Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, an \reftext{def:extended-admissible-control-2026a}{extended admissible control} $\alpha$ with values in $\mathbb{R}^{k}$, and its \reftext{def:extended-controlled-state-2026a}{controlled state} $X^{\alpha}=X+c^{\alpha}$, with notation and fixed versions as in those items. Let $m^{\mathrm f}$ and $\Pi$ be the filter process and covariance assignment of \ref{thm:kalman-bucy-filter-solution-2026a}, and let $e_t:=X_t-m^{\mathrm f}_t$ be the estimation error of \ref{thm:kalman-bucy-conditional-expectation-2026a}. Define the \textbf{extended controlled estimator}

componentwise, with the fixed versions of those items. Then:

\textbf{1. (Regularity and adaptedness)} Each component family $(\widehat X^{i}_t)_{t\in[0,T]}$ is \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous}, and for every $t\in[0,T]$ each $\widehat X^{i}_t$ is \reftext{def:almost-surely-2026a}{almost surely} equal to a $\mathcal{G}_t$-measurable \reftext{def:square-integrable-mean-square-2026a}{square-integrable} random variable, $\mathcal{G}_t$ being the observation $\sigma$-algebras of the model.

\textbf{2. (Conditional expectation)} For every $t\in[0,T]$ and every $i\in\{1,\dots,l\}$, every \reftext{def:conditional-expectation-l2-2026a}{conditional expectation} of $X^{\alpha,i}_t$ given $\mathcal{G}_t$ is almost surely equal to $\widehat X^{i}_t$.

\textbf{3. (Error invariance)} For every $t\in[0,T]$ and every $i$: $X^{\alpha,i}_t-\widehat X^{i}_t=e^{i}_t$ almost surely. Consequently, by claims 2-3 of \ref{thm:kalman-bucy-conditional-expectation-2026a}, with the \reftext{def:expectation-variance-2026a}{expectation} and \reftext{def:covariance-square-integrable-2026a}{covariance}: $\mathbb{E}[e^{i}_t]=0$ for every $i$, the matrix $\bigl(\operatorname{Cov}(e^{i}_t,e^{j}_t)\bigr)_{1\le i,j\le l}$ equals $\Pi(t)$, and the \reftext{def:independence-sigma-algebras-2026a}{$\sigma$-algebras} $\sigma(e^{1}_t,\dots,e^{l}_t)$ and $\mathcal{G}_t$ are independent.

\textbf{4. (Consistency and convergence)} If $\alpha$ is an \reftext{def:admissible-control-2026a}{admissible control}, then for every $t\in[0,T]$ the tuple $\widehat X_t$ agrees almost surely, componentwise, with the \reftext{lem:controlled-state-conditional-expectation-2026a}{controlled estimator} already defined for admissible controls. Moreover, for any approximating sequence $\bigl((\alpha^{(n)}),D\bigr)$ of $\alpha$, writing $\widehat X(\alpha^{(n)})$ for the controlled estimator of $\alpha^{(n)}$ from \ref{lem:controlled-state-conditional-expectation-2026a}: the maxima over $t\in[0,T]$ of the continuous functions $t\mapsto\sum_{i=1}^{l}\lVert\widehat X^{i}_t(\alpha^{(n)})-\widehat X^{i}_t\rVert_{2}$ tend to $0$ as $n\to\infty$, with the mean-square norm of \ref{def:square-integrable-mean-square-2026a} and the maxima existing by \ref{thm:calc-extreme-value-theorem-1d-2026c}.