Sharp Drift Linearization Error under a Control-Affine Extension

Adopt the setting of the fluctuation processes of the controlled $N$-agent dynamics: a transition-rate family $\beta$ with rate bound $B$ on $l$ states with control dimension $m$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$, a solution on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, and control $\alpha_t$, a mean-field trajectory pair $(S,A)$ for $\beta$ with horizon $T$, and the fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Assume that $\beta$ admits a twice continuously differentiable extension $(U,\bar{\beta})$ with derivative bound $K$, let $\bar{b}$ be the extended aggregate state drift of $(U,\bar{\beta})$, and let $b$ be the aggregate state drift of $\beta$, with which $\bar{b}$ agrees on $\Delta^l\times\mathbb{R}^m$ by clause (i) of the regularity of the extended aggregate state drift. Adopt the coordinate and partial-derivative notation $\partial_i$, $\partial_j\partial_i$ of the extension definition, write $\Delta^l$ for the probability simplex and $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin).

Assume moreover:

(A1) (Control set.) $\mathcal{A}$ is a nonempty convex subset of $\mathbb{R}^m$.

(A2) (Admissible values.) $A_t\in\mathcal{A}$ for every $t\in[0,T]$, and $\alpha_t(\omega)\in\mathcal{A}$ at every point $(\omega,t)\in\Omega_0\times[0,T]$.

(A3) (Affine-control extension.) $\partial_{l+k}\partial_{l+j}\bar{\beta}(\sigma,\gamma,x)=0$ for all $j,k\in\{1,\dots,m\}$, every ordered pair $(\sigma,\gamma)$ with $\sigma\neq\gamma$ in $\{1,\dots,l\}$, and every $x\in\Delta^l\times\mathcal{A}$.

Define, for $s\in[0,T]$, the real matrices $E_s$ ($l\times l$) and $\mathsf{B}_s$ ($l\times m$) by

the sans-serif $\mathsf{B}_s$ being distinct from the rate bound $B$, together with the vector $g_s=\sqrt{N}\,(b(\Sigma_s,\alpha_s)-b(S_s,A_s))\in\mathbb{R}^l$ and the residual

with the matrix-vector product and componentwise differences. (When the setting of the completion-of-squares theorem for the fluctuation cost is in force, these are its matrices $E_s$, $\mathsf{B}_s$ and residual $e_s$, by the identical defining formulas, provided the extension fixed there is the present $(U,\bar{\beta})$.) Set

Then:

1. (Exact control linearity.) For every $s\in[0,T]$ and every $a\in\mathcal{A}$, componentwise,

2. (Sharp residual bound.) At every point of $\Omega_0\times[0,T]$,