Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect

Adopt the setting and notation of Open-Loop Aggregate Solution Driven by Aggregate Transition Clocks and Existence, Uniqueness, Causality, and Measurability of the Open-Loop Aggregate Solution: natural numbers $N\ge1$, $l\ge2$, $m\ge1$, 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 $B$, the probability simplex $\Delta^l$ with its standard basis vectors $\delta_1,\dots,\delta_l$, the aggregate lattice $\mathbb{G}_N\subseteq\Delta^l$, the set $\mathcal{L}$ of transition labels $c=(\sigma,\gamma)$, which has $l(l-1)$ elements, their vectors $v_c=\delta_\gamma-\delta_\sigma$, clock families, control paths $s\mapsto a_s$, open-loop aggregate solutions with consumed clock times $\mathsf{C}^{c}$ and counters $\mathsf{N}^{c}$, and conflict-free data; coordinates of points of $\mathbb{R}^l$ carry superscripts, $\Sigma=(\Sigma^1,\dots,\Sigma^l)$, the letter $\Sigma$ denoting a generic point of $U$ in the formulas defining $\psi_c$, $g^{c}$ and $\mathcal{E}$ below and the solution path elsewhere. Write $|\cdot|$ for the Euclidean norm, $x\cdot y$ for the dot product, $\int_{[0,t]}\cdot\,ds$ for the Lebesgue integral over the compact interval $[0,t]$ (taken componentwise for maps into $\mathbb{R}^l$, and equal to $0$ for $t=0$), $\exp$ for the exponential function, and $\sqrt{\cdot}$ for the nonnegative square root; a jump time of a counting path $q$ is a real number $u>0$ with $q(u)>q(u-)$, as in that definition.

Write $\#S$ for the number of elements of a finite set $S$. Let $(U,V,\bar{\beta})$ be a twice continuously differentiable extension of $\beta$ with derivative bound $K$ (the letter $K$ is used here only for this bound, never for the stopping index of the aggregate recursion); adopt its identification of $\mathbb{R}^l\times\mathbb{R}^m$ with $\mathbb{R}^{l+m}$, its coordinates $x_1,\dots,x_{l+m}$ of a point $x=(\Sigma,\alpha)$, and its partial derivatives $\partial_i$. For a label $c=(\sigma,\gamma)$ define the label rate $\psi_c:U\times V\to\mathbb{R}$ by

$$\psi_c(\Sigma,\alpha)=\Sigma^{\sigma}\,\bar{\beta}(\sigma,\gamma,\Sigma,\alpha),$$

and, where the partial derivatives exist, its state gradient $g^{c}(\Sigma,\alpha)=(\partial_1\psi_c,\dots,\partial_l\psi_c)(\Sigma,\alpha)\in\mathbb{R}^l$ and the drift Jacobian $\mathcal{E}(\Sigma,\alpha)$, the real matrix with $l$ rows and $l$ columns and entries

$$\mathcal{E}^{\gamma\eta}(\Sigma,\alpha)=\sum_{c\in\mathcal{L}}v_c^{\gamma}\,\partial_\eta\psi_c(\Sigma,\alpha)\qquad(\gamma,\eta\in\{1,\dots,l\}),$$

so that $\mathcal{E}(\Sigma,\alpha)y=\sum_{c}v_c\,(g^{c}(\Sigma,\alpha)\cdot y)$ for $y\in\mathbb{R}^l$, with the matrix-vector product. Put

$$\Lambda_1=\sqrt{l+m}\,(B+K),\qquad \Lambda_2=\tfrac{3}{2}\,(l+m)\,K .$$

Data. Fix a clock family $p$, a control path $a$, a point $x_0\in\mathbb{G}_N$, a label $c_0=(\sigma_0,\gamma_0)$, a natural number $\mathsf{m}\ge1$ (unrelated to the control dimension $m$), and real numbers $0<u_1<u_2<\dots<u_{\mathsf{m}}$ none of which is a jump time of $p^{c_0}$. The perturbed clock family $p^{+}$ has $p^{+,c}=p^{c}$ for $c\neq c_0$ and

$$p^{+,c_0}(u)=p^{c_0}(u)+\#\{k\in\{1,\dots,\mathsf{m}\}:\ u_k\le u\}\qquad(u\ge0);$$

$p^{+,c_0}$ is a counting path (claim 1 below), so $p^{+}$ is a clock family. Fix real numbers $R\ge NBT$, $L\ge0$ and $D\ge0$ and assume the discrepancy hypothesis

$$\textbf{(D)}\qquad\bigl|p^{c}(u')-p^{c}(u)-(u'-u)\bigr|\le D\quad\text{for every }c\in\mathcal{L}\text{ and all }0\le u\le u'\le R\text{ with }u'-u\le L .$$

Assume that both $(p,a,x_0)$ and $(p^{+},a,x_0)$ are conflict-free, and let $\Sigma$ and $\Sigma^{+}$ be the open-loop aggregate solutions for these data, with consumed clock times $\mathsf{C}^{c}$, $\mathsf{C}^{+,c}$ and counters $\mathsf{N}^{c}$, $\mathsf{N}^{+,c}$. Define the response $Y_t=N(\Sigma^{+}_t-\Sigma_t)\in\mathbb{R}^l$ and the insertion count $\iota_t=\#\{k\in\{1,\dots,\mathsf{m}\}:\ u_k\le\mathsf{C}^{+,c_0}_t\}$ for $t\in[0,T]$, and put

$$A_0=\sqrt{2}\,\bigl(\mathsf{m}+l(l-1)D\bigr)\exp\bigl(\sqrt{2}\,l(l-1)\Lambda_1T\bigr).$$

1. (Label rates and the exact response identity.) $p^{+}$ is a clock family. For every label $c$ the label rate $\psi_c$ is of class $C^2$ on $U\times V$, and at every point of $\Delta^l\times\mathcal{A}$ its partial derivatives satisfy $|\partial_i\psi_c|\le B+K$ and $|\partial_j\partial_i\psi_c|\le3K$ for all $i,j\in\{1,\dots,l+m\}$; moreover $\psi_c(\Sigma,\alpha)=\Sigma^\sigma\beta(\sigma,\gamma,\Sigma,\alpha)$ on $\Delta^l\times\mathcal{A}$, so that $\mathsf{C}^{c}_t=\int_{[0,t]}N\psi_c(\Sigma_s,a_s)\,ds$ and likewise for $\Sigma^{+}$. The maps $s\mapsto Y_s$, $s\mapsto g^{c}(\Sigma_s,a_s)$ and $s\mapsto\mathcal{E}(\Sigma_s,a_s)Y_s$ are bounded on $[0,T]$ with components measurable with respect to the trace Borel $\sigma$-algebra. For every $t\in[0,T]$,

$$Y_t=v_{c_0}\,\iota_t+\sum_{c\in\mathcal{L}}v_c\Bigl(p^{c}\bigl(\mathsf{C}^{+,c}_t\bigr)-p^{c}\bigl(\mathsf{C}^{c}_t\bigr)\Bigr),\qquad 0\le\iota_t\le\mathsf{m},$$

and for every label $c$,

$$\bigl|\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t\bigr|\le\Lambda_1\int_{[0,t]}|Y_s|\,ds .$$

2. (Crude bound.) If $\Lambda_1TA_0<L$, then $|Y_t|\le A_0$ and $|\mathsf{C}^{+,c}_t-\mathsf{C}^{c}_t|\le\Lambda_1TA_0$ for every $t\in[0,T]$ and every label $c$.

3. (Linearisation defect.) If $\Lambda_1TA_0<L$, then for every $t\in[0,T]$ the defect $\mathsf{d}_t\in\mathbb{R}^l$ defined by

$$Y_t=v_{c_0}\,\iota_t+\int_{[0,t]}\mathcal{E}(\Sigma_s,a_s)\,Y_s\,ds+\mathsf{d}_t\qquad\text{satisfies}\qquad |\mathsf{d}_t|\le\sqrt{2}\,l(l-1)\Bigl(D+\frac{\Lambda_2\,T\,A_0^{2}}{N}\Bigr).$$