Well-posed infinite horizon variational problems on a compact manifold

TL;DR

This paper establishes a sufficient condition for the existence of smooth optimal solutions to infinite horizon variational problems on compact manifolds, utilizing hyperbolic dynamics and curvature properties.

Contribution

It introduces a new criterion ensuring smooth optimal synthesis for infinite horizon variational problems on compact Riemannian manifolds.

Findings

Existence of smooth optimal synthesis under the new condition

Construction of an invariant Lagrangian submanifold in the cotangent bundle

Application of hyperbolic dynamics and curvature analysis

Abstract

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.