The method of characteristics: A geometric approach to sufficient conditions for a local minimum

Abstract

So far, our focus has been on necessary conditions for optimality. The conditions of the Pontryagin maximum principle, Theorem 2.2.1, collectively form the first-order necessary conditions for optimality of a controlled trajectory (aside from the much stronger minimum condition on the Hamiltonian that generalizes the Weierstrass condition of the calculus of variations). Clearly, as in ordinary calculus, first-order conditions by themselves are no guarantee that even a local extremum is attained. High-order tests, based on second- and increasingly higher-order derivatives, like the Legendre–Clebsch conditions for singular controls, can be used to restrict the class of candidates for optimality further, but in the end, sufficient conditions need to be provided that at least guarantee some kind of local optimality. These will be the topic of the next two chapters of our text.