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

¹ Dipartimento di Matematica “F. Casorati”, Università di Pavia, and Research Associate at the IMATI – C.N.R. Pavia, via Ferrata 5, I-27100 Pavia, Italy
² Dipartimento di Matematica, Politecnico di Milano, via E. Bonardi 9, I-20133 Milano, Italy, and Alexander von Humboldt Research Fellow
³ Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
⁴ Weierstrass Institute for Applied Analysis and Stochastics, Anton-Wilhelm-Amo-Strasse 39, D-10117 Berlin, Germany

^* Corresponding author: andrea.signori@polimi.it

Received: 7 November 2024
Accepted: 18 August 2025

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.