Time discrete wave equations: Boundary observability and control

Abstract

In this paper we study the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. We prove that the projection of the solution in an appropriate filtered space is exactly controllable with uniformly bounded cost with respect to the time-step. In this way, the well-known exact-controllability property of the wave equation can be reproduced as the limit, as the time step $h\rightarrow 0$, of the controllability of projections of the time-discrete one. By duality these results are equivalent to deriving uniform observability estimates (with respect to $h\rightarrow 0$) within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. The later is established by means of a time-discrete version of the classical multiplier technique. The optimality of the order of the filtering parameter is also established, although a careful analysis of the expected velocity of propagation of time-discrete waves indicates that its actual value could be improved.