Global Existence and Uniqueness for the Kalman Covariance Riccati Equation

Let $a<b$ be real numbers and $k\ge1$ a natural number. Let $A$, $C$, and $D$ assign to each $t\in[a,b]$ real $k\times k$ matrices with entries continuous in $t$, such that every $C(t)$ and every $D(t)$ is positive semidefinite, and let $P_0$ be a positive semidefinite real $k\times k$ matrix. Integrals 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), products are matrix products, and $(\cdot)^{\top}$ is the transpose. 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 $P$ of a real $k\times k$ matrix $P(t)$ to each $t\in[a,b]$, with continuous entries, such that

Moreover, with $\Lambda$ the unique solution of the Lyapunov equation with the same data $A$, $C$, $P_0$ from Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations, every $P(t)$ is symmetric and satisfies, in the semidefinite order,

in particular, by Entry Bounds for Positive Semidefinite Matrices, every entry of $P(t)$ is bounded in absolute value by the largest diagonal entry of $\Lambda(t)$.