Optimal trajectories which contain singular arcs

Classically, the optimality of bang-bang trajectories and singular arcs have been studied separately. However, for most problems in optimal control it is necessary to consider trajectories which combine them. The author presents the basic problem, along with some preliminary observations and results. He first formalizes, for an analytic system, one way to organize singular extremals in the cotangent bundle. He then considers the concatenation of bang and singular arcs, giving an example of how to obtain structure results in a degenerate but still generic case in dimension three