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

Sylvain Ervedoza (1) , Enrique Zuazua (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 BCAM - Basque Center for Applied Mathematics

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.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Sylvain Ervedoza, Enrique Zuazua. Transmutation techniques and observability for time-discrete approximation schemes of conservative systems. Numerische Mathematik, 2015, 130 (3), pp.425--466. ⟨hal-00818793v2⟩