Afrika Matematika

, Volume 29, Issue 3–4, pp 557–574 | Cite as

Uniform interior stabilization for the finite difference fully-discretization of the 1-D wave equation

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Abstract

In this paper, we are interested to prove the uniform exponential decay of the energy for the finite difference fully-discretization of 1-D wave equation with an interior damping at \(\xi \). To this end, we decompose the fully discrete system into two subsystems: a conservative system, and a non conservative one. We show under a numerical hypothesis on \(\xi \) that a uniform observability inequality holds for a conservative system, when the mesh size tends to zero using Fourier series technique. Then, we use the observability inequality to prove the uniform exponential decay of the energy for the damped system. Finally, we describe some numerical experiments to illustrate the exponential decay of the energy.

Keywords

Interior stabilization Pointwise internal damping Uniform exponential decay Uniform interior observability Finite difference method Fully-discretization 

Mathematics Subject Classification

93D15 93B07 65N06 65N22 

References

  1. 1.
    Ammari, K., Henrot, A., Tucsnak, M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asymptot. Anal. 28, 215–240 (2001)MathSciNetMATHGoogle Scholar
  2. 2.
    Banks, H.T., Kunisch, K.: The linear regulator problem for parabolic systems. SIAM J. Control Opt. 22, 684–698 (1984)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Banks, H.T., Wang, C.: Optimal feedback control of infinite-dimensional parabolic evolution systems: approximation techniques. SIAM J. Control Optim. 27, 1182–1219 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Banks, H.T., Ito, K., Wang, B.: Exponentially stable approximations of weakly damped wave equations. Int. Ser. Numer. Math. 100, 1–33 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bouslous, H., El Boujaoui, H., Maniar, L.: Uniform boundary stabilization for the finite difference semi-discretization of 2-D wave equation. Afr. Mat. 25, 623–643 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Castro, C., Micu, S.: Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numerische Mathematik 102, 413–462 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    El Boujaoui, H., Maniar, L., Bouslous, H.: Uniform boundary stabilization for the finite difference discretization of the 1-D wave equation. Afr. Mat. (2016).  https://doi.org/10.1007/s13370-016-0406-3 MathSciNetMATHGoogle Scholar
  8. 8.
    Ervedoza, S., Zuazua, E.: Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91, 20–48 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Glowinski, R., Kinton, W., Wheeler, M.F.: A mixed finite element formulation for the boundary controllability of the wave equation. Int. J. Numer. Methods Eng. 27, 623–635 (1989)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Infante, J., Zuazua, E.: Boundary observability for the space semi-discretization of the 1-D wave equation. M2AN 33, 407–438 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 12.
    Isaacson, E., Keller, H.B.: Analysis of numerical methods. Wiley, New York (1966)MATHGoogle Scholar
  12. 13.
    Leon, L., Zuazua, E.: Boundary controllability of the finite-difference space semi-discretization of the beam equation. ESAIM Control Optim Calc. Var. 8, 827–862 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 14.
    Loreti, P., Mehrenberger, M.: An Ingham type proof for a two-grid observability theorem. ESAIM Control Optim. Calc. Var. 14, 604–631 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 15.
    Münch, A., Pazoto, A.F.: Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Control Optim. Calc. Var. 13, 265–293 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 16.
    Negreanu, M., Zuazua, E.: Uniform boundary controllability of a discrete 1-D wave equation. Syst. Control Lett. 48, 261–280 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 17.
    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, 413–418 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 18.
    Nicaise, S., Valein, J.: Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2, 425–479 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. 19.
    Nicaise, S., Valein, J.: Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization. Adv. Comput. Math. 32, 303–334 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 20.
    Rebarber, R.: Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33, 1–28 (1995)MathSciNetCrossRefMATHGoogle Scholar
  20. 21.
    Tebou, L.T., 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, 563–598 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 22.
    Tebou, L.T., Zuazua, E.: Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26, 337–365 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 23.
    Zuazua, E.: Boundary observability for the finite difference space semi-discretization of the 2-D wave equation in the square. J. Math. Pures Appl. 78, 523–563 (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 24.
    Zuazua, E.: Controllability of partial differential equations and its semi-discrete approximations. Discrete Contin. Dyn. Syst. 8, 469–513 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 25.
    Zuazua, E.: Propagation, observation, control and numerical approximation of waves. SIAM Rev. 47, 197–243 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.National School of Applied SciencesIbn Zohr UniversityAgadirMorocco

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