Null controllability of the heat equation   in unbounded domains by a finite measure control region

ESAIM: Control, Optimisation and Calculus of Variations

ESAIM: Control, Optimisation and Calculus of Variations / Volume 10 / Issue 03 / July 2004, pp 381-408
© EDP Sciences, SMAI, 2004
Published online by Cambridge University Press: 22 February 2011
DOI: http://dx.doi.org/10.1051/cocv:2004010 (About DOI)

Table of Contents - July 2004 - Volume 10, Issue 03

Research Article

Null controllability of the heat equation in unbounded domains by a finite measure control region

Article author query
cannarsa p [PubMed] [Google Scholar]
martinez p [PubMed] [Google Scholar]
vancostenoble j [PubMed] [Google Scholar]

Cannarsa, Piermarco^a1, Martinez, Patrick^a2 and Vancostenoble, Judith^a2

^a1 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy; cannarsa@mat.uniroma2.it.

^a2 Laboratoire M.I.P., UMR CNRS 5640, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31062 Toulouse Cedex 4, France; martinez@mip.ups-tlse.fr.; cancoste@mip.ups.tlse.fr.

Abstract

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically $\mathbb R_+$ or $\mathbb R^N$ . Considering an unbounded and disconnected control region of the form $\omega := \cup _n \omega _n$ , we prove two null controllability results: under some technical assumption on the control parts $\omega _n$ , we prove that every initial datum in some weighted L ² space can be controlled to zero by usual control functions, and every initial datum in L ²(Ω) can be controlled to zero using control functions in a weighted L ² space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.