Twice Continuously Differentiable Extension of Population Cost Data

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $(L,G)$ be population cost data on $l$ states with control dimension $m$, let $\Delta^l\subset\mathbb{R}^l$ be the probability simplex, and let $K\ge0$ be a real number. Points of $\mathbb{R}^l\times\mathbb{R}^m$ are written $x=(\Sigma,\alpha)$ and identified with points of Euclidean space $\mathbb{R}^{l+m}$, with coordinates $x_1,\dots,x_{l+m}$, so that $x_\gamma=\Sigma^\gamma$ for $\gamma\le l$ and $x_{l+j}=\alpha^j$ for $j\le m$; a product $U'\times V'$ of sets $U'\subseteq\mathbb{R}^l$ and $V'\subseteq\mathbb{R}^m$ is regarded as a subset of $\mathbb{R}^{l+m}$ under this identification. For a real-valued function $f$ on an open subset of a Euclidean space $\mathbb{R}^n$ and indices $i,j\in\{1,\dots,n\}$ 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 $\partial_j\partial_i f$ for the iterated partial derivative in the sense of clause 4 of that definition, namely $\partial_j$ applied to the function $\partial_i f$.

A triple $(U,\bar{L},\bar{G})$ is a twice continuously differentiable extension of the population cost data $(L,G)$ with second-derivative bound $K$ if it consists of an open set $U\subseteq\mathbb{R}^l$ with $\Delta^l\subset U$ and functions $\bar{L}:U\times\mathbb{R}^m\to\mathbb{R}$ and $\bar{G}:U\to\mathbb{R}$ such that:

1. (Extension.) $\bar{L}(\Sigma,\alpha)=L(\Sigma,\alpha)$ for all $(\Sigma,\alpha)\in\Delta^l\times\mathbb{R}^m$, and $\bar{G}(\Sigma)=G(\Sigma)$ for all $\Sigma\in\Delta^l$.

2. (Regularity.) $\bar{L}$ is of class $C^2$ on $U\times\mathbb{R}^m$, which is an open subset of $\mathbb{R}^{l+m}$ by claims 1 and 2 of Products of Euclidean Open Sets are Open, $U$ being open and $\mathbb{R}^m$ being open in $\mathbb{R}^m$; and $\bar{G}$ is of class $C^2$ on the open set $U$.

3. (Second-derivative bounds.) $|\partial_j\partial_i\bar{L}(x)|\le K$ for all $i,j\in\{1,\dots,l+m\}$ and $x\in U\times\mathbb{R}^m$, and $|\partial_{\gamma'}\partial_{\gamma}\bar{G}(\Sigma)|\le K$ for all $\gamma,\gamma'\in\{1,\dots,l\}$ and $\Sigma\in U$.

4. (Uniform continuity of second derivatives.) For every real $\varepsilon>0$ there is a real $\delta>0$ such that both of the following hold: $|\partial_j\partial_i\bar{L}(x)-\partial_j\partial_i\bar{L}(y)|\le\varepsilon$ for all $i,j\in\{1,\dots,l+m\}$ and all $x,y\in U\times\mathbb{R}^m$ whose Euclidean distance satisfies $d(x,y)\le\delta$; and $|\partial_{\gamma'}\partial_{\gamma}\bar{G}(\Sigma)-\partial_{\gamma'}\partial_{\gamma}\bar{G}(\Sigma')|\le\varepsilon$ for all $\gamma,\gamma'\in\{1,\dots,l\}$ and all $\Sigma,\Sigma'\in U$ with $d(\Sigma,\Sigma')\le\delta$.