  In optimal control theory, infimum gap means that there is a gap between the
infimum values of a given minimum problem and an extended problem, obtained by
enlarging the set of original solutions and controls. The gap phenomenon is
somewhat "dual" to the problem of the controllability of the original control
system to an extended solution. In this paper we present sufficient conditions
for the absence of an infimum gap and for controllability for a wide class of
optimal control problems subject to endpoint and state constraints. These
conditions are based on a nondegenerate version of the nonsmooth constrained
maximum principle, expressed in terms of subdifferentials. In particular, under
some new constraint qualification conditions, we prove that: (i) if an extended
minimizer is a nondegenerate normal extremal, then no gap shows up; (ii) given
an extended solution verifying the constraints, either it is a nondegenerate
abnormal extremal, or the original system is controllable to it. An application
to the impulsive extension of a free end-time, non-convex optimization problem
with control-polynomial dynamics illustrates the results.
