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Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation using a finite-difference method


Authors: Pierre Lissy and Ionel Rovenţa
Journal: Math. Comp. 88 (2019), 273-291
MSC (2010): Primary 93B05, 30E05, 65M06
DOI: https://doi.org/10.1090/mcom/3345
Published electronically: April 5, 2018
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Abstract: We consider a finite-difference semi-discrete scheme for the approximation of boundary controls for the one-dimensional wave equation. The high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. We prove that, by filtering the high frequencies of the initial data in an optimal range, we restore the uniform controllability property. Moreover, we obtain a relation between the range of filtration and the minimal time of control needed to ensure the uniform controllability. The proof is based on the moment method.


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Additional Information

Pierre Lissy
Affiliation: CEREMADE, Université Paris-Dauphine; and CNRS UMR 7534, PSL Research University, 75016 Paris, France
Email: lissy@ceremade.dauphine.fr

Ionel Rovenţa
Affiliation: Department of Mathematics, University of Craiova, Craiova 200585, Romania
Email: ionelroventa@yahoo.com

DOI: https://doi.org/10.1090/mcom/3345
Keywords: Wave equation, control approximation, moment problem, biorthogonal families.
Received by editor(s): March 27, 2017
Received by editor(s) in revised form: October 4, 2017
Published electronically: April 5, 2018
Additional Notes: The first author was partially supported by the project IFSMACS funded by the French Agence Nationale de la Recherche, 2015–2019 (Reference: ANR-15-CE40-0010).
The second author was supported by Romanian National Authority for Scientific Research CNCS - UEFISCDI research project PN-II-RU-TE-2014-4-0320.
Article copyright: © Copyright 2018 American Mathematical Society

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