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}. Adopt the notation $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$ of \reftext{lem:interval-lebesgue-toolkit-2026a}{the restricted Lebesgue measure on $[0,T]$}, and the distance $d$ between \reftext{def:admissible-control-2026a}{admissible controls} of \ref{lem:control-sequence-mean-square-limit-2026a}.

An \textbf{extended 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 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 itself, without control, and measurability means that preimages of \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} belong to $\mathcal{G}_t$;

\textbf{(ii)} there exist a \reftext{def:sequence-in-set-2026a}{sequence} $(\alpha^{(n)})_{n\in\mathbb{N}}$ of admissible controls with values in $\mathbb{R}^{k}$ and a set $D\in\mathcal{B}_{[0,T]}$ with $\lambda_{[0,T]}([0,T]\setminus D)=0$ such that the sequence is Cauchy for $d$ in the sense of \ref{lem:control-sequence-mean-square-limit-2026a}, and for every $t\in D$ and every $\kappa$ the real sequence $\bigl(\lVert\alpha^{(n),\kappa}_t-\alpha^{\kappa}_t\rVert_{2}\bigr)_{n}$ has \reftext{def:limit-sequence-real-c54-2026a}{limit} $0$, with the mean-square norm $\lVert\cdot\rVert_{2}$ of \ref{def:square-integrable-mean-square-2026a}.

A pair $\bigl((\alpha^{(n)})_{n\in\mathbb{N}},D\bigr)$ as in (ii) is called an \textbf{approximating sequence} for $\alpha$.