Global Existence for the Backward Riccati Equation under Convexity Conditions

Let $T>0$ be a real number and $l,k\ge1$ 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 continuous functions of $t$, such that every $Q(t)$ and every $R(t)$ is symmetric and every $R(t)$ is positive definite; by Invertibility of Symmetric Positive Definite Matrices 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 matrix product and transpose. Integrals below are entrywise Riemann integrals of continuous functions (existing by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; degenerate intervals by the convention of Mean-Square Riemann Integral of a Family of Random Variables). Continuity of a real-valued function on an interval is understood as continuity of a map of metric spaces, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric.

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.