Stationary Mean-Field Triple

Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$. Let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$, let $(U,V,\bar{\beta})$ be a twice continuously differentiable extension of $\beta$ with derivative bound $K$, let $\bar{b}$ be the extended aggregate state drift of $(U,V,\bar{\beta})$, let $(L,G)$ be population cost data on $l$ states with control dimension $m$, let $(W,\bar{L},\bar{G})$ be a twice continuously differentiable extension of $(L,G)$ with second-derivative bound $K_c$, let $T>0$ be a real number, and let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$. Adopt the coordinate and partial-derivative notation $\partial_i$ of the extension definitions, so that $\partial_\gamma$ with $\gamma\le l$ differentiates in the $\gamma$-th state coordinate and $\partial_{l+j}$ with $j\le m$ in the $j$-th control coordinate; the first-order partial derivatives of $\bar{b}$ used below exist and are continuous on $U\times V$ by part (i) of the regularity of the extended aggregate state drift, and those of $\bar{L}$ and $\bar{G}$ exist and are continuous by clause 2 of the cost extension definition together with clauses 1 and 2 of the $C^k$ definition. Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line.

A stationary co-state for these data is a function $P:[0,T]\to\mathbb{R}^l$, with values $P_t=(P^1_t,\dots,P^l_t)$, such that:

1. (Continuity.) Each component $t\mapsto P^\gamma_t$ ($\gamma\in\{1,\dots,l\}$) is continuous on $[0,T]$.

2. (Co-state equation.) For every $\gamma\in\{1,\dots,l\}$ the map $s\mapsto\sum_{\delta=1}^{l}\partial_\gamma\bar{b}^\delta(S_s,A_s)\,P^\delta_s-\partial_\gamma\bar{L}(S_s,A_s)$ is continuous on $[0,T]$, and for every $t\in[0,T]$

$$P^\gamma_t=-\partial_\gamma\bar{G}(S_T)+\int_t^T\Big(\sum_{\delta=1}^{l}\partial_\gamma\bar{b}^\delta(S_s,A_s)\,P^\delta_s-\partial_\gamma\bar{L}(S_s,A_s)\Big)ds,$$

where for $t<T$ the integral is the Riemann integral of the restriction of the integrand to $[t,T]$, this restriction being continuous on $[t,T]$ by claim 1 of restriction stability and hence Riemann integrable by claim 3 of the integral toolkit on a compact interval, and the integral is $0$ for $t=T$.

3. (Stationarity.) For every $t\in[0,T]$ and every $j\in\{1,\dots,m\}$:

$$\partial_{l+j}\bar{L}(S_t,A_t)=\sum_{\delta=1}^{l}\partial_{l+j}\bar{b}^\delta(S_t,A_t)\,P^\delta_t.$$

When $P$ is a stationary co-state for these data, the triple $(S,A,P)$ is called a stationary mean-field triple for $\beta$, $(L,G)$, and the chosen extensions.