  We discuss the feedback control problem for a two-dimensional two-phase Stefan problem. In our approach, we use a sharp interface representation in combination with mesh-movement to track the interface position. To attain a feedback control, we apply the linear-quadratic regulator approach to a suitable linearization of the problem. We address details regarding the discretization and the interface representation therein. Further, we document the matrix assembly to generate a non-autonomous generalized differential Riccati equation. To numerically solve the Riccati equation, we use low-rank factored and matrix-valued versions of the non-autonomous backward differentiation formulas, which incorporate implicit index reduction techniques. For the numerical simulation of the feedback controlled Stefan problem, we use a time-adaptive fractional-step-theta scheme.   We provide the implementations for the developed methods and test these in several numerical experiments. With these experiments we show that our feedback control approach is applicable to the Stefan control problem and makes this large-scale problem computable. Also, we discuss the influence of several controller design parameters, such as the choice of inputs and outputs. 
Riccati-feedback Control of a Two-dimensional Two-phase Stefan Problem