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

Free access article

Issue		ESAIM: COCV Volume 10, Number 3, July 2004


Page(s)		381 - 408
DOI		10.1051/cocv:2004010

ESAIM: COCV, July 2004, Vol. 10, pp. 381-408
DOI: 10.1051/cocv:2004010

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

Piermarco Cannarsa¹, Patrick Martinez² and Judith Vancostenoble²

¹ Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy; cannarsa@mat.uniroma2.it.
² 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.

(Received May 20, 2002. Revised July 18, 2003.)

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 L² space can be controlled to zero by usual control functions, and every initial datum in $L^2 (\Omega)$ can be controlled to zero using control functions in a weighted L² space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.

Mathematics Subject Classification. 35K05, 93B05, 93B07.

Key words: Null controllability, weighted observability inequalities.

© EDP Sciences, SMAI 2004

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