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Sharp interface control in a Penrose-Fife model. (English) Zbl 1338.49007
Summary: In this paper, we study a singular control problem for a system of PDEs describing a phase-field model of Penrose-Fife type. The main novelty of this contribution consists in the idea of forcing a sharp interface separation between the states of the system by using heat sources distributed in the domain and at the boundary. We approximate the singular cost functional with a regular one, which is based on the Legendre-Fenchel relations. Then we obtain a regularized control problem for which we compute the first order optimality conditions using an adapted penalization technique. The proof of some convergence results and the passage to the limit in these optimality conditions lead to the characterization of the desired optimal controller.

49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
90C46 Optimality conditions and duality in mathematical programming
82B26 Phase transitions (general) in equilibrium statistical mechanics
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