The State-Transition Matrix of a Linear System

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $a,b\in\mathbb{R}$ with $a<b$, let $n\in\mathbb{N}$, and let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be continuous on $[a,b]$, for $d_{\mathbb{R}}$ on $[a,b]$ and $d_{n}$ on $\mathcal{M}_{n}(\mathbb{R})$. Differentiability and derivatives of maps $[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ are as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii), taken relative to $[a,b]$, an interval containing the two distinct points $a$ and $b$ by Basic Facts about Intervals of the Real Line (v); and for a map $\Phi:[a,b]\times[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ and $s\in[a,b]$, $\Phi(\cdot,s)$ is the map $t\mapsto\Phi(t,s)$, as in Existence and Uniqueness of the State-Transition Matrix of a Linear System.

(i) The state-transition matrix of $A$ on $[a,b]$ is the map $\Phi_{A}:[a,b]\times[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ of Existence and Uniqueness of the State-Transition Matrix of a Linear System (ii): the only map $\Phi:[a,b]\times[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ such that, for every $s\in[a,b]$, $\Phi(\cdot,s)$ is differentiable, with derivative $A(t)\Phi(t,s)$ at every $t\in[a,b]$, and $\Phi(s,s)=I_{n}$.

(ii) For every $s\in[a,b]$, the map $\Phi_{A}(\cdot,s):[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $t\mapsto\Phi_{A}(t,s)$, is differentiable, with derivative $A(t)\Phi_{A}(t,s)$ at every $t\in[a,b]$, by Existence and Uniqueness of the State-Transition Matrix of a Linear System (ii).

(iii) $\Phi_{A}(s,s)=I_{n}$ for every $s\in[a,b]$, by Existence and Uniqueness of the State-Transition Matrix of a Linear System (ii).

(iv) If $\Psi:[a,b]\times[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ is a map such that, for every $s\in[a,b]$, the map $\Psi(\cdot,s):[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $t\mapsto\Psi(t,s)$, is differentiable, with derivative $A(t)\Psi(t,s)$ at every $t\in[a,b]$, and $\Psi(s,s)=I_{n}$, then $\Psi=\Phi_{A}$, by the uniqueness in Existence and Uniqueness of the State-Transition Matrix of a Linear System (ii).