## 3 Main Results
 \tag{22}\]  
By substituting Equation (22) into Equation (21), we obtain  
\[\dot{V}(t)\leq \frac{1}{2}\bar{x}^{T}(t)(I_{M}\otimes(A^{T}P+PA))\bar{x}(t) \tag{23}\] \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\] \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\] \[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\] \[\leq\frac{1}{2}\bar{x}^{T}(t)(I_{M}\otimes(A^{T}P+PA))\bar{x}(t)\] \[-\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\] \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\] \[+\gamma(t)\bar{x}^{T}(t)(L_{1}\otimes(A^{T}P+PA))\bar{x}(t)\]  
where \(\gamma(t)=\frac{\varphi(t)}{2(1-\theta)}\). In accordance with event-triggered condition (6), we have  
\[||Ke_{i}(t)||^{2}\leq\frac{\theta}{\varphi(t)+1}||K\bar{x}_{i}(t)||^{2} \tag{25}\]  
Therefore,  
\[||(L_{1}\otimes K)e(t)||^{2}\leq\frac{\theta}{\varphi(t)+1}||(L_{1}\otimes K) \bar{x}_{i}(t)||^{2}. \tag{26}\]By substituting Equation (26) into Equation (25), we discover:  
\[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]0 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]1 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]2 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]3 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]4 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]5 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]6 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]7  
According to (19), there are  
\[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]8 \[+\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P) \bar{x}(t)\]9 \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]0  
where \(k_{1}(t)=\frac{\dot{\bar{x}}(t)}{1-\bar{\xi}(t)+\bar{\delta}}\). Lemma 1 states that we can directly obtain \(V(t)\leq\frac{\delta}{1+\delta}V(0)\). Therefore,  
\[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]1  
When \(t\geq t_{f}\), we can obtain the following equation:  
\[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]2  
Using Equation (30) and the time-derivative of \(V(t)\), we obtain  
\[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]3 \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]4 \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]5 \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]6 \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]7 \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]8 \[+\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\]9 \[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]0 \[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]1 \[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]2The equation for \(k_{2}\) is given by  
\[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]3  
Consequently, it is evident that  
\[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]4  
Hence, the result is:  
\[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]5  
Thus, Theorem 2 is proved. The eigenvalues of \(L_{1}\) are known to all have positive real portions under the assumption of Condition 3 and according to Lemma 2. From Equation (17), it follows that  
\[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]6  
This implies  
\[-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x}(t)\]7  
By substituting \(x_{i}(t)=e^{At}x_{i}(0)\) for \(i\in E\), we can obtain \(\lim_{t\to\infty}\left(x_{f}(t)-\omega(t)\right)=0\)  
\[\text{where }\omega(t)=(-L_{1}^{-1}L_{2}\otimes e^{At}) \begin{bmatrix}x_{M+1 }(0)\\ \vdots\\ x_{N}(0)\end{bmatrix} .\]  
Thus, Theorem 1 is proved. **Remark 2**.