Convergence of the N-Agent Cost to the Mean-Field Cost under an Open-Loop Control

Adopt the setting and notation of the mean-square tracking proposition, and let $(L,G)$ be population cost data on $l$ states with control dimension $m$. Let $J^N[h^A]$ be the $N$-agent cost of the open-loop policy $h^A$ under $(L,G)$, and let $J^{MF}[(S),(A)]$ be the generalized mean-field cost of $(S,A)$ under $(L,G)$. Let $C$ be the bound and, for a given real $\varepsilon>0$, let $\delta>0$ be a number as in the boundedness and uniform continuity lemma for population cost data on a compact control set, applied with the control set $\mathcal{A}$.

1. (Quantitative bound.) $J^N[h^A]$ is a real number and

$$\big|J^N[h^A]-J^{MF}[(S),(A)]\big|\le(T+1)\Big(\varepsilon+\frac{2C}{\delta^2}\,\mathbb{E}\big[\Psi^2\big]\Big) .$$

2. (Convergence.) Suppose that for every natural number $N$ there are given an $N$-agent driving system and a solution as above, all with the same $(\beta_0,\beta_1)$, $\mathcal{A}$, $\tilde{\beta}$, $T$, $A$ and $S_0$, with empirical state measures $\Sigma^N$ and costs $J^N[h^A]$, and that $\mathbb{E}\big[|\Sigma^N_0-S_0|^2\big]$ converges to $0$ as $N$ increases. Then $J^N[h^A]$ converges to $J^{MF}[(S),(A)]$.