Consider a \reftext{def:linear-gaussian-state-observation-model-2026a}{linear-Gaussian state-observation model} on $[0,T]$, a control dimension $k\ge1$, and a control matrix assignment $B$ as in \ref{def:controlled-linear-gaussian-dynamics-2026a}.

\textbf{Cost data} for this system is a name for a tuple of assignments $Q$ (real $l\times l$) and $R$ (real $k\times k$) of \reftext{def:positive-semidefinite-matrix-2026a}{symmetric} matrices to each $t\in[0,T]$ and an assignment $V$ (real $l\times k$) to each $t\in[0,T]$, all with entries \reftext{def:continuity-closed-interval-c54-2026b}{continuous} in $t$, together with a symmetric real $l\times l$ matrix $F$.

For an \reftext{def:admissible-control-2026a}{admissible control} $\alpha$ with values in $\mathbb{R}^{k}$ and its \reftext{def:controlled-linear-gaussian-dynamics-2026a}{controlled state} $X^{\alpha}$, the \textbf{linear-quadratic-Gaussian cost} of $\alpha$ is the real number

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 the \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral}.

\textbf{Well-definedness.} Each expectation above is defined and finite, and the integrand is a continuous function of $t$, by claims 1-2 of \ref{lem:expected-quadratic-form-2026a}, the component families of $X^{\alpha}$ and $\alpha$ being \reftext{def:mean-square-continuous-process-2026a}{mean-square continuous}; the integral therefore exists by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}.