An Ingham type proof for a two-grid observability theorem

Paola Loreti; Michel Mehrenberger

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 14, Issue: 3, page 604-631
  • ISSN: 1292-8119

Abstract

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Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time T > 2 2 ; this time, if the observation is made in ( - T / 2 , T / 2 ) , is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I338 (2004) 413–418]. Our proof follows an Ingham type approach.

How to cite

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Loreti, Paola, and Mehrenberger, Michel. "An Ingham type proof for a two-grid observability theorem." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 604-631. <http://eudml.org/doc/90886>.

@article{Loreti2007,
abstract = { Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time $T>2\sqrt\{2\}$; this time, if the observation is made in $(-T/2,T/2)$, is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I338 (2004) 413–418]. Our proof follows an Ingham type approach. },
author = {Loreti, Paola, Mehrenberger, Michel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Uniform observability; two-grid method; Ingham type theorem; uniform observability; 1D-wave equation},
language = {eng},
month = {12},
number = {3},
pages = {604-631},
publisher = {EDP Sciences},
title = {An Ingham type proof for a two-grid observability theorem},
url = {http://eudml.org/doc/90886},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Loreti, Paola
AU - Mehrenberger, Michel
TI - An Ingham type proof for a two-grid observability theorem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/12//
PB - EDP Sciences
VL - 14
IS - 3
SP - 604
EP - 631
AB - Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time $T>2\sqrt{2}$; this time, if the observation is made in $(-T/2,T/2)$, is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I338 (2004) 413–418]. Our proof follows an Ingham type approach.
LA - eng
KW - Uniform observability; two-grid method; Ingham type theorem; uniform observability; 1D-wave equation
UR - http://eudml.org/doc/90886
ER -

References

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  1. C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1D wave equation derived from a mixed finite element method. Numer. Math.102 (2006) 413–462.  Zbl1102.93004
  2. R. Glowinski, C.H. Li and J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation (I), Dirichlet controls: Description of the numerical methods. Japan. J. Appl. Math.7 (1990) 1–76.  Zbl0699.65055
  3. A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl.68 (1989) 457–465.  Zbl0685.93039
  4. L. Ignat, Propiedades cualitativas de esquemas numéricos de aproximción de ecuaciones de difusión y de dispersión. Ph.D. thesis, Universidad Autónoma de Madrid, Spain (2006).  
  5. J.A. Infante and E. Zuazua, Boundary observability for the space discretization of the 1D wave equation. ESAIM: M2AN33 (1999) 407–438.  
  6. A.E. Ingham, Some trigonometrical inequalities with applications in the theory of series. Math. Z.41 (1936) 367–379.  Zbl0014.21503
  7. V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. Wiley, Chichester; Masson, Paris (1994).  Zbl0937.93003
  8. V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York (2005).  Zbl1094.49002
  9. J.-L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte. Masson, Paris, RMA 8 (1988).  
  10. P. Loreti and V. Valente, Partial exact controllability for spherical membranes. SIAM J. Control Optim.35 (1997) 641–653.  Zbl0879.93002
  11. S. Micu, Uniform boundary controllability of a semi-discrete 1D wave equation. Numer. Math.91 (2002) 723–766.  Zbl1002.65072
  12. S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise. SIAM J. Cont. Optim.35 (1997) 1614–1638.  Zbl0888.35017
  13. A. Münch, Family of implicit and controllable schemes for the 1D wave equation. C. R. Acad. Sci. Paris Sér. I339 (2004) 733–738.  Zbl1061.65054
  14. M. Negreanu, Numerical methods for the analysis of the propagation, observation and control of waves. Ph.D. thesis, Universidad Complutense Madrid, Spain (2003). Available at  URIhttp://www.uam.es/proyectosinv/cen/indocumentos.html
  15. M. Negreanu and E. Zuazua, Convergence of a multigrid method for the controllability of a 1D wave equation. C. R. Acad. Sci. Paris, Sér. I338 (2004) 413–418.  Zbl1038.65054
  16. M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications. SIAM J. Numer. Anal.44 (2006) 412–448.  Zbl1142.93351
  17. E. Zuazua, Propagation, observation, control and numerical approximation of waves approximated by finite difference methods. SIAM Rev.47 (2005) 197–243.  Zbl1077.65095
  18. E. Zuazua, Control and numerical approximation of the wave and heat equations, in Proceedings of the ICM 2006, Vol. III, “Invited Lectures", European Mathematical Society Publishing House, M. Sanz-Solé et al. Eds. (2006) 1389–1417.  Zbl1108.93023

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