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How to estimate observability constants of one-dimensional wave equations? Propagation versus spectral methods

By Alain Haraux, Thibault Liard and Yannick Privat


For a given bounded connected domain in $\R^n$ , the issue of computing the observability constant associated to a wave operator, an observation time $T$ and a generic observation subdomain constitutes in general a hard task, even for one-dimensional problems. In this work, we introduce and describe two methods to provide precise (and even sharp in some cases) estimates of observability constants for general one dimensional wave equations: the first one uses a spectral decomposition of the solution of the wave equation whereas the second one is based on a propagation argument along the characteristics. Both methods are extensively described and we then comment on the advantages and drawbacks of each one. The discussion is illustrated by several examples and numerical simulations. As a byproduct, we deduce from the main results estimates of the cost of control (resp. the decay rate of the energy) for several controlled (resp. damped) wave equations

Topics: Ingham's inequality, Sturm-Liouville problems, eigenvalues, wave equation, characteristics method, 35L05, 93B07, 35Q93, 35B35, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]
Publisher: Springer Verlag
Year: 2016
DOI identifier: 10.1007/s00028-016-0321-y
OAI identifier: oai:HAL:hal-01057663v2
Provided by: Hal-Diderot

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