An Optimal Control Problem Related to a Parabolic–Elliptic Chemo-repulsion System in 2D Domains

In this paper we study an optimal control problem associated to a parabolic–elliptic chemo-repulsion system with a linear production term in a two-dimensional domain. Under the injection/extract chemical substance on a control subdomain \(\varOmega _c\) ${\Omega }_c$ , we prove the existence and uniqueness of global-in-time strong solutions. Afterwards, for the optimal control problem, we prove the existence of at least one global optimal solution, and derive an optimality system via using a Lagrange multipliers theorem.