  In optimal control theory the expression infimum gap means a strictly
negative difference between the infimum value of a given minimum problem and
the infimum value of a new problem obtained by the former by extending the
original family V of controls to a larger family W. Now, for some classes of
domain-extensions -- like convex relaxation or impulsive embedding of unbounded
control problems -- the normality of an extended minimizer has been shown to be
sufficient for the avoidance of an infimum gaps. A natural issue is then the
search of a general hypothesis under which the criterium 'normality implies no
gap' holds true. We prove that, far from being a peculiarity of those specific
extensions and from requiring the convexity of the extended dynamics, this
criterium is valid provided the original family V of controls is abundant in
the extended family W. Abundance, which is stronger than the mere C^0-density
of the original trajectories in the set of extended trajectories, is a
dynamical-topological notion introduced by J. Warga, and is here utilized in a
'non-convex' version which, moreover, is adapted to differential manifolds. To
get the main result, which is based on set separation arguments, we prove an
open mapping result valid for Quasi-Differential-Quotient (QDQ) approximating
cones, a notion of 'tangent cone' resulted as a peculiar specification of H.
Sussmann's Approximate-Generalized-Differential-Quotients (AGDQ) approximating
cone.
