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 $B$, and \reftext{def:lqg-cost-functional-2026a}{cost data} $Q,V,R,F$ with every $R(t)$ \reftext{def:positive-semidefinite-matrix-2026a}{positive definite}. Suppose $Z$ is a symmetric continuous solution of the backward Riccati equation, and let $\Gamma$ be the feedback gain, both as in \ref{thm:lqg-completion-of-squares-2026a}, and let $V^{*}$ be the optimal value defined in \ref{thm:lqg-separation-theorem-2026a}. Adopt the notation $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$ and the co-null terminology of \ref{lem:control-sequence-mean-square-limit-2026a}, from \reftext{lem:interval-lebesgue-toolkit-2026a}{the restricted Lebesgue measure on $[0,T]$}.

For each \reftext{def:extended-admissible-control-2026a}{extended admissible control} $\alpha$ with values in $\mathbb{R}^{k}$, let $X^{\alpha}$ be its \reftext{def:extended-controlled-state-2026a}{controlled state}, $J[\alpha]$ its \reftext{def:extended-lqg-cost-2026a}{cost}, and $\widehat X=\widehat X(\alpha)$ its \reftext{lem:extended-controlled-state-conditional-expectation-2026a}{extended controlled estimator}, and define

with the \reftext{def:expectation-variance-2026a}{expectation}, the \reftext{def:dot-product-orthogonality-rn-2026a}{dot product} and \reftext{def:matrix-vector-product-2026a}{matrix-vector product} applied componentwise to tuples of random variables, and differences of tuples formed componentwise; each $\varphi_{\alpha}(t)$ is defined and finite by claim 1 of \ref{lem:expected-quadratic-form-2026a}, and nonnegative because every $R(t)$ is positive definite, so the random variable inside the expectation is nonnegative pointwise. Then:

\textbf{1. (Extended cost representation)} For every extended admissible control $\alpha$ and every approximating sequence $\bigl((\alpha^{(n)}),D\bigr)$ for $\alpha$: the function $\mathbf{1}_D\,\varphi_{\alpha}$ is $\mathcal{B}_{[0,T]}$-measurable with finite \reftext{def:lebesgue-integral-nonnegative-2026a}{Lebesgue integral}, this integral does not depend on the choice of approximating sequence, and

\textbf{2. (Optimality criterion)} For every extended admissible control $\alpha$: $J[\alpha]\ge V^{*}$, with equality if and only if there is a co-null set $D_0\in\mathcal{B}_{[0,T]}$ such that for every $t\in D_0$, componentwise, $\alpha_t=\Gamma(t)\widehat X_t(\alpha)$ \reftext{def:almost-surely-2026a}{almost surely}.

\textbf{3. (Attainment)} The closed-loop feedback control $\alpha^{*}$ of \ref{lem:closed-loop-feedback-control-2026a} is \reftext{def:admissible-control-2026a}{admissible}, hence extended admissible by claim 1 of \ref{lem:extended-control-convergence-2026a}, and $J[\alpha^{*}]=V^{*}$. In particular $V^{*}$ is the minimum of $J$ over all extended admissible controls with values in $\mathbb{R}^{k}$, and enlarging the class from admissible to extended admissible controls does not lower the optimal value.