Local Exact Controllability of a Parabolic System of Chemotaxis

TL;DR

This paper establishes the local exact controllability of a chemotaxis parabolic system using fixed point and null controllability techniques, with control functions bounded in $L^ Infty$.

Contribution

It introduces a novel approach combining Kakutani's fixed point theorem with maximal regularity to achieve controllability with bounded controls.

Findings

Achieved local exact controllability to trajectories for the chemotaxis system.

Demonstrated control functions are in $L^ Infty(Q)$ with explicit estimates.

Extended controllability results to nonlinear parabolic systems.

Abstract

This paper studies the controllability problem of a parabolic system of chemotaxis. The local exact controllability to trajectories of the system imposed one control force only is obtained by applying Kakutani's fixed point theorem combined with the null controllability of the associated linearized parabolic system. The control function is shown to be in $L^\infty(Q)$, which is estimated by using the methods of maximal regularity and $L^p$-$L^q$ estimates of parabolic equations.