Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$, with notation and fixed versions as there, and let $k\ge1$ be a \reftext{def:natural-numbers-2026a}{natural number}.

An \textbf{admissible control with values in $\mathbb{R}^{k}$} for the model is a \reftext{def:family-subfamily-subsets-set-2026a}{family} $\alpha=(\alpha_t)_{t\in[0,T]}$, where each $\alpha_t=(\alpha^{1}_t,\dots,\alpha^{k}_t)$ is a tuple of \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variables on $(\Omega,\mathcal{F},P)$, such that:

\textbf{(i)} for each $\kappa\in\{1,\dots,k\}$ the component family $(\alpha^{\kappa}_t)_{t\in[0,T]}$ is \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous};

\textbf{(ii)} for every $t\in[0,T]$ and every $\kappa\in\{1,\dots,k\}$, the random variable $\alpha^{\kappa}_t$ is \reftext{def:almost-surely-2026a}{almost surely} equal to a $\mathcal{G}_t$-measurable square-integrable random variable, where $\mathcal{G}_t$ is the observation $\sigma$-algebra of the model — that is, of the observation process $u$ of the model itself, without control — and measurability means that preimages of \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} belong to $\mathcal{G}_t$.