The Approximate Kalman Filter and Policy for the Controlled N-Agent Dynamics

Adopt the setting and notation of the definition of the \reftext{def:fluctuation-lqg-data-2026a}{fluctuation LQG data} of a stationary mean-field triple: natural numbers $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the \reftext{def:transition-rate-family-2026a}{transition-rate family} $\beta$ with rate bound $B$ and its \reftext{def:c2-transition-rate-extension-2026a}{extension} $(U,\bar{\beta})$, the \reftext{def:c2-population-cost-extension-2026b}{cost extension} — whose open set we here write $U_c$, freeing the letter $V$ — the horizon $T>0$, the \reftext{def:stationary-mean-field-triple-2026a}{stationary mean-field triple} $(S,A,P)$, the \reftext{def:observation-rate-family-2026a}{observation-rate family} $\tilde{\beta}$ with $\tilde{l}$ channels and rate bound $\tilde{B}$ and its \reftext{def:c2-observation-rate-extension-2026a}{extension} $(\tilde{U},\bar{\tilde{\beta}})$, and the resulting matrices $\mathcal{E}_t$, $\mathcal{B}_t$, $\tilde{\mathcal{E}}_t$, $H^{SS}_t$, $H^{SA}_t$, $H^{AS}_t$, $H^{AA}_t$, $F^\star$, $\Theta^\star_t$, $\tilde{\Theta}^\star_t$ ($t\in[0,T]$). Let $\tilde{b}$ be the \reftext{def:aggregate-observation-drift-2026a}{aggregate observation drift} of $\tilde{\beta}$, write $\tilde{b}(\Sigma)$ for the vector with components $\tilde{b}^1(\Sigma),\dots,\tilde{b}^{\tilde{l}}(\Sigma)$, and write $\Delta^l$ for the \reftext{def:probability-simplex-2026a}{probability simplex}. Vectors of $\mathbb{R}^l$, $\mathbb{R}^m$, and $\mathbb{R}^{\tilde{l}}$ are identified with one-column matrices, products of matrices are \reftext{def:product-real-matrices-2026a}{matrix products}, sums of matrices are entrywise, $(\cdot)^{\top}$ is the \reftext{def:transpose-real-matrix-2026a}{transpose}, $I$ is the \reftext{def:identity-matrix-2026a}{identity matrix} of the indicated size, inverses of square matrices are \reftext{def:inverse-matrix-invertible-real-square-matrix-2026a}{matrix inverses}, $|\cdot|$ is the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin), and $e_\upsilon$ ($\upsilon\in\{1,\dots,\tilde{l}\}$) is the $\upsilon$-th standard basis vector of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^{\tilde{l}}$. All \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integrals} of matrix- or vector-valued maps below are entrywise integrals of continuous integrands, which exist by \reftext{lem:continuous-implies-riemann-integrable-c54-2026b}{continuity} and are $0$ over degenerate intervals. Define, for $t\in[0,T]$, the symmetrized coefficient matrices

conclusion 1 below records their entries in terms of the \reftext{def:fluctuation-lqg-cost-2026b}{fluctuation Hessian coefficients}, matching the defining formulas of the coefficient matrices of the \reftext{thm:fluctuation-control-coercivity-2026a}{completion-of-squares theorem}. Assume:

\textbf{(H1)} there is a real $r>0$ such that $\sum_{i=1}^{m}\sum_{j=1}^{m}R^{ij}_t\,a^i\,a^j\ge r\,|a|^2$ for every $t\in[0,T]$ and every $a\in\mathbb{R}^m$;

\textbf{(H2)} there is a family $Z=(Z_t)_{t\in[0,T]}$ of symmetric real matrices with $l$ rows and $l$ columns, continuously differentiable in integral form as in the \reftext{lem:fluctuation-weighted-second-moment-2026a}{weighted second-moment evolution lemma}, with terminal value $Z_T=\hat{F}$ and densities

— the backward Riccati equation of hypothesis (H2) of the \reftext{thm:fluctuation-control-coercivity-2026a}{completion-of-squares theorem}; here $R_t^{-1}$ exists by conclusion 1 below, whose proof uses only (H1);

\textbf{(H3)} there is a real $\tilde{r}>0$ such that $\tilde{b}^\upsilon(S_t)\ge\tilde{r}$ for every $\upsilon\in\{1,\dots,\tilde{l}\}$ and every $t\in[0,T]$;

\textbf{(H4)} $\Pi_0$ is a symmetric \reftext{def:positive-semidefinite-matrix-2026a}{positive semidefinite} real matrix with $l$ rows and $l$ columns.

Then:

\textbf{1. (Coefficients.)} All entries of the maps $t\mapsto\mathcal{E}_t$, $t\mapsto\mathcal{B}_t$, $t\mapsto\tilde{\mathcal{E}}_t$, $t\mapsto\Theta^\star_t$, $t\mapsto\tilde{\Theta}^\star_t$, $t\mapsto Q_t$, $t\mapsto V_t$, $t\mapsto R_t$, and $t\mapsto\tilde{b}(S_t)$ are \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$. With the \reftext{def:fluctuation-lqg-cost-2026b}{fluctuation Hessian coefficients} $H_{ij}(t)$ and $F_{\gamma\delta}$ of these data, for all indices:

— the defining formulas of the coefficient matrices of the \reftext{thm:fluctuation-control-coercivity-2026a}{completion-of-squares theorem}. Each $R_t$ is symmetric \reftext{def:positive-semidefinite-matrix-2026a}{positive definite}, hence \reftext{lem:pd-inverse-2026a}{invertible}; each $\tilde{\Theta}^\star_t$ is symmetric positive definite, and its inverse is the diagonal matrix whose diagonal entries are $1/\tilde{b}^1(S_t),\dots,1/\tilde{b}^{\tilde{l}}(S_t)$; and all entries of $t\mapsto R_t^{-1}$, of $t\mapsto(\tilde{\Theta}^\star_t)^{-1}$, and of the \textbf{feedback gain} $t\mapsto\mathcal{G}_t=R_t^{-1}W_t^{\top}$ ($m$ rows, $l$ columns) are continuous on $[0,T]$.

\textbf{2. (Filter covariance and gains.)} Each $\tilde{D}_t=\tilde{\mathcal{E}}_t^{\top}(\tilde{\Theta}^\star_t)^{-1}\tilde{\mathcal{E}}_t$ is symmetric positive semidefinite with entries continuous in $t$, and each $\Theta^\star_t$ is positive semidefinite by the \reftext{lem:fluctuation-covariance-psd-2026a}{jump representation of the aggregate fluctuation covariance}. Consequently, by the \reftext{thm:riccati-global-existence-2026a}{global existence and uniqueness theorem for the Kalman covariance Riccati equation}, there is exactly one assignment $\Pi$ of a real matrix $\Pi_t$ with $l$ rows and $l$ columns to each $t\in[0,T]$, with continuous entries, such that

called the \textbf{filter covariance} of these data, and every $\Pi_t$ is symmetric positive semidefinite. All entries of the \textbf{Kalman gain} $t\mapsto\tilde{\mathcal{K}}_t=\Pi_t\,\tilde{\mathcal{E}}_t^{\top}(\tilde{\Theta}^\star_t)^{-1}$ ($l$ rows, $\tilde{l}$ columns) and of the \textbf{closed-loop matrix} $t\mapsto M_t=\mathcal{E}_t-\mathcal{B}_t\,\mathcal{G}_t-\tilde{\mathcal{K}}_t\,\tilde{\mathcal{E}}_t$ ($l$ rows, $l$ columns) are continuous on $[0,T]$.

\textbf{3. (The approximate Kalman policy.)} Let $\Phi$ and $\Psi=\Phi^{-1}$ be the fundamental solution of $M$ on $[0,T]$ and its inverse from the \reftext{thm:fundamental-solution-linear-ode-2026a}{fundamental-solution theorem}. Fix a natural number $N\ge1$. For $k\ge1$ write $\mathcal{R}_k$ for the record space with horizon $T$ (denoted there with the letter $R$) of the definition of an \reftext{def:observation-driven-control-policy-2026a}{observation-driven control policy}, and define, for $t\in[0,T]$, $\tau=(\tau_1,\dots,\tau_k)\in\mathcal{R}_k$, and $v=(v_1,\dots,v_k)\in\{1,\dots,\tilde{l}\}^k$,

where $\mathbf{1}_{\{\tau_j\le t\}}$ equals $1$ if $\tau_j\le t$ and $0$ otherwise; define $f^N_0(t)$ by the same formula with the sum omitted, and

Then $h^N=(h^N_k)_{k\ge0}$ is an observation-driven control policy with horizon $T$, control dimension $m$, and $\tilde{l}$ channels, called the \textbf{approximate Kalman policy} of these data at level $N$; and $f^N=(f^N_k)_{k\ge0}$ is an observation-driven control policy with horizon $T$, control dimension $l$, and $\tilde{l}$ channels. Moreover the family $\beta^{\sharp}$ of functions on $\Delta^l\times\mathbb{R}^{m+l}$ defined by $\beta^{\sharp}(\sigma,\gamma,\Sigma,(a,x))=\beta(\sigma,\gamma,\Sigma,a)$ for $a\in\mathbb{R}^m$ and $x\in\mathbb{R}^l$ (points of $\mathbb{R}^{m+l}$ being split as pairs) is a transition-rate family on $l$ states with control dimension $m+l$ and rate bound $B$, and the family $h^{\sharp}=(h^{\sharp}_k)_{k\ge0}$ with $h^{\sharp}_k(t,\tau,v)=\big(h^N_k(t,\tau,v),f^N_k(t,\tau,v)\big)\in\mathbb{R}^{m+l}$ is an observation-driven control policy with horizon $T$, control dimension $m+l$, and $\tilde{l}$ channels.

\textbf{4. (Realization along the controlled dynamics.)} Let $(\Omega,\mathcal{F},\mathbb{P})$ be an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} with $l$ states and $\tilde{l}$ observation channels (its probability measure written $\mathbb{P}$, the letter $P$ denoting the stationary co-state), and let state processes $\sigma^i$, observation processes $\Upsilon^\upsilon$, a control process $\alpha^{\sharp}$ with values in $\mathbb{R}^{m+l}$, and a regular event $\Omega_0$ be a \reftext{def:n-agent-controlled-dynamics-2026a}{solution of the controlled $N$-agent dynamics} on $[0,T]$ for $\beta^{\sharp}$, $\tilde{\beta}$, this driving system, and the policy $h^{\sharp}$; such solutions exist by the \reftext{thm:n-agent-dynamics-existence-2026a}{existence and uniqueness theorem for the controlled $N$-agent dynamics}. Write $\alpha_t$ for the vector of the first $m$ components of $\alpha^{\sharp}_t$, called the \textbf{approximate Kalman control}, and $\hat{\mathfrak{s}}^N_t$ for the vector of the last $l$ components of $\alpha^{\sharp}_t$, called the \textbf{approximate Kalman filter}; and at each $\omega\in\Omega_0$ let $\tilde{c}_t$ be the observation total and $\tau_1<\dots<\tau_{\tilde{c}_T}$ and $\upsilon_1,\dots,\upsilon_{\tilde{c}_T}$ the jump times of the observation total and their channels, as in condition 5 of the definition of a solution (where $\tilde{c}_t$ is written $K_t$). Then:

\textbf{(a) (Projection and policy identity.)} The state processes $\sigma^i$, the observation processes $\Upsilon^\upsilon$, the control process $\alpha$, and the regular event $\Omega_0$ are a solution of the controlled $N$-agent dynamics on $[0,T]$ for $\beta$, $\tilde{\beta}$, the same driving system, and the policy $h^N$; and conversely, if a collection of state processes, observation processes, an $\mathbb{R}^m$-valued control process, and a regular event is a solution for $\beta$, $\tilde{\beta}$, this driving system, and $h^N$, then it is indistinguishable from the projected solution above in the sense of the \reftext{thm:n-agent-dynamics-existence-2026a}{uniqueness theorem}. At every $\omega\in\Omega_0$ and every $t\in[0,T]$:

and

\textbf{(b) (Filter equation.)} At every $\omega\in\Omega_0$ and every $t\in[0,T]$, componentwise:

where for $t>0$ the integral is the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral over the compact interval} $[0,t]$ of a bounded measurable integrand whose components are continuous at every $r$ other than the finitely many jump times, the integral is $0$ for $t=0$ (so that the identity then reads $\hat{\mathfrak{s}}^N_0=0$), and the sum is $0$ when $\tilde{c}_t=0$.

\textbf{(c) (Path regularity, measurability, and bounds.)} At every $\omega\in\Omega_0$, every component path $t\mapsto\hat{\mathfrak{s}}^{N,\gamma}_t$ ($\gamma\in\{1,\dots,l\}$) is right-continuous on $[0,T]$ and continuous at every $t$ that is not one of $\tau_1,\dots,\tau_{\tilde{c}_T}$. Each map $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)\,\hat{\mathfrak{s}}^{N,\gamma}_t(\omega)$ is \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$-algebra} of the \reftext{lem:interval-lebesgue-toolkit-2026a}{trace Borel $\sigma$-algebra} on $[0,T]$ and $\mathcal{F}$, where $\mathbf{1}_{\Omega_0}$ equals $1$ on $\Omega_0$ and $0$ off $\Omega_0$. Moreover there is a real $C^\circ\ge0$, determined by $l$, $\tilde{l}$, $m$, $T$, $\tilde{B}$, and the entry bounds of $\Phi$, $\Psi$, $\tilde{\mathcal{K}}$, and $\mathcal{G}$ on $[0,T]$ — in particular the same for every $N$, every driving system, and every solution — such that at every $\omega\in\Omega_0$ and every $t\in[0,T]$: