A note on the necessity of filtering mechanism for polynomial observability of time-discrete wave equation

  • Z. Hajjej Faculty of Sciences of Monastir, University of Monastir, Avenue of Environment, 5019, Monastir, Tunisia
Keywords: observability inequality, time discretization, filtering techniques
Published online: 2017-06-07

Abstract


The problem of uniform polynomial observability was recently analyzed. It is shown that, when the continuous model is uniformly polynomially observable, it is sufficient to filter initial data to derive uniform polynomial observability inequalities for suitable time-discretization schemes. In this note, we prove that a filtering mechanism of high frequency modes is necessary to obtain uniform polynomial observability.

More precisely, we give a counterexample which proves that this latter fails without filtering the initial data for time semi-discrete approximations of the wave equation.

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