Removal Form of the Shared-Clock Insertion Response: Discrepancy on the Enlarged Clock Family, the Base-Clock Insertion Count, and the Linearisation Defect Along Either Path

Adopt the setting, notation and data of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, with the sole exception of its discrepancy hypothesis (D), which is not assumed here: the natural numbers $N\ge1$, $l\ge2$, $m\ge1$, the nonempty control set $\mathcal{A}\subseteq\mathbb{R}^m$, a subset of Euclidean space, the real numbers $B\ge0$, $T>0$, $R\ge NBT$, $L\ge0$ and $D\ge0$, the transition-rate family $\beta$ with its twice continuously differentiable extension $(U,V,\bar{\beta})$ with derivative bound $K$, the probability simplex $\Delta^l$ with its standard basis vectors $\delta_1,\dots,\delta_l$, the aggregate lattice $\mathbb{G}_N$, the set $\mathcal{L}$ of transition labels, which has $l(l-1)$ elements, with vectors $v_c$, the identification of $\mathbb{R}^l\times\mathbb{R}^m$ with $\mathbb{R}^{l+m}$ with coordinates $x_1,\dots,x_{l+m}$ and partial derivatives $\partial_i$, the label rates $\psi_c$, state gradients $g^{c}$ and drift Jacobian $\mathcal{E}$, the constants $\Lambda_1=\sqrt{l+m}\,(B+K)$ and $\Lambda_2=\tfrac32(l+m)K$, the clock family $p$, the control path $a$, the point $x_0\in\mathbb{G}_N$, the label $c_0$, the natural number $\mathsf{m}\ge1$ (unrelated to the control dimension $m$), the insertion points $0<u_1<\dots<u_{\mathsf{m}}$ (none a jump time of $p^{c_0}$), the perturbed clock family $p^{+}$, the open-loop aggregate solutions $\Sigma$ and $\Sigma^{+}$ for the conflict-free data $(p,a,x_0)$ and $(p^{+},a,x_0)$ with consumed clock times $\mathsf{C}^{c}$ and $\mathsf{C}^{+,c}$, the response $Y_t=N(\Sigma^{+}_t-\Sigma_t)$ and the insertion count $\iota_t$; also the Euclidean norm $|\cdot|$, the dot product $x\cdot y$, the Lebesgue integral over the compact interval $\int_{[0,t]}\cdot\,ds$ (componentwise for maps into $\mathbb{R}^l$, and equal to $0$ for $t=0$), the exponential function $\exp$, $\sqrt{\cdot}$ for the nonnegative square root, and the matrix-vector product. As in that lemma, coordinates of points of $\mathbb{R}^l$ carry superscripts, and the letter $\Sigma$ denotes a generic point of $U$ in the formulas defining $\psi_c$, $g^{c}$ and $\mathcal{E}$ and the solution path elsewhere. Write $\#S$ for the number of elements of a finite set $S$.

In place of the discrepancy hypothesis (D) of that lemma, which concerns the clock family $p$ and is not assumed, assume the discrepancy hypothesis on the enlarged clock family

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

Put

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

which is the constant $A_0$ of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect formed with $D+\mathsf{m}$ in place of $D$, and define the base-clock insertion count

$$\iota^{-}_t=\#\{k\in\{1,\dots,\mathsf{m}\}:\ u_k\le\mathsf{C}^{c_0}_t\}\qquad(t\in[0,T]).$$

1. (Discrepancy of the base clock family.) The clock family $p$ satisfies hypothesis (D) of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect with tolerance $D+\mathsf{m}$ in place of $D$. Consequently claims 1 and 2 of that lemma hold for the present data with $D+\mathsf{m}$ in place of $D$ and $A_0^{+}$ in place of $A_0$ (so does its claim 3, whose defect is attached to the count $\iota_t$; the defects of claim 3 below are attached to $\iota^{-}_t$ and carry the tolerance $D$ alone); in particular, 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 $c$.

2. (Removal-form response identity.) For every $t\in[0,T]$ one has $0\le\iota^{-}_t\le\mathsf{m}$ and

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

3. (Linearisation defect along either path.) Assume $\Lambda_1TA_0^{+}<L$. The maps $s\mapsto\mathcal{E}(\Sigma^{+}_s,a_s)Y_s$ and $s\mapsto\mathcal{E}(\Sigma_s,a_s)Y_s$, and $s\mapsto g^{c}(\Sigma^{+}_s,a_s)$ for every label $c$, are bounded on $[0,T]$ with components measurable with respect to the trace Borel $\sigma$-algebra, and for every $t\in[0,T]$ the defects $\mathsf{d}^{+}_t,\mathsf{d}^{-}_t\in\mathbb{R}^l$ defined by

$$Y_t=v_{c_0}\,\iota^{-}_t+\int_{[0,t]}\mathcal{E}(\Sigma^{+}_s,a_s)\,Y_s\,ds+\mathsf{d}^{+}_t,\qquad Y_t=v_{c_0}\,\iota^{-}_t+\int_{[0,t]}\mathcal{E}(\Sigma_s,a_s)\,Y_s\,ds+\mathsf{d}^{-}_t$$

satisfy

$$|\mathsf{d}^{+}_t|\le\sqrt{2}\,l(l-1)\Bigl(D+\frac{\Lambda_2\,T\,(A_0^{+})^{2}}{N}\Bigr),\qquad |\mathsf{d}^{-}_t|\le\sqrt{2}\,l(l-1)\Bigl(D+\frac{\Lambda_2\,T\,(A_0^{+})^{2}}{N}\Bigr).$$