On singularities of minimum time control-affine systems

Affine control problems arise naturally from controlled mechanical systems. Building on previous results [2, 8], we go one step further and prove that the extremal flow given by Pontrjagin maximum principle can be stratified. We also study the regularity of this flow in terms of regular-singular transition and prove that, as in the nilpotent approximation, the singularity of time minimizing extremals is logarithmic. We finally give global bounds on the number of such singularities and apply our results to the control of two and three bodies in space mechanics.