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

Cîndea, Nicolae; Fernández-Cara, Enrique; Münch, Arnaud

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1076-1108.

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.

Classification : 35L10, 65M12, 93B40
Mots clés : one-dimensional wave equation, null controllability, finite element methods, Carleman estimates

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