Optimal control for a fourth-order nonisothermal tumor growth model of Caginalp type

TL;DR

This paper investigates an optimal control problem for a complex tumor growth model involving phase-field equations, heat transfer, and nutrient dynamics, with hyperthermia as a control variable.

Contribution

It establishes the existence of optimal controls and derives first-order optimality conditions for a nonisothermal tumor growth PDE system.

Findings

Existence of optimal controls for the tumor growth model.

Derivation of first-order necessary conditions using adjoint variables.

Analysis of the control-to-state operator's differentiability.

Abstract

We study a distributed optimal control problem for a nonisothermal Caginalp-type phase-field model that describes tumour growth under thermal therapy. The PDE system couples a possibly viscous Cahn-Hilliard equation, governing the evolution of the healthy and tumor phases, with an equation for the heat balance, and a reaction-diffusion equation for the nutrient concentration. Chemotaxis and active transport effects are taken into account, and hyperthermia appears as a control variable. We introduce a suitable tracking-type cost functional and show the existence of optimal controls. Then, we analyse the differentiability of the control-to-state operator and establish necessary first-order conditions expressed through a variational inequality involving the adjoint state variables.