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SPECTRAL INEQUALITIES FOR COMBINATIONS OF HERMITE FUNCTIONS AND NULL-CONTROLLABILITY FOR EVOLUTION EQUATIONS ENJOYING GELFAND–SHILOV SMOOTHING EFFECTS

Part of: Close-to-elliptic equations and systems Nontrigonometric harmonic analysis Controllability, observability, and system structure

Published online by Cambridge University Press:  21 March 2022

Jérémy Martin  and
Karel Pravda-Starov Open the ORCID record for Karel Pravda-Starov [Opens in a new window]
Jérémy Martin
Affiliation:
Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France ([email protected])
Karel Pravda-Starov*
Affiliation:
Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
*
[email protected]
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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 us 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-self-adjoint quadratic operators, or to fractional harmonic oscillators.

Keywords

uncertainty principlesLogvinenko–Sereda-type estimatesHermite functionsnull-controllabilityquadratic equationsfractional harmonic oscillatorsGelfand–Shilov regularity

MSC classification

Primary: 93B05: Controllability
Secondary: 42C05: Orthogonal functions and polynomials, general theory 35H10: Hypoelliptic equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 6 , November 2023 , pp. 2533 - 2582
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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