Time Shift of the Mean-Field Control Problem: Restriction of a Stationary Triple, Cost Splitting, and the Dynamic Programming Principle

Adopt the setting, hypotheses and notation of the definition of the optimal value, controls and trajectories of the mean-field problem — in particular: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^{m}$, its transition-rate family $\beta$ with rate bound $B$, aggregate state drift $b$ and projected drift $\hat{b}$, 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 set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, the mean-field flow $S(z_{0},\xi)$ of claim 2 of the flow stability lemma (the two-argument flow map, distinguished throughout from the trajectory $S$ of the stationary triple below by the presence of its arguments), the mean-field cost $F$, and, for $x\in\Delta^{l}$, the optimal value $J^{*}_{x}$, the set $\mathcal{M}^{*}_{x}$ of optimal controls, and the value set written $V_{x}$ in that definition, renamed $\mathcal{V}_{x}$ here because the letter $V$ denotes the open control neighbourhood of the rate extension below. Write $|\cdot|$ for the Euclidean norm and $\int_{[a,b]}\cdot\,dt$ for the Lebesgue integral over a compact interval, taken to be $0$ when $a=b$. All of the objects above are determined by the data $(\beta_{0},\beta_{1})$ and $(L,G)$ together with the horizon, and the hypotheses just listed do not involve the horizon; for a real number $\theta>0$ we superscript by $[\theta]$ the corresponding objects of the instance with horizon $\theta$ and the same data — $\mathcal{U}^{[\theta]}_{\mathcal{A}}$, the flow $S^{[\theta]}(z_{0},\xi)$, the cost $F^{[\theta]}$, the value set $\mathcal{V}^{[\theta]}_{x}$, the optimal value $J^{*[\theta]}_{x}$ and the optimal control set $\mathcal{M}^{*[\theta]}_{x}$ — so that the unsuperscripted objects are those of the instance with horizon $T$. (The horizon-dependent auxiliary choices of the attainment theorem — a dense sequence in $L^{2}([0,\theta];\mathbb{R}^{m})$ and the associated metric on $\mathcal{U}^{[\theta]}_{\mathcal{A}}$ — exist for every horizon by the provisions recorded there, and none of the objects just named depends on them.)

Let further $(U,V,\bar{\beta})$ be a twice continuously differentiable extension of $\beta$ with derivative bound $K$ and extended aggregate state drift $\bar{b}$, let $(U_{c},\bar{L},\bar{G})$ be a twice continuously differentiable extension of $(L,G)$ with second-derivative bound $K_{c}$ (its open set, written $W$ in that definition, is written $U_{c}$ here), and let $(S,A,P)$ be a stationary mean-field triple for $\beta$, $(L,G)$ and these extensions, with horizon $T$; in particular $(S,A)$ is a mean-field trajectory pair and $P$ a stationary co-state. Adopt the mean-field Hamiltonian along the triple, $\mathcal{H}_{t}(a)=\bar{L}(S_{t},a)-\sum_{\delta=1}^{l}P^{\delta}_{t}\,\bar{b}^{\delta}(S_{t},a)$ for $t\in[0,T]$ and $a\in V$, of the quadratic growth lemma, and the fluctuation Hessian coefficients $H_{ij}(t)$ ($i,j\in\{1,\dots,l+m\}$) of these data.

Fix $t_{0}\in[0,T)$, put $T^{\sharp}=T-t_{0}\in(0,T]$, and define $S^{\sharp}:[0,T^{\sharp}]\to\Delta^{l}$, $A^{\sharp}:[0,T^{\sharp}]\to\mathcal{A}$ and $P^{\sharp}:[0,T^{\sharp}]\to\mathbb{R}^{l}$ by

The shifted data are the same $\beta$, $(L,G)$ and extensions with the horizon $T^{\sharp}$ in place of $T$.

Then the following hold.

1. (The restriction is a stationary triple.) $(S^{\sharp},A^{\sharp})$ is a mean-field trajectory pair for $\beta$ with horizon $T^{\sharp}$, and $P^{\sharp}$ is a stationary co-state for the shifted data, so that $(S^{\sharp},A^{\sharp},P^{\sharp})$ is a stationary mean-field triple for the shifted data. Every real $C_{P}$ with $\sum_{\delta=1}^{l}|P^{\delta}_{t}|\le C_{P}$ for all $t\in[0,T]$ satisfies $\sum_{\delta=1}^{l}|P^{\sharp\delta}_{t}|\le C_{P}$ for all $t\in[0,T^{\sharp}]$. Moreover, writing $\mathcal{H}^{\sharp}$ for the mean-field Hamiltonian along the shifted triple and $H^{\sharp}_{ij}$ for the fluctuation Hessian coefficients of the shifted data, for every $t\in[0,T^{\sharp}]$:

2. (Hypothesis transport.) If hypothesis (H1) of the quadratic growth lemma holds for the original data with a constant $r>0$, then it holds for the shifted data with the same $r$. If hypothesis (U) of that lemma holds for the original triple, then it holds for the shifted triple. And every real $r_{0}>0$ with $\mathcal{H}_{t}(a)-\mathcal{H}_{t}(A_{t})\ge r_{0}\,|a-A_{t}|^{2}$ for all $t\in[0,T]$ and $a\in\mathcal{A}$ satisfies $\mathcal{H}^{\sharp}_{t}(a)-\mathcal{H}^{\sharp}_{t}(A^{\sharp}_{t})\ge r_{0}\,|a-A^{\sharp}_{t}|^{2}$ for all $t\in[0,T^{\sharp}]$ and $a\in\mathcal{A}$.

3. (Flow identification and cost splitting.) The map $A$ is an admissible representative (in the sense of claim 2 of the flow stability lemma) of its class $[A]\in\mathcal{U}_{\mathcal{A}}$, and likewise $A^{\sharp}$ of $[A^{\sharp}]\in\mathcal{U}^{[T^{\sharp}]}_{\mathcal{A}}$; moreover $S=S(S_{0},[A])$ and $S^{\sharp}=S^{[T^{\sharp}]}(S_{t_{0}},[A^{\sharp}])$. The costs satisfy

where $J^{MF}$ denotes the mean-field cost of a mean-field trajectory pair, and

4. (Concatenation.) Let $\eta\in\mathcal{U}^{[T^{\sharp}]}_{\mathcal{A}}$ and let $v$ be an admissible representative of $\eta$. Define $\zeta:[0,T]\to\mathcal{A}$ by $\zeta_{s}=A_{s}$ for $s\in[0,t_{0})$ and $\zeta_{s}=v_{s-t_{0}}$ for $s\in[t_{0},T]$. Then $\zeta$ is an admissible representative of its class $[\zeta]\in\mathcal{U}_{\mathcal{A}}$; if $v'$ is another admissible representative of $\eta$ and $\zeta'$ the corresponding concatenation, then $[\zeta']=[\zeta]$; and

5. (Dynamic programming principle.) Assume $[A]\in\mathcal{M}^{*}_{S_{0}}$. Then $[A^{\sharp}]\in\mathcal{M}^{*[T^{\sharp}]}_{S_{t_{0}}}$ and