# Numerical controllability of the wave equation through primal methods and Carleman estimates

Nicolae Cîndea; Enrique Fernández-Cara; Arnaud Münch

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 4, page 1076-1108
- ISSN: 1292-8119

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topCîndea, Nicolae, Fernández-Cara, Enrique, and Münch, Arnaud. "Numerical controllability of the wave equation through primal methods and Carleman estimates." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 1076-1108. <http://eudml.org/doc/272846>.

@article{Cîndea2013,

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.},

author = {Cîndea, Nicolae, Fernández-Cara, Enrique, Münch, Arnaud},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {one-dimensional wave equation; null controllability; finite element methods; Carleman estimates; boundary null controls},

language = {eng},

number = {4},

pages = {1076-1108},

publisher = {EDP-Sciences},

title = {Numerical controllability of the wave equation through primal methods and Carleman estimates},

url = {http://eudml.org/doc/272846},

volume = {19},

year = {2013},

}

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

PB - EDP-Sciences

VL - 19

IS - 4

SP - 1076

EP - 1108

AB - 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.

LA - eng

KW - one-dimensional wave equation; null controllability; finite element methods; Carleman estimates; boundary null controls

UR - http://eudml.org/doc/272846

ER -

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