Resolvent estimates in controllability theory and applications to the discrete wave equation
Journées équations aux dérivées partielles (2009), article no. 2, 18 p.

We briefly present the difficulties arising when dealing with the controllability of the discrete wave equation, which are, roughly speaking, created by high-frequency spurious waves which do not travel. It is by now well-understood that such spurious waves can be dealt with by applying some convenient filtering technique. However, the scale of frequency in which we can guarantee that none of these non-traveling waves appears is still unknown in general. Though, using Hautus tests, which read the controllability of a given system in terms of resolvent estimates, we are able to prove that these spurious waves do not appear before some frequency scale. This document is based on the articles [12, 13, 14].

DOI: 10.5802/jedp.55
Ervedoza, Sylvain 1

1 Institut de Mathématiques de Toulouse & CNRS, Université Paul Sabatier (Toulouse 3), 118 route de Narbonne, F31062 Toulouse Cedex 9, France.
@article{JEDP_2009____A2_0,
author = {Ervedoza, Sylvain},
title = {Resolvent estimates in controllability theory and applications to the discrete wave equation},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {2},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2009},
doi = {10.5802/jedp.55},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jedp.55/}
}
TY  - JOUR
AU  - Ervedoza, Sylvain
TI  - Resolvent estimates in controllability theory and applications to the discrete wave equation
JO  - Journées équations aux dérivées partielles
PY  - 2009
PB  - Groupement de recherche 2434 du CNRS
UR  - http://archive.numdam.org/articles/10.5802/jedp.55/
UR  - https://doi.org/10.5802/jedp.55
DO  - 10.5802/jedp.55
LA  - en
ID  - JEDP_2009____A2_0
ER  - 
%0 Journal Article
%A Ervedoza, Sylvain
%T Resolvent estimates in controllability theory and applications to the discrete wave equation
%J Journées équations aux dérivées partielles
%D 2009
%I Groupement de recherche 2434 du CNRS
%U https://doi.org/10.5802/jedp.55
%R 10.5802/jedp.55
%G en
%F JEDP_2009____A2_0
Ervedoza, Sylvain. Resolvent estimates in controllability theory and applications to the discrete wave equation. Journées équations aux dérivées partielles (2009), article  no. 2, 18 p. doi : 10.5802/jedp.55. http://archive.numdam.org/articles/10.5802/jedp.55/

[1] I. Babuška and T. Strouboulis. The finite element method and its reliability. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 2001. | MR

[2] H. T. Banks, K. Ito, and C. Wang. Exponentially stable approximations of weakly damped wave equations. In Estimation and control of distributed parameter systems (Vorau, 1990), volume 100 of Internat. Ser. Numer. Math., pages 1–33. Birkhäuser, Basel, 1991. | MR | Zbl

[3] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control and Optim., 30(5):1024–1065, 1992. | MR | Zbl

[4] N. Burq and P. Gérard. Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math., 325(7):749–752, 1997. | MR | Zbl

[5] N. Burq and M. Zworski. Geometric control in the presence of a black box. J. Amer. Math. Soc., 17(2):443–471 (electronic), 2004. | MR | Zbl

[6] C. Castro and S. Micu. Boundary controllability of a linear semi-discrete 1-d wave equation derived from a mixed finite element method. Numer. Math., 102(3):413–462, 2006. | MR | Zbl

[7] C. Castro and E. Zuazua. Low frequency asymptotic analysis of a string with rapidly oscillating density. SIAM J. Appl. Math., 60(4):1205–1233 (electronic), 2000. | MR | Zbl

[8] C. Castro and E. Zuazua. Concentration and lack of observability of waves in highly heterogeneous media. Arch. Ration. Mech. Anal., 164(1):39–72, 2002. | MR | Zbl

[9] R. Dáger and E. Zuazua. Wave propagation, observation and control in $1\text{-}d$ flexible multi-structures, volume 50 of Mathématiques & Applications (Berlin). Springer-Verlag, Berlin, 2006. | MR | Zbl

[10] S. Ervedoza. Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes. ESAIM Control Optim. Calc. Var., Preprint, 2008. | Numdam | MR | Zbl

[11] S. Ervedoza. Observability in arbitrary small time for discrete approximations of conservative systems. Preprint, 2009.

[12] S. Ervedoza. Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math., 113(3):377–415, 2009. | MR | Zbl

[13] S. Ervedoza. Admissibility and observability for Schrödinger systems: Applications to finite element approximation schemes. Asymptot. Anal., To appear.

[14] S. Ervedoza, C. Zheng, and E. Zuazua. On the observability of time-discrete conservative linear systems. J. Funct. Anal., 254(12):3037–3078, June 2008. | MR | Zbl

[15] S. Ervedoza and E. Zuazua. Perfectly matched layers in 1-d: Energy decay for continuous and semi-discrete waves. Numer. Math., 109(4):597–634, 2008. | MR | Zbl

[16] S. Ervedoza and E. Zuazua. Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl., 91:20–48, 2009. | MR | Zbl

[17] S. Ervedoza and E. Zuazua. Uniform exponential decay for viscous damped systems. In Proc. of Siena “Phase Space Analysis of PDEs 2007", Special issue in honor of Ferrucio Colombini, To appear. | MR

[18] R. Glowinski. Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys., 103(2):189–221, 1992. | MR | Zbl

[19] E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equations. I, volume 8 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems. | MR | Zbl

[20] A. Haraux. Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire. J. Math. Pures Appl. (9), 68(4):457–465 (1990), 1989. | MR | Zbl

[21] A. Haraux. Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Port. Math., 46(3):245–258, 1989. | MR | Zbl

[22] L. I. Ignat and E. Zuazua. Dispersive properties of a viscous numerical scheme for the Schrödinger equation. C. R. Math. Acad. Sci. Paris, 340(7):529–534, 2005. | MR | Zbl

[23] L. I. Ignat and E. Zuazua. A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence. C. R. Math. Acad. Sci. Paris, 341(6):381–386, 2005. | MR | Zbl

[24] L. I. Ignat and E. Zuazua. Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal., 47(2):1366–1390, 2009. | MR | Zbl

[25] O. Y. Imanuvilov. On Carleman estimates for hyperbolic equations. Asymptot. Anal., 32(3-4):185–220, 2002. | MR | Zbl

[26] J.A. Infante and E. Zuazua. Boundary observability for the space semi discretizations of the 1-d wave equation. Math. Model. Num. Ann., 33:407–438, 1999. | Numdam | MR | Zbl

[27] A. E. Ingham. Some trigonometrical inequalities with applications to the theory of series. Math. Z., 41(1):367–379, 1936. | MR | Zbl

[28] V. Komornik. A new method of exact controllability in short time and applications. Ann. Fac. Sci. Toulouse Math. (5), 10(3):415–464, 1989. | Numdam | MR | Zbl

[29] V. Komornik. Exact controllability and stabilization. RAM: Research in Applied Mathematics. Masson, Paris, 1994. The multiplier method. | MR | Zbl

[30] G. Lebeau. Équations des ondes amorties. Séminaire sur les Équations aux Dérivées Partielles, 1993–1994,École Polytech., 1994. | Numdam | MR | Zbl

[31] J.-L. Lions. Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, volume RMA 8. Masson, 1988. | Zbl

[32] K. Liu. Locally distributed control and damping for the conservative systems. SIAM J. Control Optim., 35(5):1574–1590, 1997. | MR | Zbl

[33] F. Macià. The effect of group velocity in the numerical analysis of control problems for the wave equation. In Mathematical and numerical aspects of wave propagation—WAVES 2003, pages 195–200. Springer, Berlin, 2003. | MR | Zbl

[34] L. Miller. Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal., 218(2):425–444, 2005. | MR | Zbl

[35] C. S. Morawetz. Decay for solutions of the exterior problem for the wave equation. Comm. Pure Appl. Math., 28:229–264, 1975. | MR | Zbl

[36] A. Münch. A uniformly controllable and implicit scheme for the 1-D wave equation. M2AN Math. Model. Numer. Anal., 39(2):377–418, 2005. | Numdam | MR | Zbl

[37] A. Münch and A. F. Pazoto. Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Control Optim. Calc. Var., 13(2):265–293 (electronic), 2007. | Numdam | MR | Zbl

[38] M. Negreanu, A.-M. Matache, and C. Schwab. Wavelet filtering for exact controllability of the wave equation. SIAM J. Sci. Comput., 28(5):1851–1885 (electronic), 2006. | MR | Zbl

[39] M. Negreanu and E. Zuazua. Convergence of a multigrid method for the controllability of a 1-d wave equation. C. R. Math. Acad. Sci. Paris, 338(5):413–418, 2004. | MR | Zbl

[40] A. Osses. A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control. SIAM J. Control Optim., 40(3):777–800 (electronic), 2001. | MR | Zbl

[41] K. Ramdani, T. Takahashi, G. Tenenbaum, and M. Tucsnak. A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator. J. Funct. Anal., 226(1):193–229, 2005. | MR | Zbl

[42] K. Ramdani, T. Takahashi, and M. Tucsnak. Uniformly exponentially stable approximations for a class of second order evolution equations—application to LQR problems. ESAIM Control Optim. Calc. Var., 13(3):503–527, 2007. | Numdam | MR | Zbl

[43] P.-A. Raviart and J.-M. Thomas. Introduction à l’analyse numérique des équations aux dérivées partielles. Collection Mathématiques Appliquées pour la Maitrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris, 1983. | MR | Zbl

[44] L. R. Tcheugoué Tebou and E. Zuazua. Uniform boundary stabilization of the finite difference space discretization of the $1d$ wave equation. Adv. Comput. Math., 26(1-3):337–365, 2007. | MR | Zbl

[45] 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. Numer. Math., 95(3):563–598, 2003. | MR | Zbl

[46] L. N. Trefethen. Group velocity in finite difference schemes. SIAM Rev., 24(2):113–136, 1982. | MR | Zbl

[47] G. Weiss. Admissibility of unbounded control operators. SIAM J. Control Optim., 27(3):527–545, 1989. | MR | Zbl

[48] X. Zhang. Explicit observability estimate for the wave equation with potential and its application. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456(1997):1101–1115, 2000. | MR | Zbl

[49] X. Zhang, C. Zheng, and E. Zuazua. Exact controllability of the time discrete wave equation. Discrete and Continuous Dynamical Systems, 2007.

[50] E. Zuazua. Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. (9), 78(5):523–563, 1999. | MR | Zbl

[51] E. Zuazua. Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev., 47(2):197–243 (electronic), 2005. | MR | Zbl

Cited by Sources: