March  2017, 6(1): 35-58. doi: 10.3934/eect.2017003

Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

1. 

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

2. 

Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

3. 

Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

* Corresponding author: Pierluigi Colli

Received  July 2016 Revised  September 2016 Published  December 2016

This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors nonlocal model here studied consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called 'deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.

Citation: Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential. Evolution Equations & Control Theory, 2017, 6 (1) : 35-58. doi: 10.3934/eect.2017003
References:
[1]

V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566-597. doi: 10.1016/0022-247X(81)90125-6. 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]

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

[4]

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

[5]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Ann. Acad. Rom. Sci. Math. Appl., 7 (2015), 41-66. Google Scholar

[6]

P. ColliM. 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. Google Scholar

[7]

P. ColliG. GilardiP. KrejčíP. Podio-Guidugli and J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system, ESAIM Math. Model. Numer. Anal., 48 (2014), 1061-1087. doi: 10.1051/m2an/2014005. Google Scholar

[8]

P. ColliG. GilardiP. Krejčí and J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evol. Equ. Control Theory, 3 (2014), 257-275. doi: 10.3934/eect.2014.3.257. Google Scholar

[9]

P. ColliG. GilardiP. Krejčí and J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, Math. Methods Appl. Sci., 37 (2014), 1318-1324. Google Scholar

[10]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870. doi: 10.1137/110828526. Google Scholar

[11]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity, Boll. Unione Mat. Ital., 5 (2012), 495-513. Google Scholar

[12]

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

[13]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Continuous dependence for a nonstandard Cahn-Hilliard system with nonlinear atom mobility, Rend. Sem. Mat. Univ. Politec. Torino, 70 (2012), 27-52. Google Scholar

[14]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368. doi: 10.3934/dcdss.2013.6.353. Google Scholar

[15]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Differential Equations, 254 (2013), 4217-4244. doi: 10.1016/j.jde.2013.02.014. Google Scholar

[16]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, preprint, arXiv:1601.04567 [math.AP] (2016), 1-32.Google Scholar

[17]

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

[18]

P. ColliG. Gilardi and J. Sprekels, Regularity of the solution to a nonstandard system of phase field equations, Rend. Cl. Sci. Mat. Nat., 147 (2013), 3-19. Google Scholar

[19]

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

[20]

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

[21]

P. ColliG. Gilardi and J. Sprekels, On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation, J. Differential Equations, 260 (2016), 7940-7964. doi: 10.1016/j.jde.2016.02.011. Google Scholar

[22]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Math., 1 (2016), 225-260. doi: 10.3934/Math.2016.3.225. Google Scholar

[23]

S. Frigeri, M. Grasselli and J. Sprekels, Strong solutions and optimal distributed control of nonlocal Cahn-Hilliard/Navier-Stokes systems in 2D with singular potential and degenerate mobility, in preparation.Google Scholar

[24]

S. FrigeriE. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in two dimensions, SIAM J. Control Optim., 54 (2016), 221-250. doi: 10.1137/140994800. Google Scholar

[25]

M. Hinterm, T. Keil and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities, preprint, arXiv: 1506.03591 [math.AP] (2015), 1-35.Google Scholar

[26]

M. Hintermüller and D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density, SIAM J. Control Optim., 50 (2012), 388-418. doi: 10.1137/110824152. Google Scholar

[27]

M. Hintermüller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772. doi: 10.1137/120865628. Google Scholar

[28]

M. Hintermüller and D. Wegner, Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies, Isaac Newton Institute Preprint Series, (2014), 1-29. Google Scholar

[29]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118. doi: 10.1007/s11587-006-0008-8. Google Scholar

[30]

E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680. doi: 10.1137/140964308. Google Scholar

[31]

J. Simon, Compact sets in the space Lp(0, T;B), Ann. Mat. Pura. Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[32]

Q.-F. Wang and S.-i. Nakagiri, Weak solutions of Cahn-Hilliard equations having forcing terms and optimal control problems, Mathematical models in functional equations (Japanese) (Kyoto, 1999), Sūrikaisekikenkyūsho Kõkyūroku, 1128 (2000), 172-180. Google Scholar

[33]

X. Zhao and C. Liu, Optimal control of the convective Cahn-Hilliard equation, Appl. Anal., 92 (2013), 1028-1045. doi: 10.1080/00036811.2011.643786. Google Scholar

[34]

X. Zhao and C. Liu, Optimal control for the convective Cahn-Hilliard equation in 2D case, Appl. Math. Optim., 70 (2014), 61-82. doi: 10.1007/s00245-013-9234-0. Google Scholar

show all references

References:
[1]

V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566-597. doi: 10.1016/0022-247X(81)90125-6. 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]

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

[4]

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

[5]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Ann. Acad. Rom. Sci. Math. Appl., 7 (2015), 41-66. Google Scholar

[6]

P. ColliM. 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. Google Scholar

[7]

P. ColliG. GilardiP. KrejčíP. Podio-Guidugli and J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous Cahn-Hilliard system, ESAIM Math. Model. Numer. Anal., 48 (2014), 1061-1087. doi: 10.1051/m2an/2014005. Google Scholar

[8]

P. ColliG. GilardiP. Krejčí and J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evol. Equ. Control Theory, 3 (2014), 257-275. doi: 10.3934/eect.2014.3.257. Google Scholar

[9]

P. ColliG. GilardiP. Krejčí and J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, Math. Methods Appl. Sci., 37 (2014), 1318-1324. Google Scholar

[10]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870. doi: 10.1137/110828526. Google Scholar

[11]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity, Boll. Unione Mat. Ital., 5 (2012), 495-513. Google Scholar

[12]

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

[13]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Continuous dependence for a nonstandard Cahn-Hilliard system with nonlinear atom mobility, Rend. Sem. Mat. Univ. Politec. Torino, 70 (2012), 27-52. Google Scholar

[14]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368. doi: 10.3934/dcdss.2013.6.353. Google Scholar

[15]

P. ColliG. GilardiP. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Differential Equations, 254 (2013), 4217-4244. doi: 10.1016/j.jde.2013.02.014. Google Scholar

[16]

P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, preprint, arXiv:1601.04567 [math.AP] (2016), 1-32.Google Scholar

[17]

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

[18]

P. ColliG. Gilardi and J. Sprekels, Regularity of the solution to a nonstandard system of phase field equations, Rend. Cl. Sci. Mat. Nat., 147 (2013), 3-19. Google Scholar

[19]

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

[20]

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

[21]

P. ColliG. Gilardi and J. Sprekels, On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation, J. Differential Equations, 260 (2016), 7940-7964. doi: 10.1016/j.jde.2016.02.011. Google Scholar

[22]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Math., 1 (2016), 225-260. doi: 10.3934/Math.2016.3.225. Google Scholar

[23]

S. Frigeri, M. Grasselli and J. Sprekels, Strong solutions and optimal distributed control of nonlocal Cahn-Hilliard/Navier-Stokes systems in 2D with singular potential and degenerate mobility, in preparation.Google Scholar

[24]

S. FrigeriE. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in two dimensions, SIAM J. Control Optim., 54 (2016), 221-250. doi: 10.1137/140994800. Google Scholar

[25]

M. Hinterm, T. Keil and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities, preprint, arXiv: 1506.03591 [math.AP] (2015), 1-35.Google Scholar

[26]

M. Hintermüller and D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density, SIAM J. Control Optim., 50 (2012), 388-418. doi: 10.1137/110824152. Google Scholar

[27]

M. Hintermüller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772. doi: 10.1137/120865628. Google Scholar

[28]

M. Hintermüller and D. Wegner, Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies, Isaac Newton Institute Preprint Series, (2014), 1-29. Google Scholar

[29]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118. doi: 10.1007/s11587-006-0008-8. Google Scholar

[30]

E. Rocca and J. Sprekels, Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., 53 (2015), 1654-1680. doi: 10.1137/140964308. Google Scholar

[31]

J. Simon, Compact sets in the space Lp(0, T;B), Ann. Mat. Pura. Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[32]

Q.-F. Wang and S.-i. Nakagiri, Weak solutions of Cahn-Hilliard equations having forcing terms and optimal control problems, Mathematical models in functional equations (Japanese) (Kyoto, 1999), Sūrikaisekikenkyūsho Kõkyūroku, 1128 (2000), 172-180. Google Scholar

[33]

X. Zhao and C. Liu, Optimal control of the convective Cahn-Hilliard equation, Appl. Anal., 92 (2013), 1028-1045. doi: 10.1080/00036811.2011.643786. Google Scholar

[34]

X. Zhao and C. Liu, Optimal control for the convective Cahn-Hilliard equation in 2D case, Appl. Math. Optim., 70 (2014), 61-82. doi: 10.1007/s00245-013-9234-0. Google Scholar

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