Spectral inequalities for combinations of Hermite functions and null-controllability for evolution equations enjoying Gelfand-Shilov smoothing effects

TL;DR

This paper establishes spectral inequalities for Hermite functions and uses them to prove null-controllability of certain evolution equations with Gelfand-Shilov smoothing effects, under specific geometric conditions on control sets.

Contribution

It introduces new spectral inequalities for Hermite functions and applies these to demonstrate null-controllability for evolution equations with Gelfand-Shilov regularization effects.

Findings

Spectral inequalities hold for control subsets thick with respect to unbounded densities.

Null-controllability is achieved in any positive time for equations with Gelfand-Shilov smoothing.

Results apply to hypoelliptic quadratic operators and fractional harmonic oscillators.

Abstract

This work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost linearly at infinity, and provide quantitative estimates, with respect to the energy level of the Hermite functions seen as eigenfunctions of the harmonic oscillator, for the constants appearing in these spectral estimates. These spectral inequalities allow to derive the null-controllability in any positive time for evolution equations enjoying specific regularizing effects. More precisely, for a given index $\frac{1}{2} \leq \mu <1$, we deduce sufficient geometric conditions on control subsets to ensure the null-controllability of evolution equations enjoying regularizing effects in the symmetric Gelfand-Shilov space $S^{\mu}_{\mu}(\mathbb{R}^n)$. These results apply in particular to derive the null-controllability in any positive time for evolution equations associated to certain classes of hypoelliptic non-selfadjoint quadratic operators, or to fractional harmonic oscillators.