Transversality, the maximum principle, and the gap problem

Hector Sussmann

The Maximum Principle is really a sufficient condition for a certain pair of single-valued or set-valued maps to intersect transversally. Using this observation, we prove sufficient conditions for the nonexistence a gap between an optimal control problem with a class U of open-loop controls and the same problem with a smaller class V of controls which is an "abundant subset" of U the sense introduced by Jack Warga.