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 $\beta$ on $l$ states with control set $\mathcal{A}$, a nonempty subset of Euclidean space $\mathbb{R}^m$, and rate bound $B$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, 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 $(U,V,\bar{\beta})$ with derivative bound $K$, let $\bar{b}$ be the extended aggregate state drift of $(U,V,\bar{\beta})$, and let $b$ be the aggregate state drift of $\beta$, with which $\bar{b}$ agrees on $\Delta^l\times\mathcal{A}$ 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 convex.

(A2) (Admissible values.) $A_t\in\mathcal{A}$ for every $t\in[0,T]$, and $\alpha_t(\omega)\in\mathcal{A}$ at every point $(\omega,t)\in\Omega_0\times[0,T]$ (automatic in the present setting, the policy being $\mathcal{A}$-valued and the trajectory pair taking values in $\Delta^l\times\mathcal{A}$; recorded for direct use in the proof).

(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

$$E^{\delta\gamma}_s=\partial_\gamma\bar{b}^\delta(S_s,A_s),\qquad \mathsf{B}^{\delta j}_s=\partial_{l+j}\bar{b}^\delta(S_s,A_s)\qquad(\gamma,\delta\in\{1,\dots,l\},\ j\in\{1,\dots,m\}),$$

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

$$e_s=g_s-E_s\mathfrak{s}_s-\mathsf{B}_s\mathfrak{a}_s ,$$

with the matrix-vector product and componentwise differences. Set

$$c_a=3\,l^{3}\,\sqrt{m}\,K .$$

Then:

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

$$b(S_s,a)-b(S_s,A_s)=\mathsf{B}_s\,(a-A_s).$$

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

$$|e_s|\ \le\ c_a\,N^{-1/2}\,\big(|\mathfrak{s}_s|^2+|\mathfrak{s}_s|\,|\mathfrak{a}_s|\big).$$