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Numerische Mathematik

, Volume 118, Issue 4, pp 601–661 | Cite as

Uniform controllability properties for space/time-discretized parabolic equations

  • Franck BoyerEmail author
  • Florence Hubert
  • Jérôme Le Rousseau
Article

Abstract

This article is concerned with the analysis of semi-discrete-in-space and fully-discrete approximations of the null controllability (and controllability to the trajectories) for parabolic equations. We propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters. For time discretization we use θ-schemes with \({\theta \in [\frac{1}2,1]}\) . For the proofs of controllability we rely on the strategy introduced by Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) for the null-controllability of the heat equation, which is based on a spectral inequality. We obtain relaxed uniform observability estimates in both the semi-discrete and fully-discrete frameworks, and associated uniform controllability properties. For the practical computation of the control functions we follow J.-L. Lions’ Hilbert Uniqueness Method strategy, exploiting the relaxed uniform observability estimate. Algorithms for the computation of the controls are proposed and analysed in the semi-discrete and fully-discrete cases. Additionally, we prove an error bound between the fully discrete and the semi-discrete control functions. This bound is however not uniform with respect to the space discretization. The theoretical results are illustrated through numerical experimentations.

Mathematics Subject Classification (2000)

35K05 65M06 93B05 93B07 93B40 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Franck Boyer
    • 1
    Email author
  • Florence Hubert
    • 1
  • Jérôme Le Rousseau
    • 2
  1. 1.Aix-Marseille UniversitéLaboratoire d’Analyse Topologie Probabilités (LATP)Marseille cedex 13France
  2. 2.Laboratoire Mathématiques et Applications, Physique Mathématique d’OrléansUniversité d’OrléansOrléans cedex 2France

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