An Ingham type proof for a two-grid observability theorem

Mehrenberger, Michel; Loreti, Paola

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 604-631.

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.

Classification : 35L05, 65M55, 93B07
Mots clés : uniform observability, two-grid method, Ingham type theorem

[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. | MR | Zbl

[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. | MR | Zbl

[3] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl. 68 (1989) 457-465. | MR | Zbl

[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. | Numdam | MR | Zbl

[6] A.E. Ingham, Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41 (1936) 367-379. | MR | Zbl

[7] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. Wiley, Chichester; Masson, Paris (1994). | MR | Zbl

[8] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York (2005). | MR | Zbl

[9] J.-L. Lions, Contrôlabilité Exacte, Stabilisation et Perturbation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte. Masson, Paris, RMA 8 (1988). | MR | Zbl

[10] P. Loreti and V. Valente, Partial exact controllability for spherical membranes. SIAM J. Control Optim. 35 (1997) 641-653. | MR | Zbl

[11] S. Micu, Uniform boundary controllability of a semi-discrete 1D wave equation. Numer. Math. 91 (2002) 723-766. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[16] M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications. SIAM J. Numer. Anal. 44 (2006) 412-448. | MR | Zbl

[17] E. Zuazua, Propagation, observation, control and numerical approximation of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. | MR | Zbl

[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. | MR | Zbl