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On the observability of abstract time-discrete linear parabolic equations

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Abstract

This article aims at analyzing the observability properties of time-discrete approximation schemes of abstract parabolic equations \(\dot{z}+Az=0\) , where A is a self-adjoint positive definite operator with dense domain and compact resolvent. We analyze the observability properties of these diffusive systems for an observation operator \(B\in\mathfrak{L}(\mathcal{D}(A^{\nu}),\,Y)\) with ν<1/2. Assuming that the continuous system is observable, we prove uniform observability results for suitable time-discretization schemes within the class of conveniently filtered data. We also propose a HUM type algorithm to compute discrete approximations of the exact controls. Our approach also applies to sequences of operators which are uniformly observable. In particular, our results can be combined with the existing ones on the observability of space semi-discrete systems, yielding observability properties for fully discrete approximation schemes.

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Authors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin en Yvelines, 45 avenue des États Unis, 78035, Versailles Cedex, France

    Sylvain Ervedoza

  2. Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313, Valenciennes Cedex 9, France

    Julie Valein

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  1. Sylvain Ervedoza
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  2. Julie Valein
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Correspondence to Sylvain Ervedoza.

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Ervedoza, S., Valein, J. On the observability of abstract time-discrete linear parabolic equations. Rev Mat Complut 23, 163–190 (2010). https://doi.org/10.1007/s13163-009-0014-y

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