Continuity of optimal control costs and its application to weak KAM theory
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We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry–Mather problems corresponding to these systems.
Mathematics Subject Classification (2000)35F21 49L25
Andrei Agrachev was supported by PRIN and Paul W. Y. Lee was supported by the NSERC postdoctoral fellowship.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
- 3.Bernard, P., Buffoni, B.: Weak KAM pairs and Monge–Kantorovich duality. Asymptotic analysis and singularities—elliptic and parabolic PDEs and related problems. Adv. Stud. Pure Math., 47-2, pp. 397–420. Mathematical Society of Japan, Tokyo (2007)Google Scholar
- 7.Evans L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence (1998)Google Scholar
- 9.Fathi, A.: The Weak KAM Theorem in Lagrangian Dynamics, 10th prelimiary versionGoogle Scholar
- 12.Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi equation, unpublished preprint (1987)Google Scholar
- 13.Montgomery R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs, 91. American Mathematical Society, Providence (2002)Google Scholar
- 14.Villani C.: Topics in Mass Transportation. American Mathematical Society, Providence (2003)Google Scholar
- 15.Villani C.: Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer, Berlin (2009)Google Scholar
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