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In this article, we give a completely constructive proof of the observability/controllability of the wave equation on a compact manifold under optimal geometric conditions. This contrasts with the original proof of Bardos–Lebeau–Rauch [C. Bardos, G. Lebeau and J. Rauch, SIAM J. Control Optim. 30 (1992) 1024–1065], which contains two non-constructive arguments. Our method is based on the Dehman-Lebeau [B. Dehman and G. Lebeau, SIAM J. Control Optim. 48 (2009) 521–550] Egorov approach to treat the high-frequencies, and the optimal unique continuation stability result of the authors [C. Laurent and M. Léautaud. Preprint arXiv:1506.04254 (2015)] for the low-frequencies. As an application, we first give estimates of the blowup of the observability constant when the time tends to the limit geometric control time (for wave equations with possibly lower order terms). Second, we provide (on manifolds with or without boundary) with an explicit dependence of the observability constant with respect to the addition of a bounded potential to the equation.
Laurent, Camille 1 ; Léautaud, Matthieu 2
@article{COCV_2016__22_4_1097_0,
author = {Laurent, Camille and L\'eautaud, Matthieu},
title = {Uniform observability estimates for linear waves},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1097--1136},
publisher = {EDP-Sciences},
volume = {22},
number = {4},
year = {2016},
doi = {10.1051/cocv/2016046},
mrnumber = {3570496},
zbl = {1368.35163},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016046/}
}
TY - JOUR AU - Laurent, Camille AU - Léautaud, Matthieu TI - Uniform observability estimates for linear waves JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1097 EP - 1136 VL - 22 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016046/ DO - 10.1051/cocv/2016046 LA - en ID - COCV_2016__22_4_1097_0 ER -
%0 Journal Article %A Laurent, Camille %A Léautaud, Matthieu %T Uniform observability estimates for linear waves %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1097-1136 %V 22 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016046/ %R 10.1051/cocv/2016046 %G en %F COCV_2016__22_4_1097_0
Laurent, Camille; Léautaud, Matthieu. Uniform observability estimates for linear waves. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1097-1136. doi: 10.1051/cocv/2016046
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