Conditional Positivity of Quadratic Forms in Hilbert Space
Duncan H. Martin · SIAM Journal on Mathematical Analysis · 1980
If X and Y are real Hilbert spaces, $A:X \to Y$ is a bounded linear operator, and $\Gamma \subseteq Y$ is a closed convex cone, an immediate sufficient condition for a quadratic form Q on X to be positive subject to the constraint $Ax \in \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 not obvious, but is established here 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 unknown 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.