Standard necessary conditions for optimal control problems with pathwise st
ate constraints supply no useful information about minimizers in a number o
f cases of interest, e.g., when the left endpoint of state trajectories is
fixed at x(0) and x(0) lies in the boundary of the state constraint set; in
these cases a nonzero, but nevertheless trivial, set of multipliers exists
. We give conditions for the existence of nontrivial multipliers. A feature
of these conditions is that they allow nonconvex velocity sets and measura
bly time-dependent data. The proof techniques are based on refined estimate
s of the distance of a given state trajectory from the set of state traject
ories satisfying the state constraint, originating in the dynamic programmi
ng literature.