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

¹ 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
² Dpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41012 Sevilla, Spain
cara@us.es

Received: 8 February 2012
Revised: 19 December 2012

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.