25 research outputs found
Nonholonomic tangent spaces: intrinsic construction and rigid dimensions
A nonholonomic space is a smooth manifold equipped with a bracket generating family of vector fields. Its infinitesimal version is a homogeneous space of a nilpotent Lie group endowed with a dilation which measures the anisotropy of the space. We give an intrinsic construction of these infinitesimal objects and classify all rigid (i.e. not deformable) cases
Invariant Lagrange submanifolds of dissipative systems
We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) =
0, q \in M, on a compact manifold M
Jacobi fields in optimal control: one-dimensional variations
We study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piece-wise analytic in this case but may have much more singularities than the control. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighborhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps
On the Hausdorff volume in sub-Riemannian geometry
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative
of the spherical Hausdorff measure with respect to a smooth volume. We prove
that this is the volume of the unit ball in the nilpotent approximation and it
is always a continuous function. We then prove that up to dimension 4 it is
smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4
on every smooth curve) but in general not C^5. These results answer to a
question addressed by Montgomery about the relation between two intrinsic
volumes that can be defined in a sub-Riemannian manifold, namely the Popp and
the Hausdorff volume. If the nilpotent approximation depends on the point (that
may happen starting from dimension 5), then they are not proportional, in
general.Comment: Accepted on Calculus and Variations and PD
Continuity of Optimal Control Costs and its application to Weak KAM Theory
We prove continuity of certain cost functions arising from optimal control of
affine control systems. We give sharp sufficient conditions for this
continuity. As an application, we prove a version of weak KAM theorem and
consider the Aubry-Mather problems corresponding to these systems.Comment: 23 pages, 1 figures, added explanations in the proofs of the main
theorem and the exampl