Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$ with state $X$, a control dimension $k\ge1$, a control matrix assignment $B$ as in \ref{def:controlled-linear-gaussian-dynamics-2026a}, and an \reftext{def:extended-admissible-control-2026a}{extended admissible control} $\alpha$ with values in $\mathbb{R}^{k}$.

The \textbf{controlled state} of $\alpha$ is the \reftext{def:family-subfamily-subsets-set-2026a}{family} $X^{\alpha}=(X^{\alpha}_t)_{t\in[0,T]}$ of tuples defined componentwise by

where $c^{\alpha}$ is a fixed choice of a family satisfying claims 2(a) and 2(b) of \ref{lem:extended-control-convergence-2026a} for a chosen approximating sequence of $\alpha$.

\textbf{Well-definedness.} Such a family $c^{\alpha}$ exists by claim 2 of \ref{lem:extended-control-convergence-2026a}, and by claim 2(c) there any two choices (for any approximating sequences) agree \reftext{def:almost-surely-2026a}{almost surely} at every $t$, so $X^{\alpha}_t$ is determined up to almost sure equality at every $t$.

\textbf{Consistency.} If $\alpha$ is an \reftext{def:admissible-control-2026a}{admissible control}, then by claim 2(c) of \ref{lem:extended-control-convergence-2026a} and claim 2 of \ref{lem:controlled-state-superposition-2026a} the family $X^{\alpha}$ agrees almost surely at every $t$ with the \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled state} already defined for admissible controls, so the same symbol may be used for both.