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Optimal Distributed Control of a Cahn–Hilliard–Darcy System with Mass Sources

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Abstract

In this paper, we study an optimal control problem for a two-dimensional Cahn–Hilliard–Darcy system with mass sources that arises in the modeling of tumor growth. The aim is to monitor the tumor fraction in a finite time interval in such a way that both the tumor fraction, measured in terms of a tracking type cost functional, is kept under control and minimal harm is inflicted to the patient by administering the control, which could either be a drug or nutrition. We first prove that the optimal control problem admits a solution. Then we show that the control-to-state operator is Fréchet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her careful reading and helpful suggestions. The research of H. Wu is partially supported by NNSFC Grant No. 11631011 and the Shanghai Center for Mathematical Sciences.

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Authors and Affiliations

  1. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Jürgen Sprekels

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    Jürgen Sprekels

  3. School of Mathematical Sciences, Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, Shanghai, 200433, China

    Hao Wu

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  1. Jürgen Sprekels
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Sprekels, J., Wu, H. Optimal Distributed Control of a Cahn–Hilliard–Darcy System with Mass Sources. Appl Math Optim 83, 489–530 (2021). https://doi.org/10.1007/s00245-019-09555-4

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