Observability of rectangular membranes and plates on small sets

TL;DR

This paper advances the understanding of observability for rectangular membranes and plates, demonstrating that observation on small, well-chosen segments suffices for control, using recent mathematical theorems.

Contribution

It improves existing results by showing observability on arbitrarily short segments and applying modern generalizations of Ingham's theorem for rectangular membranes.

Findings

Observation on small segments is sufficient for control.

Multiple well-chosen segments strengthen observability estimates.

New theorems established for rectangular membranes using recent mathematical tools.

Abstract

Since the works of Haraux and Jaffard we know that rectangular plates may be observed by subregions not satisfying the geometrical control condition. We improve these results by observing only on an arbitrarily short segment inside the domain. The estimates may be strengthened by observing on several well-chosen segments. In the second part of the paper we establish various observability theorems for rectangular membranes by applying Mehrenberger's recent generalization of Ingham's theorem.