The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math. 102 (2006) 413-462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.
Mots clés : spectrum, observability, wave equation, semi-discrete systems, controllability, stabilization
@article{COCV_2010__16_2_298_0,
author = {Ervedoza, Sylvain},
title = {Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {298--326},
publisher = {EDP-Sciences},
volume = {16},
number = {2},
year = {2010},
doi = {10.1051/cocv:2008071},
mrnumber = {2654195},
zbl = {1192.35109},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008071/}
}
TY - JOUR AU - Ervedoza, Sylvain TI - Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 298 EP - 326 VL - 16 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008071/ DO - 10.1051/cocv:2008071 LA - en ID - COCV_2010__16_2_298_0 ER -
%0 Journal Article %A Ervedoza, Sylvain %T Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 298-326 %V 16 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008071/ %R 10.1051/cocv:2008071 %G en %F COCV_2010__16_2_298_0
Ervedoza, Sylvain. Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 298-326. doi : 10.1051/cocv:2008071. http://www.numdam.org/articles/10.1051/cocv:2008071/
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