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.

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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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