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Compactness for sub-Riemannian length–minimizers and subanalyticity. (English) Zbl 1039.53038
Compactness properties of the set of minimal geodesics with a prescribed small enough length are investigated. Notions of so-called abnormal geodesics are derived. Regularity properties of the sub-Riemannian distance function are considered. It is shown that any neighborhood of an initial point in a Riemannian manifold contains a point where the distance function is not continuously differentiable. It is substantiated that the set of minimal geodesics may be parametrized if all of them are regular. An efficient necessary condition for a admissible path to be a shortest one is obtained with the aid of the Pontryagin maximum principle and properties of the second variation. In the case of real-analytic Riemannian manifolds the author shows that only abnormal minimal geodesics may destroy subanalyticity of the sub-Riemannian distance function out of the initial point.

MSC:
53C22 Geodesics in global differential geometry
49N60 Regularity of solutions in optimal control
49J15 Existence theories for optimal control problems involving ordinary differential equations
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49K15 Optimality conditions for problems involving ordinary differential equations
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