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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Control from an interior hypersurface


Authors: Jeffrey Galkowski and Matthieu Léautaud
Journal: Trans. Amer. Math. Soc. 373 (2020), 3177-3233
MSC (2010): Primary 35L05, 93B07, 93B05, 35K05, 35P20
DOI: https://doi.org/10.1090/tran/7938
Published electronically: February 11, 2020
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Abstract: We consider a compact Riemannian manifold $ M$ (possibly with boundary) and $ \Sigma \subset M\setminus \partial M$ an interior hypersurface (possibly with boundary). We study observation and control from $ \Sigma $ for both the wave and heat equations. For the wave equation, we prove controllability from $ \Sigma $ in time $ T$ under the assumption $ (\mathcal {T}$GCC) that all generalized bicharacteristics intersect $ \Sigma $ transversally in the time interval $ (0,T)$. For the heat equation we prove unconditional controllability from $ \Sigma $. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on $ \Sigma $ under $ \mathcal {T}$GCC and unconditional exponential lower bounds on such Cauchy data.


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Additional Information

Jeffrey Galkowski
Affiliation: Department of Mathematics, University College, London, United Kingdom
Email: j.galkowski@ucl.ac.uk

Matthieu Léautaud
Affiliation: Département de Mathématiques, Université Paris-Sud, France
Email: matthieu.leautaud@math.u-psud.fr

DOI: https://doi.org/10.1090/tran/7938
Received by editor(s): March 4, 2019
Received by editor(s) in revised form: June 14, 2019
Published electronically: February 11, 2020
Additional Notes: The first author is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661.
The second author was partially supported by the Agence Nationale de la Recherche under grants GERASIC ANR-13-BS01-0007-01 and ISDEEC ANR-16-CE40-0013.
Article copyright: © Copyright 2020 by the authors under \href{https://creativecommons.org/licenses/by/4.0/}Creative Commons Attribution 4.0 International (CC BY 4.0)