The Lyapunov Differential Equation

In the setting of Vector- and Matrix-Valued Functions on a Compact Interval: Standing Conventions, let $a,b\in\mathbb{R}$ with $a<b$, and let $n\in\mathbb{N}$.

(i) Let $A,C:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be functions and let $P_{0}\in\mathcal{M}_{n}(\mathbb{R})$. Let $H:[a,b]\times\mathcal{M}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ be the function with

$$H(t,P)=A(t)P+PA(t)^{\top}+C(t)\qquad\bigl(t\in[a,b],\ P\in\mathcal{M}_{n}(\mathbb{R})\bigr).$$

The Lyapunov differential equation with data $A$, $C$ and $P_{0}$ is the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) on $[a,b]$, with $b$ for $c$, $n$ for $p$ and $q$, right-hand side $H$ and initial value $P_{0}$:

$$P'(t)=A(t)P(t)+P(t)A(t)^{\top}+C(t)\quad\bigl(t\in[a,b]\bigr),\qquad P(a)=P_{0},$$

for an unknown function $P:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$. Its solutions are the solutions of this initial value problem.

(ii) Let $A,Q:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be functions and let $F\in\mathcal{M}_{n}(\mathbb{R})$. The backward Lyapunov differential equation with data $A$, $Q$ and $F$ is the terminal value problem

$$Z'(t)=-\bigl(A(t)^{\top}Z(t)+Z(t)A(t)+Q(t)\bigr)\quad\bigl(t\in[a,b]\bigr),\qquad Z(b)=F,$$

for an unknown function $Z:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$. A solution of it is a function $Z:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ that is differentiable and satisfies both conditions.