Boundedness, Lower Semicontinuity and Attainment of the Mean-Field Cost

Adopt the setting and notation of the mean-field cost of a control from an initial state: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^{m}$, the horizon $T>0$, the population cost data $(L,G)$, the probability simplex $\Delta^{l}$, the set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, the mean-field flow $S(x_{0},\xi)$ of claim 2 of the flow stability lemma, and the mean-field cost $F(x_{0},\xi)$. Write $|\cdot|$ for the Euclidean norm on $\mathbb{R}^{k}$, for any natural number $k\ge1$, and for the absolute value on $\mathbb{R}$.

As part of the data of the rate family, $\mathcal{A}$ is a nonempty compact convex subset of $\mathbb{R}^{m}$. Assume in addition that $L$ is convex in the control on $\mathcal{A}$, that is, that for every $\Sigma\in\Delta^{l}$ the function $\mathcal{A}\to\mathbb{R}$ sending $a$ to $L(\Sigma,a)$ is convex on $\mathcal{A}$; these are the standing hypotheses on $\mathcal{A}$ and on $L$ of the running-cost lower-semicontinuity lemma.

Write $d_{\mathbb{R}^{l}}$ for the Euclidean distance on $\mathbb{R}^{l}$, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, and let $d_{\Delta}$ be its restriction to $\Delta^{l}$, a metric by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology. Fix a sequence $(w_{r})_{r\in\mathbb{N}}$ in $L^{2}([0,T];\mathbb{R}^{m})$ whose set of terms is dense in that space for the metric $d_{L^{2}}$ of the Lebesgue space — such a sequence exists because Separability of the Lebesgue Space of Square-Integrable Vector-Valued Functions provides a countable dense subset, and a nonempty countable set is the set of terms of a sequence — and let $\rho$ be the associated metric on $\mathcal{U}_{\mathcal{A}}$ furnished by claim 1 of the weak metrizability and compactness theorem. Let $X=\Delta^{l}\times\mathcal{U}_{\mathcal{A}}$ be the Cartesian product and let $d_{X}$ be the product metric on $X$, a metric by claim 1 of The Product Metric is a Metric. Regard $F$ as a function $X\to\mathbb{R}$.

Then the following hold.

1. (Boundedness.) There is a real number $C_{F}\ge0$, depending only on $L$, $G$, $\Delta^{l}$, $\mathcal{A}$ and $T$, such that $|F(x_{0},\xi)|\le C_{F}$ for every $x_{0}\in\Delta^{l}$ and every $\xi\in\mathcal{U}_{\mathcal{A}}$.

2. (Representation by the running-cost integral.) Let $x_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$, and write $S=S(x_{0},\xi)$. Then $S$ is an admissible state path in the sense of the running-cost lower-semicontinuity lemma, and

$$F(x_{0},\xi)=\Phi_{S}(\xi)+G\bigl(S_{T}\bigr),$$

where $\Phi_{S}(\xi)$ is the running-cost integral of $\xi$ along $S$ furnished by claim 1 of that lemma.

3. (Lower semicontinuity.) $F$ is lower semicontinuous on $X$ for the metric $d_{X}$.

4. (Lower semicontinuity in the control.) For every $x_{0}\in\Delta^{l}$, the function $\mathcal{U}_{\mathcal{A}}\to\mathbb{R}$ sending $\xi$ to $F(x_{0},\xi)$ is lower semicontinuous on $\mathcal{U}_{\mathcal{A}}$ for the metric $\rho$.

5. (Attainment of the minimum.) For every $x_{0}\in\Delta^{l}$ there exists $\xi^{*}\in\mathcal{U}_{\mathcal{A}}$ such that

$$F(x_{0},\xi^{*})\le F(x_{0},\xi)\qquad\text{for every }\xi\in\mathcal{U}_{\mathcal{A}} .$$