## 4 Theoretical Results
\right)E\},j\in\mathcal{M},\]which leads to  
\[\left\|\dot{X}_{c}\right\|\leq\left(\left\|\bar{A}\right\|+\left\|\bar{B}\right\| \left\|\bar{K}\right\|+\left\|\bar{B}\right\|\left\|H\right\|\right)\left\|X_{c }\right\|+\left(\left\|\bar{B}\right\|+\left\|\bar{L}\right\|\right)\left\|E \right\|\leq L_{1}\left\|X_{c}\right\|+L_{2}\left\|E\right\|. \tag{13}\]  
Similarly, one has  
\[\frac{d}{dt}\left\|e_{u_{s}}\right\|\leq\left\|e_{u_{s}}\right\|\leq\left\| \Gamma_{u}^{s}\bar{K}\dot{X}_{c}\right\|\leq L_{1,u_{s}}\left\|X_{c}\right\|+L _{2,u_{s}}\left\|E\right\|,s=1,\ldots,p, \tag{14a}\] \[\frac{d}{dt}\left\|e_{y_{t}}\right\|\leq\left\|\dot{e}_{y_{t}}\right\|\leq \left\|\Pi_{y}^{t}\tilde{C}\dot{X}_{c}\right\|\leq L_{1,y_{t}}\left\|X_{c} \right\|+L_{2,y_{t}}\left\|E\right\|,l=1,\ldots,r, \tag{14b}\]  
where \(L_{1,u_{s}}=\left\|\Pi_{u}^{s}\bar{K}\right\|L_{1}\), \(L_{2,u_{s}}=\left\|\Pi_{u}^{s}\bar{K}\right\|L_{2}\), \(L_{1,y_{t}}=\left\|\Pi_{y}^{t}\bar{\mathcal{C}}\right\|L_{1}\), and \(L_{2,y_{t}}=\left\|\Pi_{y}^{t}\bar{\mathcal{C}}\right\|L_{2}\). Denote by \(\{t_{k}\}_{k=0}^{\infty}=\underset{s=1}{p}\{t_{s}^{s}\}\cup\underset{l=1}{ \cup}\{t_{k_{l}}^{t}\}\) the overall triggering instants. By definitions, there may exist more than one node being triggered for some \(t_{k}\). In addition, since there is no Zeno phenomenon, \(\underset{k\rightarrow\infty}{\lim}t_{k}=\infty\). Initially, by contradiction, we prove the inequality \(\int_{t_{0}}^{t}E^{\mathrm{T}}\mathrm{d}\omega\leq\theta\rho_{0}^{2}\int_{t_ {0}}^{t}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) for \(t\in[t_{0},t_{1})\). If \(X_{c}(t_{0})=0\), the conclusion holds obviously. In the case \(X_{c}(t_{0})\
eq 0\), assume the positive instant \(T_{0}>t_{0}\) (it allows being infinity) as the first time when \(\int_{t_{0}}^{T_{0}}\left\|E\right\|^{2}\mathrm{d}\omega=\theta\rho_{0}^{2} \int_{t_{0}}^{T_{0}}\left\|X_{c}\right\|^{2}\mathrm{d}\omega\). Such a \(T_{0}\) exists because of \(E(t_{0})=0\). If \(T_{0}\geq t_{1}\), the inequality holds. If \(T_{0}<t_{1}\), \(\int_{t_{0}}^{t}E^{\mathrm{T}}\mathrm{d}\omega\leq\theta\rho_{0}^{2}\int_{t_ {0}}^{t}X_{c}^{\mathrm{T}}X_{c}\mathrm{d}\omega\) for \(t\in[t_{0},T_{0}]\), and, at this point, it can be proved that for \(t\in[t_{0},T_{0}]\),  
\[\int_{t_{0}}^{t}\left\|e_{u_{k}}\right\|^{2}\mathrm{d}\omega\leq\sigma_{u}^{s} \left\|\Pi_{u}^{s}\bar{K}\right\|^{2}\int_{t_{0}}^{t}\left\|X_{c}\right\|^{2} \mathrm{d}\omega,\int_{t_{0}}^{t}\left\|e_{y_{t}}\right\|^{2}\mathrm{d}\omega \leq\sigma_{y}^{t}\left\|\Pi_{y}^{t}\bar{\mathcal{C}}\right\|^{2}\int_{t_{0} }^{t}\left\|X_{c}\right\|^{2}\mathrm{d}\omega. \tag{15}\]  
To illustrate Label (15), for example, we consider \(e_{u_{1}}\). First, in the case \(T_{0}-t_{0}\leq\tau_{u}^{1}\), Label (14a) implies that  
\[\left\|e_{u_{1}}\right\|^{2} \leq\left(\int_{t_{0}}^{t}L_{1,u_{1}}\left\|X_{c}\right\|+L_{2,u_{ 1}}\left\|E\right\|\mathrm{d}\omega\right)^{2} \tag{16}\] \[\leq 2(t-t_{0})\int_{t_{0}}^{t}\left(L_{1,u_{1}}\left\|X_{c} \right\|\right)^{2}+\left(L_{2,u_{1}}\left\|E\right\|\right)^{2}\mathrm{d}\omega,\]  
where the last inequality employs the Cauchy-Schwarz inequality and the fact that \((a+b)^{2}<2a^{2}+2b^{2}\).