Twice Continuously Differentiable Affine-Controlled Rate Data

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $\Delta^l\subset\mathbb{R}^l$ be the probability simplex, let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ that is convex and compact for the topology determined by the Euclidean distance, called the control set, and let $K_0$ be a nonnegative real number. For a real-valued function $f$ on an open subset of $\mathbb{R}^l$ and indices $i,i'\in\{1,\dots,l\}$ we write $\partial_i f$ for the partial derivative of $f$ with respect to the $i$th variable, which is unambiguous wherever it exists by Uniqueness of the Partial Derivative on a Euclidean Open Set, and $\partial_{i'}\partial_i f$ for the iterated partial derivative in the sense of clause 4 of that definition, namely $\partial_{i'}$ applied to the function $\partial_i f$.

Twice continuously differentiable affine-controlled rate data on $l$ states with control set $\mathcal{A}$ and derivative bound $K_0$ is a triple $(U,\bar{\beta}_0,\bar{\beta}_1)$ consisting of an open, convex, bounded set $U\subseteq\mathbb{R}^l$ with $\Delta^l\subset U$ and two families of functions

$$\bar{\beta}_0(\sigma,\gamma,\cdot):U\to\mathbb{R},\qquad \bar{\beta}_1(\sigma,\gamma,\cdot):U\to\mathbb{R}^m,$$

indexed by the ordered pairs $(\sigma,\gamma)$ with $\sigma,\gamma\in\{1,\dots,l\}$ and $\sigma\neq\gamma$, such that the following hold for every such pair, where $\bar{\beta}_1^k$ denotes the $k$-th component of $\bar{\beta}_1$ for $k\in\{1,\dots,m\}$, and where $f$ denotes any one of the $1+m$ real-valued functions $\bar{\beta}_0(\sigma,\gamma,\cdot),\bar{\beta}_1^1(\sigma,\gamma,\cdot),\dots,\bar{\beta}_1^m(\sigma,\gamma,\cdot)$ on $U$:

1. (Nonnegativity.) $\bar{\beta}_0(\sigma,\gamma,\Sigma)+\bar{\beta}_1(\sigma,\gamma,\Sigma)\cdot\alpha\ge0$ for every $\Sigma\in\Delta^l$ and every $\alpha\in\mathcal{A}$, where $x\cdot y$ denotes the dot product.

2. (Regularity.) $f$ is of class $C^2$ on the open set $U$.

3. (Bounds.) $|f(\Sigma)|\le K_0$, $|\partial_i f(\Sigma)|\le K_0$ and $|\partial_{i'}\partial_i f(\Sigma)|\le K_0$ for all $i,i'\in\{1,\dots,l\}$ and all $\Sigma\in U$.

4. (Uniform continuity of second derivatives.) For every real $\varepsilon>0$ there is a real $\delta>0$ such that $|\partial_{i'}\partial_i f(\Sigma)-\partial_{i'}\partial_i f(\Sigma')|\le\varepsilon$ for all $i,i'\in\{1,\dots,l\}$ and all $\Sigma,\Sigma'\in U$ whose Euclidean distance satisfies $d(\Sigma,\Sigma')\le\delta$.