Affine-Controlled Transition-Rate Family

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 $\Lambda$ be a nonnegative real number, and 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.

An affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}$ and Lipschitz constant $\Lambda$ is a pair of families of functions

$$\beta_0(\sigma,\gamma,\cdot):\Delta^l\to\mathbb{R},\qquad \beta_1(\sigma,\gamma,\cdot):\Delta^l\to\mathbb{R}^m,$$

indexed by the ordered pairs $(\sigma,\gamma)$ with $\sigma,\gamma\in\{1,\dots,l\}$ and $\sigma\neq\gamma$, such that for every such pair:

1. (Nonnegativity.) $\beta_0(\sigma,\gamma,\Sigma)+\beta_1(\sigma,\gamma,\Sigma)\cdot\alpha\ge0$ for every $\Sigma\in\Delta^l$ and every $\alpha\in\mathcal{A}$, where $x\cdot y$ denotes the dot product.

2. (Lipschitz dependence on the state.) The maps $\beta_0(\sigma,\gamma,\cdot)$ and $\beta_1(\sigma,\gamma,\cdot)$ are Lipschitz with constant $\Lambda$ on $\Delta^l$.