An optimal control problem subject to strong solutions of chemotaxis-consumption models

Francisco Guillén-González, André Luiz Corrêa Vianna Filho

Published 2023-02-15Version 1

In the present work we investigate a bilinear optimal control problem associated to the following chemotaxis-consumption model in a bounded domain $\Omega \subset \mathbb{R}^3$ during a time interval $(0,T)$: $$\partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v + f v 1_{\Omega_c}$$ 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 existence of weak solutions $(u,v)$ to this model given $f \in L^q((0,T) \times \Omega)$, for some $q > 5/2$, has been proved in \cite{guillen2022optimal}. In this work the optimal control problem is studied in a strong solution setting. First we prove that the regularity criterion $u ^s,f \in L^q((0,T) \times \Omega)$ allows us to get existence and uniqueness of global-in-time strong solutions. In the sequel, we show the existence of a global optimal solution. Finally, using a Lagrange multipliers theorem, we establish first order optimality conditions for any local optimal solution, proving existence, uniqueness and regularity of the associated Lagrange multipliers.