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. CavaterraSecondo
;
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.File | Dimensione | Formato | |
---|---|---|---|
cocv230166.pdf
accesso aperto
Tipologia:
Publisher's version/PDF
Dimensione
1.05 MB
Formato
Adobe PDF
|
1.05 MB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.