The Mean-Field Cost of a Control from an Initial State

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 $\Delta^{l}$ be the probability simplex, let $T>0$ be a real number, and let $(L,G)$ be population cost data on $l$ states with control dimension $m$. Adopt the notation of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$ in the case $d=m$, including the convention of claim 5 there by which an element of that space is denoted by the same symbol as a representative of it, and let $\mathcal{U}_{\mathcal{A}}$ be the set of $\mathcal{A}$-valued controls.

Let $x_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$, the representative denoted by $\xi$ being taken so that $\xi(t)\in\mathcal{A}$ for every $t\in[0,T]$; such a representative — an admissible representative of $\xi$ — exists by claim 2 of the flow stability lemma. Let $S(x_{0},\xi)$ be the mean-field flow of the same claim, with values $S_{t}(x_{0},\xi)$; it is the map furnished by claim 1 of the existence and uniqueness theorem applied to the initial value $x_{0}$ and the control $\xi$, so that $\bigl(S(x_{0},\xi),\xi\bigr)$ is a generalized mean-field trajectory pair for $(\beta_{0},\beta_{1})$ with horizon $T$ whose value at $t=0$ is $x_{0}$.

The mean-field cost of the control $\xi$ from the initial state $x_{0}$ under $(L,G)$ is the generalized mean-field cost of that pair,

The right-hand side is unchanged if $\xi$ is replaced by another admissible representative of the same element of $\mathcal{U}_{\mathcal{A}}$: the flow is the same for every such choice by claim 2 of the flow stability lemma; the integrand of the running-cost term is integrable for each such choice, as recorded in the definition of the generalized mean-field cost, and two such representatives agree outside a set of measure zero, so the corresponding integrals agree by claim 2 of the lemma on integrals over a null set.