Optimal Distributed Control for a Cahn-Hilliard-Darcy System with Mass Sources, Unmatched Viscosities and Singular Potential
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916360028684288 |
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| author | Abatangelo, Marco Cavaterra, Cecilia Grasselli, Maurizio Wu, Hao |
| author_facet | Abatangelo, Marco Cavaterra, Cecilia Grasselli, Maurizio Wu, Hao |
| contents | 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 $\mathbf{u}$ as well as homogeneous Neumann boundary conditions for the phase function $φ$ and the chemical potential $μ$. 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 $φ$ being strictly separated from the pure phases $\pm 1$. This well-posedness result enables us to characterize the control-to-state mapping $\mathcal{S}:R \mapsto φ$. Then we obtain the existence of an optimal control, the Fréchet differentiability of $\mathcal{S}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01569 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Distributed Control for a Cahn-Hilliard-Darcy System with Mass Sources, Unmatched Viscosities and Singular Potential Abatangelo, Marco Cavaterra, Cecilia Grasselli, Maurizio Wu, Hao Optimization and Control Analysis of PDEs 35Q35, 35Q92, 49J20, 49J50, 49K20, 76D27, 76T06 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 $\mathbf{u}$ as well as homogeneous Neumann boundary conditions for the phase function $φ$ and the chemical potential $μ$. 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 $φ$ being strictly separated from the pure phases $\pm 1$. This well-posedness result enables us to characterize the control-to-state mapping $\mathcal{S}:R \mapsto φ$. Then we obtain the existence of an optimal control, the Fréchet differentiability of $\mathcal{S}$ 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. |
| title | Optimal Distributed Control for a Cahn-Hilliard-Darcy System with Mass Sources, Unmatched Viscosities and Singular Potential |
| topic | Optimization and Control Analysis of PDEs 35Q35, 35Q92, 49J20, 49J50, 49K20, 76D27, 76T06 |
| url | https://arxiv.org/abs/2308.01569 |