Abstract
We consider various time discretization schemes of abstract conservative evolution equations of the form z ˙ = A z , where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair ( A , B ) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by Burq and Zworski in [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2) (2004) 443–471 (electronic)]. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations.
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