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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 581053, 12 pages
http://dx.doi.org/10.1155/2012/581053
Research Article

Exponential Convergence of an Observer Based on Partial Field Measurements for the Wave Equation

MACS Project Team, Inria, France

Received 27 July 2012; Accepted 2 October 2012

Academic Editor: Valery Yakhno

Copyright © 2012 Dominique Chapelle et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We analyze an observer strategy based on partial—that is, in a subdomain—measurements of the solution of a wave equation, in order to compensate for uncertain initial conditions. We prove the exponential convergence of this observer under a nonstandard observability condition, whereas using measurements of the time derivative of the solution would lead to a standard observability condition arising in stabilization and exact controlabillity. Nevertheless, we directly relate our specific observability condition to the classical geometric control condition. Finally, we provide some numerical illustrations of the effectiveness of the approach.