The Linear-Quadratic-Gaussian Cost of an Extended Admissible Control

Consider a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, cost data $Q,V,R,F$, and an extended admissible control $\alpha$ with values in $\mathbb{R}^{k}$.

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

$$J[\alpha]:=\lim_{n\to\infty}J\bigl[\alpha^{(n)}\bigr],$$

the limit of the costs of the members of any approximating sequence $\bigl((\alpha^{(n)}),D\bigr)$ for $\alpha$.

Well-definedness. By claim 3 of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control, this limit exists and does not depend on the choice of approximating sequence.

Consistency. If $\alpha$ is an admissible control, then by claims 1 and 3 of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control this limit equals the cost of $\alpha$ from The Linear-Quadratic-Gaussian Cost Functional, so the same symbol $J[\alpha]$ may be used for both.