A. A. Agrachev, “Well-posed infinite horizon variational problems on a compact manifold”, Proceedings of the Steklov Institute of Mathematics, 2010, 268, 17–31
We give an effective sufficient condition for a variational problem with infinite horizon on a 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.