Volume 19, Number 4, October-December 2013
|Page(s)||1076 - 1108|
|Published online||13 August 2013|
Numerical controllability of the wave equation through primal methods and Carleman estimates
Laboratoire de Mathématiques, Université Blaise Pascal
(Clermont-Ferand 2), UMR CNRS 6620, Campus de Cézeaux, 63177
2 Dpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41012 Sevilla, Spain
Revised: 19 December 2012
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.
Mathematics Subject Classification: 35L10 / 65M12 / 93B40
Key words: One-dimensional wave equation / null controllability / finite element methods / Carleman estimates
© EDP Sciences, SMAI, 2013
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