Conjugate points and intersections of bang-bang trajectories

The cut-locus of bang-bang trajectories is the set of all points where bang-bang trajectories of different switching orders intersect. Results on time-optimal control for 3-D systems show that the cut-locus plays an essential role in questions of optimality near a reference point. Here the cut-locus is related to conjugate points. Essentially it is proved that a nontrivial cut-locus can only bifurcate if the conjugate points satisfy certain regularity conditions