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Approximation of the controls for the beam equation with vanishing viscosity


Authors: Ioan Florin Bugariu, Sorin Micu and Ionel Rovenţa
Journal: Math. Comp. 85 (2016), 2259-2303
MSC (2010): Primary 93B05, 58J45, 65N06, 30E05
DOI: https://doi.org/10.1090/mcom/3064
Published electronically: February 11, 2016
MathSciNet review: 3511282
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Abstract: We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the convergence of the approximate boundary controls corresponding to initial data in the finite energy space cannot be guaranteed. We prove that, by adding a vanishing numerical viscosity, the uniform controllability property and the convergence of the scheme is ensured.


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Additional Information

Ioan Florin Bugariu
Affiliation: Department of Mathematics, University of Craiova, 200585, Romania
Email: florin$_$bugariu$_$86@yahoo.com

Sorin Micu
Affiliation: Department of Mathematics, University of Craiova, 200585 and Institute of Mathematical Statistics and Applied Mathematics, 70700, Bucharest, Romania
Email: sd$_$micu@yahoo.com

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

DOI: https://doi.org/10.1090/mcom/3064
Keywords: Beam equation, control approximation, vanishing viscosity, moment problem biorthogonals.
Received by editor(s): January 22, 2014
Received by editor(s) in revised form: September 11, 2014, and January 6, 2015
Published electronically: February 11, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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