The Open-Loop Policy Determined by a Measurable Control

Let $T>0$ be a real number, let $m$ and $\tilde{l}$ be natural numbers, and let $A:[0,T]\to\mathbb{R}^m$ be a map whose components are measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra on the real line. Define a family $h^A=(h^A_k)_{k\ge0}$ by

$$h^A_0(t)=A_t,\qquad h^A_k(t,\tau_1,\dots,\tau_k,\upsilon_1,\dots,\upsilon_k)=A_t\quad(k\ge1),$$

for $t\in[0,T]$, ordered times $0\le\tau_1\le\dots\le\tau_k\le T$ and channels $\upsilon_1,\dots,\upsilon_k\in\{1,\dots,\tilde{l}\}$.

1. $h^A$ is an observation-driven control policy with horizon $T$, control dimension $m$ and $\tilde{l}$ channels; it is called the open-loop policy determined by $A$.

2. Let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ with $A_t\in\mathcal{A}$ for every $t\in[0,T]$. Then $h^A$ is $\mathcal{A}$-valued.

3. Let $\mathcal{A}$ and $A$ be as in claim 2, so that $h^A$ is an $\mathcal{A}$-valued policy. In the setting of the controlled $N$-agent dynamics with a transition-rate family having control set $\mathcal{A}$ and with policy $h^A$, every solution on $[0,T]$ satisfies $\alpha_t(\omega)=A_t$ for every $t\in[0,T]$ and every $\omega$ in the regular event.