Sharp interface control in a Penrose−Fife model

¹ Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy
pierluigi.colli@unipv.it
² Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie No.13, Sector 5, 050711, Bucharest, Romania
gabriela.marinoschi@acad.ro
³ Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
elisabetta.rocca@wias-berlin.de
⁴ Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy
elisabetta.rocca@unimi.it

Received: 17 March 2014
Revised: 3 December 2014

Abstract

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.