Geometric Control Methods, Sub-Riemannian Geometry and Applications – GCM

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. Indeed, while the French community on the domain is one of the most important in the world, it is also quite wide-spread. Such a project will then comfort the French leadership on the emerging field of research. 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. We plan to address problems involving both ODEs and PDEs, for which geometric control techniques open new horizons. More precisely we plan to study: -) 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 sub-elliptic 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-Carath'eodory spaces in terms of curvatures. 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. This approach was already quite successful. Indeed it allowed to solve open problems in the field, e.g., the proof that, for generic sub-Riemannian structures of rank greater than two, there are no nontrivial minimizing singular curves, the explicit construction of the hypoelliptic heat kernel for the most important 3D Lie groups, and the proof of the controllability of the bilinear Schroedinger equation with discrete spectrum, under some "generic" assumptions. Members of the project are internationally recognized experts in SRG, and more generally in geometric and optimal control and their applications, both from a theoretical and numerical point of view. They have been collaborating for many years and they have many papers in common. They also have developed a net of high level international collaborations.