Optimal control for a phase field system with a possibly singular potential

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March 2016, 6(1): 95-112. doi: 10.3934/mcrf.2016.6.95

Optimal control for a phase field system with a possibly singular potential

Pierluigi Colli ^1, , Gianni Gilardi ^1, , Gabriela Marinoschi ^2, and Elisabetta Rocca ^3,

1.	Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia
2.	"Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy (ISMMA), Calea 13 Septembrie 13, 050711 Bucharest, Romania
3.	Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin

Received October 2014 Revised July 2015 Published January 2016

Abstract
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In this paper we study a distributed control problem for a phase field system of Caginalp type with logarithmic potential. The main aim of this work would be to force the location of the diffuse interface to be as close as possible to a prescribed set. However, due to the discontinuous character of the cost functional, we have to approximate it by a regular one and, in this case, we solve the associated control problem and derive the related first order necessary optimality conditions.

Keywords: phase transition, adjoint state system., optimality conditions, optimal control, singular potentials, Phase field system.

Mathematics Subject Classification: Primary: 49J20; Secondary: 35K55, 49K20, 80A2.

Citation: Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control and Related Fields, 2016, 6 (1) : 95-112. doi: 10.3934/mcrf.2016.6.95

References:

[1]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.
[2]	V. Barbu, M. L. Bernardi, P. Colli and G. Gilardi, Optimal control problems of phase relaxation models, J. Optim. Theory Appl., 109 (2001), 557-585. doi: 10.1023/A:1017563604922.
[3]	K. N. Blazakis, A. Madzvamuse, C. C. Reyes-Aldasoro, V. Styles and C. Venkataraman, Whole cell tracking through the optimal control of geometric evolution laws, J. Comput. Phys., 297 (2015), 495-514. doi: 10.1016/j.jcp.2015.05.014.
[4]	J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. I. Mathematical Analysis, European J. Appl. Math., 2 (1991), 233-280. doi: 10.1017/S095679250000053X.
[5]	J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical Analysis, European J. Appl. Math., 3 (1992), 147-179. doi: 10.1017/S0956792500000759.
[6]	J. L. Boldrini, B. M. C. Caretta and E. Fernández-Cara, Some optimal control problems for a two-phase field model of solidification, Rev. Mat. Complut., 23 (2010), 49-75. doi: 10.1007/s13163-009-0012-0.
[7]	H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.
[8]	M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.
[9]	G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.
[10]	L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27. doi: 10.1016/j.na.2012.11.010.
[11]	L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
[12]	P. Colli, M. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1-24. doi: 10.1007/s00245-014-9250-8.
[13]	P. Colli, G. Gilardi and G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl., 434 (2016), 432-463, (see also preprint arXiv:1501.04517 [math.AP] (2015), pp. 1-32). doi: 10.1016/j.jmaa.2015.09.011.
[14]	P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with general potentials,, in preparation., ().
[15]	P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Contin. Mech. Thermodyn, 24 (2012), 437-459. doi: 10.1007/s00161-011-0215-8.
[16]	P. Colli, G. Gilardi and J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan J. Math., 80 (2012), 119-149. doi: 10.1007/s00032-012-0181-z.
[17]	P. Colli, G. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., 4 (2015), 311-325, (see also arXiv:1503.03213 [math.AP] (2015), pp. 1-18). doi: 10.1515/anona-2015-0035.
[18]	P. Colli, G. Marinoschi and E. Rocca, Sharp interface control in a Penrose-Fife model, ESAIM Control Optim. Calc. Var., (see also preprint arXiv:1403.4446 [math.AP] (2014), pp. 1-33). doi: 10.1051/cocv/2015014.
[19]	P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234. doi: 10.1137/120902422.
[20]	M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65. doi: 10.1007/BF01385847.
[21]	A. Damlamian, N. Kenmochi and N. Sato, Subdifferential operator approach to a class of nonlinear systems for Stefan problems with phase relaxation, Nonlinear Anal., 23 (1994), 115-142. doi: 10.1016/0362-546X(94)90255-0.
[22]	C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations, in Free boundary problems, Internat. Ser. Numer. Math., Birkhäuser Verlag, Basel, 95 (1990), 46-58.
[23]	M. H. Farshbaf-Shaker, A penalty approach to optimal control of Allen-Cahn variational inequalities: MPEC-view, Numer. Funct. Anal. Optim., 33 (2012), 1321-1349. doi: 10.1080/01630563.2012.672354.
[24]	M. H. Farshbaf-Shaker and C. Hecht, Optimal control of elastic vector-valued Allen-Cahn variational inequalities, WIAS Preprint, 1858 (2013), 1-20.
[25]	G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.
[26]	G. Gilardi, A. Miranville and G. Schimperna, Long-time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math. Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7.
[27]	M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.
[28]	M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.
[29]	K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458.
[30]	K.-H. Hoffmann, N. Kenmochi, M. Kubo and N. Yamazaki, Optimal control problems for models of phase-field type with hysteresis of play operator, Adv. Math. Sci. Appl., 17 (2007), 305-336.
[31]	N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising phase change problems, Nonlinear Anal., 22 (1994), 1163-1180. doi: 10.1016/0362-546X(94)90235-6.
[32]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Trans. Amer. Math. Soc., 23, Amer. Math. Soc., Providence, RI, 1968.
[33]	Ph. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 167-185. doi: 10.1017/S0308210500030663.
[34]	C. Lefter and J. Sprekels, Optimal boundary control of a phase field system modeling nonisothermal phase transitions, Adv. Math. Sci. Appl., 17 (2007), 181-194.
[35]	J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Grundlehren, Band 111, Springer-Verlag, Berlin, 1961.
[36]	A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential, Appl. Math. Lett., 24 (2011), 1003-1008. doi: 10.1016/j.aml.2011.01.016.
[37]	A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.
[38]	G. Schimperna, Abstract approach to evolution equations of phase field type and applications, J. Differential Equations, 164 (2000), 395-430. doi: 10.1006/jdeq.1999.3753.
[39]	K. Shirakawa and N. Yamazaki, Optimal control problems of phase field system with total variation functional as the interfacial energy, Adv. Differential Equations, 18 (2013), 309-350.
[40]	J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.
[41]	J. Sprekels and S. Zheng, Optimal control problems for a thermodynamically consistent model of phase-field type for phase transitions, Adv. Math. Sci. Appl., 1 (1992), 113-125.

show all references

References:

[1]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.
[2]	V. Barbu, M. L. Bernardi, P. Colli and G. Gilardi, Optimal control problems of phase relaxation models, J. Optim. Theory Appl., 109 (2001), 557-585. doi: 10.1023/A:1017563604922.
[3]	K. N. Blazakis, A. Madzvamuse, C. C. Reyes-Aldasoro, V. Styles and C. Venkataraman, Whole cell tracking through the optimal control of geometric evolution laws, J. Comput. Phys., 297 (2015), 495-514. doi: 10.1016/j.jcp.2015.05.014.
[4]	J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. I. Mathematical Analysis, European J. Appl. Math., 2 (1991), 233-280. doi: 10.1017/S095679250000053X.
[5]	J. F. Blowey and C. M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical Analysis, European J. Appl. Math., 3 (1992), 147-179. doi: 10.1017/S0956792500000759.
[6]	J. L. Boldrini, B. M. C. Caretta and E. Fernández-Cara, Some optimal control problems for a two-phase field model of solidification, Rev. Mat. Complut., 23 (2010), 49-75. doi: 10.1007/s13163-009-0012-0.
[7]	H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.
[8]	M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8.
[9]	G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.
[10]	L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27. doi: 10.1016/j.na.2012.11.010.
[11]	L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
[12]	P. Colli, M. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1-24. doi: 10.1007/s00245-014-9250-8.
[13]	P. Colli, G. Gilardi and G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl., 434 (2016), 432-463, (see also preprint arXiv:1501.04517 [math.AP] (2015), pp. 1-32). doi: 10.1016/j.jmaa.2015.09.011.
[14]	P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with general potentials,, in preparation., ().
[15]	P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Contin. Mech. Thermodyn, 24 (2012), 437-459. doi: 10.1007/s00161-011-0215-8.
[16]	P. Colli, G. Gilardi and J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan J. Math., 80 (2012), 119-149. doi: 10.1007/s00032-012-0181-z.
[17]	P. Colli, G. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., 4 (2015), 311-325, (see also arXiv:1503.03213 [math.AP] (2015), pp. 1-18). doi: 10.1515/anona-2015-0035.
[18]	P. Colli, G. Marinoschi and E. Rocca, Sharp interface control in a Penrose-Fife model, ESAIM Control Optim. Calc. Var., (see also preprint arXiv:1403.4446 [math.AP] (2014), pp. 1-33). doi: 10.1051/cocv/2015014.
[19]	P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234. doi: 10.1137/120902422.
[20]	M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65. doi: 10.1007/BF01385847.
[21]	A. Damlamian, N. Kenmochi and N. Sato, Subdifferential operator approach to a class of nonlinear systems for Stefan problems with phase relaxation, Nonlinear Anal., 23 (1994), 115-142. doi: 10.1016/0362-546X(94)90255-0.
[22]	C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations, in Free boundary problems, Internat. Ser. Numer. Math., Birkhäuser Verlag, Basel, 95 (1990), 46-58.
[23]	M. H. Farshbaf-Shaker, A penalty approach to optimal control of Allen-Cahn variational inequalities: MPEC-view, Numer. Funct. Anal. Optim., 33 (2012), 1321-1349. doi: 10.1080/01630563.2012.672354.
[24]	M. H. Farshbaf-Shaker and C. Hecht, Optimal control of elastic vector-valued Allen-Cahn variational inequalities, WIAS Preprint, 1858 (2013), 1-20.
[25]	G. Gilardi, A. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.
[26]	G. Gilardi, A. Miranville and G. Schimperna, Long-time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math. Ser. B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7.
[27]	M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.
[28]	M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277.
[29]	K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458.
[30]	K.-H. Hoffmann, N. Kenmochi, M. Kubo and N. Yamazaki, Optimal control problems for models of phase-field type with hysteresis of play operator, Adv. Math. Sci. Appl., 17 (2007), 305-336.
[31]	N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising phase change problems, Nonlinear Anal., 22 (1994), 1163-1180. doi: 10.1016/0362-546X(94)90235-6.
[32]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Trans. Amer. Math. Soc., 23, Amer. Math. Soc., Providence, RI, 1968.
[33]	Ph. Laurençot, Long-time behaviour for a model of phase-field type, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 167-185. doi: 10.1017/S0308210500030663.
[34]	C. Lefter and J. Sprekels, Optimal boundary control of a phase field system modeling nonisothermal phase transitions, Adv. Math. Sci. Appl., 17 (2007), 181-194.
[35]	J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites, Grundlehren, Band 111, Springer-Verlag, Berlin, 1961.
[36]	A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential, Appl. Math. Lett., 24 (2011), 1003-1008. doi: 10.1016/j.aml.2011.01.016.
[37]	A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.
[38]	G. Schimperna, Abstract approach to evolution equations of phase field type and applications, J. Differential Equations, 164 (2000), 395-430. doi: 10.1006/jdeq.1999.3753.
[39]	K. Shirakawa and N. Yamazaki, Optimal control problems of phase field system with total variation functional as the interfacial energy, Adv. Differential Equations, 18 (2013), 309-350.
[40]	J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.
[41]	J. Sprekels and S. Zheng, Optimal control problems for a thermodynamically consistent model of phase-field type for phase transitions, Adv. Math. Sci. Appl., 1 (1992), 113-125.

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