# Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme

## TL;DR

This paper investigates an optimal control problem for a tumor growth model based on a phase field system of Cahn-Hilliard type, employing asymptotic schemes and establishing existence and necessary conditions for optimal controls.

## Contribution

It introduces an asymptotic scheme approach to an optimal control problem for a tumor growth phase field model, including existence and characterization of optimal controls.

## Key findings

- Existence of optimal controls is established.
- Necessary optimality conditions are derived.
- The model incorporates growth conditions and smallness restrictions.

## Abstract

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard type equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, further assumptions have to be required. Mainly, we ought to impose some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide existence of optimal controls and a necessary condition that an optimal control has to satisfy has been characterized as well.