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

Published online by Cambridge University Press:  15 April 2005

Arnaud Münch
Arnaud Münch*
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, UFR de Sciences et Techniques, Université de Franche-Comté, 16, route de Gray 25030, Besançon cedex, France. [email protected]
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Abstract

This paper studies theexact controllability of a finite dimensional system obtained bydiscretizing in space and time the linear 1-D wave system with aboundary control at one extreme. It is known that usual schemesobtained with finite difference or finite element methods are notuniformly controllable with respect to the discretizationparameters h and Δt. We introduce an implicit finitedifference scheme which differs from the usual centered one byadditional terms of order h 2 and Δt 2. Using a discreteversion of Ingham's inequality for nonharmonic Fourier series andspectral properties of the scheme, we show that the associatedcontrol can be chosen uniformly bounded in L2(0,T) and in sucha way that it converges to the HUM control of the continuous wave,i.e. the minimal L 2-norm control. The results are illustratedwith several numerical experiments.

Keywords

Exact boundary controllabilitywave systemfinitedifference.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 2 , March 2005 , pp. 377 - 418
Copyright
© EDP Sciences, SMAI, 2005

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