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

Abstract

One proves that the moving interface of a two-phase Stefan problem on Ω ⊂ R 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 { σ t u ; 0 ≤ t ≤ T } of a modified Stefan problem and the main result amounts to saying that, for each Lebesgue measurable set Ω ∗ with positive measure, there is u ∈ L 2 ( ( 0 , T ) × ∂ Ω ) such that Ω ∗ ⊂ σ T u . To this aim, one uses an optimal control approach combined with Carleman’s inequality and the Kakutani fixed point theorem.