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This paper deals with the numerical computation of boundary null controls forthe 1D wave equation with a potential. The goal is to compute an approximationof controls that drive the solution from a prescribed initial state to zero ata large enough controllability time. We do not use in this work dualityarguments but explore instead a direct approach in the framework of globalCarleman estimates. More precisely, we consider the control that minimizes overthe class of admissible null controls a functional involving weighted integralsof the state and of the control. The optimality conditions show that both theoptimal control and the associated state are expressed in terms of a newvariable, the solution of a fourth-order elliptic problem defined in thespace-time domain. We first prove that, for some specific weights determined bythe global Carleman inequalities for the wave equation, this problem iswell-posed. Then, in the framework of the finite element method, we introduce afamily of finite-dimensional approximate control problems and we prove a strongconvergence result. Numerical experiments confirm the analysis. We complete ourstudy with several comments.

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