## 3 Integral-Type Edge-Event-Triggered Policy
\begin{array}{cccc}\hat{A}+\hat{B}&I_{\mathcal{N}\times n}&H\otimes I _{n}&I_{\mathcal{N}}\otimes PB_{1}&H\otimes PBK\\ I_{\mathcal{N}\times n}&-\frac{1}{\eta\|\mathbb{C}\|^{2}}I_{\mathcal{N}\times n }&0&0\\ H^{T}\otimes I_{n}&0&-\frac{1}{a\tau}I_{\mathcal{N}\times n}&0&0\\ I_{\mathcal{N}}\otimes B_{1}^{T}P&0&0&-I_{\mathcal{N}\times m}&0\\ H^{T}\otimes K^{T}B^{T}P&0&0&0&-\frac{d}{c^{2}}I_{\mathcal{N}\times n}\end{array}  \right], \tag{16}\]  
is negative definite, where \(\hat{A}=I_{\mathcal{N}}\otimes A^{T}P+PA\), \(\hat{B}=-c[HH^{T}\otimes(PBK+K^{T}B^{T}P)]\). Then, using the [PERSON] complement lemma, we obtain that  
\[\left[ \begin{array}{cccc}\hat{A}+\hat{B}+\hat{C}&I_{\mathcal{N}\times n}&H \otimes I_{n}&I_{\mathcal{N}}\otimes PB_{1}\\ I_{\mathcal{N}\times n}&-\frac{1}{\eta\|\mathbb{C}\|^{2}}I_{\mathcal{N}\times n }&0&0\\ H^{T}\otimes I_{n}&0&-\frac{1}{a\tau}I_{\mathcal{N}\times n}&0\\ I_{\mathcal{N}}\otimes B_{1}^{T}P&0&0&-I_{\mathcal{N}\times m}\end{array}  \right]<0, \tag{17}\]  
where \(\hat{C}=\frac{c^{2}}{a}(HH^{T}\otimes PBKK^{T}B^{T}P)\). Similarly, we have that  
\[\left[ \begin{array}{cccc}\hat{A}+\hat{B}+\hat{C}+I_{\mathcal{N}}\otimes PB _{1}B_{1}^{T}P+a\gamma(HH^{T}\otimes I_{n})&0\\ 0&-\frac{1}{\eta\|\mathbb{C}\|^{2}}I_{\mathcal{N}\times n}\end{array} \right]<0, \tag{18}\]  
being equivalent to \(\hat{A}+\hat{B}+\hat{C}+I_{\mathcal{N}}\otimes PB_{1}B_{1}^{T}P+a\gamma(HH^{T} \otimes I_{n})+\eta\|\mathbb{C}\|^{2}I_{\mathcal{N}\times n}<0\). i.e., \((I_{\mathcal{N}}\otimes A^{T}P)+(I_{\mathcal{N}}\otimes PA)-c(HH^{T}\otimes PBK )-c(HH^{T}\otimes K^{T}B^{T}P)\)\[+\frac{c^{2}}{\alpha}(HH^{T}\otimes PBKK^{T}B^{T}P)+a\gamma(HH^{T} \otimes I_{n})+(I_{\mathcal{N}}\otimes PB_{1}B_{1}^{T}P)+\eta\|C\|^{2}I_{ \mathcal{N}\times n}<0.\]  
fore, the above matrix is negative definite, with defined as \(-\hat{P}\). Then  
\[ \begin{split} V(t)-V(t_{0})\leq&\int_{t_{0}}^{t}-R^{ T}(s)\hat{P}R(s)+\frac{a\beta N}{1-2k_{max}}e^{-\alpha s}\,ds\\ \leq&\int_{t_{0}}^{t}-\lambda_{min}(\hat{P})\|R(s) \|^{2}+\frac{a\beta N}{1-2k_{max}}e^{-\alpha s}\,ds\end{split}  \tag{19}\]  
where \(\lambda_{min}(\hat{P})\) is the minimum eigenvalue of the matrix \(\hat{P}\). Inequality (19) yields  
\[\int_{t_{0}}^{t}\|R(s)\|^{2}\,ds\leq\frac{V(t_{0})\alpha(1-2k_{max})+a\beta Ne^ {-\alpha t_{0}}}{\alpha(1-2k_{max})\lambda_{min}(\hat{P})} \tag{20}\]  
based on the fact that \(V(t)\geq 0\), \(e^{-\alpha t}>0\), As a result, \(\int_{t_{0}}^{t}\|R(s)\|^{2}\,ds\) has an upper bound. Furthermore, we implies that \(V(t)\) is bounded, and hence, \(V^{\prime}(t)\) is also bounded. Thus, we can further have that \(\frac{d^{2}}{dt^{2}}\int_{t_{0}}^{t}\|R(s)\|^{2}\,ds\) is bounded. According to [PERSON]'s Lemma, one has that  
\[\lim_{t\to\infty}\frac{d}{dt}\int_{t_{0}}^{t}\|R(s)\|^{2}\,ds=\lim_{t\to\infty }\|R(s)\|^{2}=0. \tag{21}\]  
Therefore, the synchronization is achieved. Next, we show that no Zone behavior occurs. If there is a lower bound on the inter-event intervals of any agent in the integral-type event condition (6), Zeno-free behavior can be assured. Let us consider the derivative of \(e_{i}(t)\) over the interval \([t_{k}^{i},t_{k+1}^{i})\). \[ \begin{split} e_{i}^{\prime}(t)=&-z_{i}^{\prime}(t) \\ =&-\sum_{j\in N_{i}}(x_{i}^{\prime}(t)-x_{j}^{\prime}( t))\\ =&-Az_{i}(t)+cBK\sum_{j\in N_{i}}\left(z_{i}(t_{k}^ {i})-z_{j}(t_{k(t)}^{j})\right)-B_{1}\sum_{j\in N_{i}}\left(\phi(y_{i})-\phi( y_{j})\right)\\ \leq&\|A\|\|e_{i}(t)\|+\Xi_{k}^{i},\end{split}  \tag{22}\]  
where \(t_{k(t)}^{i}\) denotes the latest triggering time instant of agent \(j\) for \(t<t_{k}^{i}\), with \(\Xi_{k}^{i}=\max_{t\in[t_{k}^{i},t_{k+1}^{i})}\|-Az_{i}(t_{k}^{i})+cBK\sum_{j \in N_{i}}[z_{i}(t_{k}^{i})-z_{j}(t_{k(t)}^{j})]\|+\Delta\). The matrix \(B_{1}\sum_{j\in N_{i}}[\phi(y_{i})-\phi(y_{j})]\) is bounded by \(\Delta\), since \(\phi(y_{i})-\phi(y_{j})\) goes to zero as \(t\to\infty\). Then, the derivative of \(\|e_{i}(t)\|^{2}\) satisfies  
\[