Local exact controllability to positive trajectory forparabolic system of chemotaxis

Bao-Zhu  Guo; Liang  Zhang

doi:10.3934/mcrf.2016.6.143

Article Contents

2016, Volume 6, Issue 1: 143-165. Doi: 10.3934/mcrf.2016.6.143

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Local exact controllability to positive trajectory for parabolic system of chemotaxis

Bao-Zhu Guo^1, and
Liang Zhang^2,

1.
Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190
2.
Department of Mathematics, Wuhan University of Technology, Wuhan 430070

Received: February 28, 2015

Revised: May 31, 2015

Published: February 2016

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Abstract

In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani's fixed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is shown to be in $L^\infty(Q)$, which is estimated by using the methods of maximal regularity and $L^p$-$L^q$ estimate for parabolic equations.

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Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 37C25.

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