## 4 Theoretical Results
By integrating Label (16), one has  
\[\int_{t_{0}}^{t}\left\|e_{u_{1}}\right\|^{2}\mathrm{d}\omega\leq(t-t_{0})^{2} \int_{t_{0}}^{t}\left(L_{1,u_{1}}\left\|X_{c}\right\|\right)^{2}+\left(L_{2,u_ {1}}\left\|E\right\|\right)^{2}\mathrm{d}\omega. \tag{17}\]  
Since \(\int_{t_{0}}^{t}E^{\mathrm{T}}\mathrm{d}\omega\leq\theta\rho_{0}^{2}\int_{t_ {0}}^{t}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) for \(t\in[t_{0},T_{0}]\), Label (17) leads to  
\[\int_{t_{0}}^{t}\left\|e_{u_{1}}\right\|^{2}\mathrm{d}\omega\leq(t-t_{0})^{2} (L_{1,u_{1}}^{2}+\theta\rho_{0}^{2}L_{2,u_{2}}^{2})\int_{t_{0}}^{t}\left\|X_{c} \right\|^{2}\mathrm{d}\omega.\]  
Due to item 2 and the fact \(T_{0}-t_{0}\leq\tau_{u}^{1}\), \(\int_{t_{0}}^{t}\left\|e_{u_{s}}\right\|^{2}\mathrm{d}\omega\leq\sigma_{u}^{s} \left\|\Pi_{u}^{s}\bar{K}\right\|^{2}\int_{t_{0}}^{t}\left\|X_{c}\right\|^{2} \mathrm{d}\omega\) for \(t\in[t_{0},T_{0}]\). In the case \(T_{0}>\tau_{u}^{1}\), Label (15) can be obtained by the integral part of the event condition (4). In fact,  
\[\int_{t_{0}}^{t}\left\|e_{u_{1}}\right\|^{2}\mathrm{d}\omega\leq\sigma_{u}^{1} \int_{t_{0}}^{t}\left\|u_{1}\right\|^{2}\mathrm{d}\omega\leq\sigma_{u}^{1} \left\|\Pi_{u}^{1}\bar{K}\right\|^{2}\int_{t_{0}}^{t}\left\|X_{c}\right\|^{2} \mathrm{d}\omega.\]  
The similar analysis can be applied to other \(e_{u_{s}}\) and \(e_{y_{t}}\) by using Label (14a) and Label (14b). Hence, Label (15) holds for \(t\in[t_{0},T_{0}]\). Thus, according to the item 1 in the theorem, for \(t\in(t_{0},T_{0}]\),  
\[\int_{t_{0}}^{t}\left\|E\right\|^{2}\mathrm{d}\omega =\int_{t_{0}}^{t}\sum_{s=1}^{p}\left\|e_{u_{s}}\right\|^{2}+\sum_{ l=1}^{r}\left\|e_{y_{l}}\right\|^{2}\mathrm{d}\omega\] \[\frac{d}{dt}\left\|e_{u_{s}}\right\|\leq\left\|e_{u_{s}}\right\|\leq\left\| \Gamma_{u}^{s}\bar{K}\dot{X}_{c}\right\|\leq L_{1,u_{s}}\left\|X_{c}\right\|+L _{2,u_{s}}\left\|E\right\|,s=1,\ldots,p, \tag{14a}\]0 \[\frac{d}{dt}\left\|e_{u_{s}}\right\|\leq\left\|e_{u_{s}}\right\|\leq\left\| \Gamma_{u}^{s}\bar{K}\dot{X}_{c}\right\|\leq L_{1,u_{s}}\left\|X_{c}\right\|+L _{2,u_{s}}\left\|E\right\|,s=1,\ldots,p, \tag{14a}\]1  
Namely, \(\int_{t_{0}}^{T_{0}}\left\|E\right\|^{2}\mathrm{d}\omega<\theta\rho_{0}^{2} \int_{t_{0}}^{T_{0}}\left\|X_{c}\right\|^{2}\mathrm{d}\omega\), which contradicts the definition of \(T_{0}\). Therefore, \(\int_{t_{0}}^{t}\left\|E\right\|^{2}\mathrm{d}\omega\leq\theta\rho_{0}^{2} \int_{t_{0}}^{t}\left\|X_{c}\right\|^{2}\mathrm{d}\omega\) for \(t\in[t_{0},t_{1})\). Next, we consider the interval \([t_{k},t_{k+1})\). At this point, assume that Label (15) and \(\int_{t_{0}}^{t}E^{\mathrm{T}}\mathrm{E}\mathrm{d}\omega\leq\theta\rho_{0}^{2 }\int_{t_{0}}^{t}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) hold for \(t\in[t_{0},t_{k})\). Define \(T_{k}>t_{k}\) as the first time when \(\int_{t_{0}}^{T_{k}}E^{\mathrm{T}}\mathrm{E}\mathrm{d}\omega=\theta\rho_{0}^{2 }\int_{t_{0}}^{t_{k}}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\). If the event conditions in all of the nodes are triggered at \(t_{k}\), then \(\int_{t_{0}}^{t_{k}}E^{\mathrm{T}}\mathrm{E}\mathrm{d}\omega\leq\theta\rho_{0} ^{2}\int_{t_{0}}^{t_{k}}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) as well as \(E(t_{k})=0\), and, hence, \(T_{k}\) is proved to obviously exist. If there are some nodes where the event conditions are not triggered at \(t_{k}\), for example \(u_{s}\), one has \(\int_{t_{0}}^{t_{k}}\left\|e_{u_{s}}\right\|^{2}\mathrm{d}\omega<\sigma_{u}^{s }\left\|\Pi_{u}^{s}\bar{K}\right\|^{2}\int_{t_{0}}^{t_{k}}\left\|X_{c}\right\| ^{2}\mathrm{d}\omega\). As a result, \(\int_{t_{0}}^{t_{k}}E^{\mathrm{T}}\mathrm{E}\mathrm{d}\omega<\theta\rho_{0}^{2 }\int_{t_{0}}^{t}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) implies the existence of \(T_{k}\). For \(u_{s}\), if \(t_{s}^{s}=t_{k}\), then \(\int_{t_{0}}^{t_{k}}\left\|e_{u_{s}}\right\|^{2}\mathrm{d}\omega\leq\sigma_{u} ^{s}\left\|\Pi_{u}^{s}\bar{K}\right\|^{2}\int_{t_{0}}^{t_{k}}\left\|X_{c}\right\| ^{2}\mathrm{d}\omega\).