Controllability and influence of the spatial discretization of the beam equation

We study the problem of the control of a finite difference semi-discrete scheme for the 1D beam equation modelling the transverse vibrations of a beam with fixed ends. We show that for most numerical schemes, due to high frequency spurious solutions, the observability may be lost under numerical discretizations when the mesh size tends to zero. We then prove that the uniform observability property is obtained by truncating the high frequencies. We also show how spatial discretization can be performed by using our nonstandard finite difference scheme.