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 a $C^1$ map on $U$ whose partial derivatives are again $C^1$ 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) $U\times V$ is an open subset of $\mathbb{R}^{l+m}$ once $V$ is known to be open (Step 2): given $x=(\Sigma,\alpha)\in U\times V$, choose $\rho_1,\rho_2>0$ such that every point within distance $\rho_1$ of $\Sigma$ lies in $U$ and every point within distance $\rho_2$ of $\alpha$ lies in $V$; if $|x'-x|<\min(\rho_1,\rho_2)$ then by (0c) $|\Sigma'-\Sigma|<\rho_1$ and $|\alpha'-\alpha|<\rho_2$, so $x'\in U\times V$.

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

(a) State coordinates. Fix $i\in\{1,\dots,l\}$ and hold all coordinates of $x$ except $x_i$ fixed. By the slice-function lemma, the slice of $g$ in the coordinate $x_i$ is differentiable at $\Sigma$ with derivative $\partial_ig(\Sigma)$, and likewise for each $g_k$. The slice of $F$ in the coordinate $x_i$ is the finite linear combination of these slices with the constant coefficients $1,\alpha^1,\dots,\alpha^m$, so by the linear-combination rule it is differentiable with derivative $\partial_ig(\Sigma)+\sum_k\alpha^k\partial_ig_k(\Sigma)$. By the slice-function lemma again,

(b) Control coordinates. Fix $k\in\{1,\dots,m\}$ and hold all coordinates except $x_{l+k}$ fixed. The slice of $F$ in that coordinate is the map $t\mapsto c_0+g_k(\Sigma)\,t$, where $c_0=g(\Sigma)+\sum_{k'\neq k}g_{k'}(\Sigma)\alpha^{k'}$ does not depend on $t$. For every real $h\neq0$ the corresponding difference quotient is

a quantity independent of $h$; hence the slice is differentiable with derivative $g_k(\Sigma)$, and by the slice-function lemma $\partial_{l+k}F(x)=g_k(\Sigma)$. 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 $C^1$ on $U$. 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 slice in any control coordinate $x_{l+k'}$ is constant and every difference quotient vanishes, giving $\partial_{l+k'}\partial_{l+k}F=0$, the second identity of clause 4; and its slice 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 on $U\times V$, so each first partial derivative of $F$ is a $C^1$ map there. Moreover each first partial derivative of $F$ is itself continuous: 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 $|\partial_{l+k}F(x)-\partial_{l+k}F(x')|=|g_k(\Sigma)-g_k(\Sigma')|\le\sqrt{l}\,K_0|x-x'|$. Since all partial derivatives of $F$ exist on the open set $U\times V$ and are continuous there, $F$ is a $C^1$ map 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.