46 research outputs found
Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic
We study the sub-Riemannian exponential for contact distributions on
manifolds of dimension greater or equal to 5. We compute an approximation of
the sub-Riemannian Hamiltonian flow and show that the conjugate time can have
multiplicity 2 in this case. We obtain an approximation of the first conjugate
locus for small radii and introduce a geometric invariant to show that the
metric for contact distributions typically exhibits an original behavior,
different from the classical 3-dimensional case. We apply these methods to the
case of 5-dimensional contact manifolds. We provide a stability analysis of the
sub-Riemannian caustic from the Lagrangian point of view and classify the
singular points of the exponential map
Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic
International audienceWe study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small radii and introduce a geometric invariant to show that the metric for contact distributions typically exhibits an original behavior, different from the classical 3-dimensional case. We apply these methods to the case of 5-dimensional contact manifolds. We provide a stability analysis of the sub-Riemannian caustic from the Lagrangian point of view and classify the singular points of the exponential map
A bisector line field approach to interpolation of orientation fields
We propose an approach to the problem of global reconstruction of an orientation field. The method is based on a geometric model called "bisector line fields", which maps a pair of vector fields to an orientation field, effectively generalizing the notion of doubling phase vector fields. Endowed with a well chosen energy minimization problem, we provide a polynomial interpolation of a target orientation field while bypassing the doubling phase step. The procedure is then illustrated with examples from fingerprint analysis
Approximate observability and back and forth observer of a PDE model of crystallisation process
In this paper, we are interested in the estimation of Particle Size
Distributions (PSDs) during a batch crystallization process in which particles
of two different shapes coexist and evolve simultaneously. The PSDs are
estimated thanks to a measurement of an apparent Chord Length Distribution
(CLD), a measure that we model for crystals of spheroidal shape. Our main
result is to prove the approximate observability of the infinite-dimensional
system in any positive time. Under this observability condition, we are able to
apply a Back and Forth Nudging (BFN) algorithm to reconstruct the PSD
New inversion methods for the single/multi-shape CLD-to-PSD problem with spheroid particles
In this paper, we express the Chord Length Distribution (CLD) measure
associated to a given Particle Size Distribution (PSD) when particles are
modeled as suspended spheroids in a reactor. Using this approach, we propose
two methods to reconstruct the unknown PSD from its CLD. In the single-shape
case where all spheroids have the same shape, a Tikhonov regularization
procedure is implemented. In the multi-shape case, the measured CLD mixes the
contribution of the PSD associated to each shape. Then, an evolution model for
a batch crystallization process allows to introduce a Back and Forth Nudging
(BFN) algorithm, based on dynamical observers. We prove the convergence of this
method when crystals are split into two clusters: spheres and elongated
spheroids. These methods are illustrated with numerical simulations
Generic singularities of line fields on 2D manifolds
International audienceGeneric singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with respect to a given conformal structure. The singularities correspond to the zeros of the vector fields and the genericity is considered with respect to a natural topology in the space of pairs of vector fields. Line fields at generic singularities turn out to be topologically equivalent to the Lemon, Star and Monstar singularities that one finds at umbilical points
Localized bounds on log-derivatives of the heat kernel on incomplete Riemannian manifolds
Bounds on the logarithmic derivatives of the heat kernel on a compact Riemannian manifolds have been long known, and were recently extended, for the log-gradient and log-Hessian, to general complete Riemannian manifolds. Here, we further extend these bounds to incomplete Riemannan manifolds under the least restrictive condition on the distance to infinity available, for derivatives of all orders. Moreover, we consider not only the usual heat kernel associated to the Laplace-Beltrami operator, but we also allow the addition of a conservative vector field. We show that these bounds are sharp in general, even for compact manifolds, and we discuss the difficulties that arise when the operator incorporates non-conservative vector fields or when the Riemannian structure is weakened to a sub-Riemannian structure
Avoiding observability singularities in output feedback bilinear systems
Control-affine output systems generically present observability singularities, i.e. inputs that make the system unobservable. This proves to be a difficulty in the context of output feedback stabilization, where this issue is usually discarded by uniform observability assumptions for state feedback stabilizable systems. Focusing on state feedback stabilizable bilinear control systems with linear output, we use a transversality approach to provide perturbations of the stabilizing state feedback law, in order to make our system observable in any time even in the presence of singular inputs