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

In this study, we construct a sufficiency theorem for an unconstrained Lagrange fixed-endpoint problem that offers sufficient conditions for processes that do not satisfy the traditional nonsingularity assumption, i.e., the new sufficiency theorem does not impose the strengthened Legendre condition. In contrast to various generalisations of conjugate points, solutions of certain matrix Riccati equations, invariant integrals, or the Hamiltonian-Jacobi theory, the latter leverages explicitly the positivity of the second variation in the demonstration of sufficiency.