Uncertainty Principles Associated to Sets Satisfying the Geometric Control Condition

TL;DR

This paper extends uncertainty principles related to control theory, providing a new spectral inequality for functions on an annulus and applying it to derive energy decay rates for damped fractional wave equations in higher dimensions.

Contribution

It establishes a Paneah-Logvinenko-Sereda type theorem for annuli and applies it to control and damping problems in fractional wave equations.

Findings

Spectral inequality for functions with spectrum in an annulus

Energy decay rates for damped fractional wave equations

Extension to higher-dimensional and non-periodic settings

Abstract

In this paper, we study forms of the uncertainty principle suggested by problems in control theory. We obtain a version of the classical Paneah-Logvinenko-Sereda theorem for the annulus. More precisely, we show that a function with spectrum in an annulus of a given thickness can be bounded, in $L^2$-norm, from above by its restriction to a neighborhood of a GCC set, with constant independent of the radius of the annulus. We apply this result to obtain energy decay rates for damped fractional wave equations, extending the work of Malhi and Stanislavova to both the higher-dimensional and non-periodic setting.