---
abstract: 'In this article, we consider abstract linear conservative systems and their time-discrete counterparts. Our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time-discrete one. As an application, we show how observability properties for the time continuous model yield uniform (with respect to the time-step) observability results for its time-discrete approximation counterparts, provided the initial data are suitably filtered. The main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time-discrete models. Besides, using a reverse representation formula, we also prove that this estimate on the time of uniform observability for the time-discrete models is sharp. We then conclude with some general comments and open problems.'
author:
- |
    Sylvain Ervedoza$^{1,2,}$[^1]\
    [*$^{1}$ CNRS ; Institut de Mathématiques de Toulouse UMR 5219 ;*]{}\
    [*F-31062 Toulouse, France,*]{}\
    [*$^{2}$ Université de Toulouse ; UPS, INSA, INP, ISAE, UT1, UTM ; IMT ;*]{}\
    [*F-31062 Toulouse, France,*]{}\
    and Enrique Zuazua$^{3,4,}$ [^2]\
    [*$^{3}$ BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo, 14*]{}\
    [*E-48009 Bilbao, Basque Country, Spain.*]{}\
    [*$^{4}$ Ikerbasque Research Professor, Ikerbasque - Basque Foundation for Science,*]{}\
    [*E-48011 Bilbao, Basque Country, Spain.*]{}
title: 