``The aim of this paper is to give high-order conditions for a point of a reference trajectory to be interior to the reachable set. This property is linked to the minimum-time problem by the fact that if \(\widehat x\) is an optimal trajectory on \([0,T]\), then, for each \(t\in [0,T]\), \(\widehat x(t)\) belongs to the boundary of \(R(\xi_ 0,t)\), i.e. the reachable set at time \(t\) from the initial point \(\xi_ 0= \widehat x(0)\) of the optimal trajectory. Therefore, each sufficient condition for \(\widehat x(t)\) to belong to the interior of \(R(\xi_ 0,t)\) yields a necessary condition for \(\widehat x\) to be time optimal. The high-order conditions are of particular interest in the case of nonlinear systems for which the Pontryagin maximum principle may not be sufficient to single out a unique candidate and singular trajectories may appear.'' ``For controllability along a possibly nonstationary trajectory, the idea is to use some known conditions based on the relations at a point in the Lie algebra associated with the system in order to construct high-order variations of the trajectory.'' ``Our techniques can be used for both bounded and unbounded controls. Many examples are given''.