51 research outputs found
Abnormal Singular Foliations and the Sard Conjecture for generic co-rank one distributions
Given a smooth totally nonholonomic distribution on a smooth manifold, we
construct a singular distribution capturing essential abnormal lifts which is
locally generated by vector fields with controlled divergence. Then, as an
application, we prove the Sard Conjecture for rank 3 distribution in dimension
4 and generic distributions of corank 1
On the stabilization problem for nonholonomic distributions
Abstract. Let M be a smooth connected complete manifold of dimension n, and be a smooth nonholonomic distribution of rank m ≤ n on M. We prove that if there exists a smooth Riemannian metric on for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of 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 we establish fine properties of optimal trajectories
Abnormal subanalytic distributions and minimal rank Sard Conjecture
We present a description of singular horizontal curves of a totally
nonholonomic analytic distribution in term of the projections of the orbits of
some isotropic subanalytic singular distribution defined on the nonzero
annihilator of the initial distribution in the cotangent bundle. As a
by-product of our first result, we obtain, under an additional assumption on
the constructed subanalytic singular distribution, a proof of the minimal rank
Sard conjecture in the analytic case. It establishes that from a given point
the set of points accessible through singular horizontal curves of minimal
rank, which corresponds to the rank of the distribution, has Lebesgue measure
zero
Mass Transportation on Sub-Riemannian Manifolds
We study the optimal transport problem in sub-Riemannian manifolds where the
cost function is given by the square of the sub-Riemannian distance. Under
appropriate assumptions, we generalize Brenier-McCann's Theorem proving
existence and uniqueness of the optimal transport map. We show the absolute
continuity property of Wassertein geodesics, and we address the regularity
issue of the optimal map. In particular, we are able to show its approximate
differentiability a.e. in the Heisenberg group (and under some weak assumptions
on the measures the differentiability a.e.), which allows to write a weak form
of the Monge-Amp\`ere equation
Generic hyperbolicity of Aubry sets on surfaces
International audienceGiven a Tonelli Hamiltonian of class C2 on the cotangent bundle of a compact surface, we show that there is an open dense set of potentials in the C2 topology for which the Aubry set is hyperbolic in its energy level
Mass transportation with LQ cost functions
We study the optimal transport problem in the Euclidean space where the cost
function is given by the value function associated with a Linear Quadratic
minimization problem. Under appropriate assumptions, we generalize Brenier's
Theorem proving existence and uniqueness of an optimal transport map. In the
controllable case, we show that the optimal transport map has to be the
gradient of a convex function up to a linear change of coordinates. We give
regularity results and also investigate the non-controllable case
Semiconcavity results for optimal control problems admitting no singular minimizing controls
Semiconcavity results have generally been obtained for optimal control problems in absence of state constraints. In this paper, we
prove the semiconcavity of the value function of an optimal control problem with end-point constraints for which all minimizing
controls are supposed to be nonsingular
- …