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}; by \ref{lem:pd-inverse-2026a} each $R(t)^{-1}$ exists and is symmetric positive definite, with \reftext{def:continuity-closed-interval-c54-2026b}{continuous} entries by claim 1 of \ref{lem:matrix-inverse-continuity-2026a}. 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}. Let $P_0$ and $\Pi$ be the initial covariance matrix and covariance assignment of \ref{thm:kalman-bucy-filter-solution-2026a}, and for each \reftext{def:admissible-control-2026a}{admissible control} $\alpha$ let $X^{\alpha}$ be its \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled state}, $J[\alpha]$ its \reftext{def:lqg-cost-functional-2026a}{cost}, and $\widehat X(\alpha)$ its \reftext{lem:controlled-state-conditional-expectation-2026a}{controlled estimator}.

By the definition of admissibility, controls are mean-square continuous and adapted, up to \reftext{def:almost-surely-2026a}{almost sure} equality, to the observation $\sigma$-algebras $\mathcal{G}_t$ of the \emph{uncontrolled} model; optimality below is asserted within this class, and claim 3 reconciles the constraint with the controlled observations for the optimal control itself.

Define the \textbf{optimal value}

with the \reftext{def:matrix-trace-2026a}{trace}, the \reftext{def:expectation-variance-2026a}{expectation}, $\mathbb{E}[\xi]:=(\mathbb{E}[\xi^{1}],\dots,\mathbb{E}[\xi^{l}])$, the \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} of a continuous integrand, and the \reftext{def:dot-product-orthogonality-rn-2026a}{dot product}. Then:

\textbf{1. (Cost representation)} For every admissible control $\alpha$ with values in $\mathbb{R}^{k}$, the function $t\mapsto\mathbb{E}\bigl[(\alpha_t-\Gamma(t)\widehat X_t(\alpha))\cdot\bigl(R(t)(\alpha_t-\Gamma(t)\widehat X_t(\alpha))\bigr)\bigr]$ is continuous and nonnegative on $[0,T]$, and

differences of tuples being formed componentwise, with the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}.

\textbf{2. (Optimality criterion)} For every admissible control $\alpha$: $J[\alpha]\ge V^{*}$, with equality if and only if for every $t\in[0,T]$, componentwise, $\alpha_t=\Gamma(t)\widehat X_t(\alpha)$ almost surely.

\textbf{3. (Existence of an optimal control)} The closed-loop feedback control $\alpha^{*}$ of \ref{lem:closed-loop-feedback-control-2026a} is admissible, satisfies $\alpha^{*}_t=\Gamma(t)\widehat X_t(\alpha^{*})$ almost surely for every $t$, is determined by its own observations in the sense of claim 3 of \ref{lem:closed-loop-feedback-control-2026a}, and attains the optimal value: $J[\alpha^{*}]=V^{*}$.