Geodesic
Boundary controllability of the finite-difference space semi-discretizations of the beam equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 827-862

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We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.

DOI : 10.1051/cocv:2002025
Classification : 93C20, 35Q33, 65N06
Keywords: beam equation, finite difference semi-discretization, exact boundary controllability 
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     author = {Le\'on, Liliana and Zuazua, Enrique},
     title = {Boundary controllability of the finite-difference space semi-discretizations of the beam equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {827--862},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002025},
     mrnumber = {1932975},
     zbl = {1063.93025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv:2002025/}
}
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León, Liliana; Zuazua, Enrique. Boundary controllability of the finite-difference space semi-discretizations of the beam equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 827-862. doi: 10.1051/cocv:2002025

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