Optimal Distributed Control of a Cahn–Hilliard–Darcy System with Mass Sources
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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.
KeywordsCahn–Hilliard–Darcy system Distributed optimal control Necessary optimality condition
Mathematics Subject Classification35G25 49J20 49K20 49J50
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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