Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem

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 $a<b$, and let $n,q\in\mathbb{N}$. $\mathbb{R}\times\mathbb{R}^{n}$ carries the product metric of $d_{\mathbb{R}}$ and $d_{E}$, and $\mathbb{R}\times\mathcal{M}_{n\times q}(\mathbb{R})$ the product metric of $d_{\mathbb{R}}$ and $d_{n\times q}$, metrics by The Product Metric is a Metric (i). A map defined on $[a,b]$, on $[a,b]\times\mathbb{R}^{n}$ or on $[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})$ is continuous on that set for these metrics, for $d_{\mathbb{R}}$ on $\mathbb{R}$, $d_{E}$ on $\mathbb{R}^{n}$ and $d_{p\times r}$ on $\mathcal{M}_{p\times r}(\mathbb{R})$ ($p,r\in\mathbb{N}$). $\lVert\cdot\rVert$ is the Euclidean norm on vectors and the operator norm on matrices, as in Euclidean Space and Real Matrices: Standing Notation (vi).

(i) Let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ and $g:[a,b]\to\mathbb{R}^{n}$ be maps. There is exactly one map $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $F(t,x)=A(t)x+g(t)$ for all $t\in[a,b]$ and $x\in\mathbb{R}^{n}$. Likewise, let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $B:[a,b]\to\mathcal{M}_{q}(\mathbb{R})$ and $G:[a,b]\to\mathcal{M}_{n\times q}(\mathbb{R})$ be maps. There is exactly one map $F:[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})\to\mathcal{M}_{n\times q}(\mathbb{R})$ with $F(t,X)=A(t)X+XB(t)+G(t)$ for all $t\in[a,b]$ and $X\in\mathcal{M}_{n\times q}(\mathbb{R})$.

(ii) Let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ and $g:[a,b]\to\mathbb{R}^{n}$ be continuous on $[a,b]$, and let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the map of (i) for $A$ and $g$, $F(t,x)=A(t)x+g(t)$. Then $F$ is continuous on $[a,b]\times\mathbb{R}^{n}$; there is $K\in\mathbb{R}$ with $K\ge0$ and $\lVert A(t)\rVert\le K$ for every $t\in[a,b]$; for every such $K$,

$$\lVert F(t,x)-F(t,y)\rVert\le K\,\lVert x-y\rVert\qquad\bigl(t\in[a,b],\ x,y\in\mathbb{R}^{n}\bigr).$$

(iii) Let $A$, $g$ and $F$ be as in (ii). For every $\xi\in\mathbb{R}^{n}$ the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i) with right-hand side $F$ and initial value $\xi$ has exactly one solution.

(iv) Let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $B:[a,b]\to\mathcal{M}_{q}(\mathbb{R})$ and $G:[a,b]\to\mathcal{M}_{n\times q}(\mathbb{R})$ be continuous on $[a,b]$, and let $F:[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})\to\mathcal{M}_{n\times q}(\mathbb{R})$ be the map of (i) for $A$, $B$ and $G$, $F(t,X)=A(t)X+XB(t)+G(t)$. Then $F$ is continuous on $[a,b]\times\mathcal{M}_{n\times q}(\mathbb{R})$; there are $K_{A},K_{B}\in\mathbb{R}$ with $K_{A},K_{B}\ge0$, $\lVert A(t)\rVert\le K_{A}$ and $\lVert B(t)\rVert\le K_{B}$ for every $t\in[a,b]$; for all such $K_{A}$ and $K_{B}$,

$$\lVert F(t,X)-F(t,Y)\rVert\le(K_{A}+K_{B})\,\lVert X-Y\rVert\qquad\bigl(t\in[a,b],\ X,Y\in\mathcal{M}_{n\times q}(\mathbb{R})\bigr).$$

(v) Let $A$, $B$, $G$ and $F$ be as in (iv). For every $\Xi\in\mathcal{M}_{n\times q}(\mathbb{R})$ the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) with right-hand side $F$ and initial value $\Xi$ has exactly one solution.