Optimal distributed control of a Cahn--Hilliard--Darcy system with mass sources
- Sprekels, Jürgen
- 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.
- Appl. Math. Optim., 83 (2021), pp. 489--530 (published online on 24.01.2019), DOI 10.1007/s00245-019-09555-4 .