## 3 Main Results
Theorem 2: _For systems (18) with the distributed controller (14) and (15) and the event-triggering condition (22) and (23), if there exists a symmetric matrix \(Q\) and \(F=B^{T}Q\) such that the following inequalities are satisfied_  
\[-\Psi_{\sigma(t)}\bar{H}_{\sigma(t)}-\Big{(}\bar{H}_{\sigma(t)}\Big{)}^{T} \Psi_{\sigma(t)}\leq 0 \tag{48}\]  
\[0<\Psi_{\sigma(t)}\leq\beta_{1}I_{N} \tag{49}\]  
_and_  
\[2\Big{(}QA+A^{T}Q+\beta_{2}Q\Gamma\Big{)}<\beta_{3}Q \tag{50}\]  
_where \(\Psi_{\sigma(t)}\in\{\Psi_{1},\Psi_{2},\ldots,\Psi_{w}\}\), \(\beta_{2}=\frac{\theta_{\max}\big{(}\textbf{V}_{\sigma(t)},G\big{)}}{\theta_{ \min}\big{(}\textbf{V}_{\sigma(t)}\big{)}},\beta_{3}>0\), and \(\Psi_{1},\Psi_{2},\ldots,\Psi_{w}\) are the positive definite matrices, then the following inequality can be obtained:_  
\[V_{b}(t)\leq e^{\beta_{3}\big{(}t-t_{2}^{i}\big{)}}V_{b}\Big{(}t_{2l}^{ij} \Big{)} \tag{51}\]Proof. : Constructing the following Lyapunov function  
\[\Big{|}\Big{\|}\xi_{i2}\Big{(}t_{k}^{i}\Big{)}\Big{\|}-\|\xi_{i2}(t)\|\Big{|}\leq \Gamma_{\max}\{-H\otimes I_{N}\}\sqrt{\frac{\gamma_{i}(q_{1}-q_{2}\rho)}{q_{2} \rho^{-1}+q_{3}}}\|e_{i}(t)\|=\sigma c_{i}\|e_{i}(t)\| \tag{45}\]0  
The time-derivative trajectory of (52) yields  
\[ \begin{split}\dot{V}_{b}(t)&=2\delta^{T}\Big{(}\Psi_ {\sigma(t)}\otimes Q\Big{)}\Big{[}\Big{(}I_{N}\otimes A+G\otimes\Gamma-\hat{H} _{\sigma(t)}\otimes BF\Big{)}\delta-\Big{(}\hat{H}_{\sigma(t)}\otimes BF \Big{)}e\Big{]}\\ &=2\delta^{T}\Big{(}\Psi_{\sigma(t)}\otimes Q\Big{)}\big{(}I_{N} \otimes A+G\otimes\Gamma\big{)}\delta-2\delta^{T}\Big{(}\Psi_{\sigma(t)} \otimes Q\Big{)}\Big{(}\hat{H}_{\sigma(t)}\otimes BF\Big{)}\delta\\ &\quad\quad-2\delta^{T}\Big{(}\Psi_{\sigma(t)}\otimes Q\Big{)} \Big{(}\hat{H}_{\sigma(t)}\otimes BF\Big{)}e=2\delta^{T}\Psi_{\sigma(t)} \otimes(QA+A^{T}Q)\delta\\ &\quad\quad+2\delta^{T}\Big{(}\Psi_{\sigma(t)}G\otimes Q\Gamma \Big{)}\delta-\delta^{T}\Big{(}\Psi_{\sigma(t)}\hat{H}_{\sigma(t)}+\Big{(} \hat{H}_{\sigma(t)}\Big{)}^{T}\Psi_{\sigma(t)}\Big{)}\otimes QBB^{T}Q\delta \\ &\quad\quad-\delta^{T}\Big{(}\Psi_{\sigma(t)}\hat{H}_{\sigma(t)}+ \Big{(}\hat{H}_{\sigma(t)}\Big{)}^{T}\Psi_{\sigma(t)}\Big{)}\otimes QBB^{T}Qe \end{split}  \tag{53}\]  
By Lemma 1 and (53), we can obtain  
\[ \begin{split}\dot{V}_{b}(t)&\leq 2\delta^{T}\Psi_{ \sigma(t)}\otimes\big{(}QA+A^{T}Q\big{)}\delta+2\delta^{T}\Psi_{\sigma(t)} \otimes\bigg{(}\frac{\theta_{\max}\big{(}\Psi_{\sigma(t)}G\big{)}}{\theta_{ \min}\big{(}\Psi_{\sigma(t)}\big{)}}\bigg{)}Q\Gamma\\ &\quad=2\delta^{T}\bigg{\{}\Psi_{\sigma(t)}\otimes\Big{[}\big{(}QA +A^{T}Q\big{)}+\Big{(}\frac{\theta_{\max}\big{(}\Psi_{\sigma(t)}G\big{)}}{ \theta_{\min}\big{(}\Psi_{\sigma(t)}\big{)}}\Big{)}Q\Gamma\Big{]}\bigg{\}} \delta\\ &\quad=2\delta^{T}\bigg{\{}\Psi_{\sigma(t)}\otimes\big{[}\big{(}QA +A^{T}Q\big{)}+\beta_{2}Q\Gamma\big{]}\bigg{\}}\delta\end{split}  \tag{54}\]  
From the condition of (50) in Theorem 2, (54) is equivalent to  
\[\dot{V}_{b}(t)\leq\beta_{3}V_{b}(t) \tag{55}\]  
Then, the following inequality holds:  
\[V_{b}(t)\leq e^{\beta_{3}\big{(}t-t_{22}^{ij}\big{)}}V_{b}\Big{(}t_{2l}^{ij} \Big{)} \tag{56}\]  
The proof is completed. Theorem 3: _If DoS attacks satisfy Definitions 1 and 2 with \(T_{0}>0\), and conditions (14), (17) and (23) hold, then all nodes of CCPNs (1) are synchronized under DoS attacks provided that the following two conditions are satisfied:_  
_(1) \(\exists\ \theta_{1}^{*}\in(0,\beta_{3})\), and the frequency of DoS attacks \(F_{a_{(b\sigma)}}\) satisfies_  
\[F_{a_{(b\sigma)}}=\frac{N_{a_{(b\sigma)}}}{t-t_{0}}\leq\frac{\theta_{1}^{*}}{2 \ln(\mu)+(\alpha+\beta_{3})\Delta_{*}} \tag{57}\]  
_(2) \(\exists\ \tau_{a}\geq 0\), for \(\forall\ T_{0}>0\) such that_  
\[\tau_{a}>\frac{\alpha+\beta_{3}}{\alpha+\theta_{1}^{*}} \tag{58}\]  
_where \(\beta_{3}>0\), \(\mu=\max\{\theta_{\max}(P)/\theta_{\min}(Q),\ \theta_{\max}(Q)/\theta_{\min}(P)\}\)._  
Proof. : Construct the following piecewise Lyapunov function for the CCPNs (1):  
\[V=\left\{