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

TL;DR

This paper develops an optimal control framework for a tumor growth model based on the Cahn-Hilliard-Darcy system, ensuring tumor suppression while minimizing harm through control inputs like drugs or nutrition.

Contribution

It proves the existence of optimal controls, establishes differentiability of the control-to-state map, and derives first-order optimality conditions for the tumor growth control problem.

Findings

Existence of optimal controls for the tumor model.

Differentiability of the control-to-state operator.

Derivation of first-order optimality conditions.

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 Fr\'echet differentiable between suitable Banach spaces and derive the first-order necessary optimality conditions in terms of the adjoint variables and the usual variational inequality.