We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the (volume) averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function phi and the chemical potential mu. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought, for instance, as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with phi being strictly separated from the pure phases +/- 1. This well-posedness result enables us to characterize the control-to-state mapping S : R phi. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

Optimal distributed control for a Cahn–Hilliard–Darcy system with mass sources, unmatched viscosities and singular potential / M. Abatangelo, C. Cavaterra, M. Grasselli, H. Wu. - In: ESAIM. COCV. - ISSN 1292-8119. - 30:(2024), pp. 52.1-52.49. [10.1051/cocv/2024041]

Optimal distributed control for a Cahn–Hilliard–Darcy system with mass sources, unmatched viscosities and singular potential

C. Cavaterra
Secondo
;
2024

Abstract

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the (volume) averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function phi and the chemical potential mu. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought, for instance, as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with phi being strictly separated from the pure phases +/- 1. This well-posedness result enables us to characterize the control-to-state mapping S : R phi. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.
Cahn-Hilliard-Darcy system; singular potential; unmatched viscosities; strong solution; optimal control; necessary optimality condition; sufficient optimality condition;
Settore MATH-03/A - Analisi matematica
   Assegnazione Dipartimenti di Eccellenza 2023-2027 - Dipartimento di MATEMATICA 'FEDERIGO ENRIQUES'
   DECC23_012
   MINISTERO DELL'UNIVERSITA' E DELLA RICERCA

   Partial differential equations and related geometric-functional inequalities.
   MINISTERO DELL'UNIVERSITA' E DELLA RICERCA
   20229M52AS_004
2024
12-lug-2024
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1104832
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