Envelopes, high-order optimality conditions and Lie brackets

A generalization of the theory of envelopes of the classical calculus of variations to optimal control is described. This generalization extends the author´s previous work (1986) in two ways. It allows the use of quasi-extremal trajectories, i.e. trajectories that satisfy all the conditions of the Pontryagin maximum principle except for the fact that the sign of the constant λ₀ that appears in the Hamiltonian multiplying the Lagrangian is not restricted, and it makes it possible to prove nonoptimality of some `abnormal´ trajectories. The envelope technique is used to obtain an alternative proof of a theorem on the number of switchings of time-optimal bang-bang trajectories for two-vector-field systems in three dimensions