Skip to main content
SpringerLink
Log in

Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]).

Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L2(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L2-norm control. We illustrate the mathematical results with several numerical experiments.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avdonin, S.A., Ivanov, S.A.: Families of exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press, 1995

  2. Banks, H.T., Ito, K., Wang, C.: Exponentially stable approximations of weakly damped wave equations. Estimation and control of distributed parameter systems (Vorau, 1990), Internat. Ser. Numer. Math., 100, Birkhäuser, Basel, 1991, pp. 1–33

  3. Brezzi, F., Fortin, M.: Mixed and finite elements method. Springer-Verlag, New York, 1991

  4. Glowinski, R.: Ensuring well-posedness by analogy; Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103, 189–221 (1992)

    Google Scholar 

  5. Glowinski, R., Kinton, W., Wheeler, M.F.: A mixed finite element formulation for the boundary controllability of the wave equation. Int. J. Numer. Meth. Eng. 27(3), 623–636 (1989)

    Google Scholar 

  6. 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. Jap. J. Appl. Math. 7, 1–76 (1990)

    Google Scholar 

  7. Glowinski, R., Lions, J.-L.: Exact and approximate controllability for distributed parameter systems. Acta Numerica 1996, pp. 159–333

  8. Infante, J.A., Zuazua, E.: Boundary observability for the space semi-discretization of the 1-D wave equation. M2AN 33(2), 407–438 (1999)

    Google Scholar 

  9. Ingham, A.E.: Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41, 367–369 (1936)

    Google Scholar 

  10. Kappel, F., Ito, K.: The Trotter-Kato theorem and approximations of PDE's. Math. Comput. 67, 21–44 (1998)

    Google Scholar 

  11. Lions, J.-L.: Contrôlabilité exacte perturbations et stabilisation de systèmes distribués. Tome 1, Masson, Paris, 1988

  12. Micu, S.: Uniform boundary controllability of a semi-discrete 1-D wave equation. Numerische Mathematik 91, 723–768 (2002)

    Google Scholar 

  13. Negreanu, M.: PhD Thesis, Universidad Complutense de Madrid, 2004

  14. Negreanu, M., Zuazua, E.: Uniform boundary controllability of a discrete 1-D wave equation. System and Control Letters 48(3–4), 261–280 (2003)

    Google Scholar 

  15. Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: P.G. Ciarlet, J.-L. Lions (eds.), Handbook of Numerical Analysis, vol. 2, North-Holland, Amsterdam, The Netherlands, 1989

  16. Raviart, P.A., Thomas, J.-M.: Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, Paris, 1983

  17. Russell, D.L.: Controllability and stabilization theory for linear partial differential equations: recent progress and open questions. SIAM Review 20(4), 639–737 (1978)

    Google Scholar 

  18. Zuazua, E.: Boundary observability for the finite difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78, 523–563 (1999)

    Google Scholar 

  19. Zuazua, E.: Propagation, Observation, Control and Numerical Approximation of Waves approximated by finite difference methods. SIAM Review 47(2), 197–243 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040, Madrid, Spain

    Carlos Castro

  2. Facultatea de Matematica-Informatica, Universitatea din Craiova, 1100, Romania

    Sorin Micu

Authors
  1. Carlos Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sorin Micu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Castro.

Additional information

Supported by Grant BFM 2002-03345 of MCYT (Spain) and the TMR projects of the EU ``Homogenization and Multiple Scales" and ``New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulations".

Partially Supported by Grant BFM 2002-03345 of MCYT (Spain), Grant 17 of Egide-Brancusi Program and Grant 80/2005 of CNCSIS (Romania).

Rights and permissions

Reprints and permissions

About this article

Cite this article

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, 413–462 (2006). https://doi.org/10.1007/s00211-005-0651-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0651-0

Mathematics Subject Classification (2000)

Search

Navigation