Observation-Driven Control Policy

Let $m$ and $\tilde{l}$ be natural numbers with $m\ge 1$ and $\tilde{l}\ge 1$, and let $T>0$ be a real number, called the horizon. For each natural number $k\ge 1$ define the record space

a subset of Euclidean space.

An observation-driven control policy with horizon $T$, control dimension $m$, and $\tilde{l}$ channels is a family $h=(h_k)_{k\ge0}$ of functions

such that for every $k\ge 1$, every mark vector $v\in\{1,\dots,\tilde{l}\}^k$, and every component index $j\in\{1,\dots,m\}$, the real-valued map $(t,\tau)\mapsto h^j_k(t,\tau,v)$ on $[0,T]\times R_k(T)$ is measurable with respect to the $\sigma$-algebra generated by the relatively open subsets of $[0,T]\times R_k(T)$, that is, by the sets $U\cap([0,T]\times R_k(T))$ with $U$ an open subset of $\mathbb{R}^{1+k}$; and likewise each component $t\mapsto h^j_0(t)$ is measurable with respect to the $\sigma$-algebra generated by the relatively open subsets of $[0,T]$.