Optimal Boundary-Flux Control of a Sharp Moving Interface in the Classical Two-Phase Stefan Problem

In this paper, we study the optimal boundary control of solidification governed by the classical two-phase Stefan problem with a sharp moving interface. The main objective is to formulate an optimal control problem for interface motion using boundary heat-flux control. The control acts as a Neumann heat flux on a designated boundary segment and steers the interface through the Stefan condition. Using an enthalpy formulation, we prove well-posedness under boundary control and establish Lipschitz continuity of the control-to-state map and continuous dependence on the initial data. We then derive first-order necessary optimality conditions using a Lagrangian approach and propose a practical algorithm that couples a semismooth Newton method with Sequential Quadratic Programming (SQP) to compute the optimal boundary flux.