Transmutation techniques and observability for time-discrete approximation schemes of conservative systems

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. ∗The first author is partially supported by the Agence Nationale de la Recherche (ANR, France), Project CISIFS number NT09-437023, and grant MTM2011-29306 of the MICINN, Spain. Part of this work has been done while he was visiting the BCAM – Basque Center for Applied Mathematics as a Visiting Fellow. The second author is supported by the ERC Advanced Grant FP7-246775 NUMERIWAVES, the Grant PI2010-04 of the Basque Government, the ESF Research Networking Program OPTPDE and Grant MTM2011-29306 of the MICINN, Spain. †e-mail: [email protected] ‡e-mail: [email protected]