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

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$: $${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v),{\mathrm{\partial }}_{t}v-\mathrm{\Delta }v=-u^{s}v+fv{1}_{{\mathrm{\Omega }}_c},$$ with $s\ge 1$, 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 $fv{1}_{{\mathrm{\Omega }}_c}$, in a subdomain ${\mathrm{\Omega }}_c\subset \mathrm{\Omega }$. 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.