arXiv subjects

Cyril Letrouit

Publications and source records attributed to Cyril Letrouit.

21 records · Page 2Linked to original sources

arXiv2021

Observability and Controllability for the Schr{ö}dinger Equation on Quotients of Groups of Heisenberg Type

We give necessary and sufficient conditions for the controllability of a Schr\''odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\''odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\''odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

math.AP↗

arXiv2021

Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates

In this article, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the "degenerated directions" of the subelliptic structure. First, for any $γ\geq 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator $Δ_γ=\partial_x^2+|x|^{2γ}\partial_y^2$, which has step $γ+1$. We then derive consequences for the observability of the Schr\''odinger type equation $i\partial_tu-(-Δ_γ)^{s}u=0$ where $s\in N$. We identify three different cases: depending on the value of the ratio $(γ+1)/s$, observability may hold in arbitrarily small time, or only for sufficiently large times, or even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial_tu+(-Δ_γ)^su=0$ and establish a decay rate for the damped wave equation associated with $Δ_γ$.

math.AP↗

arXiv2020

From internal to pointwise control for the 1D heat equation and minimal control time

Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set $ω_{\varepsilon}=(x_{0}-\varepsilon, x_{0}+\varepsilon )$, in the limit $\varepsilon\rightarrow 0$, where $x_{0}\in (0,1)$. It is known that depending on arithmetic properties of $x_{0}$, there may exist a minimal time $T_{0}$ of pointwise control at $x_{0}$ of the heat equation. Besides, for any $\varepsilon$ fixed, the heat equation is controllable with control set $ω_{\varepsilon}$ in any time $T>0$. We relate these two phenomena. We show that the observability constant on $ω_\varepsilon$ does not converge to $0$ as $\varepsilon\rightarrow 0$ at the same speed when $T>T_{0}$ (in which case it is comparable to $\varepsilon^{1/2}$) or $T<T_{0}$ (in which case it converges faster to $0$). We also describe the behavior of optimal $L^{2}$ null-controls on $ω_{\varepsilon}$ in the limit $\varepsilon \rightarrow 0$.

math.AP↗