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

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: 20 May 2002
Revised: 18 July 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 or . Considering an unbounded and disconnected control region of the form , we prove two null controllability results: under some technical assumption on the control parts , 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 L²(Ω) 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.