In this paper, we develop a sufficiency theorem for an unconstrained Lagrange fixed-endpoint problem that provides sufficient conditions for processes that do not satisfy the traditional nonsingularity assumption, i.e., the strengthened Legendre condition is not imposed by the new sufficiency theorem. Unlike numerous generalisations of conjugate points, solutions of specific matrix Riccati equations, invariant integrals, or the Hamiltonian-Jacobi theory, the latter uses the positivity of the second variation explicitly in the demonstration of sufficiency.
Gerardo Sanchez Licea
Departamento de Matematicas, Facultad de Ciencias UNAM, Mexico D.F. 04510, Mexico.