Let $T>0$ be a \reftext{def:real-numbers-c54-2026c}{real number} and $l,k\ge1$ \reftext{def:natural-numbers-2026a}{natural numbers}. Let $A$ ($l\times l$), $B$ ($l\times k$), $Q$ ($l\times l$), $V$ ($l\times k$), and $R$ ($k\times k$) assign real matrices to each $t\in[0,T]$, all entries being \reftext{def:continuity-closed-interval-c54-2026b}{continuous} functions of $t$, such that every $Q(t)$ and every $R(t)$ is \reftext{def:positive-semidefinite-matrix-2026a}{symmetric} and every $R(t)$ is positive definite; by \ref{lem:pd-inverse-2026a} each $R(t)^{-1}$ exists and is symmetric positive definite. Let $F$ be a symmetric positive semidefinite real $l\times l$ matrix, and suppose that for every $t\in[0,T]$ the matrix

is positive semidefinite, with the \reftext{def:product-real-matrices-2026a}{matrix product} and \reftext{def:transpose-real-matrix-2026a}{transpose}. Integrals below are entrywise \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integrals} of continuous functions (existing by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}; degenerate intervals by the convention of \ref{def:mean-square-riemann-integral-2026a}).

Then there is exactly one assignment $Z$ of a real $l\times l$ matrix $Z(t)$ to each $t\in[0,T]$, with continuous entries, satisfying the backward Riccati equation

moreover every $Z(t)$ is symmetric positive semidefinite.