doi: 10.3934/dcdsb.2021110
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Optimal distributed control for a coupled phase-field system

Department of Mathematics, Jilin University, Changchun 130012, China

* Corresponding author: Changchun Liu

Received  November 2020 Revised  February 2021 Early access April 2021

Fund Project: This work is supported by the Jilin Scientific and Technological Development Program

Our aim is to consider a distributed optimal control problem for a coupled phase-field system which was introduced by Cahn and Novick-Cohen. First, we establish that the existence of a weak solution, in particular, we also obtain that a strong solution is uniqueness. Then the existence of optimal controls is proved. Finally we derive that the control-to-state operator is Fréchet differentiable and the first-order necessary optimality conditions involving the adjoint system are discussed as well.

Citation: Bosheng Chen, Huilai Li, Changchun Liu. Optimal distributed control for a coupled phase-field system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021110
References:
[1]

D. BrochetD. Hilhorst and A. Novick-Cohen, Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87.  doi: 10.1016/0893-9659(94)90118-X.  Google Scholar

[2]

J. W. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, Journal of Statistical Physics, 76 (1994), 877-909.  doi: 10.1007/BF02188691.  Google Scholar

[3]

C. CavaterraE. Rocca and H. Wu, Optimal boundary control of a simplified Ericksen-Leslie system for nematic liquid crystal flows in 2D, Arch. Ration. Mech. Anal., 224 (2017), 1037-1086.  doi: 10.1007/s00205-017-1095-2.  Google Scholar

[4]

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.  Google Scholar

[5]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with a possibly singular potential, Evol. Equ. Control Theory, 7 (2018), 95-116.  doi: 10.3934/eect.2018006.  Google Scholar

[6]

R. Dal PassoL. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Bound., 1 (1999), 199-226.  doi: 10.4171/IFB/9.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[8]

C. Kahle and K. F. Lam, Parameter identification via optimal control for a Cahn-Hilliard-chemotaxis system with a variable mobility, Appl. Math. Optim., 82 (2020), 63-104.  doi: 10.1007/s00245-018-9491-z.  Google Scholar

[9]

M. KurokibaN. Tanaka and A. Tani, Maximal attractor and inertial set for Eguchi-Oki-Matsumura equation, J. Math. Anal. Appl., 365 (2010), 638-645.  doi: 10.1016/j.jmaa.2009.06.014.  Google Scholar

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S. Li and D. Yan, On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3077-3088.  doi: 10.3934/dcdsb.2018301.  Google Scholar

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J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, (French) Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[12]

C. Liu and Z. Wang, Optimal control for a sixth order nonlinear parabolic equation, Math. Methods Appl. Sci., 38 (2015), 247-262.  doi: 10.1002/mma.3063.  Google Scholar

[13]

C. Liu and X. Zhang, Optimal distributed control for a new mechanochemical model in biological patterns, J. Math. Anal. Appl., 478 (2019), 825-863.  doi: 10.1016/j.jmaa.2019.05.057.  Google Scholar

[14]

A. Makki, A. Miranville and W. Saoud, On a Cahn-Hilliard/Allen-Cahn system coupled with a type Ⅲ heat equation and singular potentials, Nonlinear Anal., 196 (2020), 20 pp. doi: 10.1016/j.na.2020.111804.  Google Scholar

[15]

A. MiranvilleW. Saoud and R. Talhouk, Asymptotic behavior of a model for order-disorder and phase separation, Asymptot. Anal., 103 (2017), 57-76.  doi: 10.3233/ASY-171419.  Google Scholar

[16]

A. Miranville and R. Quintanilla, On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law, Math. Methods Appl. Sci., 39 (2016), 4385-4397.  doi: 10.1002/mma.3867.  Google Scholar

[17]

A. MiranvilleR. Quintanilla and W. Saoud, Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature, Commun. Pure Appl. Anal., 19 (2020), 2257-2288.  doi: 10.3934/cpaa.2020099.  Google Scholar

[18]

A. MiranvilleW. Saoud and R. Talhouk, On the Cahn-Hilliard/Allen-Cahn equations with singular potentials, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3633-3651.  doi: 10.3934/dcdsb.2018308.  Google Scholar

[19]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Phys. D, 137 (2000), 1-24.  doi: 10.1016/S0167-2789(99)00162-1.  Google Scholar

[20]

A. Novick-Cohen and L. Peres Hari, Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case, Phys. D, 209 (2005), 205-235.  doi: 10.1016/j.physd.2005.06.028.  Google Scholar

[21]

S. Rokkam, A. El-Azab, P. Millett and D. Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Model. Simul. Mater. Sci. Eng., 17 (2009), 0064002. doi: 10.1088/0965-0393/17/6/064002.  Google Scholar

[22]

T. C. Sideris, Ordinary Differential Equations and Dynamical Systems, Atlantis Studies in Differential Equations, 2, Atlantis Press, Paris, RI, 2013. doi: 10.2991/978-94-6239-021-8.  Google Scholar

[23]

A. Signori, Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305-331.  doi: 10.3934/mcrf.2019040.  Google Scholar

[24]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.  Google Scholar

[25]

R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[26]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Translated from the 2005 German original by Jürgen Sprekels, Graduate Studies in Mathematics, 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

[27]

Q. Wang and D. Yan, On the stability and transition of the Cahn-Hilliard/Allen-Cahn system, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2607-2620.  doi: 10.3934/dcdsb.2020024.  Google Scholar

[28]

Y. XiaY. Xu and C. W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835.   Google Scholar

[29]

E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅱ/A. Linear Monotone Operators, Translated from the German by the author and Leo F. Boron., Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[30]

X. ZhangH. Li and C. Liu, Optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state constraint, Applied Mathematics and Optimization, 82 (2020), 721-754.  doi: 10.1007/s00245-018-9546-1.  Google Scholar

[31]

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]

D. BrochetD. Hilhorst and A. Novick-Cohen, Finite-dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87.  doi: 10.1016/0893-9659(94)90118-X.  Google Scholar

[2]

J. W. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, Journal of Statistical Physics, 76 (1994), 877-909.  doi: 10.1007/BF02188691.  Google Scholar

[3]

C. CavaterraE. Rocca and H. Wu, Optimal boundary control of a simplified Ericksen-Leslie system for nematic liquid crystal flows in 2D, Arch. Ration. Mech. Anal., 224 (2017), 1037-1086.  doi: 10.1007/s00205-017-1095-2.  Google Scholar

[4]

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.  Google Scholar

[5]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with a possibly singular potential, Evol. Equ. Control Theory, 7 (2018), 95-116.  doi: 10.3934/eect.2018006.  Google Scholar

[6]

R. Dal PassoL. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Bound., 1 (1999), 199-226.  doi: 10.4171/IFB/9.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[8]

C. Kahle and K. F. Lam, Parameter identification via optimal control for a Cahn-Hilliard-chemotaxis system with a variable mobility, Appl. Math. Optim., 82 (2020), 63-104.  doi: 10.1007/s00245-018-9491-z.  Google Scholar

[9]

M. KurokibaN. Tanaka and A. Tani, Maximal attractor and inertial set for Eguchi-Oki-Matsumura equation, J. Math. Anal. Appl., 365 (2010), 638-645.  doi: 10.1016/j.jmaa.2009.06.014.  Google Scholar

[10]

S. Li and D. Yan, On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3077-3088.  doi: 10.3934/dcdsb.2018301.  Google Scholar

[11]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, (French) Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[12]

C. Liu and Z. Wang, Optimal control for a sixth order nonlinear parabolic equation, Math. Methods Appl. Sci., 38 (2015), 247-262.  doi: 10.1002/mma.3063.  Google Scholar

[13]

C. Liu and X. Zhang, Optimal distributed control for a new mechanochemical model in biological patterns, J. Math. Anal. Appl., 478 (2019), 825-863.  doi: 10.1016/j.jmaa.2019.05.057.  Google Scholar

[14]

A. Makki, A. Miranville and W. Saoud, On a Cahn-Hilliard/Allen-Cahn system coupled with a type Ⅲ heat equation and singular potentials, Nonlinear Anal., 196 (2020), 20 pp. doi: 10.1016/j.na.2020.111804.  Google Scholar

[15]

A. MiranvilleW. Saoud and R. Talhouk, Asymptotic behavior of a model for order-disorder and phase separation, Asymptot. Anal., 103 (2017), 57-76.  doi: 10.3233/ASY-171419.  Google Scholar

[16]

A. Miranville and R. Quintanilla, On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law, Math. Methods Appl. Sci., 39 (2016), 4385-4397.  doi: 10.1002/mma.3867.  Google Scholar

[17]

A. MiranvilleR. Quintanilla and W. Saoud, Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature, Commun. Pure Appl. Anal., 19 (2020), 2257-2288.  doi: 10.3934/cpaa.2020099.  Google Scholar

[18]

A. MiranvilleW. Saoud and R. Talhouk, On the Cahn-Hilliard/Allen-Cahn equations with singular potentials, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3633-3651.  doi: 10.3934/dcdsb.2018308.  Google Scholar

[19]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Phys. D, 137 (2000), 1-24.  doi: 10.1016/S0167-2789(99)00162-1.  Google Scholar

[20]

A. Novick-Cohen and L. Peres Hari, Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case, Phys. D, 209 (2005), 205-235.  doi: 10.1016/j.physd.2005.06.028.  Google Scholar

[21]

S. Rokkam, A. El-Azab, P. Millett and D. Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Model. Simul. Mater. Sci. Eng., 17 (2009), 0064002. doi: 10.1088/0965-0393/17/6/064002.  Google Scholar

[22]

T. C. Sideris, Ordinary Differential Equations and Dynamical Systems, Atlantis Studies in Differential Equations, 2, Atlantis Press, Paris, RI, 2013. doi: 10.2991/978-94-6239-021-8.  Google Scholar

[23]

A. Signori, Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305-331.  doi: 10.3934/mcrf.2019040.  Google Scholar

[24]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.  Google Scholar

[25]

R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2$^{nd}$ edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[26]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Translated from the 2005 German original by Jürgen Sprekels, Graduate Studies in Mathematics, 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.  Google Scholar

[27]

Q. Wang and D. Yan, On the stability and transition of the Cahn-Hilliard/Allen-Cahn system, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2607-2620.  doi: 10.3934/dcdsb.2020024.  Google Scholar

[28]

Y. XiaY. Xu and C. W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835.   Google Scholar

[29]

E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅱ/A. Linear Monotone Operators, Translated from the German by the author and Leo F. Boron., Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[30]

X. ZhangH. Li and C. Liu, Optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state constraint, Applied Mathematics and Optimization, 82 (2020), 721-754.  doi: 10.1007/s00245-018-9546-1.  Google Scholar

[31]

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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