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Numerical controllability of the wave equation through primal methods 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 wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not apply in this work the usual duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. 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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