# Quasi-Consensus of Disturbed Nonlinear Multiagent Systems with Event-Triggered Impulsive Control
## 3 Main Results
** Note that \(v>1>e^{\rho(t_{k}-t_{k-1})}\) and \(e^{l_{k}-\lambda(t_{k}-t_{k-1})}\leqslant e^{\rho(t_{k}-t_{k-1})}\) with \(\rho>-\lambda\), and thus \(t_{k}-t_{k-1}\geqslant\frac{l_{k}}{\rho+\lambda}>0\). By simple deduction, it follows from (26) and (27) that  
\[\lim\limits_{k\rightarrow\infty}(t_{k}-t_{0})\geqslant\lim\limits_{k \rightarrow\infty}\sum\limits_{i=1}^{k}\frac{l_{i}}{\rho+\lambda}=+\infty. \tag{30}\]  
**(ii) \(e^{\rho(t_{k}-t_{k-1})}\leqslant\frac{4(t_{k}-t_{k-1})}{\eta_{2}}\)**. When \(e^{l_{k}-\lambda(t_{k}-t_{k-1})}\geqslant v\), we obtain \(v\leqslant\frac{4(t_{k}-t_{k-1})}{\eta_{2}}\), i.e., \(t_{k}-t_{k-1}\geqslant\frac{\eta_{2}v}{4}>0\), thus \(t_{k}-t_{0}\geqslant\frac{k\eta_{2}v}{4}\). When \(e^{l_{k}-\lambda(t_{k}-t_{k-1})}\leqslant v\), we obtain \(t_{k}-t_{k-1}\geqslant\frac{l_{k}-\ln v}{\lambda}>0\), thus \(t_{k}-t_{0}\geqslant\sum\limits_{i=1}^{k}\frac{l_{i}-\ln v}{\lambda}\). If (26) and (27) are held, we have \(t_{k}-t_{0}\rightarrow+\infty\) as \(k\rightarrow+\infty\) when \(e^{\rho(t_{k}-t_{k-1})}\leqslant\frac{4(t_{k}-t_{k-1})}{\eta_{2}}\). **Case 2: \(\rho>0\). ** It follows from (28) that  
\[e^{l_{k}-\lambda(t_{k}-t_{k-1})}E[V(e(t_{k-1}))]+vw^{2}\leqslant e^{\rho(t_{k }-t_{k-1})}E[V(e(t_{k-1}))]+\frac{4(e^{\rho(t_{k}-t_{k-1})}-1)}{\rho\eta_{2}}w^ {2}, \tag{31}\]  
which implies that \(\min\{e^{l_{k}-\lambda(t_{k}-t_{k-1})},v\}\leqslant\max\{e^{\rho(t_{k}-t_{k-1 })},\frac{4(e^{\rho(t_{k}-t_{k-1})}-1)}{\rho\eta_{2}}\}\). **(i) \(e^{\rho(t_{k}-t_{k-1})}\leqslant\frac{4(e^{\rho(t_{k}-t_{k-1})}-1)}{\rho\eta_{2 }}\)**. **(ii) \(e^{\rho(t_{k}-t_{k-1})}\geqslant\frac{4(e^{\rho(t_{k}-t_{k-1})}-1)}{\rho\eta_{2 }}\)**. When \(e^{l_{k}-\lambda(t_{k}-t_{k-1})}\geqslant v\), we have \(v\leqslant e^{\rho(t_{k}-t_{k-1})}\), i.e., \(t_{k}-t_{k-1}\geqslant\frac{\ln v}{\rho}>0\), thus \(t_{k}-t_{0}\geqslant\sum\limits_{i=1}^{k}\frac{l_{i}-\ln v}{\lambda}\). According to (26) and (27), we can \(\text{obtain}_{k}-t_{0}\rightarrow+\infty\) as \(k\rightarrow+\infty\) when \(e^{\rho(t_{k}-t_{k-1})}\geqslant\frac{4(e^{\rho(t_{k}-t_{k-1})}-1)}{\rho\eta_{2 }}=+\infty\). **Case 3:**\(\rho=0\). It follows from (28) that  
\[e^{l_{k}-\lambda(t_{k}-t_{k-1})}E[V(e(t_{k-1}))]+vw^{2}\leqslant E[V(e(t_{k-1}))]+ \frac{4}{\eta_{2}}w^{2}(t_{k}-t_{k-1}), \tag{32}\]  
which implies that \(\min\{e^{l_{k}-\lambda(t_{k}-t_{k-1})},v\}\leqslant\max\{1,\frac{4}{\eta_{2}}( t_{k}-t_{k-1})\}\). **(i)**\(\frac{4}{\eta_{2}}(t_{k}-t_{k-1})\geqslant 1\), i.e., \(t_{k}-t_{k-1}\geqslant\frac{\eta_{2}}{4}>0\), we have \(\lim\limits_{k\to\infty}(t_{k}-t_{0})\geqslant\lim\limits_{k\to\infty}\sum \limits_{i=1}^{k}\frac{t_{k}\eta_{2}}{4}=+\infty\). **(ii)**\(\frac{4}{\eta_{2}}(t_{k}-t_{k-1})\leqslant 1\). Since \(1<v\), thus \(e^{l_{k}-\lambda(t_{k}-t_{k-1})}\leqslant 1\), i.e., \(t_{k}-t_{k-1}\geqslant\frac{l_{k}}{\lambda}>0\). We have \(\lim\limits_{k\to\infty}(t_{k}-t_{0})\geqslant\lim\limits_{k\to\infty}\sum \limits_{i=1}^{k}\frac{l_{i}}{\lambda}=+\infty\), from which (27). Based on the above discussion, for any \(\rho\), Zeno behavior is excluded in this scenario and \(t_{k}\to+\infty\) as \(k\to+\infty\). _Scenario 3:_\(\mathcal{N}\) _is generated by both forced impulses and other events_  
In this scenario, suppose that \(T_{1}>t_{0}\) is the Zeno time and there exist infinitely many events which happened in the interval \([T_{0},T_{1})\) with \(T_{0}=T_{1}-\frac{\tau_{\rm sup}}{\tau}\), where \(\tau>1\). With the assumption above, there then exists, and only exists, one impulse as the forced impulse, and we assume that it occurs at \(t=\hat{t}\). Hence, impulses are generated by \(t_{k}=t_{k}^{*}\) in (9) for any \(t\in(\hat{t},T_{1})\). However, one can obtain the fact that Zeno behavior is excluded in Scenario 2, and thus, it is a contradiction. Therefore, Zeno behavior is also excluded in this scenario. The proof is completed. **Remark 5**.