March  2018, 7(1): 95-116. doi: 10.3934/eect.2018006

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

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia Via Ferrata 1, 27100 Pavia, Italy

2. 

Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, 050711 Bucharest, Romania and Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Group of the Project PN-Ⅲ-P4-ID-PCE-2016-0372, Romania

Corresponding author: Elisabetta Rocca

Received  August 2017 Revised  October 2017 Published  January 2018

In this paper we study a distributed control problem for a phase-field system of conserved type with a possibly singular potential. We mainly handle two cases: the case of a viscous Cahn-Hilliard type dynamics for the phase variable in case of a logarithmic-type potential with bounded domain and the case of a standard Cahn-Hilliard equation in case of a regular potential with unbounded domain, like the classical double-well potential, for example. Necessary first order conditions of optimality are derived under natural assumptions on the data.

Citation: Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a conserved phase field system with a possibly singular potential. Evolution Equations & Control Theory, 2018, 7 (1) : 95-116. doi: 10.3934/eect.2018006
References:
[1]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.  Google Scholar

[2]

V. BarbuM. L. BernardiP. Colli and G. Gilardi, Optimal control problems of phase relaxation models, J. Optim. Theory Appl., 109 (2001), 557-585.  doi: 10.1023/A:1017563604922.  Google Scholar

[3]

J. L. BoldriniB. 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.  Google Scholar

[4]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud., 5 North-Holland, Amsterdam, 1973.  Google Scholar

[5]

D. BrochetD. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Differential Equations, 1 (1996), 547-578.   Google Scholar

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.  Google Scholar

[7]

G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94.  doi: 10.1093/imamat/44.1.77.  Google Scholar

[8]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[9]

C. CavaterraM. Grasselli and H. Wu, Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions, Comm. Pure Appl. Anal., 13 (2014), 1855-1890.  doi: 10.3934/cpaa.2014.13.1855.  Google Scholar

[10]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696-2721.  doi: 10.1137/140984749.  Google Scholar

[11]

P. ColliG. GilardiP. Laurençot and A. Novick-Cohen, Uniqueness and long-time behaviour for the conserved phase-field system with memory, Discrete Contin. Dynam. Systems, 5 (1999), 375-390.  doi: 10.3934/dcds.1999.5.375.  Google Scholar

[12]

P. ColliG. 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.  doi: 10.1016/j.jmaa.2015.09.011.  Google Scholar

[13]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95.  Google Scholar

[14]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Distributed optimal control problems for phase field systems with singular potential, An. Ȿtiinţ. Univ. "Ovidius" Constanţa Ser. Mat., to appear (2017). Google Scholar

[15]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994.  doi: 10.1016/j.jmaa.2014.05.008.  Google Scholar

[16]

P. ColliG. 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.  doi: 10.1515/anona-2015-0035.  Google Scholar

[17]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Optim., 73 (2016), 195-225.  doi: 10.1007/s00245-015-9299-z.  Google Scholar

[18]

P. ColliG. GilardiP. 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.  Google Scholar

[19]

P. ColliG. 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.  Google Scholar

[20]

P. ColliG. Marinoschi and E. Rocca, Sharp interface control in a Penrose-Fife model, ESAIM Control Optim. Calc. Var., 22 (2016), 473-499.  doi: 10.1051/cocv/2015014.  Google Scholar

[21]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Rend. Cl. Sci. Mat. Nat., 141 (2007), 129-161.   Google Scholar

[22]

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.  Google Scholar

[23]

K.-H. HoffmannN. KenmochiM. 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.   Google Scholar

[24]

N. Kenmochi and M. Niezgódka, Nonlinear system for non-isothermal diffusive phase separation, J. Math. Anal. Appl., 188 (1994), 651-679.  doi: 10.1006/jmaa.1994.1451.  Google Scholar

[25]

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.  Google Scholar

[26]

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.   Google Scholar

[27]

J. -L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[28]

A. Miranville, On the conserved phase-field model, J. Math. Anal. Appl., 400 (2013), 143-152.  doi: 10.1016/j.jmaa.2012.11.038.  Google Scholar

[29]

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.  Google Scholar

[30]

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.   Google Scholar

[31]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96.   Google Scholar

[32]

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.   Google Scholar

show all references

References:
[1]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.  Google Scholar

[2]

V. BarbuM. L. BernardiP. Colli and G. Gilardi, Optimal control problems of phase relaxation models, J. Optim. Theory Appl., 109 (2001), 557-585.  doi: 10.1023/A:1017563604922.  Google Scholar

[3]

J. L. BoldriniB. 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.  Google Scholar

[4]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud., 5 North-Holland, Amsterdam, 1973.  Google Scholar

[5]

D. BrochetD. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Differential Equations, 1 (1996), 547-578.   Google Scholar

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.  Google Scholar

[7]

G. Caginalp, The dynamics of a conserved phase field system: Stefan-like, Hele-Shaw, and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94.  doi: 10.1093/imamat/44.1.77.  Google Scholar

[8]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267.  doi: 10.1002/9781118788295.ch4.  Google Scholar

[9]

C. CavaterraM. Grasselli and H. Wu, Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions, Comm. Pure Appl. Anal., 13 (2014), 1855-1890.  doi: 10.3934/cpaa.2014.13.1855.  Google Scholar

[10]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696-2721.  doi: 10.1137/140984749.  Google Scholar

[11]

P. ColliG. GilardiP. Laurençot and A. Novick-Cohen, Uniqueness and long-time behaviour for the conserved phase-field system with memory, Discrete Contin. Dynam. Systems, 5 (1999), 375-390.  doi: 10.3934/dcds.1999.5.375.  Google Scholar

[12]

P. ColliG. 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.  doi: 10.1016/j.jmaa.2015.09.011.  Google Scholar

[13]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95.  Google Scholar

[14]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Distributed optimal control problems for phase field systems with singular potential, An. Ȿtiinţ. Univ. "Ovidius" Constanţa Ser. Mat., to appear (2017). Google Scholar

[15]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994.  doi: 10.1016/j.jmaa.2014.05.008.  Google Scholar

[16]

P. ColliG. 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.  doi: 10.1515/anona-2015-0035.  Google Scholar

[17]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Optim., 73 (2016), 195-225.  doi: 10.1007/s00245-015-9299-z.  Google Scholar

[18]

P. ColliG. GilardiP. 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.  Google Scholar

[19]

P. ColliG. 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.  Google Scholar

[20]

P. ColliG. Marinoschi and E. Rocca, Sharp interface control in a Penrose-Fife model, ESAIM Control Optim. Calc. Var., 22 (2016), 473-499.  doi: 10.1051/cocv/2015014.  Google Scholar

[21]

G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Rend. Cl. Sci. Mat. Nat., 141 (2007), 129-161.   Google Scholar

[22]

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.  Google Scholar

[23]

K.-H. HoffmannN. KenmochiM. 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.   Google Scholar

[24]

N. Kenmochi and M. Niezgódka, Nonlinear system for non-isothermal diffusive phase separation, J. Math. Anal. Appl., 188 (1994), 651-679.  doi: 10.1006/jmaa.1994.1451.  Google Scholar

[25]

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.  Google Scholar

[26]

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.   Google Scholar

[27]

J. -L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[28]

A. Miranville, On the conserved phase-field model, J. Math. Anal. Appl., 400 (2013), 143-152.  doi: 10.1016/j.jmaa.2012.11.038.  Google Scholar

[29]

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.  Google Scholar

[30]

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.   Google Scholar

[31]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96.   Google Scholar

[32]

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.   Google Scholar

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