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

- 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, Nancy-Université ;
- 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.

The purpose of this project consists in gathering French mathematicians working on these issues and to create a research network on sub-Riemannian geometry. We also hope, via postdoc positions and conferences, to disseminate the knowledge acquired world-wide, and to stimulate young mathematicians to work in this interdisciplinary area.

- 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 information science (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 time.
- 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 Sturm, 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.