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January & February  2009, 23(1&2): 571-604. doi: 10.3934/dcds.2009.23.571

Time discrete wave equations: Boundary observability and control


Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190


School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China


Basque Center for Applied Mathematics (BCAM), Gran Via 35, 48009 Bilbao, Spain

Received  September 2007 Revised  February 2008 Published  September 2008

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.
Citation: Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

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