# Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials

## TL;DR

This paper develops an optimal control framework for a phase field model of tumor growth that includes chemotaxis and singular potentials, providing mathematical analysis and optimality conditions.

## Contribution

It introduces a general phase field tumor growth model with chemotaxis and singular potentials, and derives optimal control conditions with rigorous mathematical analysis.

## Key findings

- Proved well-posedness of the extended tumor growth model.
- Established Fréchet differentiability of the control-to-state operator.
- Derived first-order necessary optimality conditions.

## Abstract

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fr\'echet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.