The Separation Theorem for Partial-Information Linear-Quadratic-Gaussian Control

Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line.

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; by Invertibility of Positive Definite Matrices each $R(t)^{-1}$ exists and is symmetric positive definite, with continuous entries by claim 1 of Continuity and Derivative of the Inverse of a Matrix-Valued Map. 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. Let $P_0$ and $\Pi$ be the initial covariance matrix and covariance assignment of The Kalman-Bucy Filter Equation and Its Solution, and for each admissible control $\alpha$ let $X^{\alpha}$ be its controlled state, $J[\alpha]$ its cost, and $\widehat X(\alpha)$ its controlled estimator.

By the definition of admissibility, controls are mean-square continuous and adapted, up to almost sure equality, to the observation $\sigma$-algebras $\mathcal{G}_t$ of the uncontrolled model; optimality below is asserted within this class, and claim 3 reconciles the constraint with the controlled observations for the optimal control itself.

Define the optimal value

$$V^{*}:=\operatorname{tr}\bigl(Z(0)P_0\bigr)+\mathbb{E}[\xi]\cdot\bigl(Z(0)\mathbb{E}[\xi]\bigr)+\int_0^T\Bigl(\operatorname{tr}\bigl(Z(t)\Theta(t)\bigr)+\operatorname{tr}\Bigl(\bigl(Z(t)B(t)+V(t)\bigr)R(t)^{-1}\bigl(Z(t)B(t)+V(t)\bigr)^{\top}\Pi(t)\Bigr)\Bigr)\,dt ,$$

with the trace, the expectation, $\mathbb{E}[\xi]:=(\mathbb{E}[\xi^{1}],\dots,\mathbb{E}[\xi^{l}])$, the Riemann integral of a continuous integrand, and the dot product. Then:

1. (Cost representation) For every admissible control $\alpha$ with values in $\mathbb{R}^{k}$, the function $t\mapsto\mathbb{E}\bigl[(\alpha_t-\Gamma(t)\widehat X_t(\alpha))\cdot\bigl(R(t)(\alpha_t-\Gamma(t)\widehat X_t(\alpha))\bigr)\bigr]$ is continuous and nonnegative on $[0,T]$, and

$$J[\alpha]=V^{*}+\int_0^T\mathbb{E}\Bigl[\bigl(\alpha_t-\Gamma(t)\widehat X_t(\alpha)\bigr)\cdot\Bigl(R(t)\bigl(\alpha_t-\Gamma(t)\widehat X_t(\alpha)\bigr)\Bigr)\Bigr]\,dt ,$$

differences of tuples being formed componentwise, with the matrix-vector product.

2. (Optimality criterion) For every admissible control $\alpha$: $J[\alpha]\ge V^{*}$, with equality if and only if for every $t\in[0,T]$, componentwise, $\alpha_t=\Gamma(t)\widehat X_t(\alpha)$ almost surely.

3. (Existence of an optimal control) The closed-loop feedback control $\alpha^{*}$ of Existence and Self-Consistency of the Closed-Loop Feedback Control is admissible, satisfies $\alpha^{*}_t=\Gamma(t)\widehat X_t(\alpha^{*})$ almost surely for every $t$, is determined by its own observations in the sense of claim 3 of Existence and Self-Consistency of the Closed-Loop Feedback Control, and attains the optimal value: $J[\alpha^{*}]=V^{*}$.