We consider various time discretization schemes of abstract conservative evolution equations of the form over(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. © 2008 Elsevier Inc. All rights reserved.
Ervedoza, S., Zheng, C., & Zuazua, E. (2008). On the observability of time-discrete conservative linear systems. Journal of Functional Analysis, 254(12), 3037–3078. https://doi.org/10.1016/j.jfa.2008.03.005