Sufficiency and singularity in optimal control

A sufficiency theorem for a strong minimum of a fixed-endpoint optimal control problem with equality control constraints, without imposing the standard strengthened Legendre–Clebsch condition, is derived. The proof is direct in nature as it deals explicitly with the positivity of the second variation, in contrast with possible generalizations of conjugate points, solutions to matrix-valued Riccati equations, or Hamilton–Jacobi theory. The approach is based on a generalization of the one introduced by Hestenes, in terms of directional convergent sequences of trajectories, originally posed for the problem of Bolza in the calculus of variations.