The N-Agent Cost Functional

Adopt the setting of the controlled $N$-agent dynamics: a transition-rate family $\beta$ on $l$ states with control dimension $m$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, and an $N$-agent driving system $(\Omega,\mathcal{F},P)$. Let $(L,G)$ be population cost data on $l$ states with control dimension $m$, and let $h$ be an observation-driven control policy with horizon $T$, control dimension $m$, and $\tilde{l}$ channels.

Let $(\sigma^i,\Upsilon^\upsilon,\alpha)$ be a solution of the controlled $N$-agent dynamics on $[0,T]$ with empirical state measure $\Sigma_t$; by the existence and uniqueness theorem, a solution exists, any two solutions are indistinguishable, and the expectation below is well defined in $(-\infty,+\infty]$ and has the same value for every solution. The $N$-agent cost of the policy $h$ is

where the inner integral is the Lebesgue integral over the compact interval $[0,T]$.