A stochastic interpretation of singular quadratic minimization theory-Part I: General conditions for minimality

A class of singular quadratic minimization problems is examined by reducing the performance index to a quadratic form in the control variable. A necessary and sufficient condition for minimality is that the kernel of this quadratic form be non-negative definite, or equivalently, be a covariance kernel (function). Various properties of such functions are then used to obtain different sets of necessary and sufficient conditions. The present paper deals only with those properties of covariances that do not involve differentiability or integrability conditions. The properties include non-negative definiteness, continuity, symmetry, and factorizability. A study of these properties yields natural interpretations of a number of quantities and conditions which arise in singular quadratic minimization theory.