Skip to main content

A Spectral Numerical Method to Approximate the Boundary Controllability of the Wave Equation with Variable Coefficients

  • Chapter
  • First Online:
Research in PDEs and Related Fields
  • 199 Accesses

Abstract

We present a spectral numerical method to approximate the boundary control of the wave equation with a non-constant potential. The numerical implementation is described and some numerical experiments show the efficiency of the method.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abdelli, C. Castro, Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation. ESAIM COCV 27(61) (2021)

    Google Scholar 

  2. C. Castro, S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102, 413–462 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Castro, S. Micu, Uniform boundary observability of a linear semi-discrete 1-D wave equation with variable coefficients derived from a mixed finite element method, preprint

    Google Scholar 

  4. C. Castro, S. Micu, A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Numer. Anal. 28, 186–214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Ervedoza, E. Zuazua, A systematic method for building smooth controls for smooth data. Discrete Contin. Dynam. Systems Series B 14, 1375–1401 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Ervedoza, E. Zuazua Numerical Approximation of Exact Controls for Waves (Springer, Berlin, 2013)

    Google Scholar 

  7. R. Glowinski, J.-L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numerica 3, 269–378 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, in Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, vol. 8 (1988). Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau, J. Rauch

    Google Scholar 

  9. P.-A. Raviart, J.-M. Thomas, Introduction l’analyse numrique des quations aux drives partielles (Dunod, Paris, 1998).

    Google Scholar 

  10. E. Zuazua, Exact controllability for semilinear wave equations in one space dimension. Ann. de ’IHP, Section C 10(1), 109–129 (1993)

    Google Scholar 

  11. E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Bardos, F. Bourquin, G. Lebeau, Calcul de dérivées normales et méthode de Galerkin appliqué au problème de contrôllabilité exacte. CR Acad. Sci. Paris 313(I), 757–760 (1991)

    MATH  Google Scholar 

  13. F. Bourquin, Approximation theory for the problem of exact controllability of the wave equation with boundary control. In: Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993). SIAM, Philadelphia (1993) 103–112

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carlos Castro .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Castro, C. (2022). A Spectral Numerical Method to Approximate the Boundary Controllability of the Wave Equation with Variable Coefficients. In: Ammari, K. (eds) Research in PDEs and Related Fields. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-14268-0_3

Download citation

Publish with us

Policies and ethics