Population Cost 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, and let $\mathbb{R}^m$ denote Euclidean space.

Population cost data on $l$ states with control dimension $m$ is a pair $(L,G)$ of functions

called the running cost and the terminal cost respectively, such that:

1. (Continuity.) Whenever $(\Sigma_n,\alpha_n)_{n\in\mathbb{N}}$ is a sequence in $\Delta^l\times\mathbb{R}^m$ such that the Euclidean distances $d(\Sigma_n,\Sigma)$ and $d(\alpha_n,\alpha)$ converge to $0$ for some $\Sigma\in\Delta^l$ and $\alpha\in\mathbb{R}^m$, then $L(\Sigma_n,\alpha_n)\to L(\Sigma,\alpha)$ and $G(\Sigma_n)\to G(\Sigma)$.

2. (Lower bounds.) There are real numbers $C_L$ and $C_G$ such that $L(\Sigma,\alpha)\ge -C_L$ and $G(\Sigma)\ge -C_G$ for all $\Sigma\in\Delta^l$ and $\alpha\in\mathbb{R}^m$.