The Mean-Field Cost Functional

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$, let $\beta$ be a transition-rate family on $l$ states with control dimension $m$, let $(L,G)$ be population cost data on $l$ states with control dimension $m$, let $T>0$ be a real number, and let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$.

The mean-field cost of $(S,A)$ under $(L,G)$ is the real number

where the integral is the Riemann integral of $t\mapsto L(S_t,A_t)$ on $[0,T]$; this integrand is continuous by the continuity clause of population cost data applied along the continuous components of $(S,A)$, so the integral exists by Riemann integrability of continuous functions.