Solvability and optimal control of a multi-species Cahn-Hilliard-Keller-Segel tumor growth model

TL;DR

This paper analyzes an optimal control problem for a complex multi-species tumor growth model involving nonlinear PDEs, establishing existence of optimal controls and deriving necessary optimality conditions.

Contribution

It extends previous modeling work by proving the existence of optimal controls and characterizing their first-order conditions for a nonlinear tumor growth system.

Findings

Existence of optimal controls is established.

First-order necessary optimality conditions are derived.

Solutions depend continuously on control variables.

Abstract

This paper investigates an optimal control problem associated with a two-dimensional multi-species Cahn-Hilliard-Keller-Segel tumor growth model, which incorporates complex biological processes such as species diffusion, chemotaxis, angiogenesis, and nutrient consumption, resulting in a highly nonlinear system of nonlinear partial differential equations. The modeling derivation and corresponding analysis have been addressed in a previous contribution. Building on this foundation, the scope of this study involves investigating a distributed control problem with the goal of optimizing a tracking-type cost functional. This latter aims to minimize the deviation of tumor cell location from desired target configurations while penalizing the costs associated with implementing control measures, akin to introducing a suitable medication. Under appropriate mathematical assumptions, we demonstrate that sufficiently regular solutions exhibit continuous dependence on the control variable. Furthermore, we establish the existence of optimal controls and characterize the first-order necessary optimality conditions through a suitable variational inequality.