Conditional Positivity of Quadratic Forms in Hilbert Space

• Published in: SIAM journal on mathematical analysis Vol. 11; no. 6; pp. 1047 - 1057
• Main Author: Martin, D. H.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.11.1980
• Subjects: Control theory; Decomposition; Hilbert space; Inequality; Theorems
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/0511092

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.