Proof of Existence and Self-Consistency of the Closed-Loop Feedback 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.

Write $\lVert\cdot\rVert_{2}$ for the mean-square norm of Square-Integrable Random Variables and the Mean-Square Inner Product, and let $D(t,r):=\Phi(t)\Psi(r)B(r)\Gamma(r)$. All entries of $\Phi,\Psi,B,\Gamma$ are continuous, hence bounded on $[0,T]$ by Extreme Value Theorem on a Closed Real Interval, so by the product entry bound (claim 2 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals, applied to the three successive products) there is a real $C_0$ with $|D(t,r)_{ij}|\le C_0$ for all $t,r\in[0,T]$ and all $i,j$; put $C:=l^{2}C_0$. For a family $y=(y_t)_{t\in[0,T]}$ of $l$-tuples with mean-square continuous components define

componentwise, with fixed versions of the mean-square Riemann integrals; the integrand components are mean-square continuous by claims 1-2 of Basic Properties of the Mean-Square Riemann Integral, so $\mathcal{T}y$ again has mean-square continuous components by claims 1-2 and 6 there, $m^{\mathrm f}$ having mean-square continuous components as a mean-square solution (claim 2 of The Kalman-Bucy Filter Equation and Its Solution). For two such families $y,y'$, setting $\delta(t):=\sum_i\lVert y^{i}_t-y'^{i}_t\rVert_2$ (continuous by claim 4 of Basic Properties of the Mean-Square Riemann Integral applied to differences, and the triangle inequality of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm), claims 1, 4 and 6 of Basic Properties of the Mean-Square Riemann Integral give the componentwise bound

Claim 1: existence. Define $y^{(0)}:=m^{\mathrm f}$ and $y^{(n+1)}:=\mathcal{T}y^{(n)}$. Let $d_n(t):=\sum_i\lVert y^{(n+1),i}_t-y^{(n),i}_t\rVert_2$ and $D_0:=\max_{t}d_0(t)$ (finite by continuity and Extreme Value Theorem on a Closed Real Interval). By ($*$) and induction, $d_n(t)\le D_0\,C^{n}t^{n}/n!$ for all $n,t$: the case $n=0$ is the definition, and $d_{n+1}(t)\le C\int_0^t d_n(r)\,dr\le D_0C^{n+1}t^{n+1}/(n+1)!$ by monotonicity of the integral. Since $\sum_n D_0C^{n}T^{n}/n!$ converges (it is dominated by the exponential series, cf. Basic Properties of the Exponential Function), for each fixed $t$ and $i$ the sequence $(y^{(n),i}_t)_n$ is Cauchy in mean square, hence converges to a square-integrable random variable $(\widehat X^{*}_t)^{i}$ by Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer); fix such versions. The convergence is uniform in $t$: $\sup_t\sum_i\lVert y^{(n),i}_t-(\widehat X^{*}_t)^{i}\rVert_2\le\sum_{n'\ge n}\sup_t d_{n'}(t)\to0$. A uniform mean-square limit of componentwise mean-square continuous families is componentwise mean-square continuous, by the standard three-term estimate $\lVert(\widehat X^{*}_t)^{i}-(\widehat X^{*}_s)^{i}\rVert_2\le2\sup_r\lVert y^{(n),i}_r-(\widehat X^{*}_r)^{i}\rVert_2+\lVert y^{(n),i}_t-y^{(n),i}_s\rVert_2$ with the triangle inequality of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm. Passing to the limit in $y^{(n+1)}=\mathcal{T}y^{(n)}$: by ($*$) applied to $y^{(n)}$ and $\widehat X^{*}$, $(\mathcal{T}y^{(n)})_t\to(\mathcal{T}\widehat X^{*})_t$ in mean square componentwise, while $y^{(n+1)}_t\to\widehat X^{*}_t$; mean-square limits agree almost surely, so $\widehat X^{*}_t=(\mathcal{T}\widehat X^{*})_t$ componentwise almost surely, which is the asserted fixed-point identity.

Uniqueness. If $y$ and $y'$ both have mean-square continuous components and satisfy the identity, then $\delta(t)\le C\int_0^t\delta(r)\,dr$ with $\delta$ continuous as above (families satisfying the identity may be replaced, time by time, by the almost surely equal right-hand sides without changing $\delta$), so $\delta\equiv0$: since $T>0$ by the standing hypothesis of Linear-Gaussian State-Observation Model, Gronwall's Lemma (Integral Form) applies to the continuous function $\delta$ on $[0,T]$ with $a=0$ and $b=C\ge0$, giving $\delta(t)\le0\cdot\exp(Ct)=0$, while $\delta\ge0$. Hence the families agree almost surely at each time.

Span structure and measurability. Let $\mathcal{S}_t$ denote the closed mean-square span of $\{1\}\cup\{u^{j}_r:0\le r\le t\}$; note $\mathcal{S}_r\subseteq\mathcal{S}_t$ for $r\le t$ (claim 1 of The Closed Mean-Square Span of a Family of Random Variables). We show by induction that every $y^{(n),i}_t\in\mathcal{S}_t$. For $n=0$ this is claim 3 of The Kalman-Bucy Filter Equation and Its Solution. For the step: by Mean-Square Riemann Integral of a Family of Random Variables, the integral $\int_0^t(\Psi B\Gamma y^{(n)})^{j}_r\,dr$ is a mean-square limit of Riemann sums, each of which is a finite linear combination of values $(\Psi(r)B(r)\Gamma(r)y^{(n)}_r)^{j}$, themselves finite linear combinations of the $y^{(n),i'}_r\in\mathcal{S}_r\subseteq\mathcal{S}_t$; by claim 1 of The Closed Mean-Square Span of a Family of Random Variables (closure under combinations and mean-square limits), the integral lies in $\mathcal{S}_t$, and then so does $y^{(n+1),i}_t$ ($m^{\mathrm f}$-term plus $\Phi(t)$-combination). Closure under mean-square limits gives $(\widehat X^{*}_t)^{i}\in\mathcal{S}_t$. Since $1$ and the $u^{j}_r$ ($r\le t$) are $\mathcal{G}_t$-measurable and square-integrable, claim 2 of The Closed Mean-Square Span of a Family of Random Variables shows every member of $\mathcal{S}_t$ is almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable.

Claim 2. Each component family of $\alpha^{*}$, $\alpha^{*\kappa}_t=\sum_i\Gamma_{\kappa i}(t)(\widehat X^{*}_t)^{i}$, is mean-square continuous (claims 1-2 of Basic Properties of the Mean-Square Riemann Integral) and square-integrable, and almost surely equal to a $\mathcal{G}_t$-measurable square-integrable random variable by claim 1 here and claim 2 of The Closed Mean-Square Span of a Family of Random Variables; hence $\alpha^{*}$ is an admissible control. Let $c$ be the correction process of Superposition Decomposition of the Controlled State and Observations for $\alpha^{*}$. Its defining integrand satisfies $(\Psi(r)B(r)\alpha^{*}_r)^{j}=(\Psi(r)B(r)\Gamma(r)\widehat X^{*}_r)^{j}$ almost surely at each $r$, so (integrals of almost surely equal mean-square continuous families agree almost surely, as in the proof of Superposition Decomposition of the Controlled State and Observations) $c_t=\Phi(t)\int_0^t\Psi(r)B(r)\Gamma(r)\widehat X^{*}_r\,dr$ almost surely, componentwise. Hence the controlled estimator for $\alpha^{*}$ satisfies, almost surely,

by the fixed-point identity of claim 1. Consequently $\alpha^{*}_t=\Gamma(t)\widehat X^{*}_t=\Gamma(t)\widehat X_t$ almost surely.

Claim 3. By claim 3 of Integrals Against the Controlled Observations and the Controlled Filter Equation applied to $\alpha^{*}$, $\widehat X$ satisfies the controlled filter equation, i.e. it is a mean-square solution of the linear stochastic differential equation with coefficient $A-K\tilde E$, forcing $\bigl(K(r)\tilde E(r)X^{\alpha^{*}}_r+B(r)\alpha^{*}_r\bigr)_r$, noise matrix $K\tilde\varepsilon$, and initial value $\mathbb{E}[\xi]$. By claim 2, $\widehat X_r=\widehat X^{*}_r$ and $B(r)\alpha^{*}_r=B(r)\Gamma(r)\widehat X^{*}_r$ almost surely at each $r$; substituting these almost surely equal integrand families (the rule for almost surely equal mean-square continuous integrands, as in the proof of Superposition Decomposition of the Controlled State and Observations) and combining the two drift terms $(A-K\tilde E)\widehat X^{*}_r$ and $B\Gamma\widehat X^{*}_r$ by linearity (claim 1 of Basic Properties of the Mean-Square Riemann Integral) shows that $\widehat X^{*}$ is a mean-square solution of the linear stochastic differential equation with coefficient $\check A:=A-K\tilde E+B\Gamma$, forcing $\bigl(K(r)\tilde E(r)X^{\alpha^{*}}_r\bigr)_r$, noise matrix $K\tilde\varepsilon$, and initial value $\mathbb{E}[\xi]$ (all coefficient entries continuous). Let $\check\Phi,\check\Psi$ be the fundamental solution of $\check A$ and its inverse (Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations). By claims 1-2 of Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations, componentwise and almost surely,

the last equality being the definition of the controlled observation integral in Integrals Against the Controlled Observations and the Controlled Filter Equation with $f=\check\Psi K$ (continuous entries). Let $\mathcal{S}^{*}_t$ denote the closed mean-square span of $\{1\}\cup\{u^{\alpha^{*},j}_q:0\le q\le t\}$. By claim 2 of Integrals Against the Controlled Observations and the Controlled Filter Equation, each component of $\int_0^t\check\Psi K\,du^{\alpha^{*}}_r$ is a mean-square limit of finite linear combinations of values of $u^{\alpha^{*}}$ up to time $t$, hence lies in $\mathcal{S}^{*}_t$; adding the constant tuple $\mathbb{E}[\xi]$ (a multiple of $1$) and applying $\check\Phi(t)$ keeps us in $\mathcal{S}^{*}_t$ (claim 1 of The Closed Mean-Square Span of a Family of Random Variables, and membership is preserved under almost sure equality since approximating combinations converge to any almost surely equal variable as well). Hence $(\widehat X^{*}_t)^{i}\in\mathcal{S}^{*}_t$, and therefore also $\alpha^{*\kappa}_t=\sum_i\Gamma_{\kappa i}(t)(\widehat X^{*}_t)^{i}\in\mathcal{S}^{*}_t$.

Next, with $c$ and $\gamma$ the correction processes for $\alpha^{*}$: as in claim 2, $c_t=\Phi(t)\int_0^t\Psi B\Gamma\widehat X^{*}_r\,dr$ almost surely, and the Riemann-sum argument of claim 1, now run in $\mathcal{S}^{*}_t$, shows $c^{i}_t\in\mathcal{S}^{*}_t$; the same argument applied to $\gamma_t=\int_0^t\tilde E(r)c_r\,dr$ gives $\gamma^{j}_t\in\mathcal{S}^{*}_t$ for every $j$ (using $c^{i}_r\in\mathcal{S}^{*}_r\subseteq\mathcal{S}^{*}_t$). By claim 4 of Superposition Decomposition of the Controlled State and Observations, $u^{j}_r=u^{\alpha^{*},j}_r-\gamma^{j}_r$ almost surely for $r\le t$, and the right-hand side lies in $\mathcal{S}^{*}_t$; membership passes to almost surely equal variables, so $u^{j}_r\in\mathcal{S}^{*}_t$. Finally, the generators $1$ and $u^{\alpha^{*},j}_q$ of $\mathcal{S}^{*}_t$ are $\mathcal{G}^{\alpha^{*}}_t$-measurable and square-integrable, so by claim 2 of The Closed Mean-Square Span of a Family of Random Variables every member of $\mathcal{S}^{*}_t$ is almost surely equal to a $\mathcal{G}^{\alpha^{*}}_t$-measurable random variable. This proves all assertions of claim 3. $\square$