Resolvent estimates in controllability theory and applications to the discrete wave equation

Sylvain Ervedoza[1]

  • [1] Institut de Mathématiques de Toulouse & CNRS, Université Paul Sabatier (Toulouse 3), 118 route de Narbonne, F31062 Toulouse Cedex 9, France.

Journées Équations aux dérivées partielles (2009)

  • Volume: 113, Issue: 3, page 1-18
  • ISSN: 0752-0360

Abstract

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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].

How to cite

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Ervedoza, Sylvain. "Resolvent estimates in controllability theory and applications to the discrete wave equation." Journées Équations aux dérivées partielles 113.3 (2009): 1-18. <http://eudml.org/doc/116373>.

@article{Ervedoza2009,
abstract = {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].},
affiliation = {Institut de Mathématiques de Toulouse & CNRS, Université Paul Sabatier (Toulouse 3), 118 route de Narbonne, F31062 Toulouse Cedex 9, France.},
author = {Ervedoza, Sylvain},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
number = {3},
pages = {1-18},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Resolvent estimates in controllability theory and applications to the discrete wave equation},
url = {http://eudml.org/doc/116373},
volume = {113},
year = {2009},
}

TY - JOUR
AU - Ervedoza, Sylvain
TI - Resolvent estimates in controllability theory and applications to the discrete wave equation
JO - Journées Équations aux dérivées partielles
DA - 2009/6//
PB - Groupement de recherche 2434 du CNRS
VL - 113
IS - 3
SP - 1
EP - 18
AB - 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].
LA - eng
UR - http://eudml.org/doc/116373
ER -

References

top
  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. Zbl0995.65501MR1857191
  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. Zbl0850.93719MR1155634
  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. Zbl0786.93009MR1178650
  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. Zbl0906.93008MR1483711
  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. Zbl1050.35058MR2051618
  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. Zbl1102.93004MR2207268
  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. Zbl0967.34074MR1760033
  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. Zbl1016.35003MR1921162
  9. R. Dáger and E. Zuazua. Wave propagation, observation and control in 1 - d flexible multi-structures, volume 50 of Mathématiques & Applications (Berlin). Springer-Verlag, Berlin, 2006. Zbl1083.74002MR2169126
  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. Zbl1192.35109MR2654195
  11. S. Ervedoza. Observability in arbitrary small time for discrete approximations of conservative systems. Preprint, 2009. Zbl1234.93026
  12. S. Ervedoza. Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math., 113(3):377–415, 2009. Zbl1170.93013MR2534130
  13. S. Ervedoza. Admissibility and observability for Schrödinger systems: Applications to finite element approximation schemes. Asymptot. Anal., To appear. Zbl1170.93013
  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. Zbl1143.65044MR2418618
  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. Zbl1148.65070MR2407324
  16. S. Ervedoza and E. Zuazua. Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl., 91:20–48, 2009. Zbl1163.74019MR2487899
  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. Zbl1201.35046MR2664618
  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. Zbl0763.76042MR1196839
  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. Zbl0789.65048MR1227985
  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. Zbl0685.93039MR1046761
  21. A. Haraux. Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Port. Math., 46(3):245–258, 1989. Zbl0679.93063MR1021188
  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. Zbl1063.35016MR2135236
  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. Zbl1079.65090MR2169157
  24. L. I. Ignat and E. Zuazua. Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal., 47(2):1366–1390, 2009. Zbl1192.65127MR2485456
  25. O. Y. Imanuvilov. On Carleman estimates for hyperbolic equations. Asymptot. Anal., 32(3-4):185–220, 2002. Zbl1050.35046MR1993649
  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. Zbl0947.65101MR1700042
  27. A. E. Ingham. Some trigonometrical inequalities with applications to the theory of series. Math. Z., 41(1):367–379, 1936. Zbl0014.21503MR1545625
  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. Zbl0702.93010MR1425495
  29. V. Komornik. Exact controllability and stabilization. RAM: Research in Applied Mathematics. Masson, Paris, 1994. The multiplier method. Zbl0937.93003MR1359765
  30. G. Lebeau. Équations des ondes amorties. Séminaire sur les Équations aux Dérivées Partielles, 1993–1994,École Polytech., 1994. Zbl0887.35091MR1300911
  31. J.-L. Lions. Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, volume RMA 8. Masson, 1988. Zbl0653.93002
  32. K. Liu. Locally distributed control and damping for the conservative systems. SIAM J. Control Optim., 35(5):1574–1590, 1997. Zbl0891.93016MR1466917
  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. Zbl1048.93047MR2077998
  34. L. Miller. Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal., 218(2):425–444, 2005. Zbl1122.93011MR2108119
  35. C. S. Morawetz. Decay for solutions of the exterior problem for the wave equation. Comm. Pure Appl. Math., 28:229–264, 1975. Zbl0304.35064MR372432
  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. Zbl1130.93016MR2143953
  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. Zbl1120.65101MR2306636
  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. Zbl1131.65056MR2272192
  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. Zbl1038.65054MR2057174
  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. Zbl0997.93013MR1871454
  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. Zbl1140.93395MR2158180
  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. Zbl1126.93050MR2329173
  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. Zbl0561.65069MR773854
  44. L. R. Tcheugoué Tebou and E. Zuazua. Uniform boundary stabilization of the finite difference space discretization of the 1 d wave equation. Adv. Comput. Math., 26(1-3):337–365, 2007. Zbl1119.65086MR2350359
  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. Zbl1033.65080MR2012934
  46. L. N. Trefethen. Group velocity in finite difference schemes. SIAM Rev., 24(2):113–136, 1982. Zbl0487.65055MR652463
  47. G. Weiss. Admissibility of unbounded control operators. SIAM J. Control Optim., 27(3):527–545, 1989. Zbl0685.93043MR993285
  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. Zbl0976.93038MR1809954
  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. Zbl0939.93016MR1697041
  51. E. Zuazua. Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev., 47(2):197–243 (electronic), 2005. Zbl1077.65095MR2179896

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