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

Throughout we use the order arithmetic of Elementary Order Arithmetic in an Ordered Field. Its clauses 1 and 10 are stated for strict inequalities; the corresponding statements for $\le$, and the transitivity of $\le$, follow by treating the equality case (and, for multiplication, the case of a zero multiplier) separately, and we use them under this convention without further comment. We write $\lambda=\lambda_{[0,T]}$ for the restricted Lebesgue measure on $[0,T]$.

Step 0 (the flow is an admissible state path). Let $x_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$ and write $S=S(x_{0},\xi)$. By claim 2 of the flow stability lemma we have $S_{t}\in\Delta^{l}$ for every $t\in[0,T]$ and $|S_{t}-S_{r}|\le K_{b}|t-r|$ for all $r,t\in[0,T]$. For each $\gamma\in\{1,\dots,l\}$, claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ bounds a coordinate by the norm, so $|S^{\gamma}_{t}-S^{\gamma}_{r}|\le|S_{t}-S_{r}|\le K_{b}|t-r|$, where $K_{b}$ is the constant fixed in the preamble of the flow stability lemma and appearing in its claim 2. Hence every component of $S$ is continuous on $[0,T]$, the interval carrying the restriction of the absolute value metric: given $t\in[0,T]$ and a real $\varepsilon>0$, take $\delta=\varepsilon/(K_{b}+1)$, which is positive; then every $r\in[0,T]$ with $|t-r|<\delta$ satisfies $|S^{\gamma}_{t}-S^{\gamma}_{r}|\le K_{b}\delta<\varepsilon$, covering also the case $K_{b}=0$. Every component of $S$ is therefore measurable by claim 4 of that lemma. Thus $S$ is an admissible state path in the sense of the running-cost lower-semicontinuity lemma, and the running-cost integral $\Phi_{S}(\xi)$ is defined.

Step 1 (the representation of $F$). Keep $x_{0}$, $\xi$ and $S$ as in Step 0, the symbol $\xi$ denoting also an admissible representative as in the definition of the mean-field cost; such a representative exists by claim 2 of the flow stability lemma. By that definition, $F(x_{0},\xi)$ is the generalized mean-field cost of the pair $(S,\xi)$ under $(L,G)$, that is

$$F(x_{0},\xi)=\int_{[0,T]}L\bigl(S_{t},\xi(t)\bigr)\,d\lambda(t)+G(S_{T}),$$

the integral being the Lebesgue integral over $[0,T]$, that is the integral against $\lambda$. By claim 1 of the running-cost lower-semicontinuity lemma, applied to the admissible state path $S$ and to that representative, that integral equals $\Phi_{S}(\xi)$ and its value does not depend on which admissible representative is used. Hence

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

Since $x_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$ were arbitrary in Step 0, this holds for every $x_{0}\in\Delta^{l}$ and every $\xi\in\mathcal{U}_{\mathcal{A}}$; we refer to it below as the representation of Step 1.

Claim 1. By claim 1 of the running-cost lower-semicontinuity lemma there is a real $C\ge0$, depending only on $L$, $\Delta^{l}$ and $\mathcal{A}$, with $|\Phi_{S}(\xi)|\le CT$ for every admissible state path $S$ and every $\xi\in\mathcal{U}_{\mathcal{A}}$. Since $\mathcal{A}$ is nonempty and compact, claim 1 of the boundedness lemma for population cost data gives a real $C'\ge0$ with $|G(\Sigma)|\le C'$ for every $\Sigma\in\Delta^{l}$. Let $x_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$ and write $S=S(x_{0},\xi)$; by Step 0, $S$ is an admissible state path and $S_{T}\in\Delta^{l}$, and by the representation of Step 1, $F(x_{0},\xi)=\Phi_{S}(\xi)+G(S_{T})$. By claim 5 of Properties of the Absolute Value in an Ordered Field,

$$|F(x_{0},\xi)|=\bigl|\Phi_{S}(\xi)+G(S_{T})\bigr|\le|\Phi_{S}(\xi)|+|G(S_{T})|\le CT+C' .$$

Put $C_{F}=CT+C'$, a nonnegative real number depending only on the data listed in the claim. This proves claim 1.

Claim 2. Let $x_{0}\in\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$ and write $S=S(x_{0},\xi)$. By Step 0, $S$ is an admissible state path in the sense of the running-cost lower-semicontinuity lemma, and by the representation of Step 1, $F(x_{0},\xi)=\Phi_{S}(\xi)+G(S_{T})$. This proves claim 2.

Claim 3. We apply the sequential characterization of lower semicontinuity with ambient metric space $(X,d_{X})$ and with $A=X$, so that the restricted metric is $d_{X}$ itself. By claim 3 of that lemma it suffices to verify its sequential condition at every point of $X$.

Let $(x_{0},\xi)\in X$, let $\bigl((x^{j}_{0},\xi_{j})\bigr)_{j\in\mathbb{N}}$ be a sequence in $X$ converging to $(x_{0},\xi)$ in $(X,d_{X})$, and let $\varepsilon$ be a real number with $0<\varepsilon$. By claim 1 of Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space the sequence $(x^{j}_{0})_{j\in\mathbb{N}}$ converges to $x_{0}$ in $(\Delta^{l},d_{\Delta})$ and the sequence $(\xi_{j})_{j\in\mathbb{N}}$ converges to $\xi$ in $(\mathcal{U}_{\mathcal{A}},\rho)$.

By the definition of the Euclidean distance, $d_{\Delta}(x^{j}_{0},x_{0})=|x^{j}_{0}-x_{0}|$, and this quantity is nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field, so it coincides with its own absolute value by the definition of the absolute value. Hence convergence of $(x^{j}_{0})$ to $x_{0}$ says exactly that the real sequence $\bigl(|x^{j}_{0}-x_{0}|\bigr)_{j\in\mathbb{N}}$ has limit $0$. By claim 2 of the weak metrizability and compactness theorem, convergence of $(\xi_{j})$ to $\xi$ in $\rho$ holds if and only if $\xi_{j}\rightharpoonup\xi$ in the sense of weak convergence; so the latter holds.

Write $S^{j}=S(x^{j}_{0},\xi_{j})$ and $S=S(x_{0},\xi)$; by Step 0 all of these are admissible state paths. Claim 6 of the flow stability lemma applies to $(x^{j}_{0})$, $x_{0}$, $(\xi_{j})$ and $\xi$, since $\xi\in\mathcal{U}_{\mathcal{A}}$, and yields: for every real $\eta>0$ there is $N\in\mathbb{N}$ with $|S^{j}_{t}-S_{t}|\le\eta$ for every $t\in[0,T]$ and every $j\ge N$. This is precisely the uniform-convergence hypothesis of claim 4 of the running-cost lower-semicontinuity lemma, whose other hypotheses hold because $\xi_{j}\in\mathcal{U}_{\mathcal{A}}$ and $\xi_{j}\rightharpoonup\xi$. That claim therefore gives

$$\Phi_{S}(\xi)\le\liminf_{j}\Phi_{S^{j}}(\xi_{j}),$$

the limit inferior being defined because the real sequence $\bigl(\Phi_{S^{j}}(\xi_{j})\bigr)_{j\in\mathbb{N}}$ is bounded by claim 1 of the same lemma.

Taking $t=T$ in the uniform estimate shows that the real sequence $\bigl(|S^{j}_{T}-S_{T}|\bigr)_{j\in\mathbb{N}}$ has limit $0$. Fix $a\in\mathcal{A}$, which is possible because $\mathcal{A}$ is nonempty, and consider the sequence $\bigl((S^{j}_{T},a)\bigr)_{j\in\mathbb{N}}$ in $\Delta^{l}\times\mathbb{R}^{m}$: the Euclidean distances from $S^{j}_{T}$ to $S_{T}$ converge to $0$, and those from $a$ to $a$ are $0$. Condition 1 of the population cost data therefore gives that the real sequence $\bigl(G(S^{j}_{T})\bigr)_{j\in\mathbb{N}}$ has limit $G(S_{T})$.

By claim 8 of Elementary Order Arithmetic in an Ordered Field there is a real $\varepsilon_{1}$ with $0<\varepsilon_{1}$ and $\varepsilon_{1}+\varepsilon_{1}=\varepsilon$. By claim 3 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence there is $N_{1}\in\mathbb{N}$ such that

$$\liminf_{j}\Phi_{S^{j}}(\xi_{j})-\varepsilon_{1}<\Phi_{S^{k}}(\xi_{k})\qquad\text{for every }k\ge N_{1}.$$

Combining with $\Phi_{S}(\xi)\le\liminf_{j}\Phi_{S^{j}}(\xi_{j})$ and adding $-\varepsilon_{1}$ to both sides of that inequality gives

$$\Phi_{S}(\xi)-\varepsilon_{1}<\Phi_{S^{k}}(\xi_{k})\qquad\text{for every }k\ge N_{1}.$$

By the convergence of $\bigl(G(S^{j}_{T})\bigr)$ to $G(S_{T})$ there is $N_{2}\in\mathbb{N}$ with $|G(S^{k}_{T})-G(S_{T})|<\varepsilon_{1}$ for every $k\ge N_{2}$. By claim 3 of Properties of the Absolute Value in an Ordered Field we have $-|G(S^{k}_{T})-G(S_{T})|\le G(S^{k}_{T})-G(S_{T})$, whence $-\varepsilon_{1}<G(S^{k}_{T})-G(S_{T})$ and so

$$G(S_{T})-\varepsilon_{1}<G(S^{k}_{T})\qquad\text{for every }k\ge N_{2}.$$

Let $N$ be the larger of the two natural numbers $N_{1}$ and $N_{2}$, and let $k\ge N$. Adding $G(S^{k}_{T})$ to both sides of the first displayed strict inequality and $\Phi_{S}(\xi)-\varepsilon_{1}$ to both sides of the second, using claim 1 of Elementary Order Arithmetic in an Ordered Field each time, and then chaining the two resulting strict inequalities by claim 2 of that lemma, we obtain

$$F(x_{0},\xi)-\varepsilon=\bigl(\Phi_{S}(\xi)-\varepsilon_{1}\bigr)+\bigl(G(S_{T})-\varepsilon_{1}\bigr)<\Phi_{S^{k}}(\xi_{k})+G(S^{k}_{T})=F(x^{k}_{0},\xi_{k}),$$

where the outer equalities hold by the representation of Step 1 (claim 2) and $\varepsilon_{1}+\varepsilon_{1}=\varepsilon$.

Thus the sequential condition of claim 1 of the sequential characterization holds at $(x_{0},\xi)$. Since $(x_{0},\xi)\in X$ was arbitrary, claim 3 of that lemma shows that $F$ is lower semicontinuous on $X$ for $d_{X}$. This proves claim 3.

Claim 4. Fix $x_{0}\in\Delta^{l}$ and define $F_{x_{0}}:\mathcal{U}_{\mathcal{A}}\to\mathbb{R}$ by $F_{x_{0}}(\xi)=F(x_{0},\xi)$; claim 4 asserts that $F_{x_{0}}$ is lower semicontinuous on $\mathcal{U}_{\mathcal{A}}$ for the metric $\rho$, applying the sequential characterization with ambient metric space $(\mathcal{U}_{\mathcal{A}},\rho)$ and $A=\mathcal{U}_{\mathcal{A}}$. Let $\xi\in\mathcal{U}_{\mathcal{A}}$, let $(\xi_{j})_{j\in\mathbb{N}}$ be a sequence in $\mathcal{U}_{\mathcal{A}}$ converging to $\xi$ in $(\mathcal{U}_{\mathcal{A}},\rho)$, and let $\varepsilon>0$ be real. The constant sequence with every term $x_{0}$ converges to $x_{0}$ in $(\Delta^{l},d_{\Delta})$, because $d_{\Delta}(x_{0},x_{0})=0$. Hence by claim 1 of Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space the sequence $\bigl((x_{0},\xi_{j})\bigr)_{j\in\mathbb{N}}$ converges to $(x_{0},\xi)$ in $(X,d_{X})$. By claim 3, already proved, $F$ is lower semicontinuous on $X$, so claim 1 of the sequential characterization, in its necessity direction, gives $N\in\mathbb{N}$ with $F(x_{0},\xi)-\varepsilon<F(x_{0},\xi_{j})$ for every $j\ge N$; that is, $F_{x_{0}}(\xi)-\varepsilon<F_{x_{0}}(\xi_{j})$ for every $j\ge N$. As $\xi$ was arbitrary, claims 1 and 3 of that lemma show that $F_{x_{0}}$ is lower semicontinuous on $\mathcal{U}_{\mathcal{A}}$ for $\rho$. Since $x_{0}\in\Delta^{l}$ was arbitrary, this proves claim 4.

Claim 5. Fix $x_{0}\in\Delta^{l}$ and keep the notation $F_{x_{0}}$ of the previous paragraph, which is lower semicontinuous on $\mathcal{U}_{\mathcal{A}}$ by claim 4. First, $\mathcal{U}_{\mathcal{A}}$ is nonempty: with $R=\sup_{\alpha\in\mathcal{A}}|\alpha|$ as in the lemma on affine-controlled data, a finite nonnegative real number by claim 1 there, so that $|a|\le R$ for every $a\in\mathcal{A}$, claim 1 of the control-set properties lemma shows that $\mathcal{U}_{\mathcal{A}}$ contains the class of a constant map with value in $\mathcal{A}$, and $\mathcal{A}$ is nonempty. By claim 3 of the weak metrizability and compactness theorem, $\mathcal{U}_{\mathcal{A}}$ is a compact subset of the metric space $(\mathcal{U}_{\mathcal{A}},\rho)$; the hypotheses of that theorem on $\mathcal{A}$ hold because $\mathcal{A}$ is nonempty, compact and convex.

Applying claim 2 of Semicontinuous Functions Attain Their Extrema on a Compact Set with metric space $(\mathcal{U}_{\mathcal{A}},\rho)$, with the nonempty compact subset $K=\mathcal{U}_{\mathcal{A}}$ and with the lower semicontinuous function $F_{x_{0}}$, we obtain $\xi^{*}\in\mathcal{U}_{\mathcal{A}}$ with $F_{x_{0}}(\xi^{*})\le F_{x_{0}}(\xi)$ for every $\xi\in\mathcal{U}_{\mathcal{A}}$, that is $F(x_{0},\xi^{*})\le F(x_{0},\xi)$ for every $\xi\in\mathcal{U}_{\mathcal{A}}$. This proves claim 5.