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  The minimizing problem for the length of trajectories with respect to a submetric on a distribution is considered.
  Quadratic sufficient conditions for the strong minimality of abnormal trajectories of arbitrary length are obtained.
  The results hold for distributions of arbitrary dimensions and for a broad class of submetrics, including those
  of sub-Riemannian and sub-Finsler metrics.
          PDF Text  in   10  pages   
  ERRATUM (incorrect translation)
  The first two sentences of the second paragraph on p. 369 should read:

  A point is said to provide a {\it Pontryagin minimum} if, for any $N$, it is a point of local minimum
  with respect to the norm $\|w\|_1$ on the part of the admissible set defined by the additional
  condition $\|w\|_\infty\leq N$. This type of minimum is intermediate between the classical
  weak and strong minima.
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