.

Application of advanced mathematical tools to Stefan problem control.

Abstract

One proves that the moving interface of a two-phase Stefan problem on $\ooo\subset\rr^d$, $d=1,2,3,$ is controllable at the end time $T$ by a Neumann boundary controller $u$. The phase-transition region is a mushy region $\{\sigma^u_t;\ 0\le t\le T\}$ of a modified Stefan problem and the main result amounts to saying that, for each Lebesque measurable set $\ooo^*$ with positive measure, there is $u\in L^2((0,T)\times\pp\ooo)$ such that $\ooo^*\subset\sigma^u_T.$ To this aim, one uses an optimal control approach combined with Carleman's inequality and the Kakutani fixed point theorem.