Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations

Let $a<b$ be real numbers and $k\ge1$ a natural number. Let $A$ and $C$ assign to each $t\in[a,b]$ real $k\times k$ matrices $A(t)$, $C(t)$ with entries continuous in $t$, and let $P_0$ be a 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, $(\cdot)^{\top}$ is the transpose, and $\Phi$, $\Psi=\Phi^{-1}$ are the fundamental solution of $A$ on $[a,b]$ and its inverse from Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations. 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.

1. (Existence, uniqueness, representation) 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

and it is given by

2. (Symmetry) If $P_0$ and every $C(t)$ are symmetric, then every $P(t)$ is symmetric.

3. (Positive semidefiniteness) If moreover $P_0$ and every $C(t)$ are positive semidefinite, then every $P(t)$ is positive semidefinite.