The Separation Theorem over Extended Admissible Controls

Consider a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, and cost data $Q,V,R,F$ with every $R(t)$ positive definite. Suppose $Z$ is a symmetric continuous solution of the backward Riccati equation, and let $\Gamma$ be the feedback gain, both as in Completion of Squares for the Linear-Quadratic-Gaussian Cost, and let $V^{*}$ be the optimal value defined in The Separation Theorem for Partial-Information Linear-Quadratic-Gaussian Control. Adopt the notation $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$ and the co-null terminology of Almost-Everywhere Mean-Square Limits of Cauchy Sequences of Admissible Controls, from the restricted Lebesgue measure on $[0,T]$.

For each extended admissible control $\alpha$ with values in $\mathbb{R}^{k}$, let $X^{\alpha}$ be its controlled state, $J[\alpha]$ its cost, and $\widehat X=\widehat X(\alpha)$ its extended controlled estimator, and define

$$\varphi_{\alpha}(t):=\mathbb{E}\Bigl[\bigl(\alpha_t-\Gamma(t)\widehat X_t\bigr)\cdot\Bigl(R(t)\bigl(\alpha_t-\Gamma(t)\widehat X_t\bigr)\Bigr)\Bigr]\qquad(0\le t\le T),$$

with the expectation, the dot product and matrix-vector product applied componentwise to tuples of random variables, and differences of tuples formed componentwise; each $\varphi_{\alpha}(t)$ is defined and finite by claim 1 of Expected Bilinear Forms: Trace Formula and Mean-Square Continuity, and nonnegative because every $R(t)$ is positive definite, so the random variable inside the expectation is nonnegative pointwise. Then:

1. (Extended cost representation) For every extended admissible control $\alpha$ and every approximating sequence $\bigl((\alpha^{(n)}),D\bigr)$ for $\alpha$: the function $\mathbf{1}_D\,\varphi_{\alpha}$ is $\mathcal{B}_{[0,T]}$-measurable with finite Lebesgue integral, this integral does not depend on the choice of approximating sequence, and

$$J[\alpha]=V^{*}+\int_{[0,T]}\mathbf{1}_D\,\varphi_{\alpha}\,d\lambda_{[0,T]} .$$

2. (Optimality criterion) For every extended admissible control $\alpha$: $J[\alpha]\ge V^{*}$, with equality if and only if there is a co-null set $D_0\in\mathcal{B}_{[0,T]}$ such that for every $t\in D_0$, componentwise, $\alpha_t=\Gamma(t)\widehat X_t(\alpha)$ almost surely.

3. (Attainment) The closed-loop feedback control $\alpha^{*}$ of Existence and Self-Consistency of the Closed-Loop Feedback Control is admissible, hence extended admissible by claim 1 of Convergence of Corrections and Costs along Approximating Sequences of an Extended Admissible Control, and $J[\alpha^{*}]=V^{*}$. In particular $V^{*}$ is the minimum of $J$ over all extended admissible controls with values in $\mathbb{R}^{k}$, and enlarging the class from admissible to extended admissible controls does not lower the optimal value.