Partial null controllability of parabolic linear systems

Farid Ammar  Khodja; Franz  Chouly; Michel  Duprez

doi:10.3934/mcrf.2016001

Article Contents

2016, Volume 6, Issue 2: 185-216. Doi: 10.3934/mcrf.2016001

This issue Previous Article Next Article Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales

Partial null controllability of parabolic linear systems

1.
Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16, Route de Gray, 25030 Besançon Cedex

Received: January 31, 2015

Revised: September 30, 2015

Published: May 2016

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Abstract

This paper is devoted to the partial null controllability issue of parabolic linear systems with $n$ equations. Given a bounded domain $\Omega$ in $\mathbb{R}^N$ ($N\in \mathbb{N}^*$), we study the effect of $m$ localized controls in a nonempty open subset $\omega$ only controlling $p$ components of the solution ($p,m \le n$). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled $2\times2$ parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.

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Mathematics Subject Classification: Primary: 93B05, 93B07; Secondary: 93C20, 93C05, 35K40.

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