Closing geodesics in C1C^1 topology

Given a closed Riemannian manifold, we show how to close an orbit of the geodesic flow by a small perturbation of the metric in the $C^1$ topology

On the stabilization problem for nonholonomic distributions

Let $M$ be a smooth connected and complete manifold of dimension $n$, and $\Delta$ be a smooth nonholonomic distribution of rank $m\leq n$ on $M$. We prove that, if there exists a smooth Riemannian metric on $\Delta$ for 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 nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the corresponding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.Comment: accept\'e pour publication dans J. Eur. Math. Soc. (2007), \`a para\^itre, 29 page

The intrinsic dynamics of optimal transport

The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a sufficiently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique. This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probility densities is unique.Comment: 33 pages, 4 figure

Stratified semiconcave control-Lyapunov functions and the stabilization problem

International audienceGiven a globally asymptotically controllable control system, we construct a control-Lyapunov function which is stratified semiconcave;that is, roughly speaking whose singular set has a Whitney stratification. Then we deduce the existence of smooth feedbacks which make the closed-loop system almost globally asymptotically stable

The stabilization problem on surfaces

International audienceWe briefly recall some remarkable result on the stabilization problem of driftless affine control systems on surfaces. Then we remark that an interesting answer to the stabilization problem for such control systems would be to construct what we call smooth repulsive stabilizing feedbacks. Thus we discuss the existence of such feedbacks and present a sufficient condition in the two dimensional case

On the Hausdorff Dimension of the Mather Quotient

Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics

On the convexity of injectivity domains on nonfocal manifolds

Given a smooth nonfocal compact Riemannian manifold, we show that the so-called Ma--Trudinger--Wang condition implies the convexity of injectivity domains. This improves a previous result by Loeper and Villani