Time Reversal: Terminal Value Problems as Initial Value Problems

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 $n,p,q\in\mathbb{N}$.

(i) Let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a function and let $\xi\in\mathbb{R}^{n}$. Let $G:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the function with $G(u,z)=-F(a+b-u,z)$ for all $u\in[a,b]$ and $z\in\mathbb{R}^{n}$, and, for a function $x:[a,b]\to\mathbb{R}^{n}$, let $y:[a,b]\to\mathbb{R}^{n}$ be the function with $y(u)=x(a+b-u)$ for every $u\in[a,b]$. Then $x$ is differentiable with

$$x'(t)=F\bigl(t,x(t)\bigr)\quad\bigl(t\in[a,b]\bigr),\qquad x(b)=\xi,$$

if and only if $y$ is differentiable with

$$y'(u)=G\bigl(u,y(u)\bigr)\quad\bigl(u\in[a,b]\bigr),\qquad y(a)=\xi,$$

that is, if and only if $y$ is a solution of the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i) on $[a,b]$ (with $b$ for $c$ in The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i)) with right-hand side $G$ and initial value $\xi$.

(ii) Let $F:[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ be a function and let $\Xi\in\mathcal{M}_{p\times q}(\mathbb{R})$. Let $G:[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ be the function with $G(u,Z)=-F(a+b-u,Z)$ for all $u\in[a,b]$ and $Z\in\mathcal{M}_{p\times q}(\mathbb{R})$, and, for a function $X:[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$, let $Y:[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$ be the function with $Y(u)=X(a+b-u)$ for every $u\in[a,b]$. Then $X$ is differentiable with

$$X'(t)=F\bigl(t,X(t)\bigr)\quad\bigl(t\in[a,b]\bigr),\qquad X(b)=\Xi,$$

if and only if $Y$ is differentiable with

$$Y'(u)=G\bigl(u,Y(u)\bigr)\quad\bigl(u\in[a,b]\bigr),\qquad Y(a)=\Xi,$$

that is, if and only if $Y$ is a solution of 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 $c$ in The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii)) with right-hand side $G$ and initial value $\Xi$.

(iii) Let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be continuous on $[a,b]\times\mathbb{R}^{n}$, and let $L\in\mathbb{R}$ with $L\ge0$ satisfy $\lVert F(t,v)-F(t,w)\rVert\le L\,\lVert v-w\rVert$ for all $t\in[a,b]$ and $v,w\in\mathbb{R}^{n}$. Then for every $\xi\in\mathbb{R}^{n}$ there is exactly one function $x:[a,b]\to\mathbb{R}^{n}$ that is differentiable with $x'(t)=F(t,x(t))$ for every $t\in[a,b]$ and $x(b)=\xi$. Likewise, let $F:[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ be continuous on $[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})$, and let $L\in\mathbb{R}$ with $L\ge0$ satisfy $\lVert F(t,V)-F(t,W)\rVert\le L\,\lVert V-W\rVert$ for all $t\in[a,b]$ and $V,W\in\mathcal{M}_{p\times q}(\mathbb{R})$. Then for every $\Xi\in\mathcal{M}_{p\times q}(\mathbb{R})$ there is exactly one function $X:[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$ that is differentiable with $X'(t)=F(t,X(t))$ for every $t\in[a,b]$ and $X(b)=\Xi$.