for all $t_0\in[0,T)$.

Moreover, we have control on $X$: $$\begin{aligned}
\label{L2_control_shift_piecewise_systems}
\int\limits_0^{t_0} (\dot{X}(t))^2\,dt\leq \mu_2(1+e^{\mu_1 t_0})\int\limits_{-R-rt_0+s(0)}^{R+rt_0+s(0)}\abs{u^0(x)-\bar{u}^0(x)}^2\,dx.\end{aligned}$$

-   The constants $\mu_1,\mu_2>0$ depend on $a$, $\rho$, $\norm{u}_{L^\infty}$, $\norm{\bar{u}}_{L^\infty}$, and bounds on the derivatives of $\eta$ on the range of $u$ and $\bar{u}$. In addition, $\mu_1$ depends on $C_G$ (see ), $\mbox{Lip}[\bar{u}]$, $R$, $T$, and bounds on the derivatives of $f$ on the range of $u$ and $\bar{u}$. Note that $r$ only depends on bounds on the derivatives of $f$ and $\eta$ (on the range of $u$ and $\bar{u}$).

-   As opposed to , the proof of will in fact go through whenever we have an estimate of the form $$\begin{aligned}
    \label{systems_new_estimate_remark_more_general}
    \abs{\int\limits_{x_1}^{x_2} \nabla\eta(u(x,t)|\bar{u}(x+X(t),t))G(u(\cdot,t))(x)\,dx} \leq C \int\limits_{x_1}^{x_2}\abs{\nabla\eta(u(x,t)|\bar{u}(x+X(t),t))}\,dx,\end{aligned}$$ for $x_1,x_2\in\mathbb{R}$ and some constant $C>0$. Note that $u\in L^\infty$ and implies .

-   Note that Hölder’s inequality and give control on the shift in the form of $$\begin{aligned}
    \frac{1}{t_0}\int\limits_0^{t_0}\abs{\dot{X}(t)}\,dt \leq \frac{\sqrt{\mu_2(1+e^{\mu_1 t_0})}}{\sqrt{t_0}}\norm{u^0(\cdot)-\bar{u}^0(\cdot)}_{L^2(-R-rt_0+s(0),R+rt_0+s(0))}.\end{aligned}$$

-   Note that by Property (b) of $(\mathcal{H}1)$ or $(\mathcal{H}1)^*$, condition is equivalent to the existence of a $\tilde{\rho}>0$ such that for all $t\in[0,T)$ $$\begin{aligned}
    \label{gap_local_case}
    r(t)>\tilde{\rho},\end{aligned}$$ where $r(t)$ satisfies $S^i_{\bar{u}(s(t)-,t)}(r(t))=\bar{u}(s(t)+,t)$.

The outline of the paper is as follows: in , we give our hypotheses on the system. In , we present technical lemmas. 