January  2021, 14(1): 243-271. doi: 10.3934/dcdss.2020213

Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, and Research Associates at the IMATI–C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy

2. 

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

3. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany

* Corresponding author: Jürgen Sprekels

Received  December 2018 Revised  August 2019 Published  January 2021 Early access  December 2019

Recently, the authors derived well-posedness and regularity results for general evolutionary operator equations having the structure of a Cahn–Hilliard system. The involved operators were fractional versions in the spectral sense of general linear operators that have compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions. The class of admissible double-well potentials driving the phase separation process modeled by the Cahn–Hilliard system included polynomial, logarithmic, and double obstacle nonlinearities. In a subsequent paper, distributed optimal control problems for such systems were investigated, where only differentiable polynomial and logarithmic potentials were admitted. Existence of optimizers and first-order optimality conditions were derived. In this paper, these results are complemented for nondifferentiable double obstacle nonlinearities. It is well known that for such nonlinearities standard constraint qualifications to construct Lagrange multipliers cannot be applied. To overcome this difficulty, we follow the so-called "deep quench" method, which has proved to be a powerful tool in optimal control problems with double obstacle potentials. We give a general convergence analysis of the deep quench approximation, including an error estimate, and demonstrate that its use leads to meaningful first-order necessary optimality conditions.

Citation: Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
References:
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M. Ainsworth and Z. Mao, Analysis and approximation of a fractional Cahn–Hilliard equation, SIAM J. Numer. Anal., 55 (2017), 1689-1718.  doi: 10.1137/16M1075302.

[2]

G. AkagiG. Schimperna and A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differential Equations, 261 (2016), 2935-2985.  doi: 10.1016/j.jde.2016.05.016.

[3]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 35 pp. doi: 10.1088/1361-6420/ab1299.

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H. Antil and E. Otárola, A FEM for an optimal control problem of fractional powers of elliptic operators, SIAM J. Control Optim., 53 (2015), 3432-3456.  doi: 10.1137/140975061.

[5]

H. Antil and E. Otárola, An a posteriori error analysis for an optimal control problem involving the fractional Laplacian, IMA J. Numer. Anal., 38 (2018), 198-226.  doi: 10.1093/imanum/drx005.

[6]

H. AntilE. Otárola and A. J. Salgado, A space-time fractional optimal control problem: analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.  doi: 10.1137/15M1014991.

[7]

H. AntilJ. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control and discretization, Commun. Math. Sci., 16 (2018), 1395-1426.  doi: 10.4310/CMS.2018.v16.n5.a11.

[8]

H. Antil and M. Warma, Optimal Control of Fractional Semilinear PDEs, ESAIM Control Optim. Calc. Var., 2019. doi: 10.1051/cocv/2019003.

[9]

H. Antil and M. Warma, Optimal control of the coefficient for the regional fractal $p$–Laplace equation: approximation and convergence, Math. Control Relat. Fields, 9 (2019), 1-38.  doi: 10.3934/mcrf.2019001.

[10]

V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[11]

T. Biswas, S. Dharmatti and M. T. Mohan, Pontryagins maximum principle for optimal control of the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, arXiv: 1802.08413.

[12]

T. Biswas, S. Dharmatti and M. T. Mohan, Maximum principle and data assimilation problem for the optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations, preprint, arXiv: 1803.11337.

[13]

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.

[14]

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. Ser. Math. Appl., 7 (2015), 41-66. 

[15]

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.

[16]

P. ColliG. GilardiE. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.

[17]

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.

[18]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016), 246-281. 

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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.

[20]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evol. Equ. Control Theory, 6 (2017), 35-58.  doi: 10.3934/eect.2017003.

[21]

P. ColliG. Gilardi and J. Sprekels, On a Cahn-Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl., 197 (2018), 1445-1475.  doi: 10.1007/s10231-018-0732-1.

[22]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691.  doi: 10.1137/17M1146786.

[23]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A `deep quench' approach, J. Convex Anal., 26 (2019), 485-514. 

[24]

P. ColliG. Gilardi and J. Sprekels, Well-posedness and regularity for a generalized fractional Cahn–Hilliard system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 437-478.  doi: 10.4171/RLM/855.

[25]

P. Colli, G. Gilardi and J. Sprekels, Optimal distributed control of a generalized fractional Cahn-Hilliard system, Appl. Math. Optim., (2018). doi: 10.1007/s00245-018-9540-7.

[26]

P. Colli and J. Sprekels, Optimal boundary control of a nonstandard Cahn–Hilliard system with dynamic boundary condition and double obstacle inclusions, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 151-182. 

[27]

N. Duan and X. Zhao, Optimal control for the multi-dimensional viscous Cahn–Hilliard equation, Electron. J. Differ. Equations, (2015), 13 pp.

[28]

S. FrigeriC. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, J. Nonlinear Sci., 26 (2016), 847-893.  doi: 10.1007/s00332-016-9292-y.

[29]

S. FrigeriC. G. GalM. Grasselli and J. Sprekels, Two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with variable viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), 678-727.  doi: 10.1088/1361-6544/aaedd0.

[30]

S. Frigeri, M. Grasselli and J. Sprekels, Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim., (2018), 1–33. doi: 10.1007/s00245-018-9524-7.

[31]

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.

[32]

T. Fukao and N. Yamazaki, A boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 255-280. 

[33]

C. G. Gal, On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 37 (2017), 131-167.  doi: 10.3934/dcds.2017006.

[34]

C. G. Gal, Non-local Cahn–Hilliard equations with fractional dynamic boundary conditions, European J. Appl. Math., 28 (2017), 736-788.  doi: 10.1017/S0956792516000504.

[35]

C. G. Gal, Doubly nonlocal Cahn-Hilliard equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 357-392.  doi: 10.1016/j.anihpc.2017.05.001.

[36]

C. Geldhauser and E. Valdinoci, Optimizing the fractional power in a model with stochastic PDE constraints, Adv. Nonlinear Stud., 18 (2018), 649-669.  doi: 10.1515/ans-2018-2031.

[37]

M. HintermüllerM. HinzeC. Kahle and T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn-Hilliard-Navier-Stokes system, Optim. Eng., 19 (2018), 629-662.  doi: 10.1007/s11081-018-9393-6.

[38]

M. HintermüllerT. Keil and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities, SIAM J. Control Optim., 55 (2017), 1954-1989.  doi: 10.1137/15M1025128.

[39]

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.

[40]

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.

[41]

M. Hintermüller and D. Wegner, Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies, Topological optimization and optimal transport, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 17 (2017), 40-63. 

[42]

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.

[43]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193-217. 

[44]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

[45]

J. Sprekels and E. Valdinoci, A new class of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.  doi: 10.1137/16M105575X.

[46]

T. Tachim Medjo, Optimal control of a Cahn-Hilliard-Navier-Stokes model with state constraints, J. Convex Anal., 22 (2015), 1135-1172. 

[47]

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), Surikaisekikenkyusho Kokyuroku, 1128 (2000), 172-180. 

[48]

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

[49]

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

[50]

J. Zheng, Time optimal controls of the Cahn-Hilliard equation with internal control, Optimal Control Appl. Methods, 36 (2015), 566-582.  doi: 10.1002/oca.2135.

[51]

J. Zheng and Y. Wang, Optimal control problem for Cahn–Hilliard equations with state constraint, J. Dyn. Control Syst., 21 (2015), 257-272.  doi: 10.1007/s10883-014-9259-y.

show all references

References:
[1]

M. Ainsworth and Z. Mao, Analysis and approximation of a fractional Cahn–Hilliard equation, SIAM J. Numer. Anal., 55 (2017), 1689-1718.  doi: 10.1137/16M1075302.

[2]

G. AkagiG. Schimperna and A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations, J. Differential Equations, 261 (2016), 2935-2985.  doi: 10.1016/j.jde.2016.05.016.

[3]

H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 35 pp. doi: 10.1088/1361-6420/ab1299.

[4]

H. Antil and E. Otárola, A FEM for an optimal control problem of fractional powers of elliptic operators, SIAM J. Control Optim., 53 (2015), 3432-3456.  doi: 10.1137/140975061.

[5]

H. Antil and E. Otárola, An a posteriori error analysis for an optimal control problem involving the fractional Laplacian, IMA J. Numer. Anal., 38 (2018), 198-226.  doi: 10.1093/imanum/drx005.

[6]

H. AntilE. Otárola and A. J. Salgado, A space-time fractional optimal control problem: analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.  doi: 10.1137/15M1014991.

[7]

H. AntilJ. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control and discretization, Commun. Math. Sci., 16 (2018), 1395-1426.  doi: 10.4310/CMS.2018.v16.n5.a11.

[8]

H. Antil and M. Warma, Optimal Control of Fractional Semilinear PDEs, ESAIM Control Optim. Calc. Var., 2019. doi: 10.1051/cocv/2019003.

[9]

H. Antil and M. Warma, Optimal control of the coefficient for the regional fractal $p$–Laplace equation: approximation and convergence, Math. Control Relat. Fields, 9 (2019), 1-38.  doi: 10.3934/mcrf.2019001.

[10]

V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[11]

T. Biswas, S. Dharmatti and M. T. Mohan, Pontryagins maximum principle for optimal control of the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, preprint, arXiv: 1802.08413.

[12]

T. Biswas, S. Dharmatti and M. T. Mohan, Maximum principle and data assimilation problem for the optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations, preprint, arXiv: 1803.11337.

[13]

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.

[14]

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. Ser. Math. Appl., 7 (2015), 41-66. 

[15]

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.

[16]

P. ColliG. GilardiE. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.

[17]

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.

[18]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016), 246-281. 

[19]

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.

[20]

P. ColliG. Gilardi and J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evol. Equ. Control Theory, 6 (2017), 35-58.  doi: 10.3934/eect.2017003.

[21]

P. ColliG. Gilardi and J. Sprekels, On a Cahn-Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl., 197 (2018), 1445-1475.  doi: 10.1007/s10231-018-0732-1.

[22]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691.  doi: 10.1137/17M1146786.

[23]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A `deep quench' approach, J. Convex Anal., 26 (2019), 485-514. 

[24]

P. ColliG. Gilardi and J. Sprekels, Well-posedness and regularity for a generalized fractional Cahn–Hilliard system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 437-478.  doi: 10.4171/RLM/855.

[25]

P. Colli, G. Gilardi and J. Sprekels, Optimal distributed control of a generalized fractional Cahn-Hilliard system, Appl. Math. Optim., (2018). doi: 10.1007/s00245-018-9540-7.

[26]

P. Colli and J. Sprekels, Optimal boundary control of a nonstandard Cahn–Hilliard system with dynamic boundary condition and double obstacle inclusions, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 151-182. 

[27]

N. Duan and X. Zhao, Optimal control for the multi-dimensional viscous Cahn–Hilliard equation, Electron. J. Differ. Equations, (2015), 13 pp.

[28]

S. FrigeriC. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, J. Nonlinear Sci., 26 (2016), 847-893.  doi: 10.1007/s00332-016-9292-y.

[29]

S. FrigeriC. G. GalM. Grasselli and J. Sprekels, Two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with variable viscosity, degenerate mobility and singular potential, Nonlinearity, 32 (2019), 678-727.  doi: 10.1088/1361-6544/aaedd0.

[30]

S. Frigeri, M. Grasselli and J. Sprekels, Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim., (2018), 1–33. doi: 10.1007/s00245-018-9524-7.

[31]

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.

[32]

T. Fukao and N. Yamazaki, A boundary control problem for the equation and dynamic boundary condition of Cahn–Hilliard type, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 255-280. 

[33]

C. G. Gal, On the strong-to-strong interaction case for doubly nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 37 (2017), 131-167.  doi: 10.3934/dcds.2017006.

[34]

C. G. Gal, Non-local Cahn–Hilliard equations with fractional dynamic boundary conditions, European J. Appl. Math., 28 (2017), 736-788.  doi: 10.1017/S0956792516000504.

[35]

C. G. Gal, Doubly nonlocal Cahn-Hilliard equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 357-392.  doi: 10.1016/j.anihpc.2017.05.001.

[36]

C. Geldhauser and E. Valdinoci, Optimizing the fractional power in a model with stochastic PDE constraints, Adv. Nonlinear Stud., 18 (2018), 649-669.  doi: 10.1515/ans-2018-2031.

[37]

M. HintermüllerM. HinzeC. Kahle and T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn-Hilliard-Navier-Stokes system, Optim. Eng., 19 (2018), 629-662.  doi: 10.1007/s11081-018-9393-6.

[38]

M. HintermüllerT. Keil and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities, SIAM J. Control Optim., 55 (2017), 1954-1989.  doi: 10.1137/15M1025128.

[39]

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.

[40]

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.

[41]

M. Hintermüller and D. Wegner, Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies, Topological optimization and optimal transport, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 17 (2017), 40-63. 

[42]

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.

[43]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, Evol. Equ. Control Theory, 9 (2020), 193-217. 

[44]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl.(4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

[45]

J. Sprekels and E. Valdinoci, A new class of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM J. Control Optim., 55 (2017), 70-93.  doi: 10.1137/16M105575X.

[46]

T. Tachim Medjo, Optimal control of a Cahn-Hilliard-Navier-Stokes model with state constraints, J. Convex Anal., 22 (2015), 1135-1172. 

[47]

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), Surikaisekikenkyusho Kokyuroku, 1128 (2000), 172-180. 

[48]

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

[49]

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

[50]

J. Zheng, Time optimal controls of the Cahn-Hilliard equation with internal control, Optimal Control Appl. Methods, 36 (2015), 566-582.  doi: 10.1002/oca.2135.

[51]

J. Zheng and Y. Wang, Optimal control problem for Cahn–Hilliard equations with state constraint, J. Dyn. Control Syst., 21 (2015), 257-272.  doi: 10.1007/s10883-014-9259-y.

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