Proof of Time Reversal: Terminal Value Problems as Initial Value Problems

Computations with real numbers are those of The Real Numbers as a Type (vi). By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing the two distinct points $a$ and $b$; it serves as both $I$ and $J$ of Rules for Derivatives of Vector- and Matrix-Valued Maps (xi) below.

The reflection. Let $\psi:[a,b]\to[a,b]$ be the function with $\psi(u)=a+b-u$ for every $u\in[a,b]$ of The Reflection of a Closed Interval (i). For all $u,v\in[a,b]$, $\psi(\psi(u))=u$ by The Reflection of a Closed Interval (ii) and $|\psi(u)-\psi(v)|=|u-v|$ by The Reflection of a Closed Interval (iii); and $\psi$, read as a function into $\mathbb{R}$, is differentiable, derivatives being taken relative to $[a,b]$, with $\psi'(u)=-1$ for every $u\in[a,b]$, by The Reflection of a Closed Interval (v). Moreover $\psi(a)=a+b-a=b$ and $\psi(b)=a+b-b=a$.

Step 1 (reflected functions). Let $w,\tilde{w}:[a,b]\to\mathbb{R}^{n}$ be functions with $\tilde{w}(u)=w(\psi(u))$ for every $u\in[a,b]$, and let $w$ be differentiable. Then $\tilde{w}=w\circ\psi$, the two functions having the same values (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)). Let $u\in[a,b]$. As $w$ is differentiable at $\psi(u)\in[a,b]$, Rules for Derivatives of Vector- and Matrix-Valued Maps (xi), with $[a,b]$ for $I$ and $J$, $\psi$ for $\psi$ and $u$ for $s_{0}$, shows that $\tilde{w}$ is differentiable at $u$ with

$$\tilde{w}'(u)=\psi'(u)\,w'\bigl(\psi(u)\bigr)=(-1)\,w'\bigl(\psi(u)\bigr)=-w'\bigl(\psi(u)\bigr),$$

the last step by Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (iv), points of $\mathbb{R}^{n}$ being maps $[n]\to\mathbb{R}$ with the operations of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv). So $\tilde{w}$ is differentiable. In the same way, by the matrix half of Rules for Derivatives of Vector- and Matrix-Valued Maps (xi) and by Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (iv) with the operations of Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv): if $W,\tilde{W}:[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$ are functions with $\tilde{W}(u)=W(\psi(u))$ for every $u\in[a,b]$ and $W$ is differentiable, then $\tilde{W}$ is differentiable with $\tilde{W}'(u)=-W'(\psi(u))$ for every $u\in[a,b]$.

Proof of (i). First, we describe $G$ and $y$. By Projection, Evaluation and Currying Maps (i), there are functions $\pi_{1}:[a,b]\times\mathbb{R}^{n}\to[a,b]$ and $\pi_{2}:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $\pi_{1}(u,z)=u$ and $\pi_{2}(u,z)=z$ for all $u\in[a,b]$ and $z\in\mathbb{R}^{n}$. By Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), applied first to $\pi_{1}$ and $\psi$ and then to $\psi\circ\pi_{1}$, $\pi_{2}$ and $F$, there is a function $K:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $K(u,z)=F(\psi(\pi_{1}(u,z)),\pi_{2}(u,z))=F(a+b-u,z)$ for all $u\in[a,b]$ and $z\in\mathbb{R}^{n}$. By Maps Formed Pointwise from Vector and Matrix Operations (i), with $[a,b]\times\mathbb{R}^{n}$ for $S$, $n$ for $q$ and $K$ for $x$, there is exactly one function $[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ whose value at every point $(u,z)$ is $-K(u,z)=-F(a+b-u,z)$; this function is $G$. For a function $x:[a,b]\to\mathbb{R}^{n}$, the composite $x\circ\psi:[a,b]\to\mathbb{R}^{n}$ is, by Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), the only function whose value at every $u\in[a,b]$ is $x(\psi(u))=x(a+b-u)$; this function is $y$.

Now let $x$ and $y$ be as in (i). For $t\in[a,b]$, $y(\psi(t))=x(\psi(\psi(t)))=x(t)$.

Suppose that $x$ is differentiable with $x'(t)=F(t,x(t))$ for every $t\in[a,b]$ and $x(b)=\xi$. By Step 1, with $x$ for $w$ and $y$ for $\tilde{w}$, $y$ is differentiable, and for $u\in[a,b]$

$$y'(u)=-x'\bigl(\psi(u)\bigr)=-F\bigl(\psi(u),x(\psi(u))\bigr)=-F\bigl(a+b-u,y(u)\bigr)=G\bigl(u,y(u)\bigr);$$

and $y(a)=x(\psi(a))=x(b)=\xi$.

Conversely, suppose that $y$ is differentiable with $y'(u)=G(u,y(u))$ for every $u\in[a,b]$ and $y(a)=\xi$. By Step 1, with $y$ for $w$ and $x$ for $\tilde{w}$ (as $x(t)=y(\psi(t))$), $x$ is differentiable, and for $t\in[a,b]$

$$x'(t)=-y'\bigl(\psi(t)\bigr)=-G\bigl(\psi(t),y(\psi(t))\bigr)=-\Bigl(-F\bigl(\psi(\psi(t)),x(t)\bigr)\Bigr)=F\bigl(t,x(t)\bigr),$$

using $-(-v)=v$ (Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii)); and $x(b)=y(\psi(b))=y(a)=\xi$.

Finally, by Solution of an Initial Value Problem on $[a,c]$ (i) and Solution of an Initial Value Problem on $[a,c]$ (iii), with $b$ for $c$, a solution of the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i) with right-hand side $G$ and initial value $\xi$ is exactly a function $[a,b]\to\mathbb{R}^{n}$ that is differentiable relative to $[a,b]$ with $y'(u)=G(u,y(u))$ for every $u\in[a,b]$ and $y(a)=\xi$.

Proof of (ii). The proof of (i) applies word for word with $\mathcal{M}_{p\times q}(\mathbb{R})$, $F$, $G$, $\Xi$, $X$, $Y$ and $Z$ in place of $\mathbb{R}^{n}$, $F$, $G$, $\xi$, $x$, $y$ and $z$, with Maps Formed Pointwise from Vector and Matrix Operations (i) applied to $K$ in the role of $f$ (with $p$ and $q$ as here) for the existence of $G$, with the matrix half of Step 1, with $-(-V)=V$ for matrices (Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii)), and with Solution of an Initial Value Problem on $[a,c]$ (ii) and The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) in place of Solution of an Initial Value Problem on $[a,c]$ (i) and The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i).

Proof of (iii). We treat the vector case; the matrix case is the same with $\mathcal{M}_{p\times q}(\mathbb{R})$, $d_{p\times q}$, (ii) and The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems (iii) in place of $\mathbb{R}^{n}$, $d_{E}$, (i) and The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems (i), $d_{p\times q}$ being a metric by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (i) with $d_{p\times q}(V,W)=\lVert V-W\rVert$ by The Distance between Real Matrices (i), and with $\lVert-V\rVert=\lVert(-1)V\rVert=|-1|\,\lVert V\rVert=\lVert V\rVert$ for $V\in\mathcal{M}_{p\times q}(\mathbb{R})$, by Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (iv) with the operations of Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv) and by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products (v), in place of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v).

Let $G$ be the function of (i). For all $v,w\in\mathbb{R}^{n}$,

$$(-v)-(-w)=(-v)+w=w+(-v)=w-v=-(v-w)$$

by Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii) and Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i), and $\lVert-v\rVert=\lVert v\rVert$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v). Hence, for $u,u'\in[a,b]$ and $v,w\in\mathbb{R}^{n}$, applying these to the points $F(\psi(u'),w)$ and $F(\psi(u),v)$ and using $G(u',w)=-F(\psi(u'),w)$ and $G(u,v)=-F(\psi(u),v)$,

$$G(u',w)-G(u,v)=-\Bigl(F\bigl(\psi(u'),w\bigr)-F\bigl(\psi(u),v\bigr)\Bigr),\qquad \lVert G(u',w)-G(u,v)\rVert=\bigl\lVert F\bigl(\psi(u'),w\bigr)-F\bigl(\psi(u),v\bigr)\bigr\rVert.\tag{$\ast$}$$

$G$ is Lipschitz. For $u\in[a,b]$ and $v,w\in\mathbb{R}^{n}$, by $(\ast)$ with $u'=u$ and by the hypothesis on $F$ at $\psi(u)\in[a,b]$, $\lVert G(u,w)-G(u,v)\rVert=\lVert F(\psi(u),w)-F(\psi(u),v)\rVert\le L\,\lVert w-v\rVert$; exchanging $v$ and $w$, $\lVert G(u,v)-G(u,w)\rVert\le L\,\lVert v-w\rVert$.

$G$ is continuous. Let $(u,v)\in[a,b]\times\mathbb{R}^{n}$ and let $\varepsilon\in\mathbb{R}$ with $\varepsilon>0$. As $F$ is continuous on $[a,b]\times\mathbb{R}^{n}$, it is continuous at $(\psi(u),v)$ relative to $[a,b]\times\mathbb{R}^{n}$ (Continuous Map Between Metric Spaces (ii)), which gives $\delta\in\mathbb{R}$ with $\delta>0$ such that $d_{E}\bigl(F(s,w),F(\psi(u),v)\bigr)<\varepsilon$ for every $(s,w)\in[a,b]\times\mathbb{R}^{n}$ whose product-metric distance from $(\psi(u),v)$ is less than $\delta$. Let $(u',w)\in[a,b]\times\mathbb{R}^{n}$ have product-metric distance less than $\delta$ from $(u,v)$. By The Product Metric is a Metric (iii), $|u-u'|<\delta$ and $d_{E}(v,w)<\delta$, with $d_{\mathbb{R}}(s,s')=|s-s'|$ as in Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv). As $|\psi(u)-\psi(u')|=|u-u'|<\delta$, the same clause shows that $(\psi(u'),w)$ has distance less than $\delta$ from $(\psi(u),v)$, and by $(\ast)$ and Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii),

$$d_{E}\bigl(G(u',w),G(u,v)\bigr)=\bigl\lVert F\bigl(\psi(u'),w\bigr)-F\bigl(\psi(u),v\bigr)\bigr\rVert=d_{E}\bigl(F(\psi(u'),w),F(\psi(u),v)\bigr)<\varepsilon .$$

So $G$ is continuous at $(u,v)$ relative to $[a,b]\times\mathbb{R}^{n}$, and hence continuous on $[a,b]\times\mathbb{R}^{n}$ by Continuous Map Between Metric Spaces (ii).

Existence and uniqueness. Let $\xi\in\mathbb{R}^{n}$. By The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems (i), with $G$ for $F$ and $L$ for $L$, the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i) with right-hand side $G$ and initial value $\xi$ has exactly one solution $y_{0}$. Let $x_{0}:[a,b]\to\mathbb{R}^{n}$ be the function with $x_{0}(t)=y_{0}(\psi(t))$ for every $t\in[a,b]$. The function $u\mapsto x_{0}(\psi(u))$ has the values $y_{0}(\psi(\psi(u)))=y_{0}(u)$, so it is $y_{0}$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)); by (i), applied to $x_{0}$ and $y_{0}$, $x_{0}$ is differentiable with $x_{0}'(t)=F(t,x_{0}(t))$ for every $t\in[a,b]$ and $x_{0}(b)=\xi$. Let $x_{1},x_{2}:[a,b]\to\mathbb{R}^{n}$ both be differentiable with $x_{i}'(t)=F(t,x_{i}(t))$ for every $t\in[a,b]$ and $x_{i}(b)=\xi$, and let $y_{1},y_{2}$ be the functions of (i) for $x_{1},x_{2}$, so $y_{i}(u)=x_{i}(\psi(u))$. By (i), $y_{1}$ and $y_{2}$ are solutions of the initial value problem with right-hand side $G$ and initial value $\xi$, so $y_{1}=y_{2}=y_{0}$. As $x_{i}(t)=y_{i}(\psi(t))$ for every $t\in[a,b]$, as shown in the proof of (i), $x_{1}(t)=y_{1}(\psi(t))=y_{2}(\psi(t))=x_{2}(t)$ for every $t\in[a,b]$, and $x_{1}=x_{2}$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)).