# An Ingham type proof for a two-grid observability theorem

Michel Mehrenberger; Paola Loreti

ESAIM: Control, Optimisation and Calculus of Variations (2008)

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

## Access Full Article

top## Abstract

top## How to cite

topMehrenberger, Michel, and Loreti, Paola. "An Ingham type proof for a two-grid observability theorem." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2008): 604-631. <http://eudml.org/doc/244699>.

@article{Mehrenberger2008,

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. I 338 (2004) 413–418]. Our proof follows an Ingham type approach.},

author = {Mehrenberger, Michel, Loreti, Paola},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {uniform observability; two-grid method; Ingham type theorem; 1D-wave equation},

language = {eng},

number = {3},

pages = {604-631},

publisher = {EDP-Sciences},

title = {An Ingham type proof for a two-grid observability theorem},

url = {http://eudml.org/doc/244699},

volume = {14},

year = {2008},

}

TY - JOUR

AU - Mehrenberger, Michel

AU - Loreti, Paola

TI - An Ingham type proof for a two-grid observability theorem

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2008

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. I 338 (2004) 413–418]. Our proof follows an Ingham type approach.

LA - eng

KW - uniform observability; two-grid method; Ingham type theorem; 1D-wave equation

UR - http://eudml.org/doc/244699

ER -

## References

top- [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. Zbl1102.93004MR2207268
- [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. Zbl0699.65055MR1039237
- [3] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire. J. Math. Pures Appl. 68 (1989) 457–465. Zbl0685.93039MR1046761
- [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: M2AN 33 (1999) 407–438. Zbl0947.65101MR1700042
- [6] A.E. Ingham, Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367–379. Zbl0014.21503MR1545625
- [7] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. Wiley, Chichester; Masson, Paris (1994). Zbl0937.93003MR1359765
- [8] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York (2005). Zbl1094.49002MR2114325
- [9] J.-L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte. Masson, Paris, RMA 8 (1988). Zbl0653.93002MR953547
- [10] P. Loreti and V. Valente, Partial exact controllability for spherical membranes. SIAM J. Control Optim. 35 (1997) 641–653. Zbl0879.93002MR1436643
- [11] S. Micu, Uniform boundary controllability of a semi-discrete 1D wave equation. Numer. Math. 91 (2002) 723–766. Zbl1002.65072MR1912914
- [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. Zbl0888.35017MR1466919
- [13] A. Münch, Family of implicit and controllable schemes for the 1D wave equation. C. R. Acad. Sci. Paris Sér. I 339 (2004) 733–738. Zbl1061.65054MR2110946
- [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 http://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. I 338 (2004) 413–418. Zbl1038.65054MR2057174
- [16] M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications. SIAM J. Numer. Anal. 44 (2006) 412–448. Zbl1142.93351MR2217389
- [17] E. Zuazua, Propagation, observation, control and numerical approximation of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. Zbl1077.65095MR2179896
- [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. Zbl1108.93023MR2275734

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.