Well-posed infinite horizon variational problems on a compact manifold

We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i. e. a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis we construct a well-projected to M invariant Lagrange submanifold of the extremals' flow in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics

Rolling balls and Octonions

In this semi-expository paper we disclose hidden symmetries of a classical nonholonomic kinematic model and try to explain geometric meaning of basic invariants of vector distributions

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

Spectrum of the Second Variation

Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study the asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation. In the case of the least action principle for the harmonic oscillator, we obtain the classical Euler identity n=1(1 - x(2)/(n)(2)) = sin x/x. The general case may serve as a rich source of new nice identities