Optimal control related to weak solutions of a chemotaxis-consumption model

Francisco Guillén-González, André Luiz Corrêa Vianna Filho

Published 2022-11-26Version 1

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain $\Omega\subset \mathbb{R}^3$: $$\partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v + f v 1_{\Omega_c}$$ with $s \geq 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$, $u$ being the cell density, $v$ the chemical concentration and $f$ the bilinear control acting in a subdomain $\Omega_c \subset \Omega$. The main novelty of this work is to consider only weak solutions satisfying an energy inequality. We prove the existence of such type of weak solutions and the existence of optimal control subject to bounded controls. Finally, we discuss the relation between the considered control problem and two other related ones that might be of interest.