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  We consider the problem of finding length minimizing trajectories w.r.t. a submetric on a k-distribution in an n-space.
  By a submetric we mean a positive sublinear functional  on the distribution; in particular, it can be a sub-Riemannian
  or sub-Finsler metric. We consider the case of so-called abnormal trajectories, which is today a subject of thorough
  attention of specialists. Setting the problem in an optimal control form, we apply sufficient conditions of some quadratic
  order for the strong minimality of singular extremals, obtained recently by the author on the basis of his and
  A.A. Milyutin's  preceeding research.     For any tuple of Lagrange multipliers we consider the second variation of the
  corresponding Lagrange function,  and using this family of second variations, we consider the notion of  quadratically
  rigid  trajectories, proposed by Milyutin.  (He proved that the quadratic rigidity of a trajectory is sufficient for its rigidity.)
  Our main result is that

         every quadratically rigid trajectory is a strict strong length minimizer
         for any choice of strictly convex submetric on the distribution.

  (Actually, the submetric can possess a certain weaker property than its strict convexity.)
  This strengthens the known results of other authors (Montgomery, Liu-Sussmann, Petrov, and Agrachev-Sarychev).
  Also, similar results are obtained for arbitrary nonstrictly convex submetrics.  In this case our conditions guarantee not
  the strong minimality, but the L_1-minimality of the trajectory w.r.t. the velocity.   Both these results are valid for
  distributions of arbitrary dimension and for trajectories of arbitrary length  (not only for their small pieces).


           PDF  Text  in   80 pages,   

           Russian original:     Квадратичные достаточные условия сильной минимальности анормальных
  субримановых геодезических -- Итоги науки и техники. Современная мат-ка и ее приложения, т. 65
  (Труды конференции, посвященной 90-летию Л.С.Понтрягина, т. 4, Оптимальное управление),
  М., 2000, с. 5—89.
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