March  2022, 27(3): 1789-1825. doi: 10.3934/dcdsb.2021110

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 Published  March 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (3) : 1789-1825. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

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