A uniformly controllable and implicit scheme for the 1-D wave equation

Arnaud Münch

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 2, page 377-418
  • ISSN: 0764-583X

Abstract

top
This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using a discrete version of Ingham's inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-norm control. The results are illustrated with several numerical experiments.

How to cite

top

Münch, Arnaud. "A uniformly controllable and implicit scheme for the 1-D wave equation." ESAIM: Mathematical Modelling and Numerical Analysis 39.2 (2010): 377-418. <http://eudml.org/doc/194266>.

@article{Münch2010,
abstract = { This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using a discrete version of Ingham's inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-norm control. The results are illustrated with several numerical experiments. },
author = {Münch, Arnaud},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Exact boundary controllability; wave system; finite difference.},
language = {eng},
month = {3},
number = {2},
pages = {377-418},
publisher = {EDP Sciences},
title = {A uniformly controllable and implicit scheme for the 1-D wave equation},
url = {http://eudml.org/doc/194266},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Münch, Arnaud
TI - A uniformly controllable and implicit scheme for the 1-D wave equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 2
SP - 377
EP - 418
AB - This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using a discrete version of Ingham's inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-norm control. The results are illustrated with several numerical experiments.
LA - eng
KW - Exact boundary controllability; wave system; finite difference.
UR - http://eudml.org/doc/194266
ER -

References

top
  1. M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation: a numerical study. COCV3 (1998) 163–212.  Zbl1052.93501
  2. H.T. Banks, K. Ito and Y. Wang, Exponentially stable approximations of weakly damped wave equations. Ser. Num. Math., Birkhäuser 100 (1990) 1–33.  Zbl0850.93719
  3. F. Bourquin, Numerical methods for the control of flexible structures. J. Struct. Control8 (2001).  
  4. C. Castro and S. Micu, Boundary controllability of a semi-discrete linear 1-D wave equation with mixed finite elements. SIAM J. Numer. Anal., submitted.  Zbl1102.93004
  5. C. Castro, S. Micu and A. Münch, Boundary controllability of a semi-discrete linear 2-D wave equation with mixed finite elements, submitted.  Zbl1139.93005
  6. I. Charpentier and Y. Maday, Identification numérique de contrôles distribués pour l'équation des ondes. C.R. Acad. Sci. Paris Sér. I322 (1996) 779–784.  Zbl0847.65043
  7. G.C. Cohen, Higher-order Numerical Methods for Transient Wave Equations. Scientific Computation, Springer (2002).  Zbl0985.65096
  8. R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys.103 (1992) 189–221.  Zbl0763.76042
  9. R. Glowinski, C.H. Li and J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet Controls: Description of the numerical methods. Japan J. Appl. Math.7 (1990) 1–76.  Zbl0699.65055
  10. R. Glowinski, W. Kinton and M.F. Wheeler, A mixed finite element formulation for the boundary controllability of the wave equation. Int. J. Numer. Methods Engrg.27 (1989) 623–636.  Zbl0711.65084
  11. G.H. Golub and C. Van Loan, Matrix Computations. Johns Hopkins Press, Baltimore (1989).  
  12. J.A. Infante and E. Zuazua, Boundary observability for the space-discretizations of the 1-D wave equation. ESAIM: M2AN33 (1999) 407–438.  Zbl0947.65101
  13. A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z.41 (1936) 367–369.  Zbl0014.21503
  14. V. Komornik, Exact controllability and Stabilization - The multiplier method. J. Wiley and Masson (1994).  Zbl0937.93003
  15. S. Krenk, Dispersion-corrected explicit integration of the wave equation. Comp. Methods Appl. Mech. Engrg.191 (2001) 975–987.  Zbl1009.76054
  16. J.L. Lions, Contrôlabilité exacte – Pertubations et stabilisation de systèmes distribués, Tome 1, Masson, Paris (1988).  Zbl0653.93002
  17. S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation. Numer. Math.91 (2002) 723–728.  Zbl1002.65072
  18. A. Münch, Family of implicit schemes uniformly controllable for the 1-D wave equation. C.R. Acad. Sci. Paris Sér. I339 (2004) 733–738.  Zbl1061.65054
  19. A. Münch and A.F. Pazoto, Uniform stabilization of a numerical approximation of a locally damped wave equation. ESAIM: COCV, submitted.  Zbl1120.65101
  20. M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1-D wave equation. Systems Control Lett.48 (2003) 261–280.  Zbl1157.93324
  21. M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications. C.R. Acad. Sci. Paris Sér. I338 (2004) 281–286.  Zbl1040.93030
  22. M. Negreanu and E. Zuazua, Convergence of a multi-grid method for the controlabillity of the 1-D wave equation. C.R. Acad. Sci. Paris, Sér. I338 (2004) 413–418.  Zbl1038.65054
  23. P.A. Raviart and J.M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, Paris (1983).  Zbl0561.65069
  24. D.L. Russell, Controllability and stabilization theory for linear partial differential equations: recent progress and open questions. SIAM Rev.20 (1978) 639–737.  Zbl0397.93001
  25. J.M. Urquiza, Contrôle d'équations des ondes linéaires et quasilinéaires. Ph.D. Thesis Université de Paris VI (2000).  
  26. E. Zuazua, Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures. Appl.78 (1999) 523–563.  Zbl0939.93016

Citations in EuDML Documents

top
  1. Nicolae Cîndea, Enrique Fernández-Cara, Arnaud Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates
  2. Arnaud Münch, Ademir Fernando Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation
  3. Sylvain Ervedoza, Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes
  4. Farah Abdallah, Serge Nicaise, Julie Valein, Ali Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications
  5. Sylvain Ervedoza, Resolvent estimates in controllability theory and applications to the discrete wave equation

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.