Observability of dispersive equations from line segments on the torus

TL;DR

This paper studies the conditions under which linear dispersive equations on the torus can be observed from line segments in space-time, providing criteria based on slopes and employing graph theory for complex cases.

Contribution

It establishes new observability conditions for dispersive equations from line segments, including both qualitative and quantitative results, using graph theory techniques.

Findings

One line segment observability follows from Ingham's inequality.

Two line segments observability relies on graph theory analysis.

Results apply to higher order Schrödinger and linear KdV equations.

Abstract

We investigate the observability of a general class of linear dispersive equations on the torus $\mathbb{T}$. We take one line segment or two line segments in space-time region as the observable set. We give the characteristic on the slopes of the line segments to guarantee the qualitative observability and quantitative observability respectively. The one line segment case, is simple, follows directly from the Ingham's inequality. However, the two line segments case is difficult, the statement of results and the proof rely heavily on the language of graph theory. We also apply our results to (higher order) Schr\"{o}dinger equations and the linear KdV equation.