Control Policy with Values in a Prescribed Subset of Euclidean Space

Let $m$ and $\tilde{l}$ be natural numbers with $m\ge1$ and $\tilde{l}\ge1$, let $T>0$ be a real number, and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^{m}$. Let $h=(h_{k})_{k\ge0}$ be an observation-driven control policy with horizon $T$, control dimension $m$ and $\tilde{l}$ channels, and let $R_{k}(T)$ be the record spaces of that definition.

The policy $h$ is $\mathcal{A}$-valued if

$$h_{0}(t)\in\mathcal{A}\quad\text{for every }t\in[0,T],$$

and, for every natural number $k\ge1$,

$$h_{k}(t,\tau,v)\in\mathcal{A}\quad\text{for every }t\in[0,T],\ \tau\in R_{k}(T)\text{ and }v\in\{1,\dots,\tilde{l}\}^{k}.$$