ABSTRACT

We consider a finite-difference semi-discrete scheme for the approximation of controls of one-dimensional evolution problems of hyperbolic type. In general, the continuous problem is controllable but the high frequency numerical spurious oscillations, introduced by the numerical scheme, lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data or if we add a numerical vanishing viscosity, we restore the uniform controllability. Moreover, in the case of boundary controls of the wave equation we obtain a relation between the range of filtration and the minimal time of control needed to ensure the uniform controllability property. The main strategy consist of proving the equivalence between a control problem and a problem of moments and constructing explicit solutions to the later problem by using biorthogonal sequences.