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 $B$, \reftext{def:lqg-cost-functional-2026a}{cost data} $Q,V,R,F$, and an \reftext{def:extended-admissible-control-2026a}{extended admissible control} $\alpha$ with values in $\mathbb{R}^{k}$.

The \textbf{linear-quadratic-Gaussian cost} of $\alpha$ is the real number

the \reftext{def:limit-sequence-real-c54-2026a}{limit} of the \reftext{def:lqg-cost-functional-2026a}{costs} of the members of any approximating sequence $\bigl((\alpha^{(n)}),D\bigr)$ for $\alpha$.

\textbf{Well-definedness.} By claim 3 of \ref{lem:extended-control-convergence-2026a}, this limit exists and does not depend on the choice of approximating sequence.

\textbf{Consistency.} If $\alpha$ is an \reftext{def:admissible-control-2026a}{admissible control}, then by claims 1 and 3 of \ref{lem:extended-control-convergence-2026a} this limit equals the cost of $\alpha$ from \ref{def:lqg-cost-functional-2026a}, so the same symbol $J[\alpha]$ may be used for both.