A uniformly controllable and implicit scheme for the 1-D wave equation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, pp. 377-418.

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 $\Delta t$. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order ${h}^{2}$ and $\Delta {t}^{2}$. 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 ${L}^{2}\left(0,T\right)$ and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal ${L}^{2}$-norm control. The results are illustrated with several numerical experiments.

DOI: 10.1051/m2an:2005012
Classification: 35L05,  65M60,  93B05
Keywords: exact boundary controllability, wave system, finite difference
@article{M2AN_2005__39_2_377_0,
author = {M\"unch, Arnaud},
title = {A uniformly controllable and implicit scheme for the {1-D} wave equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {377--418},
publisher = {EDP-Sciences},
volume = {39},
number = {2},
year = {2005},
doi = {10.1051/m2an:2005012},
zbl = {1130.93016},
mrnumber = {2143953},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2005012/}
}
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 - Modélisation Mathématique et Analyse Numérique
PY  - 2005
DA  - 2005///
SP  - 377
EP  - 418
VL  - 39
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2005012/
UR  - https://zbmath.org/?q=an%3A1130.93016
UR  - https://www.ams.org/mathscinet-getitem?mr=2143953
UR  - https://doi.org/10.1051/m2an:2005012
DO  - 10.1051/m2an:2005012
LA  - en
ID  - M2AN_2005__39_2_377_0
ER  - 
%0 Journal Article
%A Münch, Arnaud
%T A uniformly controllable and implicit scheme for the 1-D wave equation
%J ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
%D 2005
%P 377-418
%V 39
%N 2
%I EDP-Sciences
%U https://doi.org/10.1051/m2an:2005012
%R 10.1051/m2an:2005012
%G en
%F M2AN_2005__39_2_377_0
Münch, Arnaud. A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, pp. 377-418. doi : 10.1051/m2an:2005012. http://archive.numdam.org/articles/10.1051/m2an:2005012/

[1] M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation: a numerical study. COCV 3 (1998) 163-212. | Numdam | Zbl

[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. | Zbl

[3] F. Bourquin, Numerical methods for the control of flexible structures. J. Struct. Control 8 (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.

[5] C. Castro, S. Micu and A. Münch, Boundary controllability of a semi-discrete linear 2-D wave equation with mixed finite elements, submitted.

[6] I. Charpentier and Y. Maday, Identification numérique de contrôles distribués pour l'équation des ondes. C.R. Acad. Sci. Paris Sér. I 322 (1996) 779-784. | Zbl

[7] G.C. Cohen, Higher-order Numerical Methods for Transient Wave Equations. Scientific Computation, Springer (2002). | MR | Zbl

[8] R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189-221. | Zbl

[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. | Zbl

[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. | Zbl

[11] G.H. Golub and C. Van Loan, Matrix Computations. Johns Hopkins Press, Baltimore (1989). | MR | Zbl

[12] J.A. Infante and E. Zuazua, Boundary observability for the space-discretizations of the 1-D wave equation. ESAIM: M2AN 33 (1999) 407-438. | EuDML | Numdam | Zbl

[13] A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367-369. | EuDML | Zbl

[14] V. Komornik, Exact controllability and Stabilization - The multiplier method. J. Wiley and Masson (1994). | MR | Zbl

[15] S. Krenk, Dispersion-corrected explicit integration of the wave equation. Comp. Methods Appl. Mech. Engrg. 191 (2001) 975-987. | Zbl

[16] J.L. Lions, Contrôlabilité exacte - Pertubations et stabilisation de systèmes distribués, Tome 1, Masson, Paris (1988). | Zbl

[17] S. Micu, Uniform boundary controllability of a semi-discrete 1-D wave equation. Numer. Math. 91 (2002) 723-728. | Zbl

[18] A. Münch, Family of implicit schemes uniformly controllable for the 1-D wave equation. C.R. Acad. Sci. Paris Sér. I 339 (2004) 733-738. | Zbl

[19] A. Münch and A.F. Pazoto, Uniform stabilization of a numerical approximation of a locally damped wave equation. ESAIM: COCV, submitted. | Numdam | Zbl

[20] M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1-D wave equation. Systems Control Lett. 48 (2003) 261-280. | Zbl

[21] M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications. C.R. Acad. Sci. Paris Sér. I 338 (2004) 281-286. | Zbl

[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. I 338 (2004) 413-418. | Zbl

[23] P.A. Raviart and J.M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, Paris (1983). | Zbl

[24] D.L. Russell, Controllability and stabilization theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639-737. | Zbl

[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. | Zbl

Cited by Sources: