# Optimal control of stochastic phase-field models related to tumor growth

## TL;DR

This paper develops a stochastic phase-field model for tumor growth, proving well-posedness and analyzing optimal control strategies for drug administration using advanced mathematical techniques.

## Contribution

It introduces a novel stochastic model coupling tumor dynamics with nutrient diffusion and establishes optimal control conditions for treatment strategies.

## Key findings

- Proved strong well-posedness of the stochastic system
- Established existence of optimal control strategies
- Derived first-order optimality conditions

## Abstract

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.