Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Simultaneous temperature and flux controllability for heat equations with memory

Authors: S. Avdonin and L. Pandolfi
Journal: Quart. Appl. Math. 71 (2013), 339-368
MSC (2010): Primary 76A10, 93C05, 47N70
DOI: https://doi.org/10.1090/S0033-569X-2012-01287-7
Published electronically: October 22, 2012
MathSciNet review: 3087427
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Abstract: It is known that, in the case of the heat equation with memory, temperature can be controlled to an arbitrary square integrable target provided that the system evolves for a sufficiently long time. The control is the temperature on the boundary. In this paper we consider heat equations with memory (one-dimensional space variable) and we first show that when the control is square integrable, then the flux is square integrable too. Then we prove that both temperature and flux can be simultaneously controlled to a pair of independent targets, both square integrable. This solves a problem first raised by Renardy.

The method of proof relies on moment theory, and one of the contributions of this paper is the identification of $ \mathcal {L} $-bases and Riesz bases especially suited to heat equations with memory, which so appear to be endowed with a very rich bases structure.

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Additional Information

S. Avdonin
Affiliation: Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403-2598, USA
Email: s.avdonin@alaska.edu

L. Pandolfi
Affiliation: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: luciano.pandolfi@polito.it

DOI: https://doi.org/10.1090/S0033-569X-2012-01287-7
Received by editor(s): July 27, 2011
Published electronically: October 22, 2012
Additional Notes: Supported in part by the National Science Foundation, grant ARC 0724860. This paper was partly written while the first author visited the Dipartimento di Matematica, Politecnico di Torino, as a visiting professor supported by GNAMPA-INDAM
Supported in part by Italian MURST and by the project “Groupement de Recherche en Contrôle des EDP entre la France et l’Italie (CONEDP)”. This paper fits into the research programs of GNAMPA-INDAM
Article copyright: © Copyright 2012 Brown University

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