Filtration techniques for the uniform controllability of semidiscrete hyperbolic equations

Abstract

In this article we describe two methods of filtration that can be useful in approximating the controls of hyperbolic linear equations. It has been remarked that a space semidiscretization of a hyperbolic equation introduces spurious high oscillations due to which the property of uniform controllability can be lost. We analyze the case of finite element discretization of the one-dimensional wave equation. Firstly, we address the problem of uniformly controlling the projection of the discrete solutions over a subspace of finite dimension. Secondly, we show that an appropriate filtration of the high eigenfrequencies of the discrete initial data enables us to restore the uniform controllability property of the whole solution. The main tools used in this article are the equivalence between a control problem and a problem of moments and the possibility to construct explicit solutions to the later problem by using biorthogonal sequences.