Abstract
This article deals with the approximation of the boundary controls of a 1-D linear equation modeling the transversal vibrations of a hinged beam using a finite-difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh size and the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the result of the approximation process. We prove that the convergence of the scheme is ensured if the continuous initial data are sufficiently regular or if the highest frequencies of their discretization have been filtered out. In both cases, the minimal weighted \(L^2\)-norm discrete controls are shown to be convergent to the corresponding continuous one when the mesh size tends to zero.
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Acknowledgments
The first author was partially supported by Grant PN-II-ID-PCE-2011-3-0257 of the Romanian National Authority for Scientific Research, CNCS UEFISCDI and by Grant MTM2011-29306 funded by MICINN (Spain). The second and the third authors were partially supported by Grant PN-II-ID-PCE-2011-3-0257 of the Romanian National Authority for Scientific Research, CNCS UEFISCDI. The authors wish to thank the anonymous referees for their interesting and constructive suggestions which have greatly improved the first version of this paper.
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Micu, S., Rovenţa, I. & Temereancă, L.E. Approximation of the controls for the linear beam equation. Math. Control Signals Syst. 28, 12 (2016). https://doi.org/10.1007/s00498-016-0161-x
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DOI: https://doi.org/10.1007/s00498-016-0161-x
Keywords
- Beam equation
- Boundary control approximation
- HUM controls
- Filtering
- Regularity
- Moment problem
- Biorthogonals