Continuous Mean-Field Trajectory Pairs are Generalized Mean-Field Trajectory Pairs

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 projected extension, let $(L,G)$ be population cost data on $l$ states with control dimension $m$, and let $T>0$ be a real number.

Let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$ such that $A_t\in\mathcal{A}$ for every $t\in[0,T]$.

Then $(S,A)$ is a generalized mean-field trajectory pair for $(\beta_0,\beta_1)$ with horizon $T$, and its generalized mean-field cost under $(L,G)$ equals its mean-field cost under $(L,G)$.