## 3 Stability Analysis under Resilient Event-Triggered Rule and DoS Attacks  
### Stability Criterion  
Firstly, the primary goal of this subsection is to find the stability criteria for MJCPSs (1) under resilient event-triggered rule (8) and DoS attack. Then, we perform a comprehensive feasibility analysis on condition (8) to demonstrate the theoretical validity. **Theorem 1**. : _Given positive scalars \(\sigma\), \(\alpha\), \(\tau_{p}\), \(\tau_{q}\), and \(\gamma\), \(\xi_{i}\in\mathbb{R}_{>0}\), and the dual-mode security controller gains \(K_{i}^{a}\) and \(K_{i}^{b}\). Under the proposed resilient event-triggered rule (8), MJCPSs (1) are stochastically stable in the presence of intermittent DoS attacks and have an \(H_{\infty}\) disturbanceattenuation level \(\gamma\) if there exist the positive definite matrices \(P_{i}>0\), \(Q_{i}^{a}>0\), \(Q_{i}^{b}>0\), \(S_{a}>0\), \(S_{b}>0\), \(R_{a}>0\), \(R_{b}>0\), \(\Theta_{a}>0\), \(\Theta_{b}>0\), and the real matrix \(\tilde{R}_{b}\) such that the following conditions are satisfied:_  
\[\Omega=\left[ \begin{array}{ccc}\Omega_{c}&\Omega_{1}&\Omega_{2}\\ *&\Omega_{3}&0\\ *&*&\Omega_{4}\end{array} \right],\ \mathcal{R}=\left[ \begin{array}{ccc}R_{b}& \tilde{R}_{b}\\ *&R_{b}\end{array} \right]>0, \tag{13}\]  
\[\sum_{j=1}^{S}\pi_{ij}Q_{j}^{a}<S_{a},\ \sum_{j=1}^{S}\pi_{ij}Q_{j}^{b}<S_{b}, \tag{14}\]  
_where \(\mathrm{diag}\{\Omega_{3},\Omega_{4}\}=-\mathrm{diag}\{\tilde{\tau}R_{a}, \tilde{\tau}R_{b},\tilde{\alpha}\tilde{\tau}R_{a},\tilde{\alpha}\tilde{\tau}R_ {b},I,\tilde{\alpha}I\}\),_  
\[\Omega_{c}=\left[ \begin{array}{cccc}\varphi_{11}&\tau R_{a}&\varphi_{13}&0& \varphi_{15}&\varphi_{16}\\ *&\varphi_{22}&\varphi_{23}&\tilde{\tau}\tilde{R}_{b}&0&0\\ *&*&\tilde{\varphi}_{33}&\varphi_{34}&-\sigma\Theta_{b}&0\\ *&*&*&\varphi_{44}&0&0\\ *&*&*&*&\sigma\Theta_{b}-\Theta_{a}&0\\ *&*&*&*&*&-\gamma^{2}I\end{array} \right],\]  
\[\Omega_{1}=\left[ \begin{array}{cccc}A_{i}^{T}R_{a}&A_{i}^{T}R_{b}&0\\ 0&0&0\\ \tilde{R}_{i}^{T}\tilde{B}_{i}^{uT}R_{a}&\tilde{K}_{i}^{T}\tilde{B}_{i}^{uT}R_ {b}&\tilde{K}_{i}^{T}\tilde{B}_{i}^{uT}R_{a}\\ 0&0&0&0\\ -\tilde{K}_{i}^{T}\tilde{B}_{i}^{uT}R_{a}&-\tilde{K}_{i}^{T}\tilde{B}_{i}^{uT}R _{b}&-\tilde{K}_{i}^{T}\tilde{B}_{i}^{uT}R_{a}\\ B_{i}^{uT}R_{a}&B_{i}^{uT}R_{b}&0\end{array} \right],\]  
\[\Omega_{2}=\left[ \begin{array}{cccc}0&C_{i}^{T}&0\\ 0&0&0\\ \tilde{K}_{i}^{T}\tilde{B}_{i}^{uT}R_{b}&\tilde{K}_{i}^{T}\tilde{D}_{i}^{uT}& \tilde{K}_{i}^{T}\tilde{D}_{i}^{uT}\\ 0&0&0\end{array} \right],\]  
_with_  
\[\varphi_{11} =Q_{i}^{a}+\tau_{p}S_{a}+\tilde{\tau}S_{b}+A_{i}^{T}P_{i}+P_{i}A_{i }+\sum_{j=1}^{S}\pi_{ij}P_{j}\] \[-\tilde{\tau}R_{a},\ \tilde{\varphi}_{33}=-2\varphi_{23}+\sigma \Theta_{b},\ \tilde{\alpha}=\tilde{\alpha}^{-1}=\alpha\tilde{\alpha},\] \[\varphi_{13} =-\varphi_{15}=P_{i}\tilde{B}_{i}^{u}\tilde{K}_{i},\ \varphi_{16}=P_{i}B_{i}^{u},\ \tilde{\tau}=\tau_{q}-\tau_{p},\] \[\varphi_{22} =Q_{i}^{b}-Q_{i}^{a}-\tilde{\tau}R_{a}-\tilde{\tau}R_{b},\ \tilde{\tau}=\hat{\tau}^{-1},\ \tilde{\tau}=\tau_{p}^{-1},\] \[\varphi_{23} =\varphi_{34}=\tau R_{b}-\tilde{\tau}\tilde{R}_{b},\ \varphi_{44}=-Q_{i}^{b}-\tilde{\tau}R_{b}.\]  
Firstly, consider a stochastic Lyapunov-Krasovskii functional as follows:  
\[ \begin{split} V(x(t),r(t))&=V_{1}(x(t),r(t))+V_{2}(x(t),r(t)) \\ &+V_{3}(x(t),r(t))+V_{4}(x(t),r(t)),\end{split}