Bilinear optimal control for weak solutions of the Keller-Segel logistic model in $2D$ domains

TL;DR

This paper studies an optimal control problem for a Keller-Segel model with logistic growth in 2D, focusing on bilinear control acting on the chemical equation, allowing for weak solutions and singular controls.

Contribution

It introduces a novel approach to optimal control of chemotaxis models with weak solutions and singular controls, including existence and optimality conditions.

Findings

Existence of an optimal control is established.

Necessary optimality system is derived.

Control can be singular while maintaining weak solutions.

Abstract

An optimal control problem associated to the Keller-Segel with logistic reaction system will be studied in $2D$ domains. The control acts in a bilinear form only in the chemical equation. The existence of optimal control and a necessary optimality system are deduced. The main novelty is that control can be rather singular and the state (cell density $u$ and the chemical concentration $v$) remains only in a weak setting, which is not usual in the literature to solve optimal control problems subject to chemotaxis models (see e.g. Guill\'en-Gonz\'alez et al. ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 29, 21 pp).