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Mathematical Control and Related Fields (MCRF)
 

Partial null controllability of parabolic linear systems

Pages: 185 - 216, Volume 6, Issue 2, June 2016      doi:10.3934/mcrf.2016001

 
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Farid Ammar Khodja - Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16, Route de Gray, 25030 Besançon Cedex, France (email)
Franz Chouly - Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16, Route de Gray, 25030 Besançon Cedex, France (email)
Michel Duprez - Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16, Route de Gray, 25030 Besançon Cedex, France (email)

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.

Keywords:  Controllability, observability, Kalman condition, moment method, parabolic systems.
Mathematics Subject Classification:  Primary: 93B05, 93B07; Secondary: 93C20, 93C05, 35K40.

Received: February 2015;      Revised: October 2015;      Available Online: April 2016.

 References