We consider a simple classical model for the coexistence of phases in a two-phase system, where the transition between the two phases is driven by the absolute temperature. The problem is described by an energy balance equation and a singular differential inclusion characterizing the dynamics of the phase transition. The controllability problem consists in finding an external heat source which drives the solution to a prescribed final state in finite time. The goal of this talk is to show that a solution to the problem can be constructed by a two-step approximation in an optimal control system with suitably chosen cost functional and penalty approximation. This is a joint work with Chiara Gavioli from Modena.

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