About this Journal Submit a Manuscript Table of Contents
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.

Linked References

  1. D. G. Luenberger, “An introduction to observers,” IEEE Transactions on Automatic Control, vol. 16, no. 6, pp. 596–602, 1971. View at Publisher · View at Google Scholar
  2. P. Moireau, D. Chapelle, and P. Le Tallec, “Joint state and parameter estimation for distributed mechanical systems,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 6-8, pp. 659–677, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. D. Auroux and J. Blum, “A nudging-based data assimilation method: the back and forth nudging (BFN) algorithm,” Nonlinear Processes Geophysics, vol. 15, no. 2, pp. 305–319, 2008. View at Publisher · View at Google Scholar
  4. K. Ramdani, M. Tucsnak, and G. Weiss, “Recovering the initial state of an infinite-dimensional system using observers,” Automatica, vol. 46, no. 10, pp. 1616–1625, 2010. View at Publisher · View at Google Scholar
  5. L. R. Tcheugoué Tébou and E. Zuazua, “Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity,” Numerische Mathematik, vol. 95, no. 3, pp. 563–598, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. P. Moireau, D. Chapelle, and P. Le Tallec, “Filtering for distributed mechanical systems using position measurements: perspectives in medical imaging,” Inverse Problems, vol. 25, no. 3, Article ID 035010, 25 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. C. Bardos, G. Lebeau, and J. Rauch, “Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation des problèmes hyperboliques,” Rendiconti del Seminario Matematico del Universita Politecnico Torino, Fascicolo Speciale (Hyperbolic Equations), pp. 12–31, 1988.
  8. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar
  9. K. Liu, “Locally distributed control and damping for the conservative systems,” SIAM Journal on Control and Optimization, vol. 35, no. 5, pp. 1574–1590, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. X. Zhang, “Explicit observability estimate for the wave equation with potential and its application,” Proceedings: Mathematical, Physical and Engineering Sciences, vol. 456, no. 1997, pp. 1101–1115, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. P. Gérard, “Microlocal defect measures,” Communications in Partial Differential Equations, vol. 16, no. 11, pp. 1761–1794, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. N. Burq and G. Lebeau, “Mesures de défaut de compacité, application au système de lamé,” Annales Scientifiques de l'École Normale Supérieure, vol. 34, no. 6, pp. 817–870, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. N. Burq and P. Gérard, Contrôle optimal des équations aux derivées partielles, Cours de l'Ecole Polytechnique, 2002.
  14. D. Chapelle, N. Cîndea, and P. Moireau, “Improving convergence in numerical analysis using observers—the wave-like equation case,” Mathematical Models and Methods in Applied Sciences. In press. View at Publisher · View at Google Scholar
  15. S. Ervedoza and E. Zuazua, “Uniformly exponentially stable approximations for a class of damped systems,” Journal de Mathématiques Pures et Appliquées, vol. 91, no. 1, pp. 20–48, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH