Nonnegativity of a Quadratic Functional

The question of the nonnegativity of an integral $\smallint _{t_0 }^{t_f } q(x,u)dt$, where $q(x,u)$ is a quadratic form defined on solutions of the linear system \[\dot x = Ax + Bu,\quad x(t_0 ) = 0,\] arises in optimal control, in optimal filtering, and in system passivity. This paper derives sufficient conditions and necessary conditions for such an integral to be nonnegative. The conditions have the same form, and the “gap” between them can be considered as small. The existing theory for particular classes of $q(x,u)$ is recovered as a special case.