# Vanishing parameter for an optimal control problem modeling tumor growth

## TL;DR

This paper investigates the asymptotic behavior of an optimal control problem modeling tumor growth, focusing on the vanishing of relaxation parameters in a coupled Cahn-Hilliard and reaction-diffusion system.

## Contribution

It extends previous models by analyzing the limit as relaxation parameters tend to zero, confirming the existence of optimal controls and deriving first-order optimality conditions.

## Key findings

- Existence of optimal control confirmed in the vanishing parameter limit
- Characterization of first-order optimality conditions established
- Asymptotic schemes used to analyze parameter limits

## Abstract

A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn-Hilliard equation for the phase variable (relative concentration of the tumor), coupled with a reaction-diffusion equation for the nutrient. The cost functional is of standard tracking-type and the control variable models the intensity with which it is possible to dispense a medication. The model we deal with presents two small and positive parameters which are introduced in previous contributions as relaxation terms. Here, starting from the already investigated optimal control problem for the relaxed model, we aim at confirming the existence of optimal control and characterizing the first-order optimality condition, via asymptotic schemes, when one of the two occurring parameters goes to zero.