Sharp interface control in a Penrose−Fife model
ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 473-499.

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.

Received:
DOI: 10.1051/cocv/2015014
Classification: 49J20, 82B26, 90C46
Keywords: Optimal control problems, Penrose−Fife model, sharp interface
Colli, Pierluigi 1; Marinoschi, Gabriela 2; Rocca, Elisabetta 3, 4

1 Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy
2 Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie No.13, Sector 5, 050711, Bucharest, Romania
3 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
4 Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy
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Colli, Pierluigi; Marinoschi, Gabriela; Rocca, Elisabetta. Sharp interface control in a Penrose−Fife model. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 473-499. doi : 10.1051/cocv/2015014. http://archive.numdam.org/articles/10.1051/cocv/2015014/

V. Barbu, Optimal Control of Variational Inequalities. In vol. 100 of Res. Notes Math. Pitman, Advanced Publishing Program, Boston, MA (1984). | MR | Zbl

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monogr. Math. Springer, New York (2010). | MR | Zbl

P. Colli and Ph. Laurençot, Weak solutions to the Penrose−Fife phase field model for a class of admissible heat flux laws. Phys. D 111 (1998) 311–334. | DOI | MR | Zbl

P. Colli and J. Sprekels, On a Penrose−Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type. Ann. Mat. Pura Appl. 169 (1995) 269–289. | DOI | MR | Zbl

P. Colli and J. Sprekels, Stefan problems and the Penrose−Fife phase field model. Adv. Math. Sci. Appl. 7 (1997) 911–934. | MR | Zbl

P. Colli, Ph. Laurençot and J. Sprekels, Global solution to the Penrose−Fife phase field model with special heat flux laws, Variations of domain and free-boundary problems in solid mechanics (Paris, 1997) 181–188; Vol. 66. of Solid Mech. Appl. Kluwer Acad. Publ., Dordrecht (1999). | MR

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst. 25 (2009) 63–81. | DOI | MR | Zbl

M. Fabrizio, A. Favini and G. Marinoschi, An optimal control problem for a singular system of solid-liquid transition. Numer. Funct. Anal. Optim. 31 (2010) 989–1022. | DOI | MR | Zbl

E. Feireisl and G. Schimperna, Large time behaviour of solutions to Penrose−Fife phase change models. Math. Methods Appl. Sci. 28 (2005) 2117–2132. | DOI | MR | Zbl

W. Horn, J. Sokołowski and J. Sprekels, A control problem with state constraints for a phase-field model. Control Cybernet 25 (1996) 1137–1153. | Zbl

W. Horn, J. Sprekels and S. Zheng, Global existence of smooth solutions to the Penrose−Fife model for Ising ferromagnets. Adv. Math. Sci. Appl. 6 (1996) 227–241. | Zbl

A. Ito and N. Kenmochi, Inertial set for a phase transition model of Penrose−Fife type. Adv. Math. Sci. Appl. 10 (2000) 353–374; Correction in Adv. Math. Sci. Appl. 11 (2001) 481. | Zbl

Ph. Laurençot, Étude de quelques problèmes aux dérivées partielles non linéaires. Thèse de l’Université de France-Comté, Besançon (1993).

Ph. Laurençot, Solutions to a Penrose−Fife model of phase-field type. J. Math. Anal. Appl. 185 (1994) 262–274. | DOI | Zbl

Ph. Laurençot, Weak solutions to a Penrose−Fife model for phase transitions. Adv. Math. Sci. Appl. 5 (1995) 117–138. | Zbl

J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969). | Zbl

A. Miranville, E. Rocca, G. Schimperna and A. Segatti, The Penrose−Fife phase-field model with coupled dynamic boundary conditions. Discrete Contin. Dyn. Syst. 34 (2014) 4259–4290. | DOI | Zbl

J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris – Academia, Praha (1967). | Zbl

L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13 (1959) 115–162. | Numdam | Zbl

O. Penrose and P.C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Phys. D 43 (1990) 44–62. | DOI | Zbl

O. Penrose and P.C. Fife, On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field mode. Phys. D 69 (1993) 107–113. | DOI | Zbl

E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems. Phys. D 192 (2004) 279–307. | DOI | Zbl

G. Schimperna, Global and exponential attractors for the Penrose−Fife system. Math. Models Methods Appl. Sci. 19 (2009) 969–991. | DOI | Zbl

G. Schimperna, A. Segatti and S. Zelik, Asymptotic uniform boundedness of energy solutions to the Penrose−Fife model. J. Evol. Equ. 12 (2012) 863–890. | DOI | Zbl

J. Sprekels and S.M. Zheng, Optimal control problems for a thermodynamically consistent model of phase-field type for phase transitions. Adv. Math. Sci. Appl. 1 (1992) 113–125. | Zbl

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