Numerische Mathematik

, Volume 113, Issue 3, pp 377–415 | Cite as

Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes

  • Sylvain Ervedoza


In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of \({\ddot u+A_0 u=0 }\), where A 0 is an unbounded self-adjoint positive definite operator with compact resolvent. To address this problem, we present a new spectral approach based on several spectral criteria for admissibility and observability of such systems. Our approach provides very general admissibility and observability results for finite element approximation schemes of \({\ddot u+A_{0}u =0}\), which stand in any dimension and for any regular mesh (in the sense of finite elements). Our results can be combined with previous works to derive admissibility and observability properties for full discretizations of \({\ddot u+A_0 u=0}\). We also present applications of our results to controllability and stabilization problems.

Mathematics Subject Classification (2000)

35L05 35L90 65J10 93B07 93B05 93B40 93D15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Banks, H.T., Ito, K., Wang, C.: Exponentially stable approximations of weakly damped wave equations. In: Estimation and control of distributed parameter systems (Vorau, 1990) Internat. Ser. Numer. Math., vol. 100, pp. 1–33. Birkhäuser, Basel (1991)Google Scholar
  2. 2.
    Bardos C., Lebeau G., Rauch J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024–1065 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burq N., Gérard P.: Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325(7), 749–752 (1997)zbMATHGoogle Scholar
  4. 4.
    Burq N., Zworski M.: Geometric control in the presence of a black box. J. Am. Math. Soc. 17(2), 443–471 (2004) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Castro C., Micu S.: 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)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Castro C., Zuazua E.: Low frequency asymptotic analysis of a string with rapidly oscillating density. SIAM J. Appl. Math. 60(4), 1205–1233 (2000) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Castro C., Zuazua E.: Concentration and lack of observability of waves in highly heterogeneous media. Arch. Ration. Mech. Anal. 164(1), 39–72 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dáger, R., Zuazua, E.: Wave propagation, observation and control in 1−d flexible multi-structures. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 50. Springer, Berlin (2006)Google Scholar
  9. 9.
    Ervedoza, S.: Observability of the mixed finite element method for the 1d wave equation on nonuniform meshes. ESAIM Control Optim. Calc. Var. (2009, to appear)Google Scholar
  10. 10.
    Ervedoza, S.: Admissibility and observability for Schrödinger systems: applications to finite element approximation schemes. Asymptot. Anal. (2009, to appear)Google Scholar
  11. 11.
    Ervedoza, S.: Observability in arbitrary small time for discrete approximations of conservative systems (2009, preprint)Google Scholar
  12. 12.
    Ervedoza S., Zheng C., Zuazua E.: On the observability of time-discrete conservative linear systems. J. Funct. Anal. 254(12), 3037–3078 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ervedoza, S., Zuazua, E.: Uniform exponential decay for viscous damped systems. To appear in Proc. of Siena Phase Space Analysis of PDEs 2007. Special issue in honor of Ferrucio ColombiniGoogle Scholar
  14. 14.
    Ervedoza S., Zuazua E.: Perfectly matched layers in 1-d: Energy decay for continuous and semi- discrete waves. Numer. Math. 109(4), 597–634 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ervedoza S., Zuazua E.: Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91, 20–48 (2009)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Glowinski R.: Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103(2), 189–221 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Glowinski R., Li C.H., Lions J.-L.: A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods. Jpn. J. Appl. Math. 7(1), 1–76 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Haraux, A.: Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire. J. Math. Pures Appl. (9), 68(4):457–465 (1990), 1989Google Scholar
  19. 19.
    Haraux A.: Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Port. Math. 46(3), 245–258 (1989)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Imanuvilov O.Y.: On Carleman estimates for hyperbolic equations. Asymptot. Anal. 32(3-4), 185–220 (2002)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Infante J.A., Zuazua E.: Boundary observability for the space semi discretizations of the 1-d wave equation. Math. Model. Num. Ann. 33, 407–438 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ingham A.E.: Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41(1), 367–379 (1936)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Komornik, V.: Exact controllability and stabilization. RAM: Research in Applied Mathematics. Masson, Paris (1994). [The multiplier method]Google Scholar
  24. 24.
    Lebeau, G.: Équations des ondes amorties. Séminaire sur les Équations aux Dérivées Partielles, 1993–1994,École Polytech (1994)Google Scholar
  25. 25.
    Lions, J.-L.: Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, vol. RMA 8. Masson (1988)Google Scholar
  26. 26.
    Liu K.: Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35(5), 1574–1590 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Macià, F.: 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, pp. 195–200. Springer, Berlin (2003)Google Scholar
  28. 28.
    Miller L.: Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218(2), 425–444 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Miller, L.: Private communication (2009)Google Scholar
  30. 30.
    Münch, A., Pazoto, A.F.: Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Control Optim. Calc. Var. 13(2), 265–293 (2007) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Negreanu M., Matache A.-M., Schwab C.: Wavelet filtering for exact controllability of the wave equation. SIAM J. Sci. Comput. 28(5), 1851–1885 (2006) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Negreanu M., Zuazua E.: 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)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Osses A.: A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control. SIAM J. Control Optim. 40(3), 777–800 (2001) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Ramdani K., Takahashi T., Tenenbaum G., Tucsnak M.: A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator. J. Funct. Anal. 226(1), 193–229 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Ramdani K., Takahashi T., Tucsnak M.: 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)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Raviart, P.-A., Thomas, J.-M.: 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)Google Scholar
  37. 37.
    Tcheugoué Tebou L.R., Zuazua E.: Uniform boundary stabilization of the finite difference space discretization of the 1d wave equation. Adv. Comput. Math. 26(1-3), 337–365 (2007)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Tcheugoué Tébou L.R., Zuazua E.: 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)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Trefethen L.N.: Group velocity in finite difference schemes. SIAM Rev. 24(2), 113–136 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. Advanced Texts, vol. XI. Springer, Birkhäuser (2009)Google Scholar
  41. 41.
    Weiss G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27(3), 527–545 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Young R.M.: An Introduction to Nonharmonic Fourier Series, 1st edn. Academic Press Inc., San Diego (2001)zbMATHGoogle Scholar
  43. 43.
    Zhang X.: 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)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Zuazua E.: 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)zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Zuazua E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de VersaillesUniversité de Versailles Saint-Quentin-en-YvelinesVersailles CedexFrance

Personalised recommendations