We describe precisely, under generic conditions, the contact of the accessibility set at time $T$ with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-riemannian system of rank 2. As a consequence we obtain in sub-riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\gamma $ into two sectors, bordered by the first Pontryagin’s cone along $\gamma $, called the ${\mathrm{L}}^{\infty}$-sector and the ${\mathrm{L}}^{2}$-sector. Moreover we...

We describe precisely, under generic conditions, the contact of
the accessibility set at time with an abnormal direction,
first for a single-input affine control system with constraint on
the control, and then as an
application for a sub-Riemannian system of rank 2. As a
consequence we obtain in sub-Riemannian geometry a new
splitting-up of the sphere near an abnormal minimizer
into two sectors, bordered by the first Pontryagin's cone along
, called the L-sector and the
L-sector.
Moreover we...

L’objectif de ce travail est de faire quelques remarques géométriques et des calculs préliminaires pour construire l’arc atmosphérique optimal d’une navette spatiale (problème de rentrée sur Terre ou programme d’exploration de Mars). Le système décrivant les trajectoires est de dimension 6, le contrôle est l’angle de gîte cinématique et le coût est l’intégrale du flux thermique. Par ailleurs il y a des contraintes sur l’état (flux thermique, accélération normale et pression dynamique). Notre étude...

Let $M$ be a smooth connected complete manifold of dimension $n$, and $\Delta $ be a smooth nonholonomic distribution of rank $m\le n$ on $M$. We prove that if there exists a smooth Riemannian metric on1for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $\Delta $ on $M$. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of...

The aim of this article is to make some geometric remarks and
some preliminary calculations in order to construct the optimal
atmospheric arc of a spatial shuttle (problem of reentry on Earth
or Mars Sample Return
project). The system describing the trajectories is in
dimension 6, the control is the bank angle and the cost is the
total thermal flux. Moreover there are state constraints (thermal
flux, normal acceleration and dynamic pressure). Our study is
mainly geometric and is founded on the...

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.
In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $\Omega \subset {\mathbb{R}}^{n}$, with Dirichlet boundary conditions. The observation is done on a subset $\omega $ of Lebesgue measure $\left|\omega \right|=L\left|\Omega \right|$, where $L\in (0,1)$ is fixed. We denote by ${\mathcal{U}}_{L}$ the...

Euler and Lagrange proved the existence of five equilibrium points in the circular restricted three-body problem. These equilibrium points are known as the Lagrange points (Euler points or libration points) ${L}_{1},...,{L}_{5}$. The existence of families of periodic and quasi-periodic orbits around these points is well known (see [, , , , ]). Among them, halo orbits are 3-dimensional periodic orbits diffeomorphic to circles. They are the first kind of the so-called Lissajous orbits. To be selfcontained, we first provide...

We consider the wave and Schrödinger equations on a bounded open connected subset $\Omega $ of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset $\omega $ of $\Omega $ during a time interval $[0,T]$ with $T>0$. It is well known that, if the pair $(\omega ,T)$ satisfies the Geometric Control Condition ($\omega $ being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be...

This paper is a proceedings version of [6], in which we state a Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we establish some properties of the Quantum Limits (QL).
We consider a sub-Riemannian (sR) metric on a compact 3D manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We state a QE theorem for the eigenfunctions...

The aim of this article is to present algorithms to compute the first
conjugate time along a smooth extremal curve, where the trajectory
ceases to be optimal. It is based on recent theoretical developments
of geometric optimal control, and the article contains a review
of second order optimality conditions.
The computations are related to a test
of positivity of the intrinsic second order derivative or a test of
singularity of the extremal flow. We derive an algorithm called COTCOT
(Conditions...

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