摘要: If X and Y are real Hilbert spaces (A : X approaches Y is a bounded linear operator, and gamma is contained in set Y is a closed convex cone) an immediate sufficient condition for a quadratic form Q on X to be positive subject to ... 展开 If X and Y are real Hilbert spaces (A : X approaches Y is a bounded linear operator, and gamma is contained in set Y is a closed convex cone) an immediate sufficient condition for a quadratic form Q on X to be positive subject to the constraint Ax is an element of gamma, is that Q be decomposable as a sum Q(x) = C(Ax) + S(x), where C is a quadratic form on Y which is positive on gamma, and S is positive definite on X. The necessity of such a decomposition is established for a class of quadratic forms which commonly occur in variational problems - the Legendre forms. The proof furnishes formulas for C and S which are explicit apart from the occurrence of an unkown scalar. The usefulness of the result is illustrated by the determination of the focal (conjugate) time of a linear-quadratic control problem with inequality constraints on the final state. 收起