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Numerical controllability of the wave equation through primalmethods and Carleman estimates

Published online by Cambridge University Press:  13 August 2013

Nicolae Cîndea ,
Enrique Fernández-Cara  and
Arnaud Münch
Nicolae Cîndea
Affiliation:
Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferand 2), UMR CNRS 6620, Campus de Cézeaux, 63177 Aubière, France. nicolae.cindea@math.univ-bpclermont.fr; arnaud.munch@math.univ-bpclermont.fr
Enrique Fernández-Cara
Affiliation:
Dpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41012 Sevilla, Spain; cara@us.es
Arnaud Münch
Affiliation:
Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferand 2), UMR CNRS 6620, Campus de Cézeaux, 63177 Aubière, France. nicolae.cindea@math.univ-bpclermont.fr; arnaud.munch@math.univ-bpclermont.fr
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Abstract

This paper deals with the numerical computation of boundary null controls for the 1D waveequation with a potential. The goal is to compute approximations of controls that drivethe solution from a prescribed initial state to zero at a large enough controllabilitytime. We do not apply in this work the usual duality arguments but explore instead adirect approach in the framework of global Carleman estimates. More precisely, we considerthe control that minimizes over the class of admissible null controls a functionalinvolving weighted integrals of the state and the control. The optimality conditions showthat both the optimal control and the associated state are expressed in terms of a newvariable, the solution of a fourth-order elliptic problem defined in the space-timedomain. We first prove that, for some specific weights determined by the global Carlemaninequalities for the wave equation, this problem is well-posed. Then, in the framework ofthe finite element method, we introduce a family of finite-dimensional approximate controlproblems and we prove a strong convergence result. Numerical experiments confirm theanalysis. We complete our study with several comments.

Keywords

One-dimensional wave equationnull controllabilityfinite element methodsCarleman estimates
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1076 - 1108
Copyright
© EDP Sciences, SMAI, 2013

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