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 lambda /sub 0/ 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.<>