arXiv subjects

Jérémy Martin

Publications and source records attributed to Jérémy Martin.

10 recordsLinked to original sources

arXiv2024

On the dimension of observable sets for the heat equation

We consider the heat equation on a bounded $C^1$ domain in $\mathbb{R}^n$ with Dirichlet boundary conditions. The primary aim of this paper is to prove that the heat equation is observable from any measurable set with a Hausdorff dimension strictly greater than $n - 1$. The proof relies on a novel spectral estimate for linear combinations of Laplace eigenfunctions, achieved through the propagation of smallness for solutions to Cauchy-Riemann systems as established by Malinnikova, and uses the Lebeau-Robbiano method. While this observability result is sharp regarding the Hausdorff dimension scale, our secondary goal is to construct families of sets with dimensions less than $n - 1$ from which the heat equation is still observable.

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arXiv2024

Quantitative propagation of smallness and spectral estimates for the Schrödinger operator

In this paper, we investigate quantitative propagation of smallness properties for the Schrödinger operator on a bounded domain in $\mathbb R^d$. We extend Logunov, Malinnikova's results concerning propagation of smallness for $A$-harmonic functions to solutions of divergence elliptic equations perturbed by a bounded zero order term. We also prove similar results for gradient of solutions to some particular equations. This latter result enables us to follow the recent strategy of Burq, Moyano for the obtaining of spectral estimates on rough sets for the Schrödinger operator. Applications to observability estimates and to the null-controllability of associated parabolic equations posed on compact manifolds or the whole euclidean space are then considered.

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arXiv2023

Global stabilization of the cubic defocusing nonlinear Schrödinger equation on the torus

In this article, we prove the (uniform) global exponential stabilization of the cubic defocusing Schrödinger equation on the torus d-dimensional torus, for d=1, 2 or 3, with a linear damping localized in a subset of the torus satisfying some geometrical assumptions. In particular, this answers an open question of Dehman, Gérard, Lebeau from 2006. Our approach is based on three ingredients. First, we prove the well-posedness of the closed-loop system in Bourgain spaces. Secondly, we derive new Carleman estimates for the nonlinear equation by directly including the cubic term in the conjugated operator. Thirdly, by conjugating with energy estimates and Morawetz multipliers method, we then deduce quantitative observability estimates leading to the uniform exponential decay of the total energy of the system. As a corollary of the global stabilization result, we obtain an upper bound of the minimal time of the global null-controllability of the nonlinear equation by using a stabilization procedure and a local null-controllability result.

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arXiv2023

Observability estimates for the Schr{ö}dinger equation in the plane with periodic bounded potentials from measurable sets

The goal of this article is to obtain observability estimates for Schr{ö}dinger equations in the plane R 2. More precisely, considering a 2$π$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the evolution equation i$\partial$tu = --$Δ$u + V (x)u, is observable from any 2$π$Z 2-periodic measurable set, in any small time T > 0. We then extend Ta{ü}ffer's recent result [T{ä}u22] in the two-dimensional case to less regular observable sets and general bounded periodic potentials. The methodology of the proof is based on the use of the Floquet-Bloch transform, Strichartz estimates and semiclassical defect measures for the obtention of observability inequalities for a family of Schr{ö}dinger equations posed on the torus R 2 /2$π$Z 2 .

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arXiv2023

Approximate null-controllability with uniform cost for the hypoelliptic Ornstein-Uhlenbeck equations

We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on $\mathbb R^n$ is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of non-autonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on $\mathbb R^n$, for which the approximate null-controllability with uniform cost is ensured by the notion of thickness, which is stronger that the integral thickness condition considered in the present work. Examples of those parabolic equations are the fractional heat equations associated with the operator $(-Δ)^s$, in the regime $s\geq1/2$. Our strategy also allows to characterize the approximate null-controllability with uniform cost from moving control supports for this class of fractional heat equations.

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arXiv2022

Spectral Inequalities For Anisotropic Shubin Operators

In this paper, new spectral inequalities for finite combinations of eigenfunctions of anisotropic Shubin operators are presented. Given a subset $ω$ and an energy level, we provide an explicit control of the ratio of the L 2 (R d)-norm over the L 2 ($ω$)-norm with respect to the energy level. The proofs are based on recent uncertainty principles holding in Gelfand-Shilov spaces and Bernstein type estimates deduced from quantitative smoothing effects proved by Paul Alphonse. These spectral inequalities allow to derive the null-controllability in any positive time from any control subset with positive Lebesgue measure of evolution equations associated to anisotropic Shubin operators, except for the harmonic oscillator.

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arXiv2021

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

We prove that the thickness property is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space $\mathbb R^n$. These equations are associated with operators of the form $F(\vert D_x\vert)$, the function $F:[0,+\infty)\rightarrow\mathbb R$ being continuous and bounded from below. We also provide explicit feedbacks and constants associated with these stabilization properties. The notion of thickness is known to be a necessary and sufficient condition for the null-controllability of the fractional heat equations associated with the functions $F(t) = t^{2s}$ in the case $s>1/2$. Our results apply in particular for this class of equations, but also for the half heat equation associated with the function $F(t) = t$, which is the most diffusive fractional heat equation for which null-controllability is known to fail from general thick control supports.

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arXiv2021

Uncertainty principles in Gelfand-Shilov spaces and null-controllability

We provide new uncertainty principles for functions in a general class of Gelfand-Shilov spaces. These results apply, in particular, with the classical Gelfand-Shilov spaces as well as for spaces of functions with weighted Hermite expansions. Thanks to these uncertainty principles, we derive null-controllability results for evolution equations with adjoint systems enjoying smoothing effects in specific Gelfand-Shilov spaces. More precisely, we consider control subsets which are thick with respect to a quasi linearly growing density and establish sufficient conditions on the growth of the density to ensure null-controllability of these evolution equations.

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arXiv2020

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

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 $\frac{1}{2} \leq μ<1$, 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 $S^μ_μ(\mathbb{R}^n)$. 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.

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arXiv2020

Geometric conditions for the exact controllability of fractional free and harmonic Schrödinger equations

We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko-Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel-Rotach formula.

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