A condition of Jacobi type for a quadratic form to be nonnegative on a finite-faced cone

The author considers the problem of nonnegativity and positive definiteness of a Legendre quadratic form on a family of cones in Hilbert space that are the intersection of a fixed finite-faced cone with a monotonically increasing family of subspaces. This problem arises in the investigation to second order of an extremal in the Lagrange–Mayer–Bolza problem of classical variational calculus in the presence of finitely many additional constraints in the form of end and integral inequalities when there is only one set of Lagrange multipliers for the given extremal. It is shown that a complete answer can be given in terms of solutions of the corresponding Euler–Jacobi equation.
Bibliography: 5 titles.