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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 equation, null controllability, finite element methods, Carleman estimates
@article{COCV_2013__19_4_1076_0,
author = {C{\^\i}ndea, Nicolae and Fern\'andez-Cara, Enrique and M\"unch, Arnaud},
title = {Numerical controllability of the wave equation through primal methods and {Carleman} estimates},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1076--1108},
publisher = {EDP-Sciences},
volume = {19},
number = {4},
year = {2013},
doi = {10.1051/cocv/2013046},
mrnumber = {3182682},
zbl = {1292.35162},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013046/}
}
TY - JOUR AU - Cîndea, Nicolae AU - Fernández-Cara, Enrique AU - Münch, Arnaud TI - Numerical controllability of the wave equation through primal methods and Carleman estimates JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 1076 EP - 1108 VL - 19 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013046/ DO - 10.1051/cocv/2013046 LA - en ID - COCV_2013__19_4_1076_0 ER -
%0 Journal Article %A Cîndea, Nicolae %A Fernández-Cara, Enrique %A Münch, Arnaud %T Numerical controllability of the wave equation through primal methods and Carleman estimates %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 1076-1108 %V 19 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013046/ %R 10.1051/cocv/2013046 %G en %F COCV_2013__19_4_1076_0
Cîndea, Nicolae; Fernández-Cara, Enrique; Münch, Arnaud. Numerical controllability of the wave equation through primal methods and Carleman estimates. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1076-1108. doi: 10.1051/cocv/2013046
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