An Euler-Lagrange inclusion for optimal control problems

A new first-order necessary condition is proved for nonsmooth, nonlinear optimal control problems with general endpoint constraints and for which the velocity set may be possibly nonconvex. It is in the nature of a generalization of the Euler-Lagrange equation of the calculus of variations to optimal control. It resembles the weak form of the maximum principle but it is distinct from it because it employs a “total” generalized gradient instead of the customary product of partial generalized gradients. The optimality condition is shown to be sufficient for optimality when it is specialized to apply to normal, convex problems. A counterexample illustrates that, for such problems, the maximum principle is not a sufficient condition