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

Published online by Cambridge University Press:  15 June 2004

Piermarco Cannarsa ,
Patrick Martinez  and
Judith Vancostenoble
Piermarco Cannarsa
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy; cannarsa@mat.uniroma2.it.
Patrick Martinez
Affiliation:
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.
Judith Vancostenoble
Affiliation:
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.
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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 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.

Keywords

Null controllabilityweighted observability inequalities.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 3 , July 2004 , pp. 381 - 408
Copyright
© EDP Sciences, SMAI, 2004

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