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

How to estimate observability constants of one-dimensional wave equations? Propagation versus... For a given bounded connected domain in $${{\mathbbm{R}}^n}$$ R n , the issue of computing the observability constant associated with 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 by-product, 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

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

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References (27)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-016-0321-y
Publisher site
See Article on Publisher Site

Abstract

For a given bounded connected domain in $${{\mathbbm{R}}^n}$$ R n , the issue of computing the observability constant associated with 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 by-product, 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.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Mar 5, 2016

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