On the optimality of parametrized families of bang-bang trajectories

Using a field of broken extremals we show that a flow of bang-bang extremals can only lose local optimality at switching surfaces while it remains locally optimal if a switching surface is crossed transversally. We also show that optimality of the flow ceases at the switching surface if the trajectories exhibit a transversal "fold", i.e., if the two flows (before and after the switch) point to opposite sides of the switching surface.