Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$, a control dimension $k\ge1$, and a control matrix assignment $B$ as in \ref{def:controlled-linear-gaussian-dynamics-2026a}, with notation and fixed versions as in those items; adopt the distance $d$ of \ref{lem:control-sequence-mean-square-limit-2026a} and the mean-square norm $\lVert\cdot\rVert_{2}$ of \ref{def:square-integrable-mean-square-2026a}. Then:

\textbf{1. (Consistency of the class)} Every \reftext{def:admissible-control-2026a}{admissible control} $\beta$ with values in $\mathbb{R}^{k}$ is an \reftext{def:extended-admissible-control-2026a}{extended admissible control} with values in $\mathbb{R}^{k}$, and the constant sequence $\beta^{(n)}:=\beta$ together with $D:=[0,T]$ is an approximating sequence for it.

Now let $\alpha$ be an extended admissible control with values in $\mathbb{R}^{k}$ and let $\bigl((\alpha^{(n)}),D\bigr)$ be an approximating sequence for $\alpha$. For each $n$ let $c^{(n)}$ denote the correction process of \ref{lem:controlled-state-superposition-2026a} for the admissible control $\alpha^{(n)}$, with fixed versions as there.

\textbf{2. (Convergence of the corrections)} There exists a \reftext{def:family-subfamily-subsets-set-2026a}{family} $c^{\alpha}=(c^{\alpha}_t)_{t\in[0,T]}$ of tuples $c^{\alpha}_t=(c^{\alpha,1}_t,\dots,c^{\alpha,l}_t)$ of \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variables such that:

(a) each component family $(c^{\alpha,i}_t)_{t\in[0,T]}$ is \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous}, $c^{\alpha,i}_0=0$ \reftext{def:almost-surely-2026a}{almost surely}, and for every $t\in[0,T]$ each $c^{\alpha,i}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable, $\mathcal{G}_t$ being the observation $\sigma$-algebras of the model;

(b) the function $t\mapsto\sum_{i=1}^{l}\lVert c^{(n),i}_t-c^{\alpha,i}_t\rVert_{2}$ is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$ for each $n$, its maximum over $[0,T]$ exists by \ref{thm:calc-extreme-value-theorem-1d-2026c}, and these maxima tend to $0$ as $n\to\infty$;

(c) if $\bigl((\beta^{(n)}),D'\bigr)$ is any approximating sequence for $\alpha$ (possibly the same one), with correction processes $b^{(n)}$, and $\tilde c$ is any family satisfying (a) and (b) for that sequence, then for every $t\in[0,T]$ and every $i$: $\tilde c^{\,i}_t=c^{\alpha,i}_t$ almost surely; in particular, if $\alpha$ is itself admissible, then $c^{\alpha,i}_t$ is almost surely equal to the $i$-th component at $t$ of the correction process of \ref{lem:controlled-state-superposition-2026a} for $\alpha$.

\textbf{3. (Convergence of the costs)} Let $Q,V,R,F$ be \reftext{def:lqg-cost-functional-2026a}{cost data} for this system, and let $J[\alpha^{(n)}]$ be the cost of $\alpha^{(n)}$. Then the real sequence $\bigl(J[\alpha^{(n)}]\bigr)_{n}$ has a \reftext{def:limit-sequence-real-c54-2026a}{limit}; this limit is the same for every approximating sequence of $\alpha$; and if $\alpha$ is itself admissible, the limit equals $J[\alpha]$.