Uniform observability estimates for linear waves^∗,∗∗

¹ CNRS UMR 7598 and UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
laurent@ann.jussieu.fr
² Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, Bâtiment Sophie Germain, 75205 Paris cedex 13, France.
leautaud@math.univ-paris-diderot.fr

Received: 6 January 2016
Accepted: 7 June 2016

Abstract

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.