Spectral estimates for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations

TL;DR

This paper develops new spectral inequalities for Hermite functions, enabling null-controllability results for hypoelliptic quadratic equations with various control sets, extending previous uncertainty principles and control theory results.

Contribution

It establishes explicit spectral inequalities for finite Hermite function combinations, leading to null-controllability of hypoelliptic equations with measurable control regions.

Findings

Spectral inequality for Hermite functions with explicit constants.

Null-controllability for hypoelliptic quadratic equations in positive time.

Extension of control results to measurable and open control subsets.

Abstract

Some recent works have shown that the heat equation posed on the whole Euclidean space is null-controllable in any positive time if and only if the control subset is a thick set. This necessary and sufficient condition for null-controllability is linked to some uncertainty principles as the Logvinenko-Sereda theorem which give limitations on the simultaneous concentration of a function and its Fourier transform. In the present work, we prove new uncertainty principles for finite combinations of Hermite functions and establish an analogue of the Logvinenko-Sereda theorem with an explicit control of the constant with respect to the energy level of the Hermite functions as eigenfunctions of the harmonic oscillator for thick control subsets. This spectral inequality allows to derive the null-controllability in any positive time from thick control regions for parabolic equations associated with a general class of hypoelliptic non-selfadjoint quadratic differential operators. More generally, the spectral inequality for finite combinations of Hermite functions is actually shown to hold for any measurable control subset of positive Lebesgue measure, and some quantitative estimates of the constant with respect to the energy level are given for two other classes of control subsets including the case of non-empty open control subsets.