Observability of dispersive equations from line segments on the torus

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.