ANR Programme BLANC EDITION 2009
Geometric Control Methods, Sub-Riemannian Geometry and Applications
- Partner 1: Université de Toulon et du Var
- Partner 2: INRIA Nancy - Grand Est
- Partner 3: École Polytechnique
- Partner 4: Université de Nice Sophia Antipolis
Postdoc: Marco Caponigro, from february 2010.
- Jean-Paul Gauthier,
coordinator of the project, coordinator of the partner 1,
Université du Sud Toulon-Var;
- Bernard Bonnard, member of the partner 2,
Université de Bourgogne;
- Ugo Boscain, coordinator of the
partner 3, École Polytechnique ;
- Jean-Baptiste Caillau, member of the partner 2,
Université de Bourgogne;
- Thomas Chambrion, member of the partner 2,
- Gregoire Charlot, member of the partner 2,
Université Joseph Fourier (Grenoble 1);
Yacine Chitour, member of the partner 3,
Université Paris Sud;
- Frédéric Jean, member of the partner 3,
École Nationale Supérieure de Techniques Avancées;
- Ludovic Rifford ,
coordinator of the partner 4, Université Nice Sophia-Antipolis;
- Séverine Rigot,
member of the partner 4, Université Nice Sophia-Antipolis;
- Mario Sigalotti, coordinator of the
partner 2, INRIA Nancy - Grand Est.
Several fundamental problems stemming from robotics, vision and quantum
physics can efficiently be
modeled in the framework of Geometric Control. The study and analysis of
these problems can then be
ranked as research questions of sub-Riemannian geometry (SRG for short).
The purpose of this project consists in gathering French mathematicians
working on these issues and
to create a research network on sub-Riemannian geometry.
hope, via postdoc positions and
conferences, to disseminate the knowledge acquired world-wide, and to
stimulate young mathematicians
to work in this interdisciplinary area.
We plan to address problems involving both ODEs and PDEs, for which
geometric control techniques
open new horizons. More precisely we plan to study:
The approach we are proposing to tackle these scientific challenges is
based on techniques developed
in the framework of sub-Riemannian geometry and geometric control
theory, partially by the members
of the team themselves.
Problems in quantum control such as controllability properties of
the Schroedinger equation, motion
planning on Lie groups, optimal transfer between energy levels etc...
These problems have applications
in nuclear magnetic resonance (especially in medicine) and in quantum
(as in the realization of quantum gates for quantum computers).
Non-isotropic diffusion processes modeled by a heat equation whose
evolution operator is a subelliptic
Laplacian. This is a very old problem, that recently gathered refreshed
interest after the
papers of Petitot and Citti-Sarti that recognized that phenomena of
non-isotropic diffusion are key
ingredients in models of the functional architecture of the human visual
cortex V1. This problem
involves some very interesting questions related to geometric measure
theory and sophisticated
techniques of noncommutative harmonic analysis.
Problems of motion planning. Nonholonomic systems attract the
attention of the scientific community
for the theoretical challenges arising from the research on the control
of these systems and for
their relevance in applications such as robotics and quantum control. In
particular, the problem
of generating feasible trajectories joining two system configurations
(referred to as nonholonomic motion planning) has been solved for
specific classes of driftless systems by effective techniques.
However, there does not exist any general solution to the motion
planning problem at the present
Mass Transportation problems in sub-Riemannian geometry and more
generally in geometric control
theory. These problems have applications for any optimization transport
problem with nonholonomic
or holonomic constraints. Furthermore, using an approach "à la
Lott, Villani", the
study of optimal transport problems on sub-Riemannian manifolds may lead
to a better understanding
of Carnot-Carathéodory spaces in terms of curvatures.
Keywords: Geometric control theory, Sub-Riemannian geometry,
Carnot Caratheodory distance,
Conjugate points, Cut locus, Controllability of the bilinear Schroedinger equation, Hypoelliptic
heat equation, Quantum Control.