Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-30T20:04:45.869Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary controllability of the finite-difference spacesemi-discretizations of the beam equation

Published online by Cambridge University Press:  15 August 2002

Liliana León  and
Enrique Zuazua
Liliana León
Affiliation:
Laboratório de Ciências Matemáticas, Universidade Estadual do Norte Fluminense, 28015 Rio de Janeiro, Brazil; [email protected].
Enrique Zuazua
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma, Cantoblanco, 28049 Madrid, Spain; [email protected].
Get access

Abstract

We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.

Keywords

Beam equationfinite difference semi-discretizationexact boundary controllability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 827 - 862
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, J. and Slemrod, M., Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems. Comm. Pure Appl. Math. 37 (1979) 555-587. CrossRef
E. Crépeau, Exact Controllability of the Boussinesq Equation on a Bounded Domain. Adv. Differential Equations (to appear).
Haraux, A., Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl. 68 (1989) 457-465.
Ingham, A.E., Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1967) 367-379. CrossRef
Infante, J.A. and Zuazua, E., Boundary observability for the space semi-discretizations of the 1-d wave equation. Math. Model. Numer. Anal. 33 (1999) 407-438. CrossRef
E. Isaacson and H.B. Keller, Analysis of numerical methods. John Wiley and Sons (1966).
V. Komornik, Exact controllability and stabilization: The multiplier method. Masson and John Wiley, RAM 36 (1994).
Lebeau, G., Contrôle de l' équation Schrödinger. J. Math. Pures Appl. 71 (1992) 267-291.
L. León, Controle Exato da Equaç ao da Viga 1-D Semi-discretizada no Espaço por Diferenças Finitas, Ph.D. Thesis. Instituto de Matemática, Universidade Federal de Rio de Janeiro (2001).
J.L. Lions, Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués, Tome 1. Masson, RMA 8, Paris (1988).
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vols. 1 and 2. Dunod, Paris (1968).
S. Micu, Uniform Boundary Controllability of a Semi-Discrete 1-D Wave Equation. Numer. Math. (to appear).
Micu, S. and Zuazua, E., Boundary controllability of a linear hybrid system arising in the control of noise. SIAM J. Control Optim. 35 (1997) 1614-1637. CrossRef
J. Simon, Compact sets in the space L P (0,T,B). Ann. Mat. Pura Appl. CXLVI (1987) 65-96.
J.C. Strikwerda, Finite difference schemes and partial differential equation. Chapman and Hall (1995).
J.W. Thomas, Numerical partial differential equations; finite difference methods. Springer, Texts Appl. Math. 22 (1995).
R.M. Young, An introduction to nonharmonic Fourier series. Academic Press, Pure Appl. Math. A Series of Monographs and Textbooks (1980).
Zuazua, E., Boundary observability for the finite space semi-discretization of the 2-d wave equation in the square. J. Math. Pures Appl. 78 (1999) 523-563. CrossRef
E. Zuazua, Contrôlabilité exacte en un temps arbitrairement petit de quelques modèles de plaques. Appendix I in [] (1988) 465-491.