Structure of the Optimal Control Set and Separation of Near-Optimal Controls

Adopt the setting, hypotheses and notation of the optimal value, controls and trajectories of the mean-field problem: the initial state $x_{0}\in\Delta^{l}$, the optimal value $J^{*}_{x_{0}}$, the set $\mathcal{M}^{*}_{x_{0}}$ of optimal controls, the mean-field flow $S(z_{0},\xi)$ of claim 2 of the flow stability lemma, the mean-field cost $F$, the metric $\rho$ on $\mathcal{U}_{\mathcal{A}}$, and the product metric space $(X,d_{X})$ with $X=\Delta^{l}\times\mathcal{U}_{\mathcal{A}}$ of the boundedness, lower semicontinuity and attainment theorem. Write $|\cdot|$ for the Euclidean norm on $\mathbb{R}^{l}$ and let $\mathbb{N}$ be the natural numbers.

For $z_{0}\in\Delta^{l}$ and $\xi,\zeta\in\mathcal{U}_{\mathcal{A}}$ put

$$\Psi(z_{0},\xi,\zeta)=\sup\bigl\{\,\bigl|S_{t}(z_{0},\xi)-S_{t}(x_{0},\zeta)\bigr|\;:\;t\in[0,T]\,\bigr\}.$$

Then the following hold.

1. (Structure of the optimal control set.) $\mathcal{M}^{*}_{x_{0}}=\{\xi\in\mathcal{U}_{\mathcal{A}}:F(x_{0},\xi)\le J^{*}_{x_{0}}\}$; the set $\mathcal{M}^{*}_{x_{0}}$ is nonempty; it is closed in the topological space whose underlying set is $\mathcal{U}_{\mathcal{A}}$ and whose open sets are the subsets open in $(\mathcal{U}_{\mathcal{A}},\rho)$; and it is a compact and sequentially compact subset of $(\mathcal{U}_{\mathcal{A}},\rho)$.

2. (Deviation from the optimal set.) For all $z_{0}\in\Delta^{l}$ and $\xi,\zeta\in\mathcal{U}_{\mathcal{A}}$ the supremum $\Psi(z_{0},\xi,\zeta)$ exists. Moreover, for all $z_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$ there is $\zeta^{\dagger}\in\mathcal{M}^{*}_{x_{0}}$ with

$$\Psi(z_{0},\xi,\zeta^{\dagger})\le\Psi(z_{0},\xi,\zeta)\qquad\text{for every }\zeta\in\mathcal{M}^{*}_{x_{0}},$$

and the value $\Psi(z_{0},\xi,\zeta^{\dagger})$ is the same for every $\zeta^{\dagger}\in\mathcal{M}^{*}_{x_{0}}$ with this property. Write $D(z_{0},\xi)$ for that common value. Then $0\le D(z_{0},\xi)$ for all $z_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$, and $D(x_{0},\xi^{*})=0$ for every $\xi^{*}\in\mathcal{M}^{*}_{x_{0}}$.

3. (Comparison estimate.) Let $z_{0},z_{0}'\in\Delta^{l}$ and $\xi,\xi'\in\mathcal{U}_{\mathcal{A}}$. Then the supremum

$$\Theta=\sup\bigl\{\,\bigl|S_{t}(z_{0},\xi)-S_{t}(z_{0}',\xi')\bigr|\;:\;t\in[0,T]\,\bigr\}$$

exists and $\bigl|D(z_{0},\xi)-D(z_{0}',\xi')\bigr|\le\Theta$.

4. (Sequential continuity of the deviation.) Let $\bigl((z^{j}_{0},\xi_{j})\bigr)_{j\in\mathbb{N}}$ be a sequence in $X$ that converges to $(z_{0},\xi)\in X$ in $(X,d_{X})$. Then the real sequence $\bigl(D(z^{j}_{0},\xi_{j})\bigr)_{j\in\mathbb{N}}$ has limit $D(z_{0},\xi)$.

5. (Separation of near-optimal controls.) For every real $\varepsilon>0$ there is a real $\eta>0$ such that every $\xi\in\mathcal{U}_{\mathcal{A}}$ satisfying $F(x_{0},\xi)\le J^{*}_{x_{0}}+\eta$ satisfies $D(x_{0},\xi)<\varepsilon$.