Determining the Sufficiency in Optimal Control without the Strengthened Condition of Legendre

In this paper we derive a sufficiency theorem of an unconstrain ed fixed-endpoint problem of Lagrange which provides sufficient c for processes which do not satisfy the standard assumption of nonsingularity, that is, the new sufficiency theorem does not impose th e strengthened condition of Legendre. The proof of the sufficiency resul t is direct in nature since the former uses explicitly the positivity of the second variation, in contrast with possible generalizations of conjugate points, solutions of certain matrix Riccati equations, invariant integrals, or the Hamiltonian-Jacobi theory.