Quadratic Growth of the Mean-Field Hamiltonian in the Control along a Stationary Mean-Field Triple

Let $l$, $m$, $\mathcal{A}$, $\beta$ with rate bound $B$, $(U,V,\bar{\beta})$ with derivative bound $K$, $\bar{b}$, $(L,G)$, $(U_c,\bar{L},\bar{G})$ with second-derivative bound $K_c$, $T>0$, and $(S,A)$ be as in the definition of a stationary mean-field triple (the open set of the cost extension, written $W$ in that definition, is written $U_c$ here, so that $\bar{L}$ is defined on $U_c\times\mathbb{R}^m$ and $\bar{G}$ on $U_c$), and let $P$ be a stationary co-state for these data. Adopt the coordinate and partial-derivative notation $\partial_i$, $\partial_j\partial_i$ of the extension definitions: the partial derivatives of orders one and two of each $\bar{b}^\delta$ exist and are continuous on $U\times V$ by part (i) of the regularity of the extended aggregate state drift, and those of $\bar{L}$ on $U_c\times\mathbb{R}^m$ by clause 2 of the cost extension definition together with clauses 1 and 2 of the $C^k$ definition. Let $H_{ij}(t)$, for $t\in[0,T]$ and $i,j\in\{1,\dots,l+m\}$, be the fluctuation Hessian coefficients of these data. Write $\mathbb{R}$ for the real numbers, $d$ for the Euclidean distance, $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin), and $\Delta^l$ for the probability simplex. 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. Fix a real number $C_P$ with $\sum_{\delta=1}^{l}|P^\delta_t|\le C_P$ for every $t\in[0,T]$; such a number exists because each component of $P$ is continuous on $[0,T]$ by clause 1 of the co-state definition and therefore attains a maximum and a minimum on $[0,T]$ by the extreme value theorem.

Define the mean-field Hamiltonian along the triple as the function on $[0,T]\times V$ given by

$$\mathcal{H}_t(a)=\bar{L}(S_t,a)-\sum_{\delta=1}^{l}P^\delta_t\,\bar{b}^\delta(S_t,a)\qquad(t\in[0,T],\ a\in V),$$

which is well defined because $S_t\in\Delta^l\subset U\cap U_c$ and $V\subseteq\mathbb{R}^m$; the script letter $\mathcal{H}$ is distinct from the Hessian coefficients $H_{ij}$. Define, for $t\in[0,T]$, the real $m\times m$ matrix $R_t$ with entries

$$R^{ij}_t=\tfrac{1}{4}\big(H_{l+i,l+j}(t)+H_{l+j,l+i}(t)\big)\qquad(i,j\in\{1,\dots,m\}),$$

and write $a\cdot R_ta=\sum_{i,j=1}^{m}R^{ij}_ta^ia^j$ for $a\in\mathbb{R}^m$; the displayed formula agrees with the definition of $R_t$ in the completion-of-squares theorem, and only that formula and the wording of its hypothesis (H1) are borrowed from that theorem, not its setting.

Hypotheses. (A) The control set $\mathcal{A}$ is convex and compact for the topology determined by the Euclidean distance. (H1) There is a real number $r>0$ with $a\cdot R_ta\ge r|a|^2$ for every $t\in[0,T]$ and every $a\in\mathbb{R}^m$ (hypothesis (H1) of the completion-of-squares theorem). (U) For every $t\in[0,T]$, the point $A_t$ is the unique minimizer of $\mathcal{H}_t$ on $\mathcal{A}$: $\mathcal{H}_t(a)>\mathcal{H}_t(A_t)$ for every $a\in\mathcal{A}$ with $a\neq A_t$.

Then:

(a) (Continuity.) Regarding $[0,T]\times V$ as a subset of Euclidean space $\mathbb{R}^{1+m}$, the map $(t,a)\mapsto\mathcal{H}_t(a)$ is continuous at every point of $[0,T]\times V$, and so is the map $(t,a)\mapsto\mathcal{H}_t(A_t)$; this uses none of the hypotheses (A), (H1), (U).

(b) (Stationarity and the Hessian in the control.) For every $t\in[0,T]$ and $j\in\{1,\dots,m\}$,

$$\partial_{l+j}\bar{L}(S_t,A_t)-\sum_{\delta=1}^{l}P^\delta_t\,\partial_{l+j}\bar{b}^\delta(S_t,A_t)=0,$$

and for every $h\in\mathbb{R}^m$,

$$\tfrac{1}{2}\sum_{i=1}^{m}\sum_{j=1}^{m}H_{l+i,l+j}(t)\,h^ih^j=h\cdot R_th .$$

This uses none of the hypotheses (A), (H1), (U).

(c) (Upper quadratic bound.) Assume that $\mathcal{A}$ is convex (only the convexity half of (A) is used). Put $M_2=K_c+3\,l\,K\,C_P$ and $C_{\mathcal{H}}=\tfrac{1}{2}(l+m)M_2$. Then for every $t\in[0,T]$ and every $a\in\mathcal{A}$,

$$\big|\mathcal{H}_t(a)-\mathcal{H}_t(A_t)\big|\le C_{\mathcal{H}}\,|a-A_t|^2 .$$

(d) (Quadratic growth.) Assume (A), (H1), and (U). Then there is a real number $r_0>0$ such that

$$\mathcal{H}_t(a)-\mathcal{H}_t(A_t)\ \ge\ r_0\,|a-A_t|^2\qquad\text{for every }t\in[0,T]\text{ and every }a\in\mathcal{A}.$$