A Generalized Legendre-Clebsch Condition for the Singular Cases of Optimal Control

For certain optimal control problems, some of the extremal trajectories generated by simultaneous solution of the state and adjoint equations may include arcs of a special character, called “singular” arcs. The optimality of singular arcs has been the subject of considerable uncertainty, since the classical criteria are inapplicable or inconclusive. This uncertainty has recently been reduced by the discovery of additional necessary conditions for the optimality of singular arcs. The principal result of this paper is a general statement and proof of these conditions, in the form of a “generalized Legendre-Clebsch condition” which reduces to the classical Legendre-Clebsch condition when applied to nonsingular arcs, and gives additional necessary conditions when applied to singular arcs. Other results include a classification of the possible singular arcs, a useful extension of the conventional optimal-control formalism (by the introduction of “generalized Hamiltonians” and “generalized control transformations”), and some interesting variational formulae.