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} $Q,V,R,F$ such that additionally every $R(t)$ is \reftext{def:positive-semidefinite-matrix-2026a}{positive definite}; by \ref{lem:pd-inverse-2026a} each $R(t)^{-1}$ exists and is symmetric positive definite, and $t\mapsto R(t)^{-1}$ has \reftext{def:continuity-closed-interval-c54-2026b}{continuous} entries by claim 1 of \ref{lem:matrix-inverse-continuity-2026a}.

Suppose $Z$ assigns to each $t\in[0,T]$ a symmetric real $l\times l$ matrix $Z(t)$ with entries continuous in $t$, satisfying the \textbf{backward Riccati equation}

with entrywise \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integrals} of continuous functions (existing by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}; degenerate intervals by the convention of \ref{def:mean-square-riemann-integral-2026a}), the \reftext{def:product-real-matrices-2026a}{matrix product}, and the \reftext{def:transpose-real-matrix-2026a}{transpose}. Define the \textbf{feedback gain}

a real $k\times l$ matrix assignment with continuous entries.

Then for every \reftext{def:admissible-control-2026a}{admissible control} $\alpha$ with values in $\mathbb{R}^{k}$, with \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled state} $X^{\alpha}$ and \reftext{def:lqg-cost-functional-2026a}{cost} $J[\alpha]$: the function $t\mapsto\mathbb{E}\bigl[(\alpha_t-\Gamma(t)X^{\alpha}_t)\cdot\bigl(R(t)(\alpha_t-\Gamma(t)X^{\alpha}_t)\bigr)\bigr]$ is continuous on $[0,T]$, and

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 initial covariance matrix $P_0$ of claim 1 of \ref{thm:kalman-bucy-filter-solution-2026a}, $\Theta$ from the model, the \reftext{def:dot-product-orthogonality-rn-2026a}{dot product} and \reftext{def:matrix-vector-product-2026a}{matrix-vector product} applied componentwise to tuples, and differences of tuples formed componentwise.