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

$$F(x)=\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)=g(\Sigma)+\sum_{k=1}^mg_k(\Sigma)\,\alpha^k\qquad(x=(\Sigma,\alpha)\in U\times V),$$

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

$$|f(\Sigma)-f(\Sigma')|\le\sqrt{l}\,K_0\,|\Sigma-\Sigma'| .$$

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

$$|\beta_1(\Sigma)-\beta_1(\Sigma')|\le\sqrt{m}\,\sqrt{l}\,K_0\,|\Sigma-\Sigma'|=\Lambda|\Sigma-\Sigma'| .$$

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

$$\big|t\alpha+(1-t)\alpha'-\big(ta+(1-t)a'\big)\big|\le t|\alpha-a|+(1-t)|\alpha'-a'|<s+\varepsilon<1 .$$

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

$$L=\partial_ig(\Sigma)+\sum_{k=1}^m\alpha^k\,\partial_ig_k(\Sigma),\qquad C=1+\sum_{k=1}^m|\alpha^k| ,$$

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

$$\left|\frac{g(\Sigma^{(h)})-g(\Sigma)}{h}-\partial_ig(\Sigma)\right|<\varepsilon' ,$$

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,

$$\frac{F(x^{(i,h)})-F(x)}{h}-L=\left(\frac{g(\Sigma^{(h)})-g(\Sigma)}{h}-\partial_ig(\Sigma)\right)+\sum_{k=1}^m\alpha^k\left(\frac{g_k(\Sigma^{(h)})-g_k(\Sigma)}{h}-\partial_ig_k(\Sigma)\right),$$

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:

$$\partial_iF(x)=\partial_ig(\Sigma)+\sum_{k=1}^m\alpha^k\,\partial_ig_k(\Sigma).$$

(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

$$\frac{F(x^{(l+k,h)})-F(x)}{h}=\frac{g_k(\Sigma)\,h}{h}=g_k(\Sigma),$$

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 $k\in\{1,\dots,m\}$,

$$\partial_{i'}\partial_iF(x)=\partial_{i'}\partial_ig(\Sigma)+\sum_{k=1}^m\alpha^k\,\partial_{i'}\partial_ig_k(\Sigma),\qquad \partial_{l+k}\partial_iF(x)=\partial_ig_k(\Sigma),$$

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 $U\times V$,

$$|\partial_iF|\le K_0+\sqrt{m}\,K_0(R+1)=K,\qquad |\partial_{l+k}F|=|g_k|\le K_0\le K,$$

and, for the second derivatives,

$$|\partial_{i'}\partial_iF|\le K_0+\sqrt{m}\,K_0(R+1)=K,\qquad |\partial_{l+k}\partial_iF|=|\partial_ig_k|\le K_0\le K,\qquad |\partial_{i'}\partial_{l+k}F|=|\partial_{i'}g_k|\le K_0\le K,$$

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

$$\eta=\frac{\varepsilon}{2\big(1+\sqrt{m}(R+1)\big)},\qquad \delta_2=\frac{\varepsilon}{2\big(\sqrt{m}K_0+\sqrt{l}K_0+1\big)} .$$

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,

$$|\partial_{i'}\partial_iF(x)-\partial_{i'}\partial_iF(x')|\le\big|\partial_{i'}\partial_ig(\Sigma)-\partial_{i'}\partial_ig(\Sigma')\big|+|(c-c')\cdot\alpha|+|c'\cdot(\alpha-\alpha')|\le\eta+\sqrt{m}\,\eta\,(R+1)+\sqrt{m}\,K_0\,\delta\le\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon ,$$

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

$$|\partial_{l+k}\partial_iF(x)-\partial_{l+k}\partial_iF(x')|=|\partial_ig_k(\Sigma)-\partial_ig_k(\Sigma')|\le\sqrt{l}\,K_0|\Sigma-\Sigma'|\le\sqrt{l}\,K_0\,\delta\le\frac{\varepsilon}{2}\le\varepsilon ,$$

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),

$$|\partial_iF(x)-\partial_iF(x')|\le\sqrt{l}\,K_0|\Sigma-\Sigma'|+\sqrt{m}\,\sqrt{l}\,K_0|\Sigma-\Sigma'|(R+1)+\sqrt{m}\,K_0|\alpha-\alpha'|\le\Big(\sqrt{l}\,K_0\big(1+\sqrt{m}(R+1)\big)+\sqrt{m}\,K_0\Big)|x-x'| ,$$

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

$$F(x)-F(x')=\big(g(\Sigma)-g(\Sigma')\big)+\sum_{k=1}^m\big(g_k(\Sigma)-g_k(\Sigma')\big)\alpha^k+\sum_{k=1}^mg_k(\Sigma')\big(\alpha^k-\alpha'^k\big)$$

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)

$$|F(x)-F(x')|\le\Big(\sqrt{l}\,K_0\big(1+\sqrt{m}\,(R+1)\big)+\sqrt{m}\,K_0\Big)|x-x'| ,$$

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.