American Institute of Mathematical Sciences

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

 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

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.
Citation: Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control & 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, (1976). Google Scholar [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. doi: 10.1023/A:1017563604922. Google Scholar [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. doi: 10.1016/j.jcp.2015.05.014. Google Scholar [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. doi: 10.1017/S095679250000053X. Google Scholar [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. doi: 10.1017/S0956792500000759. Google Scholar [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. doi: 10.1007/s13163-009-0012-0. Google Scholar [7] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland Math. Stud. 5, 5 (1973). Google Scholar [8] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8. Google Scholar [9] G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar [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. doi: 10.1016/j.na.2012.11.010. Google Scholar [11] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. Google Scholar [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. doi: 10.1007/s00245-014-9250-8. Google Scholar [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. doi: 10.1016/j.jmaa.2015.09.011. Google Scholar [14] P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with general potentials,, in preparation., (). Google Scholar [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. doi: 10.1007/s00161-011-0215-8. Google Scholar [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. doi: 10.1007/s00032-012-0181-z. Google Scholar [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. doi: 10.1515/anona-2015-0035. Google Scholar [18] P. Colli, G. Marinoschi and E. Rocca, Sharp interface control in a Penrose-Fife model,, ESAIM Control Optim. Calc. Var., (2014), 1. doi: 10.1051/cocv/2015014. Google Scholar [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. doi: 10.1137/120902422. Google Scholar [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. doi: 10.1007/BF01385847. Google Scholar [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. doi: 10.1016/0362-546X(94)90255-0. Google Scholar [22] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations,, in Free boundary problems, 95 (1990), 46. Google Scholar [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. doi: 10.1080/01630563.2012.672354. Google Scholar [24] M. H. Farshbaf-Shaker and C. Hecht, Optimal control of elastic vector-valued Allen-Cahn variational inequalities,, WIAS Preprint, 1858 (2013), 1. Google Scholar [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. doi: 10.3934/cpaa.2009.8.881. Google Scholar [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. doi: 10.1007/s11401-010-0602-7. Google Scholar [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. doi: 10.3934/dcds.2010.28.67. Google Scholar [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. doi: 10.4171/ZAA/1277. Google Scholar [29] K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification,, Numer. Funct. Anal. Optim., 13 (1992), 11. doi: 10.1080/01630569208816458. Google Scholar [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. Google Scholar [31] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising phase change problems,, Nonlinear Anal., 22 (1994), 1163. doi: 10.1016/0362-546X(94)90235-6. Google Scholar [32] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Trans. Amer. Math. Soc., 23 (1968). Google Scholar [33] Ph. Laurençot, Long-time behaviour for a model of phase-field type,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 167. doi: 10.1017/S0308210500030663. Google Scholar [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. Google Scholar [35] J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites,, Grundlehren, (1961). Google Scholar [36] A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential,, Appl. Math. Lett., 24 (2011), 1003. doi: 10.1016/j.aml.2011.01.016. Google Scholar [37] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar [38] G. Schimperna, Abstract approach to evolution equations of phase field type and applications,, J. Differential Equations, 164 (2000), 395. doi: 10.1006/jdeq.1999.3753. Google Scholar [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. Google Scholar [40] J. Simon, Compact sets in the space $L^p(0,T; B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [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. Google Scholar

show all references

References:
 [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Noordhoff, (1976). Google Scholar [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. doi: 10.1023/A:1017563604922. Google Scholar [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. doi: 10.1016/j.jcp.2015.05.014. Google Scholar [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. doi: 10.1017/S095679250000053X. Google Scholar [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. doi: 10.1017/S0956792500000759. Google Scholar [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. doi: 10.1007/s13163-009-0012-0. Google Scholar [7] H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert,, North-Holland Math. Stud. 5, 5 (1973). Google Scholar [8] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Springer, (1996). doi: 10.1007/978-1-4612-4048-8. Google Scholar [9] G. Caginalp, An analysis of a phase field model of a free boundary,, Arch. Rational Mech. Anal., 92 (1986), 205. doi: 10.1007/BF00254827. Google Scholar [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. doi: 10.1016/j.na.2012.11.010. Google Scholar [11] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. Google Scholar [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. doi: 10.1007/s00245-014-9250-8. Google Scholar [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. doi: 10.1016/j.jmaa.2015.09.011. Google Scholar [14] P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with general potentials,, in preparation., (). Google Scholar [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. doi: 10.1007/s00161-011-0215-8. Google Scholar [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. doi: 10.1007/s00032-012-0181-z. Google Scholar [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. doi: 10.1515/anona-2015-0035. Google Scholar [18] P. Colli, G. Marinoschi and E. Rocca, Sharp interface control in a Penrose-Fife model,, ESAIM Control Optim. Calc. Var., (2014), 1. doi: 10.1051/cocv/2015014. Google Scholar [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. doi: 10.1137/120902422. Google Scholar [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. doi: 10.1007/BF01385847. Google Scholar [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. doi: 10.1016/0362-546X(94)90255-0. Google Scholar [22] C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations,, in Free boundary problems, 95 (1990), 46. Google Scholar [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. doi: 10.1080/01630563.2012.672354. Google Scholar [24] M. H. Farshbaf-Shaker and C. Hecht, Optimal control of elastic vector-valued Allen-Cahn variational inequalities,, WIAS Preprint, 1858 (2013), 1. Google Scholar [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. doi: 10.3934/cpaa.2009.8.881. Google Scholar [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. doi: 10.1007/s11401-010-0602-7. Google Scholar [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. doi: 10.3934/dcds.2010.28.67. Google Scholar [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. doi: 10.4171/ZAA/1277. Google Scholar [29] K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification,, Numer. Funct. Anal. Optim., 13 (1992), 11. doi: 10.1080/01630569208816458. Google Scholar [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. Google Scholar [31] N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising phase change problems,, Nonlinear Anal., 22 (1994), 1163. doi: 10.1016/0362-546X(94)90235-6. Google Scholar [32] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Trans. Amer. Math. Soc., 23 (1968). Google Scholar [33] Ph. Laurençot, Long-time behaviour for a model of phase-field type,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 167. doi: 10.1017/S0308210500030663. Google Scholar [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. Google Scholar [35] J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes Aux Limites,, Grundlehren, (1961). Google Scholar [36] A. Miranville and R. Quintanilla, A type III phase-field system with a logarithmic potential,, Appl. Math. Lett., 24 (2011), 1003. doi: 10.1016/j.aml.2011.01.016. Google Scholar [37] A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials,, Math. Methods Appl. Sci., 27 (2004), 545. doi: 10.1002/mma.464. Google Scholar [38] G. Schimperna, Abstract approach to evolution equations of phase field type and applications,, J. Differential Equations, 164 (2000), 395. doi: 10.1006/jdeq.1999.3753. Google Scholar [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. Google Scholar [40] J. Simon, Compact sets in the space $L^p(0,T; B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [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. Google Scholar
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