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

Throughout, $\sqrt{\cdot}$ is the nonnegative square root, and for a subset $E$ of $[0,T]$ we write $\mathbf{1}_E$ for the function on $[0,T]$ equal to $1$ on $E$ and to $0$ elsewhere. Integrals over a compact subinterval are those of the restricted Lebesgue measure space, an integral over $[0,0]$ being $0$ as in Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control. We use without further comment three facts about nonstrict inequalities between real numbers, each obtained from the corresponding strict statement in Elementary Order Arithmetic in an Ordered Field together with the case in which the inequality in question is an equality: that multiplying both sides by a nonnegative number preserves the inequality (claim 10 there, together with the further trivial case in which the multiplier is $0$, when the two products are both $0$), that adding a fixed number to both sides preserves it (claim 1 there), and that $\le$ is transitive (claim 2 there, in the form combining a nonstrict with a strict inequality). We also use that for $x\in\Delta^l$ and each $\sigma$ one has $0\le x^\sigma\le1$, the lower bound by the definition of the simplex and the upper bound by claim 6 of Properties of Finite Sums applied to the sum $\sum_\sigma x^\sigma=1$ of nonnegative terms.

Claim 1. Fix $\gamma$ and $\Sigma\in\Delta^l$. By claim 1 of The Projected Extension of an Affine-Controlled Transition-Rate Family the number $K_1$ is a finite nonnegative real, and by its definition there as a supremum, $|\beta_1(\sigma,\gamma,\Sigma)|\le K_1$ for every ordered pair of distinct states and every $\Sigma\in\Delta^l$. By claim 3 of that lemma,

a sum of $l-1$ terms. As noted above $0\le\Sigma^\sigma\le1$ for every $\sigma$, so by claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, together with the fact that the absolute value of a nonnegative real number is that number itself,

the final inequality by multiplying $|\beta_1(\sigma,\gamma,\Sigma)|\le K_1$ by the nonnegative number $\Sigma^\sigma$, multiplying $\Sigma^\sigma\le1$ by the nonnegative number $K_1$, and using transitivity of $\le$, and likewise for the second term of each summand. By claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, applied by induction on the number of summands, the norm of the displayed sum is at most the sum of the norms of the summands, and the norm of each summand is at most $K_1+K_1$ by the same claim. Hence $|b^\gamma_1(\Sigma)|\le(l-1)(K_1+K_1)=C_1$.

For the second assertion let $\alpha,\alpha'\in\mathcal{A}$. By claim 3 of The Projected Extension of an Affine-Controlled Transition-Rate Family, $b^\gamma(\Sigma,\alpha)=b^\gamma_0(\Sigma)+b^\gamma_1(\Sigma)\cdot\alpha$ and similarly for $\alpha'$, so by the distributivity of the dot product,

Thus $b^\gamma_1(\Sigma)\cdot(\alpha'-\alpha)$ is the $\gamma$-th coordinate of the vector $b(\Sigma,\alpha')-b(\Sigma,\alpha)$ of $\mathbb{R}^l$, so by claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and then claim 4 of The Projected Extension of an Affine-Controlled Transition-Rate Family,

Finally, by claims 5 and 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $|\alpha'-\alpha|\le|\alpha'|+|\alpha|\le R+R$, using that $R$ is by claim 1 of The Projected Extension of an Affine-Controlled Transition-Rate Family a finite real number bounding $|\alpha|$ on $\mathcal{A}$. Multiplying by the nonnegative number $K_2$ gives $K_2|\alpha'-\alpha|\le 2RK_2=L_1$.

Claim 2. Existence of an admissible representative. Let $\xi\in\mathcal{U}_{\mathcal{A}}$. By the definition of the control set there are $v\in\mathcal{L}^{2}([0,T];\mathbb{R}^{m})$ with $[v]=\xi$ and $N\in\mathcal{B}$ with $\lambda(N)=0$ such that $v(s)\in\mathcal{A}$ for every $s\in D:=[0,T]\setminus N$. The set $\mathcal{A}$ is nonempty by the definition of an affine-controlled transition-rate family; fix $a_0\in\mathcal{A}$ and define $u(s)=v(s)$ for $s\in D$ and $u(s)=a_0$ for $s\in N$. Then $u(s)\in\mathcal{A}$ for every $s\in[0,T]$. Each component of $u$ equals $v^{i}\mathbf{1}_{D}+a_0^{i}\mathbf{1}_{N}$ and so is measurable, $\mathbf{1}_D$ and $\mathbf{1}_N$ being measurable because $D,N\in\mathcal{B}$. Moreover $|u|^{2}\mathbf{1}_{D}=|v|^{2}\mathbf{1}_{D}$, so two applications of claim 6 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval to the nonnegative measurable functions $|u|^{2}$ and $|v|^{2}$ and the co-null set $D$ give

so $u\in\mathcal{L}^{2}([0,T];\mathbb{R}^{m})$. Since $u$ and $v$ agree on $D$ and $\lambda(N)=0$, claim 2 of the definition of $L^{2}$ gives $[u]=[v]=\xi$. So $u$ is an admissible representative of $\xi$.

Independence of the representative. Let $u$ and $\tilde{u}$ be admissible representatives of $\xi$. Since $[u]=[\tilde{u}]$, claim 2 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval provides $N\in\mathcal{B}$ with $\lambda(N)=0$ and $u(s)=\tilde{u}(s)$ for every $s\in[0,T]\setminus N$. Write $S=S^{u}$ and fix $\gamma$ and $t\in[0,T]$. If $t=0$ the two integrals below are both $0$. Suppose $t>0$. As recorded in the definition of a generalized mean-field trajectory pair, the maps $s\mapsto\hat{b}^\gamma(S_s,u(s))$ and $s\mapsto\hat{b}^\gamma(S_s,\tilde{u}(s))$ are measurable on $[0,t]$ and bounded there, hence integrable. They agree at every $s\in[0,t]$ outside $N\cap[0,t]$. Now $N=P\cap[0,T]$ for some Borel subset $P$ of the real line by claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, so $N\cap[0,t]=P\cap[0,t]$ lies in $\mathcal{B}_{[0,t]}$; and its restricted Lebesgue measure equals its Lebesgue measure, which is at most $\lambda(N)=0$ by the monotonicity in claim 2 of Basic Properties of a Measure. So $N\cap[0,t]$ is null in $[0,t]$, and claim 2 of Integrals of Functions Vanishing or Agreeing off a Null Set on a Compact Interval gives

Hence $S$ is a continuous map $[0,T]\to\mathbb{R}^l$ with $S^\gamma_t=x_0^\gamma+\int_{[0,t]}\hat{b}^\gamma(S_s,\tilde{u}(s))\,ds$ for every $\gamma$ and every $t$. Claim 1 of Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control, applied to the initial value $x_0$ and the control $\tilde{u}$, asserts that there is exactly one such map, and that map is $S^{\tilde{u}}$; therefore $S^{\tilde{u}}=S$.

The remaining assertions of claim 2 are contained in claims 1 and 3 of Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control.

Claim 3. Fix $\gamma$ and write $\psi^{i}(s)=b^{\gamma,i}_1(S_s)$ for the $i$-th component. Each component of $S$ is continuous on $[0,T]$ by claim 2, hence measurable by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions. The map $\Sigma\mapsto b^{\gamma,i}_1(\Sigma)$ is sequentially continuous on $\Delta^l$, in the sense that $\bigl(b^{\gamma,i}_1(\Sigma_k)\bigr)_{k\in\mathbb{N}}$ has limit $b^{\gamma,i}_1(\Sigma)$ whenever $\Sigma_k,\Sigma\in\Delta^l$ and the real sequence $(|\Sigma_k-\Sigma|)_{k\in\mathbb{N}}$ has limit $0$. Indeed, write $\epsilon_k=|\Sigma_k-\Sigma|$, so that the real sequence $(\epsilon_k)_k$ has limit $0$. A constant real sequence has its value as limit, directly from the definition of a limit; we use this together with claim 1 of Arithmetic of Limits of Real Sequences in the form: if $(y_k-y)_k$ has limit $0$ then $(y_k)_k$ has limit $y$, since $y_k=(y_k-y)+y$.

For each $\sigma$ we have $|\Sigma^\sigma_k-\Sigma^\sigma|\le\epsilon_k$ by claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $(\Sigma^\sigma_k-\Sigma^\sigma)_k$ has limit $0$ by claim 3 of Order Properties of Limits of Real Sequences, and hence $(\Sigma^\sigma_k)_k$ has limit $\Sigma^\sigma$. Likewise, claim 2 of Affine-Controlled Transition-Rate Family makes $\beta_1(\sigma,\gamma,\cdot)$ Lipschitz with constant $\Lambda$, so claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ gives

The sequence $(\Lambda\epsilon_k)_k$ has limit $\Lambda\cdot0=0$ by claim 3 of Arithmetic of Limits of Real Sequences, so $\bigl(\beta^{i}_1(\sigma,\gamma,\Sigma_k)-\beta^{i}_1(\sigma,\gamma,\Sigma)\bigr)_k$ has limit $0$ by claim 3 of Order Properties of Limits of Real Sequences, and therefore $\bigl(\beta^{i}_1(\sigma,\gamma,\Sigma_k)\bigr)_k$ has limit $\beta^{i}_1(\sigma,\gamma,\Sigma)$. The displayed formula for $b^\gamma_1$ expresses $b^{\gamma,i}_1$ as a finite sum of products and differences of these quantities, so claims 1, 2 and 3 of Arithmetic of Limits of Real Sequences, applied finitely many times, give the assertion. Therefore $\psi^{i}$ is measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable.

By claim 1 above together with claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ we have $|\psi^{i}(s)|\le|b^\gamma_1(S_s)|\le C_1$ for every $s\in[0,T]$, whence $(\psi^{i})^{2}\le C_1^{2}$ pointwise. The constant function with value $C_1^{2}$ is a simple function whose standard representation takes the single value $C_1^{2}$ on $[0,T]$, so its integral is $C_1^{2}\lambda([0,T])=C_1^{2}T$ by claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. The monotonicity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral therefore gives $\int_{[0,T]}(\psi^{i})^{2}\,d\lambda\le C_1^{2}T<\infty$. The $i$-th component of the map $s\mapsto\mathbf{1}_{[0,t]}(s)b^\gamma_1(S_s)$ is $\mathbf{1}_{[0,t]}\psi^{i}$, which is measurable and satisfies $(\mathbf{1}_{[0,t]}\psi^{i})^{2}\le(\psi^{i})^{2}$ pointwise; so its square has finite integral by the monotonicity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral. Summing over $i$ and using claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ together with the linearity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral gives $\int_{[0,T]}|\mathbf{1}_{[0,t]}b^\gamma_1(S_\cdot)|^{2}\,d\lambda<\infty$. Hence that map is square-integrable and $w^{\gamma,t}$ is a well-defined element of $H$.

Claim 4. Let $u$ and $u'$ be admissible representatives of $\xi$ and $\xi'$. Fix $t\in[0,T]$ and $\gamma$. Since $S_s$ and $S'_s$ lie in $\Delta^l$, claim 6 of The Projected Extension of an Affine-Controlled Transition-Rate Family gives $\hat{b}=b$ at these arguments, so by claim 2 and claim 1 of Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control,

the splitting being the linearity in claim 2 of Linearity and Monotonicity of the Lebesgue Integral. By claim 1 the last integrand equals $b^\gamma_1(S_s)\cdot(u'(s)-u(s))$. Applying claim 2 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval to $[0,t]$ and to $[0,T]$, whose zero extensions to the real line coincide, that last integral equals

by claim 4 of the definition of the pairing and the bilinearity recorded in claim 4 of The Lebesgue Space of Square-Integrable Vector-Valued Functions is a Real Inner Product Space.

Let $G_t\in\mathbb{R}^l$ be the vector with components $g^\gamma_t(\xi')$. Collecting the components of the display and using claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ twice, then Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval and claim 4 of The Projected Extension of an Affine-Controlled Transition-Rate Family,

Fix $\gamma$ and put $c=g^\gamma_t(\xi')$, so that $|c|\le G$ and hence $0\le G$ by claim 1 of Properties of the Absolute Value in an Ordered Field. By claim 3 of that lemma, $-|c|\le c\le|c|$, so $c\le G$ by transitivity, and adding $-G-|c|$ to both sides of $|c|\le G$ gives $-G\le-|c|$ and hence $-G\le c$, again by transitivity. Therefore $G-c$ and $G+c$ are both nonnegative, so their product $G^{2}-c^{2}$ is nonnegative, by claim 5 of Elementary Order Arithmetic in an Ordered Field when both factors are positive and trivially when one of them is $0$. Adding $c^{2}$ to both sides gives $c^{2}\le G^{2}$.

Summing these $l$ termwise bounds, claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ together with claims 2 and 5 of Properties of Finite Sums gives $|G_t|^{2}=\sum_\gamma\bigl(g^\gamma_t(\xi')\bigr)^{2}\le l\,G^{2}=(\sqrt{l}\,G)^{2}$.

We record an elementary fact used here and below: if $p,q$ are real numbers with $0\le p$, $0\le q$ and $p^{2}\le q^{2}$, then $p\le q$. Suppose not, so that $q<p$; then $0\le q<p$ gives $0<p$. Claim 10 of Elementary Order Arithmetic in an Ordered Field, applied to $q<p$ with the positive multiplier $p$, gives $qp<p^{2}$. Multiplying $q<p$ by $q$ gives $q^{2}\le qp$: this is an equality if $q=0$, and a strict inequality by claim 10 of the same lemma if $0<q$. Claim 2 of Elementary Order Arithmetic in an Ordered Field then yields $q^{2}<p^{2}$, contradicting $p^{2}\le q^{2}$. Applying this fact with $p=|G_t|$ and $q=\sqrt{l}\,G$, both nonnegative, gives $|G_t|\le\sqrt{l}\,G$.

Put $\phi(t)=|S'_t-S_t|$ and $a=|x_0'-x_0|+\sqrt{l}\,G$, so that $\phi(t)\le a+\Lambda_b\int_{[0,t]}\phi(s)\,ds$ for every $t\in[0,T]$. For $r,t\in[0,T]$, claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 2 give

and symmetrically with $r$ and $t$ interchanged, so $\phi$ is Lipschitz with constant $2K_b$, hence continuous by A Lipschitz Map is Uniformly Continuous and measurable by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions. Taking $r=0$ in the displayed inequality and using $\phi(0)=|x_0'-x_0|$, which holds because $S_0=x_0$ and $S'_0=x_0'$ by claim 2, gives $\phi(t)\le|x_0'-x_0|+2K_bT$ for every $t\in[0,T]$; thus $\phi$ is bounded. Since $\Lambda_b\ge0$, Gronwall's Lemma for Bounded Measurable Functions applies and yields $\phi(t)\le a\,e^{\Lambda_bt}$ for every $t\in[0,T]$. Finally $0\le a$ and $\Lambda_bt\le\Lambda_bT$, and by claims 2 and 4 of Basic Properties of the Exponential Function the exponential function is positive and strictly increasing, so $e^{\Lambda_bt}\le e^{\Lambda_bT}$, with strict inequality when $\Lambda_bt<\Lambda_bT$ and equality when $\Lambda_bt=\Lambda_bT$; multiplying by the nonnegative number $a$ gives $\phi(t)\le a\,e^{\Lambda_bT}$.

Claim 5. Let $v$ be an admissible representative of $\zeta$ and $u$ one of $\xi$, and let $r,t\in[0,T]$; by symmetry we may assume $r\le t$. By the bilinearity in claim 4 of The Lebesgue Space of Square-Integrable Vector-Valued Functions is a Real Inner Product Space,

Write $h$ for the integrand. The factor $\mathbf{1}_{[0,t]}-\mathbf{1}_{[0,r]}$ takes values in $\{0,1\}$ and vanishes outside $[r,t]$, so by claim 1 the integrand satisfies $|h(s)|\le L_1\mathbf{1}_{[r,t]}(s)$ for every $s$. By claim 3 of Properties of the Absolute Value in an Ordered Field we have $-L_1\mathbf{1}_{[r,t]}\le h\le L_1\mathbf{1}_{[r,t]}$ pointwise, so the monotonicity and linearity in claim 2 of Linearity and Monotonicity of the Lebesgue Integral give

the last equality because $L_1\mathbf{1}_{[r,t]}$ is a simple function whose standard representation takes the value $L_1$ on $[r,t]$ and $0$ elsewhere. If $r<t$ then $\lambda([r,t])=t-r$ by claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval applied to the interval $[r,t]$, whose restricted Lebesgue measure agrees with $\lambda$ on subsets of $[r,t]$; if $r=t$ then $[r,t]$ is a single point and both sides are $0$. In either case $|g^\gamma_t(\zeta)-g^\gamma_r(\zeta)|\le L_1|t-r|$. The constant $L_1=2RK_2$ was defined from $R$, $l$ and $K_1$ alone.

Claim 6. A finite grid. Let $d_{\mathbb{R}}$ be the absolute-value metric on $\mathbb{R}$ and let $d_0$ be its restriction to $[0,T]$. Then $d_0$ is a metric on $[0,T]$: for $x,y,z\in[0,T]$ the four required conditions, namely $0\le d_0(x,y)$, that $d_0(x,y)=0$ holds exactly when $x=y$, that $d_0(x,y)=d_0(y,x)$, and that $d_0(x,z)\le d_0(x,y)+d_0(y,z)$, are the corresponding conditions for $d_{\mathbb{R}}$ evaluated at points of $[0,T]$, and those hold because $d_{\mathbb{R}}$ is a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Line. The set $[0,T]$ is sequentially compact in $([0,T],d_0)$: given a sequence in $[0,T]$, A Closed Interval is Sequentially Compact in the Real Line applied with $0\le T$ provides a subsequence converging in $(\mathbb{R},d_{\mathbb{R}})$ to a point of $[0,T]$, and convergence is a condition on the distances alone, which are unchanged on passing to $d_0$. By Compactness and Sequential Compactness Agree for Subsets of a Metric Space applied in $([0,T],d_0)$ with the subset $[0,T]$, that set is compact, and by A Compact Subset of a Metric Space is Totally Bounded it is totally bounded in $([0,T],d_0)$. So for every real $\delta>0$ there is a finite set $F\subseteq[0,T]$ such that every $t\in[0,T]$ satisfies $|t-a|<\delta$ for some $a\in F$; and $F$ is nonempty because $0\in[0,T]$.

The estimate. Let $\varepsilon>0$ be real. The number $E=e^{\Lambda_bT}$ is positive, so $\eta=\varepsilon E^{-1}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field and claim 5 there. By claim 8 of that lemma choose $\theta>0$ with $\theta+\theta=\eta$. Since $l\ge2$ we have $\sqrt{l}>0$; put $\kappa=\theta\,(\sqrt{l})^{-1}>0$, and by two further applications of claim 8 choose $\kappa_2>0$ with $\kappa_2+\kappa_2+\kappa_2+\kappa_2=\kappa$. Put $L=L_1+1$, which is positive since $L_1\ge0$, and $\delta=\kappa_2L^{-1}>0$. Let $F$ be a finite nonempty subset of $[0,T]$ as above for this $\delta$.

For each $a\in F$ and each $\gamma\in\{1,\dots,l\}$, claim 3 of Arithmetic of Limits of Real Sequences and weak convergence give that the real sequence $\bigl(g^\gamma_a(\xi_j)\bigr)_{j\in\mathbb{N}}=\bigl(\langle\xi_j,w^{\gamma,a}\rangle_{L^{2}}-\langle\xi,w^{\gamma,a}\rangle_{L^{2}}\bigr)_{j\in\mathbb{N}}$ has limit $0$; so there is $N_{a,\gamma}\in\mathbb{N}$ with $|g^\gamma_a(\xi_j)|\le\kappa_2$ for every $j\ge N_{a,\gamma}$. There are finitely many pairs $(a,\gamma)$, so finitely many such choices are made and, the order on $\mathbb{N}$ being total by claims 1, 2 and 3 of Properties of the Order on the Natural Numbers, an induction on the number of pairs yields $N_1\in\mathbb{N}$ with $N_{a,\gamma}\le N_1$ for all of them. Since the real sequence $(|x^j_0-x_0|)_{j\in\mathbb{N}}$ has limit $0$, there is $N_2\in\mathbb{N}$ with $|x^j_0-x_0|\le\theta$ for every $j\ge N_2$. Let $N$ be whichever of $N_1,N_2$ is the larger.

Let $j\ge N$, let $\gamma\in\{1,\dots,l\}$ and let $r\in[0,T]$. Choose $a\in F$ with $|r-a|<\delta$. By claim 5 of Properties of the Absolute Value in an Ordered Field and claim 5 above,

Since $\kappa_2+\kappa_2\le\kappa$, the number $G=\kappa$ satisfies the hypothesis of claim 4 for the pair $(x^j_0,\xi_j)$ against $(x_0,\xi)$. Claim 4 therefore gives, for every $t\in[0,T]$,

using $\sqrt{l}\,\kappa=\theta$ and, for the second inequality, that multiplying a nonstrict inequality by the nonnegative number $E$ preserves it. This is the required estimate.