Proof of The Twice Continuously Differentiable Extension Determined by Affine-Controlled Rate Data

Throughout we fix an ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$ in $\{1,\dots,l\}$ and abbreviate $g=\bar{\beta}_0(\sigma,\gamma,\cdot)$ and $g_k=\bar{\beta}_1^k(\sigma,\gamma,\cdot)$ for $k\in\{1,\dots,m\}$; these are real-valued functions on $U$. All estimates obtained below hold for every such pair, of which there are finitely many. We write

the second equality being the coordinate formula for the dot product. We use throughout the norm properties: clause 1 (the square of the norm is the sum of the squares of the coordinates), clause 4 ($|x_i|\le|x|$), clause 5 (absolute homogeneity) and clause 6 (the triangle inequality). By the definition of the rate data, $U$ is open, convex and bounded with $\Delta^l\subset U$, the control set $\mathcal{A}$ is nonempty, convex and compact, each of $g,g_1,\dots,g_m$ is of class $C^2$ on $U$ — by clauses 2 and 3 of that definition this means that each is of class $C^1$ on $U$ and that each of its partial derivatives $\partial_i$, unambiguous by Uniqueness of the Partial Derivative on a Euclidean Open Set, is again of class $C^1$ on $U$ — and all of $|g|,|g_k|$, their first partial derivatives and their second partial derivatives are bounded by $K_0$ on $U$.

Step 0. Elementary bounds.

(0a) If $n\ge1$, $y\in\mathbb{R}^n$ and $C\ge0$ is a real number with $|y_i|\le C$ for every $i$, then $|y|\le\sqrt{n}\,C$. Indeed $y_i^2=|y_i|^2\le C^2$ by monotonicity of squaring (clause 2), so $|y|^2=\sum_iy_i^2\le nC^2=(\sqrt{n}\,C)^2$, and since $|y|\ge0$ and $\sqrt{n}\,C\ge0$ the same clause, read in the other direction, gives $|y|\le\sqrt{n}\,C$.

(0b) $1\le\sqrt{l}$ and $1\le\sqrt{m}$: since $1\le l$ and $1^2=1$, $(\sqrt{l})^2=l$, clause 2 of the same lemma applied to the nonnegative reals $1$ and $\sqrt{l}$ gives $1\le\sqrt{l}$; likewise for $m$.

(0c) For $x=(\Sigma,\alpha)$ and $x'=(\Sigma',\alpha')$ in $\mathbb{R}^{l+m}$ we have $|\Sigma-\Sigma'|\le|x-x'|$ and $|\alpha-\alpha'|\le|x-x'|$, because $|\Sigma-\Sigma'|^2\le|\Sigma-\Sigma'|^2+|\alpha-\alpha'|^2=|x-x'|^2$ and both quantities are nonnegative, so clause 2 of the monotonicity lemma applies; similarly for the second inequality.

(0d) Once $V$ is known to be open (Step 2), $U\times V$ is an open subset of $\mathbb{R}^{l+m}$ by claim 2 of Products of Euclidean Open Sets are Open, the identification of $\mathbb{R}^l\times\mathbb{R}^m$ with $\mathbb{R}^{l+m}$ being the one fixed in the statement.

Step 1. The restriction is an affine-controlled family (clause 1).

Requirement 1 of an affine-controlled transition-rate family is exactly clause 1 of the rate-data definition, restricted to $\Delta^l\times\mathcal{A}$.

For requirement 2, let $f$ be any one of $g,g_1,\dots,g_m$ and let $\Sigma,\Sigma'\in\Delta^l$. Since $\Delta^l\subset U$ and $U$ is convex, the segment joining $\Sigma$ to $\Sigma'$ lies in $U$. The map $f$ is $C^1$ on the open set $U$ with $|\partial_if|\le K_0$ there, so clause (i) of the multivariate Taylor lemma, applied with $n=l$ and $M_1=K_0$, gives

In particular $|\beta_0(\Sigma)-\beta_0(\Sigma')|\le\sqrt{l}\,K_0|\Sigma-\Sigma'|\le\Lambda|\Sigma-\Sigma'|$, using $1\le\sqrt{m}$ from (0b). Applying the displayed bound to each component $g_k$ and then (0a) in $\mathbb{R}^m$ gives

Thus both maps are Lipschitz with constant $\Lambda$ on $\Delta^l$, and since $\mathcal{A}$ is nonempty, convex and compact, $(\beta_0,\beta_1)$ is an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}$ and Lipschitz constant $\Lambda$. By clause 1 of the lemma on affine-controlled data, $R$ and $B$ are finite nonnegative real numbers, and by its clause 2 the associated $\beta$ is a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$. This proves clause 1.

Step 2. The set $V$ (clause 2).

$\mathcal{A}\subseteq V$. For $\alpha\in\mathcal{A}$ the set whose infimum defines $\rho(\alpha)$ contains the value $|\alpha-\alpha|=0$ and consists of nonnegative numbers, so $\rho(\alpha)=0<1$.

Openness. Let $\alpha\in V$ and put $s=\rho(\alpha)<1$ and $\varrho=1-s>0$. If $|\alpha'-\alpha|<\varrho$ then clause 3 of the nonexpansiveness lemma gives $\rho(\alpha')\le|\alpha'-\alpha|+\rho(\alpha)<\varrho+s=1$, so $\alpha'\in V$. Hence $V$ is open.

The bound $|\alpha|\le R+1$, and boundedness. Let $\alpha\in V$, put $s=\rho(\alpha)<1$ and $\varepsilon=(1-s)/2>0$. By clause 4 of the approximation property of the infimum there is $a\in\mathcal{A}$ with $|\alpha-a|<s+\varepsilon<1$. Since $|a|\le R$ by the definition of $R$ as a supremum, the triangle inequality gives $|\alpha|\le|a|+|\alpha-a|\le R+1$. In particular every point of $V$ lies within distance $R+1$ of the origin, so $V$ is a bounded set.

Convexity. Let $\alpha,\alpha'\in V$ and let $t$ be real with $0\le t\le1$. Put $s=\max\{\rho(\alpha),\rho(\alpha')\}<1$ and $\varepsilon=(1-s)/2>0$, and apply the approximation property of the infimum separately to $\rho(\alpha)$ and to $\rho(\alpha')$ to obtain $a,a'\in\mathcal{A}$ with $|\alpha-a|<\rho(\alpha)+\varepsilon\le s+\varepsilon$ and $|\alpha'-a'|<\rho(\alpha')+\varepsilon\le s+\varepsilon$. Since $\mathcal{A}$ is convex, $ta+(1-t)a'\in\mathcal{A}$, and by absolute homogeneity and the triangle inequality

As $\rho(t\alpha+(1-t)\alpha')$ is a lower bound for the distances to points of $\mathcal{A}$, it follows that $\rho(t\alpha+(1-t)\alpha')<1$, that is, $t\alpha+(1-t)\alpha'\in V$. This proves clause 2, and with it (0d).

Step 3. The partial derivatives of $\bar{\beta}$ (clause 4).

Let $x=(\Sigma,\alpha)\in U\times V$.

Admissible increments. By (0d) the set $U\times V$ is open, so there is a real $r>0$ such that every $y\in\mathbb{R}^{l+m}$ with $\sum_{k=1}^{l+m}(y_k-x_k)^2<r^2$ lies in $U\times V$. For $j\in\{1,\dots,l+m\}$ and real $h$ write $x^{(j,h)}$ for the point obtained from $x$ by replacing its $j$th coordinate $x_j$ by $x_j+h$; then $\sum_{k}(x^{(j,h)}_k-x_k)^2=h^2$, so every real $h$ with $|h|<r$ has $x^{(j,h)}\in U\times V$, by clause 2 of monotonicity of squaring.

(a) State coordinates. Fix $i\in\{1,\dots,l\}$, and note that for real $h$ the point $x^{(i,h)}$ is $(\Sigma^{(h)},\alpha)$, where $\Sigma^{(h)}\in\mathbb{R}^l$ is obtained from $\Sigma$ by replacing $\Sigma^i$ by $\Sigma^i+h$; in particular the control coordinates are unchanged. Since $g$ and each $g_k$ are of class $C^1$ on the open set $U$, their partial derivatives with respect to the $i$th variable exist at $\Sigma$. Put

so that $C\ge1>0$. Let $\varepsilon>0$ be real and put $\varepsilon'=\varepsilon\cdot(2C)^{-1}$, positive by claims 5, 7 and 8 of the order arithmetic of an ordered field. By the definition of the partial derivative there are reals $\delta_0,\delta_1,\dots,\delta_m>0$ such that every real $h$ with $0<|h|<\delta_0$ satisfies $\Sigma^{(h)}\in U$ and

and, for each $k\in\{1,\dots,m\}$, every real $h$ with $0<|h|<\delta_k$ satisfies $\Sigma^{(h)}\in U$ and the same inequality with $g$ and $\partial_ig(\Sigma)$ replaced by $g_k$ and $\partial_ig_k(\Sigma)$. Let $\delta$ be the least of the finitely many positive reals $r,\delta_0,\delta_1,\dots,\delta_m$, obtained by repeated application of claim 9 of that lemma, so that $\delta>0$. For real $h$ with $0<|h|<\delta$ we then have $x^{(i,h)}\in U\times V$, and, the control coordinates of $x^{(i,h)}$ and $x$ agreeing,

whose modulus is at most $\varepsilon'+\sum_{k=1}^m|\alpha^k|\,\varepsilon'=C\varepsilon'=\varepsilon\cdot2^{-1}<\varepsilon$, by claim 5 of the properties of the absolute value in an ordered field (the triangle inequality, extended to the $m+1$ summands by induction on their number) together with claim 4 of that lemma (multiplicativity), and by claim 8 of the order lemma. As $\varepsilon>0$ was arbitrary, the partial derivative of $F$ with respect to the $i$th variable exists at $x$ with value $L$, and by Uniqueness of the Partial Derivative on a Euclidean Open Set no other value is possible:

(b) Control coordinates. Fix $k\in\{1,\dots,m\}$. For real $h$ with $0<|h|<r$ the point $x^{(l+k,h)}$ lies in $U\times V$ and differs from $x$ only in the control coordinate $\alpha^k$, which is replaced by $\alpha^k+h$; hence

so the difference quotient is exactly $g_k(\Sigma)$ and its distance to $g_k(\Sigma)$ is $0<\varepsilon$ for every real $\varepsilon>0$. Therefore the partial derivative of $F$ with respect to the $(l+k)$th variable exists at $x$ with value $g_k(\Sigma)$, and $\partial_{l+k}F(x)=g_k(\Sigma)$. This is the first identity of clause 4.

(c) Second derivatives. The function $\partial_iF$ computed in (a) has exactly the form treated in (a) and (b), with $g,g_k$ replaced by $\partial_ig,\partial_ig_k$, which are again of class $C^1$ on $U$ because $g$ and the $g_k$ are of class $C^2$ there. Hence, for $i,i'\in\{1,\dots,l\}$ and $k\in\{1,\dots,m\}$,

the second identity being the third identity of clause 4. Finally $\partial_{l+k}F(x)=g_k(\Sigma)$ depends only on $\Sigma$, so its difference quotients in any control coordinate $x_{l+k'}$ all vanish, giving $\partial_{l+k'}\partial_{l+k}F=0$, the second identity of clause 4; and the argument of (a), applied to the single function $g_k$ in a state coordinate $x_{i'}$, gives $\partial_{i'}\partial_{l+k}F(x)=\partial_{i'}g_k(\Sigma)$. This proves clause 4.

Step 4. Derivative bounds.

By the bounds in the rate-data definition and (0a) in $\mathbb{R}^m$, the vector $\partial_i\bar{\beta}_1(\sigma,\gamma,\Sigma)$ with components $\partial_ig_k(\Sigma)$ satisfies $|\partial_i\bar{\beta}_1(\sigma,\gamma,\Sigma)|\le\sqrt{m}\,K_0$, and similarly for the vectors formed by $\partial_{i'}\partial_ig_k(\Sigma)$ and by $g_k(\Sigma)$. Using clause 2 of the present lemma ($|\alpha|\le R+1$ on $V$) and the Cauchy-Schwarz inequality, the identities of Step 3 give, at every point of $U\times V$,

and, for the second derivatives,

and $|\partial_{l+k'}\partial_{l+k}F|=0\le K$, where $K_0\le K$ because $\sqrt{m}(R+1)\ge0$. This is requirement 3 of a twice continuously differentiable extension.

Step 5. Uniform continuity of the second derivatives, and regularity.

Let $\varepsilon>0$ be real and put

Requirement 4 of the rate-data definition, applied to each of the finitely many functions $g,g_1,\dots,g_m$ attached to each of the finitely many ordered pairs $(\sigma,\gamma)$, yields a real $\delta_1>0$ such that all second partial derivatives of all these functions change by at most $\eta$ between any two points of $U$ at distance at most $\delta_1$. Put $\delta=\min\{\delta_1,\delta_2\}>0$ and let $x=(\Sigma,\alpha)$ and $x'=(\Sigma',\alpha')$ be points of $U\times V$ with $|x-x'|\le\delta$; by (0c), $|\Sigma-\Sigma'|\le\delta$ and $|\alpha-\alpha'|\le\delta$.

For $i,i'\in\{1,\dots,l\}$ let $c$ and $c'$ be the vectors of $\mathbb{R}^m$ with components $\partial_{i'}\partial_ig_k(\Sigma)$ and $\partial_{i'}\partial_ig_k(\Sigma')$ respectively. By (0a), $|c-c'|\le\sqrt{m}\,\eta$ and $|c'|\le\sqrt{m}\,K_0$. Hence, by Step 3(c), the triangle inequality and Cauchy-Schwarz,

using the choices of $\eta$ and $\delta_2$. For the mixed second derivatives, Step 3(c) and clause (i) of the Taylor lemma applied to $g_k$ (whose first partial derivatives are bounded by $K_0$ on the convex set $U$) give

and the same bound for $|\partial_{i'}\partial_{l+k}F(x)-\partial_{i'}\partial_{l+k}F(x')|$; here the Taylor lemma is applied to the function $\partial_ig_k$, which is $C^1$ on $U$ with second partial derivatives of $g_k$ as its partial derivatives, all bounded by $K_0$. Finally $\partial_{l+k'}\partial_{l+k}F$ vanishes identically, so its increment is $0\le\varepsilon$. This is requirement 4 of a twice continuously differentiable extension.

The estimates just obtained show in particular that every second partial derivative of $F$ is continuous at every point of $U\times V$: given a real $\varepsilon>0$, the bound just proved, applied with $\varepsilon\cdot2^{-1}$ in place of $\varepsilon$, supplies a real $\delta>0$ such that any two points of $U\times V$ at distance at most $\delta$ have their second partial derivatives of $F$ within $\varepsilon\cdot2^{-1}<\varepsilon$ of each other, by claim 8 of the order arithmetic of an ordered field. Each first partial derivative of $F$ is continuous as well: by Step 3(a), the triangle inequality, Cauchy-Schwarz and clause (i) of the Taylor lemma applied to $g$ and to each $g_k$ (all with first partial derivatives bounded by $K_0$ on the convex set $U$), together with (0a) and (0c),

and $|\partial_{l+k}F(x)-\partial_{l+k}F(x')|=|g_k(\Sigma)-g_k(\Sigma')|\le\sqrt{l}\,K_0|x-x'|$. Each of these two bounds has the form $|\partial_jF(x)-\partial_jF(x')|\le C_1|x-x'|$ with a nonnegative real constant $C_1$, so for a real $\varepsilon>0$ the choice $\delta=\varepsilon\cdot\big(2(C_1+1)\big)^{-1}$ makes the left-hand side smaller than $\varepsilon$ whenever $|x-x'|<\delta$; hence every $\partial_jF$ is continuous at every point of $U\times V$.

The function $F$ itself is continuous there too. Writing

and bounding the first group by clause (i) of the Taylor lemma, the second by Cauchy-Schwarz together with (0a), that same clause and $|\alpha|\le R+1$, and the third by Cauchy-Schwarz together with (0a) and $|g_k|\le K_0$, we obtain from (0c)

and the same choice of $\delta$ as above applies.

So $F$ is continuous at every point of the open set $U\times V$ and all of its partial derivatives exist and are continuous there, which by clauses 1 and 3 of the definition of a $C^k$ map makes $F$ of class $C^1$ on $U\times V$; and each $\partial_jF$ is likewise continuous with all of its own partial derivatives existing and continuous, hence of class $C^1$ on $U\times V$. By clause 2 of that definition, $F$ is of class $C^2$ on $U\times V$; this is requirement 2.

Step 6. Conclusion (clause 3).

The set $U$ is open, convex and bounded with $\Delta^l\subset U$ by the rate-data definition, and $V$ is open, convex and bounded with $\mathcal{A}\subseteq V$ by Step 2. For $(\Sigma,\alpha)\in\Delta^l\times\mathcal{A}$ the defining formulas for $\bar{\beta}$ and for $\beta$ agree, since $\beta_0,\beta_1$ are the restrictions of $\bar{\beta}_0,\bar{\beta}_1$; this is requirement 1. Requirements 2, 3 and 4 were established in Steps 5, 4 and 5 respectively. Hence $(U,V,\bar{\beta})$ is a twice continuously differentiable extension of $\beta$ with derivative bound $K$, which is clause 3.