Existence and Self-Consistency of the Closed-Loop Feedback Control

Consider the setting of Completion of Squares for the Linear-Quadratic-Gaussian Cost: a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, cost data $Q,V,R,F$ with every $R(t)$ positive definite, a symmetric continuous solution $Z$ of the backward Riccati equation as there, and the feedback gain $\Gamma$ defined there. Let $\Phi$ and $\Psi=\Phi^{-1}$ be the fundamental solution of $A$ and its inverse from Variation of Constants for Linear Systems, and let $m^{\mathrm f}$ be the filter process of The Kalman-Bucy Filter Equation and Its Solution. Then:

1. (Closed-loop estimator) There is a family $\widehat X^{*}=(\widehat X^{*}_t)_{t\in[0,T]}$ of $l$-tuples of square-integrable random variables, with mean-square continuous component families, satisfying, componentwise and almost surely,

$$\widehat X^{*}_t=m^{\mathrm f}_t+\Phi(t)\int_0^t\Psi(r)B(r)\Gamma(r)\,\widehat X^{*}_r\,dr\qquad(0\le t\le T),$$

with the mean-square Riemann integral taken componentwise; any two families of $l$-tuples of square-integrable random variables with mean-square continuous component families satisfying this identity agree almost surely at each time. Moreover each $(\widehat X^{*}_t)^{i}$ lies in the closed mean-square span of the family consisting of the constant $1$ and the values $u^{j}_r$ ($1\le j\le\tilde l$, $0\le r\le t$), and is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable.

2. (The feedback control and its estimator) The family $\alpha^{*}=(\alpha^{*}_t)_{t\in[0,T]}$ with $\alpha^{*}_t:=\Gamma(t)\widehat X^{*}_t$ (componentwise matrix-vector product) is an admissible control with values in $\mathbb{R}^{k}$, and its controlled estimator $\widehat X$ satisfies $\widehat X_t=\widehat X^{*}_t$ almost surely, componentwise, for every $t$; consequently $\alpha^{*}_t=\Gamma(t)\widehat X_t$ almost surely for every $t$.

3. (Self-consistency) Let $u^{\alpha^{*}}$ and $\mathcal{G}^{\alpha^{*}}_t$ be the controlled observations and controlled observation $\sigma$-algebras for $\alpha^{*}$. Then for every $t\in[0,T]$: each $(\widehat X^{*}_t)^{i}$, each $u^{j}_r$ with $0\le r\le t$, and each $\alpha^{*\kappa}_t$ lies in the closed mean-square span of the family consisting of the constant $1$ and the values $u^{\alpha^{*},j}_q$ ($1\le j\le\tilde l$, $0\le q\le t$), and each is almost surely equal to a $\mathcal{G}^{\alpha^{*}}_t$-measurable random variable. In particular the feedback control is determined by its own observations.