Proof of Linearisation of a Perturbed Controlled Aggregate Flow Along a Comparison Pair: Exact Variation-of-Constants Identity and Residual Bound

Throughout, $C^2$ and $C^1$ refer to C^k Maps on a Euclidean Open Set; by claim 1 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, each $\psi_c$ is of class $C^2$ on the open set $U\times V\subseteq\mathbb{R}^{l+m}$, its partial derivatives satisfy $|\partial_i\psi_c|\le B+K$ and $|\partial_j\partial_i\psi_c|\le3K$ at every point of $\Delta^l\times\mathcal{A}$, and $\psi_c(\Sigma,\alpha)=\Sigma^\sigma\beta(\sigma,\gamma,\Sigma,\alpha)$ on $\Delta^l\times\mathcal{A}$ for $c=(\sigma,\gamma)$. By clause 2 of C^k Maps on a Euclidean Open Set each $\partial_i\psi_c$ is of class $C^1$ on $U\times V$, and $\psi_c$ and its first partial derivatives are continuous at every point of $U\times V$ by clause 1 (see also claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous); such a function $F$ is sequentially continuous at every $x\in U\times V$: given a sequence $x_n\to x$ and $\varepsilon>0$, continuity at $x$ provides $\delta>0$ with $|F(x')-F(x)|\le\varepsilon$ whenever $d(x',x)<\delta$, and $d(x_n,x)<\delta$ for all large $n$. We use that $|v_c|=\sqrt{2}$ (as $v_c=\delta_\gamma-\delta_\sigma$ with $\sigma\neq\gamma$ has exactly two nonzero coordinates, equal to $1$ and $-1$), that $\mathcal{L}$ has $l(l-1)$ elements, that $|\Sigma|\le1$ for $\Sigma\in\Delta^l$ (since $|\Sigma|^{2}=\sum_\sigma(\Sigma^\sigma)^{2}\le(\sum_\sigma\Sigma^\sigma)^{2}=1$, the coordinates being nonnegative with sum $1$), so that any two points of $\Delta^l$ are at distance at most $2$, and that $\Delta^l$ is convex: if $\Sigma,\Sigma'\in\Delta^l$ and $\tau\in[0,1]$, the coordinates of $\Sigma+\tau(\Sigma'-\Sigma)=(1-\tau)\Sigma+\tau\Sigma'$ are nonnegative and sum to $(1-\tau)+\tau=1$, so the point lies in $\Delta^l$; hence for $\Sigma,\Sigma'\in\Delta^l$ and $\alpha\in\mathcal{A}$ the segment between $(\Sigma,\alpha)$ and $(\Sigma',\alpha)$ lies in $\Delta^l\times\mathcal{A}\subseteq U\times V$.

Claim 1. For $\gamma\in\{1,\dots,l\}$ the $\gamma$-th component of $v_c$ is $1$ if $c=(\sigma,\gamma)$ for some $\sigma\neq\gamma$, $-1$ if $c=(\gamma,\gamma')$ for some $\gamma'\neq\gamma$, and $0$ otherwise; hence on $\Delta^l\times\mathcal{A}$

$$b^{\gamma}(\Sigma,\alpha)=\sum_{\sigma:\sigma\neq\gamma}\Sigma^\sigma\beta(\sigma,\gamma,\Sigma,\alpha)-\sum_{\gamma':\gamma'\neq\gamma}\Sigma^\gamma\beta(\gamma,\gamma',\Sigma,\alpha),$$

which is the $\gamma$-th component of the aggregate state drift. The identity $\mathcal{E}(\Sigma,\alpha)z=\sum_cv_c(g^{c}(\Sigma,\alpha)\cdot z)$ is recalled from the setting of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, where it follows from the entry formula $\mathcal{E}^{\gamma\eta}=\sum_cv_c^{\gamma}\partial_\eta\psi_c$. Since $0\le\psi_c\le B$ on $\Delta^l\times\mathcal{A}$ (the coordinate $\Sigma^\sigma$ lies in $[0,1]$ and $0\le\beta\le B$ by clause 1 of Transition-Rate Family), the triangle inequality gives $|b|\le\sum_c|v_c|\psi_c\le\sqrt{2}\,l(l-1)B$; the bound $|g^{c}|\le\sqrt{l}(B+K)$ follows from $|\partial_i\psi_c|\le B+K$ for the $l$ components; and $|\mathcal{E}z|\le\sum_c|v_c|\,|g^{c}\cdot z|\le\sqrt{2}\,l(l-1)\sqrt{l}(B+K)|z|$ by the Cauchy-Schwarz inequality. For measurable $x'$ and $a'$ the map $u\mapsto(x'_u,a'_u)\in\mathbb{R}^{l+m}$ is measurable componentwise, and $\psi_c$, $\partial_i\psi_c$ are continuous on the open set $U\times V$ containing its values, so $u\mapsto\psi_c(x'_u,a'_u)$ and $u\mapsto\partial_i\psi_c(x'_u,a'_u)$ are measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable (sequential continuity having been noted above); finite linear combinations of these are measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and the bounds just proved show boundedness. The absolute value of a measurable function is measurable (claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and the Euclidean norm of a bounded measurable $\mathbb{R}^l$-valued map is bounded measurable (Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval), so $u\mapsto\mathsf{d}_u$ is bounded measurable, with $\mathsf{d}_u\le l(l-1)\cdot2\sqrt{l}(B+K)$.

Claim 2. Measurability and boundedness of $\rho$ follow from claim 1 and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions (the entries of $\mathcal{E}^{\star}$ are continuous, hence bounded measurable by claim 4 of Measurability of Countable Suprema, Bounded Pointwise Limits, Monotone Functions, and Continuous Functions and Continuous Real-Valued Functions on a Compact Interval are Bounded, and $e$ is bounded measurable with $|e_u|\le2$ by the preliminary remarks). Fix $u\in[0,T]$ and a label $c$. Apply part (ii) of Multivariate Taylor Expansion with Uniform Second-Order Remainder to $f=\psi_c$ on $W=U\times V$ with $n=l+m$, at the points $(y_u,a_u)$ and $(x_u,a_u)$, whose segment lies in $\Delta^l\times\mathcal{A}$ by convexity, with $h=(e_u,0)$, $|h|=|e_u|$ and $M_2=3K$: since $h_i=0$ for $i>l$,

$$\bigl|\psi_c(x_u,a_u)-\psi_c(y_u,a_u)-g^{c}(y_u,a_u)\cdot e_u\bigr|\le\tfrac12(l+m)\,3K\,|e_u|^{2}=\Lambda_2|e_u|^{2}.$$

Next apply part (i) of the same lemma to $f=\partial_i\psi_c$ ($i\le l$), of class $C^1$ on $W$, at the points $(S_u,a_u)$ and $(y_u,a_u)$ with $h=(y_u-S_u,0)$ and $M_1=3K$: $|\partial_i\psi_c(y_u,a_u)-\partial_i\psi_c(S_u,a_u)|\le\sqrt{l+m}\,3K\,|y_u-S_u|$, so that, summing the squares of the $l$ components, $|g^{c}(y_u,a_u)-g^{c}(S_u,a_u)|\le\sqrt{l}\sqrt{l+m}\,3K\,|y_u-S_u|=\Lambda_3|y_u-S_u|$. Now, by claim 1,

$$\rho_u=\sum_{c}v_c\Bigl(\psi_c(x_u,a_u)-\psi_c(y_u,a_u)-g^{c}(y_u,a_u)\cdot e_u\Bigr)+\sum_{c}v_c\Bigl(\bigl(g^{c}(y_u,a_u)-g^{c}(S_u,\mathsf{A}_u)\bigr)\cdot e_u\Bigr),$$

and $|(g^{c}(y_u,a_u)-g^{c}(S_u,\mathsf{A}_u))\cdot e_u|\le\bigl(\Lambda_3|y_u-S_u|+|g^{c}(S_u,a_u)-g^{c}(S_u,\mathsf{A}_u)|\bigr)|e_u|$ by the triangle inequality and the Cauchy-Schwarz inequality. Taking norms, using $|v_c|=\sqrt{2}$ and summing over the $l(l-1)$ labels gives the stated bound.

Claim 3. Put $z_t=e_t-\mathfrak{m}_t$, a bounded measurable map. Subtracting the two integral equations of the statement and using linearity of the integral,

$$z_t=e_0+\int_{[0,t]}\bigl(b(x_u,a_u)-b(y_u,a_u)\bigr)\,du=e_0+\int_{[0,t]}\bigl(\mathcal{E}^{\star}_u z_u+g_u\bigr)\,du,\qquad g_u:=\mathcal{E}^{\star}_u\mathfrak{m}_u+\rho_u,$$

since $b(x_u,a_u)-b(y_u,a_u)=\mathcal{E}^{\star}_ue_u+\rho_u=\mathcal{E}^{\star}_uz_u+\mathcal{E}^{\star}_u\mathfrak{m}_u+\rho_u$ by the definition of $\rho$ and claim 1 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product. Here $g$ is bounded measurable (claims 1 and 2). By claims 2 and 3 of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution applied with $k=l$, $A(u)=\mathcal{E}^{\star}_u$, $\xi=e_0$ and this $g$, the bounded measurable solution $z$ is given by $z_t=\Phi(t)e_0+\int_{[0,t]}\Phi^{\mathcal{E}}(t,u)g_u\,du$, where $\Phi(t)=\Phi^{\mathcal{E}}(t,0)$ because $\Psi(0)=I_l$ by claim 2 of Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations. Adding $\mathfrak{m}_t$ and splitting the integral by linearity gives the identity of claim 3; the integrands $u\mapsto\Phi^{\mathcal{E}}(t,u)\mathcal{E}^{\star}_u\mathfrak{m}_u$ and $u\mapsto\Phi^{\mathcal{E}}(t,u)\rho_u$ are bounded measurable by claim 2 of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution.

Claim 4. By claim 3, $e_t-\Phi^{\mathcal{E}}(t,0)e_0-L_t(\mathfrak{m})=\int_{[0,t]}\Phi^{\mathcal{E}}(t,u)\rho_u\,du$, whose norm is at most $\int_{[0,t]}|\Phi^{\mathcal{E}}(t,u)\rho_u|\,du\le\bar{\Phi}^{2}\int_{[0,t]}|\rho_u|\,du$ by Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval, the bound $|\Phi^{\mathcal{E}}(t,u)y|\le\bar{\Phi}^{2}|y|$ of the statement, and monotonicity of the integral.