Cost Limit Along the Approximate Kalman Policy

Adopt the setting, hypotheses \textbf{(H1)}--\textbf{(H4)}, and notation of the \reftext{lem:approximate-kalman-policy-2026a}{approximate Kalman filter and policy lemma}, for the \reftext{def:fluctuation-lqg-data-2026a}{fluctuation LQG data} of the \reftext{def:stationary-mean-field-triple-2026a}{stationary mean-field triple} $(S,A,P)$: in particular the \reftext{def:c2-population-cost-extension-2026b}{cost extension} of the \reftext{def:population-cost-data-2026a}{population cost data} $(L,G)$ is part of the data, the family $Z=(Z_t)_{t\in[0,T]}$ is the Riccati family of (H2), $W_t=Z_t\mathcal{B}_t+\tfrac12V_t$, $R_t$ is symmetric positive definite with continuous inverse (conclusion 1), $\Theta^\star_t$ is the state noise covariance of the LQG data, $\Pi_0$ is the matrix of (H4), and $\Pi=(\Pi_t)_{t\in[0,T]}$ is the filter covariance of conclusion 2. For each natural number $N\ge1$, let $h^N$ be the approximate Kalman policy at level $N$, fix an \reftext{def:n-agent-driving-system-2026a}{$N$-agent driving system} and a projected \reftext{def:n-agent-controlled-dynamics-2026a}{solution} as in conclusion 4 of the policy lemma --- a solution of the controlled $N$-agent dynamics for $\beta$, $\tilde{\beta}$, and $h^N$ --- with empirical state measure $\Sigma^N_t$, approximate Kalman control $\alpha^N_t$, approximate Kalman filter $\hat{\mathfrak{s}}^N_t$, \reftext{def:n-agent-fluctuation-processes-2026a}{fluctuation processes} $\mathfrak{s}^N_t=\sqrt{N}(\Sigma^N_t-S_t)$ and $\mathfrak{a}^N_t=\sqrt{N}(\alpha^N_t-A_t)$, and \reftext{lem:kalman-filter-error-covariance-2026a}{filter error} $\varepsilon^N_t=\mathfrak{s}^N_t-\hat{\mathfrak{s}}^N_t$ with filter error covariance $\Pi^{N}_t$. Let $J^N[h^N]$ be the \reftext{def:n-agent-cost-2026a}{$N$-agent cost} of this solution, let $J^{MF}=J^{MF}[(S),(A)]$ be the \reftext{def:mean-field-cost-2026a}{mean-field cost}, and set $\zeta_N=N(\mathbb{E}[\Sigma^N_0]-S_0)\in\mathbb{R}^l$ with the componentwise \reftext{def:expectation-variance-2026a}{expectation}, as in the \reftext{thm:n-agent-cost-expansion-2026b}{second-order cost expansion}. For real matrices $A'$ and $B'$ with $l$ rows and $l$ columns write

and adopt the entry notation $x\cdot My$, $(MM')^{pq}$, $(M^{\top})^{qp}$ of the \reftext{thm:fluctuation-control-coercivity-2026b}{completion-of-squares theorem}. Assume the two initial-condition hypotheses:

\textbf{(I1)} for all $\gamma,\delta\in\{1,\dots,l\}$, $\ \mathbb{E}\big[\mathfrak{s}^{N,\gamma}_0\mathfrak{s}^{N,\delta}_0\big]\to\Pi^{\gamma\delta}_0$ as $N\to\infty$;

\textbf{(I2)} $\ \sup\big\{\mathbb{E}\big[|\mathfrak{s}^N_0|^4\big]:N\ge1\big\}<\infty$.

Then the map $t\mapsto Z_t\cdot\Theta^\star_t+(W_tR_t^{-1}W_t^{\top})\cdot\Pi_t$ is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on $[0,T]$, each $N(J^N[h^N]-J^{MF})+\sum_{\gamma=1}^{l}P^\gamma_0\zeta^\gamma_N$ is a well-defined real number, and

the integral being the \reftext{lem:interval-lebesgue-toolkit-2026a}{Lebesgue integral over the compact interval} $[0,T]$ of the continuous integrand (which \reftext{lem:riemann-lebesgue-integral-agree-2026a}{agrees} with its Riemann integral).