Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls

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 $\beta$ be its projected extension, and adopt from that lemma the real numbers $R$, $K_1$, $B$ and $\Lambda_b$, the aggregate state drift $b$, the projected drift $\hat{b}$, and the coefficient maps $b^\gamma_0:\Delta^l\to\mathbb{R}$ and $b^\gamma_1:\Delta^l\to\mathbb{R}^m$ of claim 3 there, where $\Delta^l$ is the probability simplex. Let $T>0$ be a real number and write $K_b=2\sqrt{l}\,(l-1)B$.

Adopt the notation of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$ in the case $d=m$: write $H=L^{2}([0,T];\mathbb{R}^{m})$ with its pairing $\langle\cdot,\cdot\rangle_{L^{2}}$, and let $\mathcal{U}_{\mathcal{A}}\subseteq H$ be the set of $\mathcal{A}$-valued controls. Write $\lambda=\lambda_{[0,T]}$ and $\mathcal{B}=\mathcal{B}_{[0,T]}$ for the restricted Lebesgue measure and its $\sigma$-algebra on $[0,T]$. Write $|\cdot|$ for the Euclidean norm and $x\cdot y$ for the dot product. For $t\in[0,T]$ let $\mathbf{1}_{[0,t]}$ be the function on $[0,T]$ equal to $1$ on $[0,t]$ and to $0$ elsewhere. Set

Then the following hold.

1. (Bounds on the affine coefficient.) For every $\gamma\in\{1,\dots,l\}$ and every $\Sigma\in\Delta^l$ we have $|b^\gamma_1(\Sigma)|\le C_1$, and

2. (Admissible representatives and the flow.) Every $\xi\in\mathcal{U}_{\mathcal{A}}$ has a representative $u$ with $u(t)\in\mathcal{A}$ for every $t\in[0,T]$; such a $u$ is called an admissible representative of $\xi$. Let $x_0\in\Delta^l$ and $\xi\in\mathcal{U}_{\mathcal{A}}$. For an admissible representative $u$ of $\xi$, let $S^{u}$ be the map furnished by claim 1 of the existence and uniqueness theorem applied to the initial value $x_0$ and the control $u$. Then $S^{u}$ does not depend on the choice of admissible representative $u$ of $\xi$; write $S(x_0,\xi)$ for it, with values $S_t(x_0,\xi)$. It satisfies $S_t(x_0,\xi)\in\Delta^l$ for every $t$, $S_0(x_0,\xi)=x_0$, and $|S_t(x_0,\xi)-S_r(x_0,\xi)|\le K_b|t-r|$ for all $r,t\in[0,T]$.

3. (The control-perturbation functionals.) Let $x_0\in\Delta^l$ and $\xi\in\mathcal{U}_{\mathcal{A}}$, and write $S=S(x_0,\xi)$. For $\gamma\in\{1,\dots,l\}$ and $t\in[0,T]$, the map $[0,T]\to\mathbb{R}^m$ sending $s$ to $\mathbf{1}_{[0,t]}(s)\,b^\gamma_1(S_s)$ is square-integrable; let $w^{\gamma,t}\in H$ denote its class. For $\zeta\in\mathcal{U}_{\mathcal{A}}$ put

4. (Quantitative stability.) In the setting of claim 3, let $x_0'\in\Delta^l$ and $\xi'\in\mathcal{U}_{\mathcal{A}}$ and write $S'=S(x_0',\xi')$. Let $G$ be a real number with $|g^\gamma_r(\xi')|\le G$ for every $\gamma\in\{1,\dots,l\}$ and every $r\in[0,T]$. Then

the square root being the nonnegative square root.

5. (Equicontinuity in time.) In the setting of claim 3, for every $\zeta\in\mathcal{U}_{\mathcal{A}}$, every $\gamma\in\{1,\dots,l\}$ and all $r,t\in[0,T]$,

The constant $L_1$ depends only on the rate family and $\mathcal{A}$, not on $\zeta$, on $\xi$ or on $x_0$.

6. (Sequential stability.) Let $(x^j_0)_{j\in\mathbb{N}}$ be a sequence in $\Delta^l$ and let $x_0\in\Delta^l$ be such that the real sequence $(|x^j_0-x_0|)_{j\in\mathbb{N}}$ has limit $0$. Let $(\xi_j)_{j\in\mathbb{N}}$ be a sequence in $\mathcal{U}_{\mathcal{A}}$ and let $\xi\in\mathcal{U}_{\mathcal{A}}$ be such that $\xi_j\rightharpoonup\xi$ in the sense of weak convergence. Then, writing $S^j=S(x^j_0,\xi_j)$ and $S=S(x_0,\xi)$, for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ such that