## 3 Integral-Type Edge-Event-Triggered Policy
\begin{split} V^{\prime}(t)=& R^{T}(t)\Big{(}(I_{N} \otimes A^{T}P)+(I_{N}\otimes PA)-c(HH^{T}\otimes PBK)-c(HH^{T}\otimes K^{T} B^{T}P)\Big{)}R(t)\\ &-R^{T}(t)(2c(H\otimes PBK))e(t)+R^{T}(t)(2(H\otimes PB_{1})) \Phi(y)\\ \leq& R^{T}(t)\Big{(}(I_{N}\otimes A^{T}P)+(I_{N} \otimes PA)-c(HH^{T}\otimes PBK)-c(HH^{T}\otimes K^{T}B^{T}P)\\ &+\frac{c^{2}}{a}(HH^{T}\otimes PBKK^{T}B^{T}P)\Big{)}R(t)+ae^{T} (t)e(t)+R^{T}(t)[2(H\otimes PB_{1})]\Phi(y),\end{split}  \tag{10}\]  
where we use the fact that \(-R^{T}[2(H\otimes PBK)]e\leq\frac{1}{a}R^{T}(HH^{T}\otimes PBKK^{T}B^{T}P)]R+ae^{T}e\) for any positive real constant \(a\) to derive the above inequality. However, \(a\) has to be fixed due to the product of \(a\) and \(X\). In addition, it follows from the event condition (6) that  
\[\int_{t_{0}}^{t}\|e(s)\|^{2}\,ds\leq\frac{1}{1-2k_{max}}\int_{t_{0}}^{t}2k_{ max}\|z(s)\|^{2}+\beta Ne^{-\alpha s}\,ds, \tag{11}\]  
Noting that \(k_{max}=max_{i\in\{1,2,\ldots,N\}}k_{i}<\frac{1}{2}\) and \(z(t)=(H^{T}\otimes I_{n})R(t)\). Then, integrating (10) from \(t_{0}\) to \(t\), one gets  
\[ \begin{split} V(t)-V(t_{0})\leq&\int_{t_{0}}^{t}R^{T}( s)\big{(}(I_{N}\otimes A^{T}P)+(I_{N}\otimes PA)-c(HH^{T}\otimes PBK)-c(HH^{T} \otimes K^{T}B^{T}P)\\ &+\frac{c^{2}}{a}(HH^{T}\otimes PBKK^{T}B^{T}P)\big{)}R(s)+ae^{T} (s)e(s)+R^{T}(s)(2(H\otimes PB_{1}))\Phi(y)\,ds\\ \leq&\int_{t_{0}}^{t}R^{T}(s)\Big{(}(I_{N}\otimes A^{T }P)+(I_{N}\otimes PA)-c(HH^{T}\otimes PBK)-c(HH^{T}\otimes K^{T}B^{T}P)\\ &+\frac{c^{2}}{a}(HH^{T}\otimes PBKK^{T}B^{T}P)+\frac{2 ak_{max}}{1 -2k_{max}}(E^{T}E\otimes I_{n})\Big{)}R(s)+R^{T}(s)(2(H\otimes PB_{1}))\Phi(y) \\ &+\frac{a\beta N}{1-2k_{max}}e^{-as}\,ds.\end{split}  \tag{12}\]Similarly, by the fact that \(R^{T}(2(H\otimes PB_{1}))\Phi(y)\leq R^{T}(I_{\mathcal{N}}\otimes PB_{1}B_{1}^{T}P)R +\left((H\otimes I_{m})\Phi(y)\right)^{T}((H\otimes I_{m})\Phi(y))\), we have  
\[V(t)-V(t_{0})\leq \int_{t_{0}}^{t}R^{T}(s)\bigg{(}(I_{\mathcal{N}}\otimes A^{T}P)+( I_{\mathcal{N}}\otimes PA)-c(HH^{T}\otimes PBK)-c(HH^{T}\otimes K^{T}B^{T}P)\] \[+\frac{c^{2}}{a}(HH^{T}\otimes PBKK^{T}B^{T}P)+\frac{2 ak_{max}}{1- 2k_{max}}(HH^{T}\otimes I_{n})+(I_{\mathcal{N}}\otimes PB_{1}B_{1}^{T}P)]\bigg{)}R (s) \tag{13}\] \[+((H\otimes I_{m})\Phi(y))^{T}((H\otimes I_{m})\Phi(y))+\frac{a \beta N}{1-2k_{max}}e^{-as}\,ds,\]  
and then, based on Lemma 1, we get  
\[V(t)-V(t_{0})\leq \int_{t_{0}}^{t}R^{T}(s)\bigg{(}((I_{\mathcal{N}}\otimes A^{T}P) +(I_{\mathcal{N}}\otimes PA)-c(HH^{T}\otimes PBK)-c(HH^{T}\otimes K^{T}B^{T}P)\] \[+\frac{c^{2}}{a}(HH^{T}\otimes PBKK^{T}B^{T}P)+\frac{2 ak_{max}}{1 -2k_{max}}(HH^{T}\otimes I_{n})+(I_{\mathcal{N}}\otimes PB_{1}B_{1}^{T}P) \tag{14}\] \[+\eta\|\mathbb{C}\|^{2}I_{\mathcal{N}\times n}\bigg{)}R(s)+\frac {a\beta N}{1-2k_{max}}e^{-as}\,ds.\]  
The following, we will prove that \((I_{\mathcal{N}}\otimes A^{T}P)+(I_{\mathcal{N}}\otimes PA)-c(HH^{T}\otimes PBK )-c(HH^{T}\otimes K^{T}B^{T}P)+\frac{c^{2}}{a}(HH^{T}\otimes PBKK^{T}B^{T}P)+ \frac{2 ak_{max}}{1-2k_{max}}(HH^{T}\otimes I_{n})+(I_{\mathcal{N}}\otimes PB _{1}B_{1}^{T}P)+\eta\|\mathbb{C}\|^{2}I_{\mathcal{N}\times n}<0\) holds. On the other hand, from (8), we have that  
\[\left[ \begin{array}{cccc}A+B&I_{\mathcal{N}}\otimes X&H\otimes X&I_{ \mathcal{N}}\otimes B_{1}&H\otimes BY\\ I_{\mathcal{N}}\otimes X&-\frac{1}{\eta\|\mathbb{C}\|^{2}}I_{\mathcal{N}\times n }&0&0\\ H^{T}\otimes X&0&-\frac{1}{4\tau}I_{\mathcal{N}\times n}&0&0\\ I_{\mathcal{N}}\otimes B_{1}^{T}&0&0&-I_{\mathcal{N}\times m}&0\\ H^{T}\otimes Y^{T}B^{T}&0&0&0&-\frac{d}{c^{2}}I_{\mathcal{N}}\otimes X^{2} \end{array} \right], \tag{15}\]  
is a negative definite matrix, due to the fact that \(-X^{2}\leq I_{n}-2X\) and \((I_{n}-X)^{2}=I_{n}-2X+X^{2}\geq 0\). By premultipling and postmultipling the matrix inequality with \(\text{diag}(I_{\mathcal{N}}\otimes P,I_{\mathcal{N}\times n},I_{\mathcal{N} \times n},I_{\mathcal{N}\times m},I_{\mathcal{N}\times m},I_{\mathcal{N}} \otimes P)\), along with \(K=YX^{-1}\). \[\left[