A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations

Nicolae Cindea (1) , Arnaud Munch (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

This paper deals with the numerical computation of null controls for the 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. In [\textit{Cindea, Fernandez-Cara \& Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013}], a so called primal method is described leading to a strongly convergent approximation of boundary controls : the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and his adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case.

Mots clés: Linear wave equation; null controllability; finite element methods; Mixed formulation

Domaines: Optimisation et contrôle [math.OC]

Nicolae Cindea, Arnaud Munch. A mixed formulation for the direct approximation of the control of minimal $L^2$-norm for linear type wave equations. 2013. ⟨hal-00853767⟩