Geodesic
Well-posed infinite horizon variational problems on a~compact manifold
Trudy Matematicheskogo Instituta imeni V.A. Steklova, 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. 
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     author = {A. A. Agrachev},
     title = {Well-posed infinite horizon variational problems on a~compact manifold},
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     url = {http://geodesic.mathdoc.fr/item/TM_2010_268_a2/}
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A. A. Agrachev. Well-posed infinite horizon variational problems on a~compact manifold. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. I, Tome 268 (2010), pp. 24-39. http://geodesic.mathdoc.fr/item/TM_2010_268_a2/

[1] Agrachev A. A., “Krivizna i giperbolichnost gamiltonovykh sistem”, Tr. MIAN, 256, 2007, 31–53 | MR | Zbl

[2] Agrachev A. A., Chittaro F. C., “Smooth optimal synthesis for infinite horizon variational problems”, ESAIM: Control Optim. and Calc. Var., 15 (2009), 173–188 | DOI | MR

[3] Pesin Ya. B., Lektsii po teorii chastichnoi giperbolichnosti i ustoichivoi ergodichnosti, MTsNMO, M., 2006, 142 pp.

[4] Wojtkowski M. P., “Magnetic flows and Gaussian thermostats on manifolds of negative curvature”, Fund. math., 163 (2000), 177–191 | MR | Zbl