The Optimal Value, the Optimal Controls and the Optimal Trajectories of the Mean-Field Problem

Adopt the setting, hypotheses and notation of the boundedness, lower semicontinuity and attainment theorem for the mean-field cost: 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 $(L,G)$ 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 with the metric $\rho$, the mean-field flow $S(z_{0},\xi)$, for $z_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$, of claim 2 of the flow stability lemma, and the mean-field cost $F$.

Let $x_{0}\in\Delta^{l}$, called the initial state of the problem. Put

$$V_{x_{0}}=\bigl\{F(x_{0},\xi)\;:\;\xi\in\mathcal{U}_{\mathcal{A}}\bigr\}\subseteq\mathbb{R}.$$

By claim 5 of that theorem there is at least one element of $\mathcal{U}_{\mathcal{A}}$, so $V_{x_{0}}$ is nonempty; and by claim 1 of that theorem $|F(x_{0},\xi)|\le C_{F}$ for every $\xi\in\mathcal{U}_{\mathcal{A}}$, so by claim 6 of Properties of the Absolute Value in an Ordered Field we have $-C_{F}\le F(x_{0},\xi)$ for every such $\xi$, that is, the number $-C_{F}$ is a lower bound of $V_{x_{0}}$. Hence the infimum of $V_{x_{0}}$ exists by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below and is unique by Uniqueness of the Supremum and of the Infimum.

The optimal mean-field value from $x_{0}$ is the real number

$$J^{*}_{x_{0}}=\inf V_{x_{0}}.$$

The set of optimal mean-field controls from $x_{0}$ is

$$\mathcal{M}^{*}_{x_{0}}=\bigl\{\xi\in\mathcal{U}_{\mathcal{A}}\;:\;F(x_{0},\xi)=J^{*}_{x_{0}}\bigr\}.$$

The set of optimal mean-field trajectories from $x_{0}$ is

$$\mathcal{S}^{*}_{x_{0}}=\bigl\{S(x_{0},\xi)\;:\;\xi\in\mathcal{M}^{*}_{x_{0}}\bigr\}.$$