Mean-Square Tracking of the Mean-Field Trajectory under an Open-Loop Control

Let $(\beta_0,\beta_1)$ be an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^m$, let $\beta$ be its transition-rate family, with rate bound $B$, aggregate state drift $b$ and state-Lipschitz constant $\Lambda_b$, and let $\Delta^l$ be the probability simplex, with $|\cdot|$ the Euclidean norm on $\mathbb{R}^l$. Let $\tilde{\beta}$ be an observation-rate family on $l$ states with $\tilde{l}$ channels, let $T>0$ be a real number, let $N$ be a natural number, and let $(\Omega,\mathcal{F},P)$ be an $N$-agent driving system.

Let $A:[0,T]\to\mathcal{A}$ be a map whose components are measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$, let $h^A$ be the open-loop policy determined by $A$, and let $(\sigma^i,\Upsilon^\upsilon,\alpha)$ be a solution of the controlled $N$-agent dynamics on $[0,T]$ for $\beta$, $\tilde{\beta}$, this driving system and the policy $h^A$, with empirical state measure $\Sigma$. Let $S_0\in\Delta^l$ and let $(S,A)$ be the generalized mean-field trajectory pair with value $S_0$ at $t=0$ provided by the existence and uniqueness theorem.

Let $D$ be the set of dyadic partition points of $[0,T]$ and let $\Omega_*$ be any event with the properties listed in the martingale bound for the empirical state measure, with $\mathbf{1}_{\Omega_*}$ the function equal to $1$ on $\Omega_*$ and $0$ elsewhere. Then

$$\Psi=\sup_{t\in D}\big(|\Sigma_t-S_t|\,\mathbf{1}_{\Omega_*}\big)$$

is a random variable with $0\le\Psi\le2$, it satisfies $\Psi(\omega)=\sup_{t\in[0,T]}|\Sigma_t(\omega)-S_t|$ for every $\omega\in\Omega_*$, and

$$\mathbb{E}\big[\Psi^2\big]\le2\,e^{2\Lambda_bT}\Big(\mathbb{E}\big[|\Sigma_0-S_0|^2\big]+\frac{8\,l\,(l-1)\,B\,T}{N}\Big) .$$