Time discrete wave equations: Boundary observability and control

In this paper we study the exact boundary controllability of atrapezoidal time discrete wave equation in a bounded domain. Weprove that the projection of the solution in an appropriate filteredspace is exactly controllable with uniformly bounded cost withrespect to the time-step. In this way, the well-knownexact-controllability property of the wave equation can bereproduced as the limit, as the time step $h\rightarrow 0$, of thecontrollability of projections of the time-discrete one. By dualitythese results are equivalent to deriving uniform observabilityestimates (with respect to $h\rightarrow 0$) within a class ofsolutions of the time-discrete problem in which the high frequencycomponents have been filtered. The later is established by means ofa time-discrete version of the classical multiplier technique. Theoptimality of the order of the filtering parameter is alsoestablished, although a careful analysis of the expected velocity ofpropagation of time-discrete waves indicates that its actual valuecould be improved.