Uniform boundary controllability of a semi-discrete 1-D wave equation

A numerical scheme for the controlled semi-discrete I-D wave equation is considered. We analyze the convergence of the boundary controls of the semi-discrete equations to a control of the continuous wave equation when the mesh size tends to zero. We prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded controls and any weak limit of this sequence is a control for the continuous problem. The number of the eliminated frequencies depends on the mesh size and the regularity of the continuous initial data. The case of the HUM controls is also discussed.