On the optimal control of viscous Cahn–Hilliard systems with hyperbolic relaxation of the chemical potential

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Abstract

In this paper, we study an optimal control problem for a viscous Cahn–Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence, and regularity are known. We show Fréchet differentiability of the associated control-to- state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.

Keywords

Optimal control / Cahn–Hilliard system / hyperbolic relaxation / first-order optimality conditions / asymptotic analysis

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References

[1] P. Colli, J. Sprekels, F. Tröltzsch, Optimality conditions for sparse optimal control of viscous Cahn-Hilliard systems with logarithmic potential, Appl. Math. Optim., 90 (2024), 47. https://doi.org/10.1007/s00245-024-10187-6

[2] M. Conti, S. Gatti, A. Miranville, A perturbation of the Cahn-Hilliard equation with logarithmic nonlinearity, J. Differential Equations, 382 (2024), 50-76. https://doi.org/10.1016/j.jde.2023.11.013

[3] G. Gilardi, E. Rocca, A. Signori, Well-posedness and optimal control for a viscous Cahn-Hilliard- Oono system with dynamic boundary conditions. Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 3573-3605. https://doi.org/10.3934/dcdss.2023127

[4] G. Gilardi, A. Signori, J. Sprekels, Nutrient control for a viscous Cahn-Hilliard-Keller-Segel model with logistic source describing tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 3552-3572. https://doi.org/10.3934/dcdss.2023123

[5] K. Kagawa, M. Ȏtani, Asymptotic limits of viscous Cahn-Hilliard equation with homogeneous Dirichlet boundary condition, J. Math. Anal. Appl., 512 (2022), 126106. https://doi.org/10.1016/j.jmaa.2022.126106

[6] A. Bonfoh, The viscous Cahn-Hilliard equation with inertial term, Nonlinear Anal., 74 (2011), 946-964. https://doi.org/10.1016/j.na.2010.09.049

[7] A. Bonfoh, Existence and continuity of inertial manifolds for the hyperbolic relaxation of the viscous Cahn-Hilliard equation, Appl. Math. Optim., 84 (2021), 3339-3416. https://doi.org/10.1007/s00245-021-09749-9

[8] H. Chen, Error estimates for the scalar auxiliary variable (SAV) schemes to the viscous Cahn-Hilliard equation with hyperbolic relaxation, J. Math. Anal. Appl., 499 (2021), 125002. https://doi.org/10.1016/j.jmaa.2021.125002

[9] X. Chen, L. Ma, X. Yang, Error analysis of second-order IEQ numerical schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation, Comput. Math. Appl., 152 (2023), 112-128. https://doi.org/10.1016/j.camwa.2023.10.003

[10] D. Dor, A. Miranville, M. Pierre, Hyperbolic relaxation of the viscous Cahn-Hilliard equation with a symport term for biological applications, Math. Methods Appl. Sci., 47 (2024), 5999-6035. https://doi.org/10.1002/mma.9904

[11] A. Savostianov, S. Zelik, Global well-posedness and attractors for the hyperbolic Cahn-Hilliard- Oono equation in the whole space, Math. Models Methods Appl. Sci., 26 (2016), 1357-1384. https://doi.org/10.1142/S0218202516500329

[12] Z. Şen, A. Khanmamedov, The Cahn-Hilliard/Allen-Cahn equation with inertial and proliferation terms, J. Math. Anal. Appl., 530 (2024), 127736. https://doi.org/10.1016/j.jmaa.2023.127736

[13] P. Colli, J. Sprekels, Hyperbolic relaxation of the chemical potential in the viscous Cahn-Hilliard equation, preprint, arXiv:2408.16332.

[14] P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546. https://doi.org/10.1088/1361-6544/aa6e5f

[15] P. Colli, G. Gilardi, G. Marinoschi, Solvability and sliding mode control for the viscous Cahn- Hilliard system with a possibly singular potential, Math. Control Relat. Fields, 11 (2021), 905-934. https://doi.org/10.3934/mcrf.2020051

[16] P. Colli, G. Gilardi, J. Sprekels, Recent results on well-posedness and optimal control for a class of generalized fractional Cahn-Hilliard systems, Control Cybernet., 48 (2019), 153-197.

[17] J. Simon, Compact sets in the space L^p(0,T; B) , Ann. Mat. Pura Appl., 146 (1987), 65-96. https://doi.org/10.1007/BF01762360

[18] J. Sprekels, F. Tröltzsch, Second-order sufficient conditions for sparse optimal control of singular Allen-Cahn systems with dynamic boundary conditions, Discrete Cont. Dyn. Systems Ser. S, 16 (2023), 3784-3812. https://doi.org/10.3934/dcdss.2023163