A grazing guard contact has square-root rather than linear root-location sensitivity

The exact trajectory is $x(t)=0$, so the exact guard is $h(t)=(t-1{)}^2$. It has an interior root at $\tau =1$ and $h^{\mathrm{\prime }}(1)=0$. The guard does not cross either declared orientation, so this is a grazing guard contact, not an enabled hybrid event under [F1].

For any $\eta >0$, take the dense reconstruction $U_{\eta }(t)=-\eta$. Its uniform state error is $\eta$, while its guard $g(t,U_{\eta }(t))=(t-1{)}^2-\eta$ has zero residual at $\widehat{\tau }=1+\sqrt{\eta }$. Thus $\epsilon =0$ and $\mathrm{\mid }\widehat{\tau }-\tau \mathrm{\mid }=\sqrt{\eta }$.

If a fixed $C$ gave the refuted root-location bound, it would require $\sqrt{\eta }\le C\eta$ for every $\eta >0$, equivalently $1\mathrm{/}\sqrt{\eta }\le C$. The left side is unbounded as $\eta \downarrow 0$, so no such constant exists. Here $h^{\mathrm{\prime }}(1)=0$, so the positive derivative margin required by [F2] is absent; the calculation concerns sensitivity of a guard contact, not the time of an enabled event.