WIAS Preprint No. 2548, (2018)

Optimal distributed control of a Cahn--Hilliard--Darcy system with mass sources


  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604
  • Wu, Hao

2010 Mathematics Subject Classification

  • 35G25 49J20 49K20 49J50


  • Cahn--Hilliard--Darcy system, distributed optimal control, necessary optimality condition




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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