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

¹ Department of Mathematics, University of Trento Via Sommarive 14, 38123 Povo (Trento), Italy.
² Department of Mathematics, University of Pavia, and IMATI - C.N.R. Via Ferrata 5, 27100 Pavia, Italy.
³ Faculty of Mathematics, University of Vienna Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

^* Corresponding author: luca.scarpa@univie.ac.at

Received: 2 August 2019
Accepted: 23 April 2020

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.