Quadratic Expansion Bounds for the Mean-Field To-Go Value along a Stationary Mean-Field Triple

Adopt the setting, hypotheses and notation of the time-shift lemma for the mean-field control problem: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states — whose control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ is nonempty, compact and convex as part of its data — and its transition-rate family $\beta$, the horizon $T>0$, the population cost data $(L,G)$ with $L$ convex in the control on $\mathcal{A}$, the probability simplex $\Delta^{l}$, the value-layer objects of the optimal-solution-set definition together with their horizon-$\theta$ instances superscripted by $[\theta]$ — in particular the sets $\mathcal{U}_{\mathcal{A}}$ and $\mathcal{U}^{[\theta]}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, the mean-field costs $F$ and $F^{[\theta]}$, and the optimal values $J^{*}_{x}$ and $J^{*[\theta]}_{x}$ — the rate extension $(U,V,\bar{\beta})$ with derivative bound $K$, the cost extension $(U_{c},\bar{L},\bar{G})$ with second-derivative bound $K_{c}$, and the stationary mean-field triple $(S,A,P)$ with horizon $T$. The variable $t_{0}$, fixed in the time-shift lemma, is not fixed here: for each $t_{0}\in[0,T)$ we write $T^{\sharp}=T-t_{0}$ and $S^{\sharp}$, $A^{\sharp}$, $P^{\sharp}$ for the shifted maps of that lemma — all depending on $t_{0}$, left implicit in the notation — and invoke its conclusions for that $t_{0}$. Write $|\cdot|$ for the Euclidean norm and $\int_{[a,b]}\cdot\,dt$ for the Lebesgue integral over a compact interval.

Fix a real $C_{P}$ with $\sum_{\delta=1}^{l}|P^{\delta}_{t}|\le C_{P}$ for all $t\in[0,T]$, which exists by clause 1 of the co-state definition and the extreme value theorem, and adopt the constants $M_{2}=K_{c}+3\,l\,K\,C_{P}$ and $C_{2}=\tfrac{1}{2}(l+m)M_{2}$ of part (b) of the first-order expansion lemma for the mean-field cost and the constant $\Lambda_{b}$ of the affine rate-family lemma. Define

$$C_{up}=\Big(C_{2}\,T+\tfrac{l\,K_{c}}{2}\Big)\,e^{2\Lambda_{b}T},$$

with the exponential function.

Then the following hold.

1. (Upper support bound.) For every $t_{0}\in[0,T)$ and every $x\in\Delta^{l}$,

$$J^{*[T^{\sharp}]}_{x}\ \le\ \int_{[t_{0},T]}L(S_{t},A_{t})\,dt+G(S_{T})-\sum_{\gamma=1}^{l}P^{\gamma}_{t_{0}}\,\big(x^{\gamma}-S^{\gamma}_{t_{0}}\big)+C_{up}\,\big|x-S_{t_{0}}\big|^{2}.$$

2. (To-go comparison inequality.) Assume $[A]\in\mathcal{M}^{*}_{S_{0}}$, and assume:

(TG) (First-order regularity of the to-go value along the triple.) There are real numbers $\varepsilon_{tg}>0$ and $C_{tg}\ge0$ such that for every $t_{0}\in[0,T)$ and every $x\in\Delta^{l}$ with $|x-S_{t_{0}}|\le\varepsilon_{tg}$,

$$J^{*[T^{\sharp}]}_{x}\ \ge\ J^{*[T^{\sharp}]}_{S_{t_{0}}}-\sum_{\gamma=1}^{l}P^{\gamma}_{t_{0}}\,\big(x^{\gamma}-S^{\gamma}_{t_{0}}\big)-C_{tg}\,\big|x-S_{t_{0}}\big|^{2}.$$

Then for every $t_{0}\in[0,T)$, every $x\in\Delta^{l}$ with $|x-S_{t_{0}}|\le\varepsilon_{tg}$, and every $\xi\in\mathcal{U}^{[T^{\sharp}]}_{\mathcal{A}}$,

$$F^{[T^{\sharp}]}(x,\xi)\ \ge\ \int_{[t_{0},T]}L(S_{t},A_{t})\,dt+G(S_{T})-\sum_{\gamma=1}^{l}P^{\gamma}_{t_{0}}\,\big(x^{\gamma}-S^{\gamma}_{t_{0}}\big)-C_{tg}\,\big|x-S_{t_{0}}\big|^{2}.$$