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

TL;DR

This paper investigates an optimal control problem for a chemotaxis-consumption model in three dimensions, establishing existence of optimal controls within a weak solution framework despite potential non-uniqueness of solutions.

Contribution

It proves the existence of optimal controls for a chemotaxis model under weak solution assumptions, addressing challenges of non-uniqueness and bounded controls.

Findings

Existence of optimal control under weak solution framework.

Optimal controls are bounded and satisfy an energy inequality.

Relation to other control problems with non-unique solutions is discussed.

Abstract

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 $(u,v)$, being $u$ the cell density, $v$ the chemical concentration and $f$ the control acting in the $v$-equation through the bilinear term $f \,v\, 1_{\Omega_c}$, in a subdomain $\Omega_c \subset \Omega$. We address the existence of optimal control restricted to a weak solution setting, where, in particular, uniqueness of state $(u,v)$ given a control $f$ is not clear. Then by considering weak solutions satisfying an adequate energy inequality, we prove the existence of optimal control subject to uniformly bounded controls. Finally, we discuss the relation between the considered control problem and two other related ones, where the existence of optimal solution can not be proved.