A Review on the Analysis and Optimal Control of Chemotaxis-Consumption Models
André Luiz Corrêa Vianna Filho, Francisco Guillén-González
TL;DR
This paper surveys the analysis, numerical approximation, and optimal control of the chemotaxis-consumption model $u_t-\Delta u=-\nabla\cdot(u\nabla v)$, $v_t-\Delta v=-u^s v$ with $s\ge 1$, under Neumann boundary conditions and nonnegative initial data, focusing on a bilinear control entering the chemical equation via $f v 1_{\Omega_c}$. It develops a weak- formulation framework by introducing $z=\sqrt{v+\alpha}$ to derive energy estimates, enabling global weak solutions in 3D and global strong solutions in 2D, and it analyzes time-discretization schemes that preserve positivity and mass. It also extends the analysis to controlled problems, proving existence of weak (and under a regularity criterion, strong) controlled solutions and establishing global optimal controls on truncated sets, along with adjoint systems and explicit optimality conditions for unconstrained controls. Collectively, the work provides a rigorous bridge between analytical theory, stable numerical schemes, and bilinear optimal control for chemotaxis-driven consumption processes, with potential applications in biofilm growth, immune response, and tumor modeling.
Abstract
In the present review we focus on the chemotaxis-consumption model $\partial_t u - Δu = - \nabla \cdot (u \nabla v)$ and $\partial_t v - Δv = - u^s v$ in $(0,T) \times Ω$, for any fixed $s \geq 1$, endowed with isolated boundary conditions and nonnegative initial conditions, where $(u,v)$ model cell density and chemical signal concentration. Our objective is to present an overview of the related literature and latest results on the aforementioned model concerning the following three distinct research lines we have obtained in [12,24-26]: the mathematical analysis, the numerical analysis and the related optimal control theory with a bilinear control acting on the chemical equation.