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: filtering the high frequencies, controlling projections on subspaces where the high frequencies have been filtered; 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.
Mots clés : beam equation, finite difference semi-discretization, exact boundary controllability
@article{COCV_2002__8__827_0, 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://www.numdam.org/articles/10.1051/cocv:2002025/} }
TY - JOUR AU - León, Liliana AU - Zuazua, Enrique TI - Boundary controllability of the finite-difference space semi-discretizations of the beam equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 827 EP - 862 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002025/ DO - 10.1051/cocv:2002025 LA - en ID - COCV_2002__8__827_0 ER -
%0 Journal Article %A León, Liliana %A Zuazua, Enrique %T Boundary controllability of the finite-difference space semi-discretizations of the beam equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 827-862 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2002025/ %R 10.1051/cocv:2002025 %G en %F COCV_2002__8__827_0
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. http://www.numdam.org/articles/10.1051/cocv:2002025/
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