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

Let $(U,\bar{\beta}_0,\bar{\beta}_1)$ be twice continuously differentiable affine-controlled rate data on $l$ states with control set $\mathcal{A}$ and derivative bound $K_0$, and let $\Delta^l$ be the probability simplex. For each ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$ in $\{1,\dots,l\}$ let $\beta_0(\sigma,\gamma,\cdot)$ and $\beta_1(\sigma,\gamma,\cdot)$ denote the restrictions of $\bar{\beta}_0(\sigma,\gamma,\cdot)$ and $\bar{\beta}_1(\sigma,\gamma,\cdot)$ to $\Delta^l$, and let $\bar{\beta}_1^k$ denote the $k$-th component of $\bar{\beta}_1$ for $k\in\{1,\dots,m\}$. Write $|\cdot|$ for the Euclidean norm, $x\cdot y$ for the dot product, and $\sqrt{\cdot}$ for the nonnegative square root. Put

$$R=\sup_{\alpha\in\mathcal{A}}|\alpha|,\qquad \Lambda=\sqrt{l}\,\sqrt{m}\,K_0,\qquad K=K_0\big(1+\sqrt{m}\,(R+1)\big),$$

let

$$V=\{\alpha\in\mathbb{R}^m:\ \rho(\alpha)<1\},$$

where $\rho(\alpha)$ denotes the distance from $\alpha$ to $\mathcal{A}$, and define

$$\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)=\bar{\beta}_0(\sigma,\gamma,\Sigma)+\bar{\beta}_1(\sigma,\gamma,\Sigma)\cdot\alpha\qquad(\Sigma\in U,\ \alpha\in V).$$

1. (Restriction.) $(\beta_0,\beta_1)$ is an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}$ and Lipschitz constant $\Lambda$. Let $\beta$ and $B$ denote the transition-rate family and the rate bound determined by $(\beta_0,\beta_1)$ as in the lemma on affine-controlled data.

2. (Control neighbourhood.) $V$ is an open, convex, bounded subset of Euclidean space $\mathbb{R}^m$ with $\mathcal{A}\subseteq V$, and $|\alpha|\le R+1$ for every $\alpha\in V$.

3. (Extension.) $(U,V,\bar{\beta})$ is a twice continuously differentiable extension of $\beta$ with derivative bound $K$.

4. (Affine dependence on the control.) Points of $U\times V$ are written $x=(\Sigma,\alpha)$ with coordinates indexed as in that definition, and $\partial_i$ denotes the partial derivative with respect to the $i$th variable, that is, with respect to the coordinate $x_i$, which is unambiguous wherever it exists by Uniqueness of the Partial Derivative on a Euclidean Open Set. For every ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$, all $k,k'\in\{1,\dots,m\}$, every $i\in\{1,\dots,l\}$ and every $(\Sigma,\alpha)\in U\times V$,

$$\partial_{l+k}\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)=\bar{\beta}_1^k(\sigma,\gamma,\Sigma),\qquad \partial_{l+k'}\partial_{l+k}\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)=0,\qquad \partial_{l+k}\partial_i\bar{\beta}(\sigma,\gamma,\Sigma,\alpha)=\partial_i\bar{\beta}_1^k(\sigma,\gamma,\Sigma).$$