The Generalized Mean-Field Cost Functional

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 $(L,G)$ be population cost data on $l$ states with control dimension $m$, let $T>0$ be a real number, and let $(S,A)$ be a generalized mean-field trajectory pair for $(\beta_0,\beta_1)$ with horizon $T$.

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

$$J^{MF}[(S),(A)]=\int_{[0,T]}L(S_t,A_t)\,dt+G(S_T),$$

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

The integrand is measurable and bounded, so the integral exists and is a real number; this follows from measurability of sequentially continuous functions of measurable Euclidean maps and from boundedness of population cost data on a compact control set.