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

TL;DR

This paper establishes a representation formula linking continuous and discrete solutions of conservative systems, enabling uniform observability results for discrete models based on continuous system properties, and proves the sharpness of these estimates.

Contribution

It introduces a novel representation formula for solutions of conservative systems and their discrete approximations, facilitating the transfer of observability properties between them.

Findings

Representation formula connecting continuous and discrete solutions

Uniform observability results for discrete models derived from continuous properties

Sharpness of the observability time estimate for discrete models

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.