  This paper studies the regularity of the minimum time function, $T(\cdot)$,
for a control system with a general closed target, taking the state equation in
the form of a differential inclusion. Our first result is a sensitivity
relation which guarantees the propagation of the proximal subdifferential of
$T$ along any optimal trajectory. Then, we obtain the local $C^2$ regularity of
the minimum time function along optimal trajectories by using such a relation
to exclude the presence of conjugate times.
