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 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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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2010, Vol. 268, pp. 24–39.
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Agrachev, A.A. Well-posed infinite horizon variational problems on a compact manifold. Proc. Steklov Inst. Math. 268, 17–31 (2010). https://doi.org/10.1134/S0081543810010037
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DOI: https://doi.org/10.1134/S0081543810010037