Geodesic
Well-posed infinite horizon variational problems on a~compact manifold
Informatics and Automation, Differential equations and topology. I, Tome 268 (2010), pp. 24-39

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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. 
@article{TRSPY_2010_268_a2,
     author = {A. A. Agrachev},
     title = {Well-posed infinite horizon variational problems on a~compact manifold},
     journal = {Informatics and Automation},
     pages = {24--39},
     publisher = {mathdoc},
     volume = {268},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_268_a2/}
}
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A. A. Agrachev. Well-posed infinite horizon variational problems on a~compact manifold. Informatics and Automation, Differential equations and topology. I, Tome 268 (2010), pp. 24-39. http://geodesic.mathdoc.fr/item/TRSPY_2010_268_a2/