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

Adopt the setting and notation of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect as far as the following objects are concerned: natural numbers $l\ge2$ and $m\ge1$, a nonempty subset $\mathcal{A}$ of Euclidean space $\mathbb{R}^m$, real numbers $B\ge0$ and $T>0$, a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate bound $B$, a twice continuously differentiable extension $(U,V,\bar{\beta})$ of $\beta$ with derivative bound $K$, the probability simplex $\Delta^l$, the set $\mathcal{L}$ of transition labels $c=(\sigma,\gamma)$ with their vectors $v_c=\delta_\gamma-\delta_\sigma$, the label rates $\psi_c$, state gradients $g^{c}$ and drift Jacobian $\mathcal{E}$ on $U\times V$, and the constant $\Lambda_2=\tfrac32(l+m)K$. Put

$$\Lambda_3=3K\sqrt{l(l+m)},\qquad \Lambda_{\mathcal{E}}=\sqrt{2}\,l(l-1)\sqrt{l}\,(B+K),$$

and define the drift $b:\Delta^l\times\mathcal{A}\to\mathbb{R}^l$ by $b(\Sigma,\alpha)=\sum_{c\in\mathcal{L}}v_c\,\psi_c(\Sigma,\alpha)$. Write $|\cdot|$ for the Euclidean norm, $x\cdot y$ for the dot product, $My$ for the matrix-vector product, and $\int_{[0,t]}\cdot\,du$ for the componentwise Lebesgue integral over the compact interval $[0,t]$ (equal to $0$ for $t=0$); a map from $[0,T]$ into a Euclidean space is bounded measurable when its components are bounded and measurable with respect to the trace Borel $\sigma$-algebra on $[0,T]$ and the Borel $\sigma$-algebra of the real line, and measurable when its components are measurable in this sense.

Comparison pair. Let $S:[0,T]\to\Delta^l$ and $\mathsf{A}:[0,T]\to\mathcal{A}$ be measurable maps such that every entry of $u\mapsto\mathcal{E}^{\star}_u:=\mathcal{E}(S_u,\mathsf{A}_u)$ is continuous on $[0,T]$, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric. Let $\Phi^{\mathcal{E}}(t,u)$ ($t,u\in[0,T]$) be the two-parameter fundamental solution of $u\mapsto\mathcal{E}^{\star}_u$ on $[0,T]$, and let $\bar{\Phi}\ge0$ be a real number with $|\Phi^{\mathcal{E}}(t,u)y|\le\bar{\Phi}^{2}|y|$ for all $t,u\in[0,T]$ and $y\in\mathbb{R}^l$; such a number exists by claim 1 of that lemma.

Perturbed and unperturbed paths. Let $a:[0,T]\to\mathcal{A}$ be a measurable map (the control path), let $x,y:[0,T]\to\Delta^l$ be measurable maps, and let $\mathfrak{m}:[0,T]\to\mathbb{R}^l$ be bounded measurable (the perturbation; the fraktur letter is unrelated to the control dimension $m$), such that for every $t\in[0,T]$

$$x_t=x_0+\int_{[0,t]}b(x_u,a_u)\,du+\mathfrak{m}_t,\qquad y_t=y_0+\int_{[0,t]}b(y_u,a_u)\,du,$$

the integrands being bounded measurable by claim 1 (taking $t=0$ shows $\mathfrak{m}_0=0$). Put $e_t=x_t-y_t$ and define the residual, the control-gradient discrepancy and the linear response by

$$\rho_u=b(x_u,a_u)-b(y_u,a_u)-\mathcal{E}^{\star}_u\,e_u,\qquad \mathsf{d}_u=\sum_{c\in\mathcal{L}}\bigl|g^{c}(S_u,a_u)-g^{c}(S_u,\mathsf{A}_u)\bigr|,\qquad L_t(\mathfrak{m})=\mathfrak{m}_t+\int_{[0,t]}\Phi^{\mathcal{E}}(t,u)\,\mathcal{E}^{\star}_u\,\mathfrak{m}_u\,du .$$

1. (The drift and its measurable compositions.) $b$ is the aggregate state drift of $\beta$; recall from the adopted setting that $\mathcal{E}(\Sigma,\alpha)z=\sum_{c\in\mathcal{L}}v_c\,(g^{c}(\Sigma,\alpha)\cdot z)$ for $(\Sigma,\alpha)\in U\times V$ and $z\in\mathbb{R}^l$. For all $(\Sigma,\alpha)\in\Delta^l\times\mathcal{A}$ and $z\in\mathbb{R}^l$: $|b(\Sigma,\alpha)|\le\sqrt{2}\,l(l-1)B$, $|g^{c}(\Sigma,\alpha)|\le\sqrt{l}\,(B+K)$ and $|\mathcal{E}(\Sigma,\alpha)z|\le\Lambda_{\mathcal{E}}|z|$. For every measurable $\Delta^l$-valued $x'$ and measurable $\mathcal{A}$-valued $a'$ on $[0,T]$, the maps $u\mapsto b(x'_u,a'_u)$, $u\mapsto g^{c}(x'_u,a'_u)$ and the entries of $u\mapsto\mathcal{E}(x'_u,a'_u)$ are bounded measurable; in particular $u\mapsto\mathsf{d}_u$ is bounded measurable with $0\le\mathsf{d}_u\le2l(l-1)\sqrt{l}\,(B+K)$.

2. (Residual bound.) $u\mapsto\rho_u$ is bounded measurable, and for every $u\in[0,T]$

$$|\rho_u|\ \le\ \sqrt{2}\,l(l-1)\,\Lambda_2\,|e_u|^{2}+\sqrt{2}\,l(l-1)\,\Lambda_3\,|y_u-S_u|\,|e_u|+\sqrt{2}\,\mathsf{d}_u\,|e_u| .$$

3. (Exact variation-of-constants identity.) For every $t\in[0,T]$,

$$e_t=\Phi^{\mathcal{E}}(t,0)\,e_0+L_t(\mathfrak{m})+\int_{[0,t]}\Phi^{\mathcal{E}}(t,u)\,\rho_u\,du ,$$

all integrands being bounded measurable.

4. (Linearisation bound.) For every $t\in[0,T]$,

$$\bigl|e_t-\Phi^{\mathcal{E}}(t,0)\,e_0-L_t(\mathfrak{m})\bigr|\ \le\ \bar{\Phi}^{2}\int_{[0,t]}|\rho_u|\,du .$$