A Review on the Analysis and Optimal Control of Chemotaxis-Consumption Models

In the present review we focus on the chemotaxis-consumption model ${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v)$ and ${\mathrm{\partial }}_{t}v-\mathrm{\Delta }v=-u^{s}v$ in $(0,T)\times \mathrm{\Omega }$, for any fixed $s\ge 1$, 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.