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

TL;DR

This paper studies an optimal control problem for a chemotaxis-consumption model, establishing existence, uniqueness, and optimality conditions for strong solutions in a three-dimensional bounded domain.

Contribution

It extends previous weak solution results to strong solutions, proving existence, uniqueness, and deriving first order optimality conditions for the control problem.

Findings

Existence and uniqueness of global strong solutions under regularity conditions.

Existence of a global optimal control solution.

Derivation of first order optimality conditions using Lagrange multipliers.

Abstract

We consider 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 [F. Guill\'en-Gonz\'alez and A. L. Corr\^ea Vianna Filho, Optimal Control Related to Weak Solutions of a Chemotaxis-Consumption Model, arXiv:2211.14612, 2022]. In this paper, we study a related optimal control problem in the strong solution setting. First, imposing the regularity criterion $u ^s \in L^q((0,T) \times \Omega)$ ($q > 5/2$) for a given weak solution, we prove existence and uniqueness of global-in-time strong solutions. Then, the existence of a global optimal solution can be deduced. 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.