Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density

TL;DR

This paper develops uncertainty principles for Hermite functions with geometric conditions on sensor sets, leading to null-controllability results for harmonic oscillator equations even with sparse sensor distributions.

Contribution

It introduces a geometric criterion ensuring norm equivalence for Hermite functions and applies it to establish null-controllability with sensor sets of decaying density.

Findings

Uncertainty principles for Hermite functions with geometric criteria

Null-controllability of harmonic oscillator from sparse sensor sets

Sensor sets with sub-exponential decay density are effective

Abstract

We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset $S\subset \RR^d$ ensuring that the $L^2$-seminorm associated to $S$ is equivalent to the full $L^2$-norm on $\RR^d$ when restricted to the space of Hermite functions up to a given degree. We give precise estimates how the equivalence constant depends on this degree and on geometric parameters of $S$. From these estimates we deduce that the parabolic equation whose generator is the harmonic oscillator is null-controllable from $S$. In all our results, the set $S$ may have sub-exponentially decaying density and, in particular, finite volume. We also show that bounded sets are not efficient in this context.