## 3 Main Results
\begin{split}\dot{V}_{a}(t)&\leq-q_{1}\big{\|}\dot{ \xi}_{2}\big{\|}^{2}+2q_{2}\|e\|\big{\|}\dot{\xi}_{2}\big{\|}+q_{3}\|e\|^{2}- \varepsilon_{2}\delta^{T}\delta\\ &\leq-q_{1}\big{\|}\dot{\xi}_{2}\big{\|}^{2}+2q_{2}\bigg{(}\frac{ 1}{2\rho}\|e\|^{2}+\frac{\rho}{2}\|\dot{\xi}_{2}\|^{2}\bigg{)}+q_{3}\|e\|^{2}- \varepsilon_{2}\delta^{T}\delta\\ &\leq-\sum\limits_{i=1}^{N}\bigg{[}(q_{1}-q_{2}\rho)\big{\|}\dot{ \xi}_{2}\big{\|}^{2}-\bigg{(}\frac{q_{2}}{\rho}+q_{3}\bigg{)}\|e_{i}\|^{2} \bigg{]}-\varepsilon_{2}\delta^{T}\delta\end{split}  \tag{30}\]  
We have \(\dot{V}_{a}(t)<0\), and we can obtain  
\[q_{1}-q_{2}\rho>0,\|e_{i}\|^{2}<\frac{\gamma_{i}(q_{1}-q_{2}\rho)}{q_{2}\rho^ {-1}+q_{3}}\big{\|}\dot{\xi}_{2}\big{\|}^{2} \tag{31}\]  
Then, we have  
\[\dot{V}_{a}(t)\leq-\alpha V_{a}(t) \tag{32}\]  
where \(\alpha=\frac{\varepsilon_{2}}{\theta_{\min}(P)}\). By Lyapunov stability theory and Definition 1, one can obtain that all nodes of CCPNs (1) are synchronized under the distributed controller (14) without DoS attacks. The proof is completed. The Zeno and singular triggering behaviors are the most common failure behaviors in the ETC strategy, in which the Zeno behavior is defined as multiple transmissions in a continuous time, and singular triggering behavior is defined as no further trigger after a single trigger. The Zeno and singular triggering behaviors can cause the proposed event-triggered strategy to be ineffective, so these two behaviors must be excluded to ensure the efficient transmission of system information. Then, the following Lemmas show that Zeno behavior is excluded and no node exhibits singular triggering behavior. For CCPNs (1) without DoS attacks, under the event-triggered controller (14), all nodes will not have Zeno behavior if the inter-event time intervals satisfy  
\[\tau>\max\big{\{}b_{1},b_{2}\big{\}}^{-1}\ln\Bigg{\{}\sqrt{\frac{\gamma_{i}(q_ {1}-q_{2}\rho)}{q_{2}\rho^{-1}+q_{3}}}+1\Bigg{\}} \tag{33}\]  
where \(b_{1}=\Gamma_{\max}\{I_{N}\otimes BK\}\), and  
\(b_{2}=\max\{\Gamma_{\max}\{I_{N}\otimes A+H\otimes BK\}+\Gamma_{\max}\{H \otimes BK+G\otimes\Gamma+I_{N}\otimes A\}\}\). Proof. : Let \(e_{i}(t)=x_{i}\big{(}t_{k}^{i}\big{)}-x_{i}(t)\), then we obtain  
\[ \begin{split}\dot{e}_{i}(t)&=Ax_{i}\big{(}t_{k}^{i }\big{)}-Ax_{i}(t)-\sum\limits_{j=1}^{N}g_{ij}\Gamma x_{i}(t)-Bu_{i}(t)=A\big{(} x_{i}\big{(}t_{k}^{i}\big{)}-x_{i}(t)\big{)}\\ &-\sum\limits_{j=1}^{N}g_{ij}\Gamma x_{i}(t)-BK\sum\limits_{j=1}^{ N}a_{ij}^{0}\big{(}x_{j}\big{(}t_{k}^{i}\big{)}-x_{i}\big{(}t_{k}^{i}\big{)}\big{)}- BKb_{i}^{0}\big{(}x_{N+1}(t)-x_{i}\big{(}t_{k}^{i}\big{)}\big{)}\\ &=A\big{(}x_{i}\big{(}t_{k}^{i}\big{)}-x_{i}(t)\big{)}-\sum\limits _{j=1}^{N}g_{ij}\Gamma x_{i}(t)-BK\sum\limits_{j=1}^{N}a_{ij}^{0}\big{(}x_{j} (t)-x_{i}(t)\big{)}\\ &-BK\sum\limits_{j=1}^{N}a_{ij}^{0}\big{(}e_{j}(t)-e_{i}(t)\big{)} -BKb_{i}^{0}(x_{N+1}(t)-x_{i}(t))+BKb_{i}^{0}e_{i}(t)\end{split}  \tag{34}\]  
The system (34) can be rewritten as  
\[\dot{e}=[(I_{N}\otimes A)-(G\otimes\Gamma)+(H\otimes BK)]e-(H\otimes BK)\delta \tag{35}\]  
We have  
\[\frac{d}{dt}\Bigg{(}\frac{\|e_{i}\|}{\|\dot{\xi}_{2}\|}\Bigg{)}\leq\frac{\|e_ {i}\|}{\|\dot{\xi}_{2}\|}+\frac{\|e_{i}\|\Big{\|}\dot{\xi}_{i2}\Big{\|}}{\| \dot{\xi}_{2}\|}\Bigg{\|} \tag{36}\]It follows from (36) that  
\[\|\dot{e}_{i}\|\leq\|(I_{N}\otimes A)+(G\otimes\Gamma)+(H\otimes BK)\|\|e_{i}\|- \|(H\otimes BK)\|\|\delta_{i}\| \tag{37}\]  
From (36) and (37), it yields  
\[ \begin{split}&\frac{d}{dt}\left(\frac{\|e_{i}\|}{\|\dot{\xi}_{i2}\|} \right)\\ &\leq\frac{\|(I_{N}\otimes A)+(G\otimes\Gamma)+(H\otimes BK)\| \|e_{i}\|}{\|\dot{\xi}_{i2}\|}-\frac{\|(H\otimes BK)\|\|\delta_{i}\|}{\|\dot{ \xi}_{i2}\|}+\frac{\|e_{i}\|\left\|\dot{\xi}_{i2}\right\|}{\|\dot{\xi}_{i2}\| }\\ &\leq\|(I_{N}\otimes A)+(H\otimes BK)\|\frac{\|e_{i}\|}{\|\dot{ \xi}_{i2}\|}+\|(I_{N}\otimes BK)\|+(G\otimes\Gamma)\frac{\|e_{i}\|}{\|\dot{\xi }_{i2}\|}\\ &\qquad\qquad\qquad\qquad+\|(H\otimes BK)\|\frac{\|e_{i}\|}{\|\dot {\xi}_{i2}\|}+(I_{N}\otimes A)\frac{\|e_{i}\|}{\|\dot{\xi}_{i2}\|}\end{split}