## 3 Main Results
\] \[= \tilde{x}^{T}(t)((c(L^{0}+(L^{0})^{T})+2 cD^{0})\otimes I_{n}) \tilde{v}(t)+\tilde{x}^{T}[f(x(t),v(t))-f(\tilde{x}(t),\tilde{v}(t))]\] \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\] \[+\tilde{v}^{T}(t)[f(x(t),v(t))-f(\tilde{x}(t),\tilde{v}(t))]- \tilde{v}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+\tilde{v}(t))-c \tilde{v}^{T}(t)e^{0}(t)\] \[\leq \tilde{x}^{T}(t)((c(L^{0}+(L^{0})^{T})+2 cD^{0})\otimes I_{n}) \tilde{v}(t)+(l_{1}+\frac{l_{2}}{2})\tilde{x}^{T}(t)\tilde{x}(t)+\frac{l_{2}} {2}\tilde{v}^{T}(t)\tilde{v}(t)\] \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})\tilde{x}(t)- \tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})\tilde{v}(t)-c\tilde{x}^{T}(t)e ^{0}(t)+\tilde{v}^{T}(t)\tilde{v}(t)\] \[+\frac{l_{1}}{2}\tilde{x}^{T}(t)\tilde{x}(t)+(l_{2}+\frac{l_{1}}{ 2})\tilde{v}^{T}(t)\tilde{v}(t)-\tilde{v}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n}) \tilde{x}(t)\] \[-\tilde{v}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})\tilde{v}(t)-c \tilde{v}^{T}(t)e^{0}(t)\] \[= \tilde{x}^{T}(t)[((\frac{3l_{1}+l_{2}}{2})I_{N}-c\tilde{L}^{0}- cD^{0})\otimes I_{n}]\tilde{x}(t)-c\tilde{x}^{T}e^{0}(t)-c\tilde{v}^{T}e^{0}(t)\] \[+\tilde{v}^{T}(t)[((\frac{3l_{2}+l_{1}}{2}+1)I_{N}-c\tilde{L}^{0} -cD^{0})\otimes I_{n}]\tilde{v}(t)\]  
According to the event-triggered condition, we can obtain  
\[-c\tilde{x}^{T}(t)e^{0}(t) =-c\sum_{i=1}^{N}\tilde{x}_{i}^{T}(t)e_{i}^{0}(t)\leq c\sum_{i=1} ^{N}||\tilde{x}_{i}(t)||\cdot||e_{i}^{0}(t)||\leq\frac{c}{2}(\sum_{i=1}^{N} \tilde{x}_{i}^{T}(t)\tilde{x}_{i}(t)+\sum_{i=1}^{N}(e_{i}^{0}(t))^{T}e_{i}^{0} (t))\] \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]0 \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]1 \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]2 \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]3  
Similarly, we can obtain  
\[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]4  
Thus, we have \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]5 \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]6 \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]7 \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]8 \[-\tilde{x}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+ \tilde{v}(t))-c\tilde{x}^{T}(t)e^{0}(t)+\theta^{T}(t)\tilde{v}(t)\]9 \[+\tilde{v}^{T}(t)[f(x(t),v(t))-f(\tilde{x}(t),\tilde{v}(t))]- \tilde{v}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+\tilde{v}(t))-c \tilde{v}^{T}(t)e^{0}(t)\]0 \[+\tilde{v}^{T}(t)[f(x(t),v(t))-f(\tilde{x}(t),\tilde{v}(t))]- \tilde{v}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+\tilde{v}(t))-c \tilde{v}^{T}(t)e^{0}(t)\]1 \[+\tilde{v}^{T}(t)[f(x(t),v(t))-f(\tilde{x}(t),\tilde{v}(t))]- \tilde{v}^{T}(t)(c(L^{0}+D^{0})\otimes I_{n})(\tilde{x}(t)+\tilde{v}(t))-c \tilde{v}^{T}(t)e^{0}(t)\]2 Let \[\Psi= \begin{bmatrix}c\bar{L}^{0}+cD^{0}-(\delta+\frac{c}{2})I_{N}-2c\alpha(H^ {0})^{T}H^{0}&0\\ 0&c\bar{L}^{0}+cD^{0}-(\delta+\frac{c}{2})I_{N}-2c\alpha(H^{0})^{T}H^{0}\end{bmatrix} \] According to condition (1), we can have \[\dot{V}_{0}(t)\leq-\lambda_{min}(\Psi)\tilde{\sigma}^{T}(t)\tilde{\xi}(t)\] For the convenience of the following discussion, we let \[\Omega= \begin{bmatrix}c(L^{0}+(L^{0})^{T})+2 cD^{0}&I_{N}\\ I_{N}&I_{N}\end{bmatrix}