Boundary controllability of the finite-difference space semi-discretizations of the beam equation

¹ Laboratório de Ciências Matemáticas, Universidade Estadual do Norte Fluminense, 28015 Rio de Janeiro, Brazil; lalm@uenf.br.
² Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma, Cantoblanco, 28049 Madrid, Spain; enrique.zuazua@uam.es.

Received: 2 November 2001

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.