## 3 Main Results  
We propose an ETC data update strategy to guarantee that all nodes of CCPNs are synchronized in this section. Then, we solve the topologies change caused by DoS attacks by a distributed controller. **Theorem 1**. : _The all nodes of CCPNs (1) without DoS attacks can be synchronized based on the event-triggered distributed controller (14) with \(K=B^{T}P\) satisfying_  
\[A^{T}P+PA-2\varepsilon_{\min}P\Gamma-2\varepsilon_{\min}PBB^{T}P+\varepsilon_{ \min}I_{N}<0 \tag{21}\]  
_and the triggering function_  
\[\mathfrak{F}\big{(}e_{i},\hat{\xi}_{i2}\big{)}{=}\|e_{i}\|-\sqrt{\frac{\gamma_{ i}(q_{1}-q_{2}\rho)}{q_{2}\rho^{-1}+q_{3}}}\|\hat{\xi}_{i2}\| \tag{22}\]  
_Moreover, the event-triggering time is defined as_  
\[\tau_{k}^{i}=\inf\Bigl{\{}t>t_{k}^{i}\big{|}\mathfrak{F}\big{(}e_{i},\hat{\xi} _{i2}\big{)}=0\,\Bigr{\}},\hskip 14.226378 ptk=1,2,\ldots \tag{23}\]  
_where \(\rho>0\), \(0<\gamma_{i}<1\) and \(q_{1}-q_{2}\rho>0\), \(\varepsilon_{\min}=\max\{\theta(G),\theta(H)\}\), \(0<\varepsilon_{1}<\varepsilon_{\min}\), \(\varepsilon_{\min}=\varepsilon_{1}+\varepsilon_{2}\), \(q_{1}=\min\bigl{\{}\varepsilon_{1}\Gamma_{\min}\bigl{(}H^{-2}\bigr{)}\bigr{\}}\), \(q_{2}=\max\bigl{\{}\Gamma_{\max}\bigl{\{}H^{-1}\otimes 2\varepsilon_{1}I_{N}-I_{N} \otimes 2 PBB^{T}P\bigr{\}}\bigr{\}}>0\), \(q_{3}=\max\bigl{\{}\Gamma_{\max}\bigl{\{}I_{N}\otimes 2 PBB^{T}P\bigr{\}}- \varepsilon_{1}\bigr{\}}>0\)._  
Proof. : The following Lyapunov function is constructed for systems (19)  
\[V_{a}(t)=\delta^{T}(I_{N}\otimes P)\delta \tag{24}\]  
From (16), one can obtain  
\[ \begin{array}{l}\hat{\xi}_{i2}(t)=\sum\limits_{j=1}^{N}a_{ij}^{0}\bigl{(}x_{ j}\bigl{(}t_{k}^{i}\bigr{)}-x_{i}\bigl{(}t_{k}^{i}\bigr{)}\bigr{)}+b_{i}^{0} \bigl{(}x_{N+1}\bigl{(}t_{k}^{i}\bigr{)}-x_{i}\bigl{(}t_{k}^{i}\bigr{)}\bigr{)} \\ \hskip 14.226378 pt=-(H\otimes I_{N})\bigl{(}x_{i}\bigl{(}t_{k}^{i}\bigr{)}-x_{ N+1}(t)\bigr{)}\end{array}  \tag{25}\]  
and  
\[\delta_{i}(t)=x_{i}(t_{k}^{i})-e_{i}(t)-x_{N+1}(t) \tag{26}\]  
Combining (25) and (26), we have  
\[\delta=-\Bigl{(}H^{-1}\otimes I_{N}\Bigr{)}\hat{\xi}_{2}-e \tag{27}\]  
The time derivatives of (24) yield  
\[ \begin{array}{l}\dot{V}_{a}(t)=2\delta^{T}(I_{N}\otimes P)\delta\\ \hskip 14.226378 pt=2\delta^{T}(I_{N}\otimes P)[(I_{N}\otimes A+G\otimes\Gamma- H\otimes BK)\delta-(H\otimes BK)e]\\ \hskip 14.226378 pt=\delta^{T}\bigl{(}A^{T}P+PA+G\otimes 2P\Gamma-H\otimes 2 PBB^{T}P \bigr{)}\delta-\delta^{T}\bigl{(}H\otimes 2 PBB^{T}P\bigr{)}e\\ \hskip 14.226378 pt\leq-\varepsilon_{\min}\delta^{T}\delta-\delta^{T}\bigl{(}H \otimes 2 PBB^{T}P\bigr{)}e\\ \hskip 14.226378 pt\leq\hskip 14.226378 pt-(\varepsilon_{1}+\varepsilon_{2}) \bigl{\{}\frac{\delta^{T}_{2}}{2}\bigl{(}H^{-2}\otimes I_{N}\bigr{)}\hat{\xi} _{2}^{2}+2\xi_{2}^{T}\bigl{(}H^{-1}\otimes I_{N}\bigr{)}e+e^{T}e\bigr{\}}\\ \hskip 14.226378 pt-e^{T}\bigl{(}I_{N}\otimes 2 PBB^{T}P\bigr{)}\hat{\xi}_{2}+e^{T} \bigl{(}I_{N}\otimes 2 PBB^{T}P\bigr{)}e\\ \hskip 14.226378 pt\leq-\varepsilon_{1}\xi_{2}^{T}\bigl{(}H^{-2}\otimes I_{N} \bigr{)}\hat{\xi}_{2}+2\varepsilon_{1}\xi_{2}^{T}\bigl{(}H^{-1}\otimes I_{N} \bigr{)}e-\varepsilon_{1}e^{T}e\\ \hskip 14.226378 pt-e^{T}\bigl{(}I_{N}\otimes 2 PBB^{T}P\bigr{)}\hat{\xi}_{2}+e^{T} \bigl{(}I_{N}\otimes 2 PBB^{T}P\bigr{)}e-\varepsilon_{2}\delta^{T}\delta\end{array}  \tag{28}\]  
Combine with \(M^{T}N\leq\|M\|\|N\|\), \(M^{T}N\leq\frac{1}{2H}\|M\|^{2}+\frac{h}{2}\|N\|^{2}\), then we have  
\[ \begin{array}{l}-\varepsilon_{1}\hat{\xi}_{2}^{T}\bigl{(}H^{-2}\otimes I_{N} \bigr{)}\hat{\xi}_{2}\leq-q_{1}\bigl{\|}\hat{\xi}_{2}\bigr{\|}^{2},-\varepsilon_ {1}e^{T}e+e^{T}\bigl{(}I_{N}\otimes 2 PBB^{T}P\bigr{)}e\leq q_{3}\|e\|^{2}\\ 2\varepsilon_{1}\hat{\xi}_{2}^{T}\bigl{(}H^{-1}\otimes I_{N}\bigr{)}e-e^{T} \bigl{(}I_{N}\otimes 2 PBB^{T}P\bigr{)}\hat{\xi}_{2}\leq 2q_{2}\|e\|\|\hat{\xi}_{2}\| \end{array}  \tag{29}\]Inequality (28) can be rewritten as  
\[