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; let $k\ge1$ be a \reftext{def:natural-numbers-2026a}{natural number}; let $B$ assign to each $t\in[0,T]$ a real $l\times k$ matrix $B(t)$ whose entries are \reftext{def:continuity-closed-interval-c54-2026b}{continuous} functions of $t$; and let $\alpha=(\alpha_t)_{t\in[0,T]}$ be an \reftext{def:admissible-control-2026a}{admissible control with values in $\mathbb{R}^{k}$} for the model.

The controlled system with these data consists of:

\textbf{The controlled state} $X^{\alpha}=(X^{\alpha}_t)_{t\in[0,T]}$: a fixed choice of versions of a \reftext{def:linear-sde-mean-square-solution-2026a}{mean-square solution} of the linear stochastic differential equation with data $(A,g,\varepsilon,\xi,W)$, where $g$ is the forcing family $g_r:=B(r)\,\alpha_r$ ($0\le r\le T$), formed componentwise with the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}.

\textbf{The controlled observation process} $u^{\alpha}=(u^{\alpha}_t)_{t\in[0,T]}$, $u^{\alpha}_t=(u^{\alpha,1}_t,\dots,u^{\alpha,\tilde l}_t)$: a fixed choice of versions of

where the first integral is the \reftext{def:mean-square-riemann-integral-2026a}{mean-square Riemann integral}, the last are \reftext{thm:vector-wiener-integral-gaussian-2026a}{Wiener integrals} for which we fix the same versions as fixed for the observation process $u$ of the model, and integrals over the degenerate interval at $t=0$ are $0$ by the conventions of \ref{def:mean-square-riemann-integral-2026a} and \ref{def:ito-integral-2026a}.

\textbf{The controlled observation $\sigma$-algebras} $\mathcal{G}^{\alpha}_t:=\sigma\bigl(u^{\alpha,j}_r:1\le j\le\tilde l,\ 0\le r\le t\bigr)$ ($0\le t\le T$), with the \reftext{def:independence-sigma-algebras-2026a}{$\sigma$-algebra generated by a family of random variables}, determined by the fixed choice of versions of $u^{\alpha}$.

\textbf{Well-definedness.} Each component family of the forcing $g$ is \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous} by claims 1-2 of \ref{lem:mean-square-riemann-integral-properties-2026a}, so a mean-square solution exists and any two agree almost surely at each time by \ref{thm:linear-sde-variation-of-constants-2026a}; the integrand family $\bigl((\tilde E(r)X^{\alpha}_r)^{j}\bigr)_{r}$ is mean-square continuous by the same claims, so the first integral exists by \ref{lem:mean-square-riemann-integral-existence-2026a}.