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 an invariant Lagrangian submanifold (well-projected to $M$) of the flow of extremals in the cotangent bundle $T^*M$. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics.