## 3 Stability Analysis under Resilient Event-Triggered Rule and DoS Attacks
### Stability Criterion
\right]\]  
with \(\varphi_{33}=-2\varphi_{23}=2\tilde{\tau}\tilde{R}_{b}-2\tilde{\tau}R_{b}\) and other parameters, which are given in Theorem 1. Then, we will show that MJCPs(1) have an \(H_{\infty}\) disturbance attenuation level \(\gamma\) for any non-zero \(\omega(t)\in\mathcal{L}_{2}[0,+\infty)\). Define \(\Lambda(t):=\mathfrak{L}V(x(t),r(t))+\mathbb{E}\{z^{T}(t)z(t)-\gamma^{2}\omega ^{T}(t)\omega(t)\}\). Based on Definition 2, it can be derived that  
\[ \begin{split}\Lambda(t)&\leq\eta^{T}(t)\big{[}\Omega_{b}+\zeta_ {1}^{T}(\tau_{p}R_{a}+\hat{\tau}R_{b})\zeta_{1}+\zeta_{3}^{T}\zeta_{3}\\ &\quad+\zeta_{2}^{T}(\tilde{\alpha}\tau_{p}R_{a}+\tilde{\alpha} \hat{\tau}R_{b})\zeta_{2}+\tilde{\alpha}\zeta_{4}^{T}\zeta_{4}\big{]}\eta(t), \end{split}  \tag{24}\]  
where \(\zeta_{3}=\left[C_{i},0,\tilde{D}_{i}^{u}\tilde{K}_{i},0,-\tilde{D}_{i}^{u} \tilde{K}_{i},0\right]\), \(\zeta_{4}=\left[0,0,\tilde{D}_{i}^{u}\tilde{K}_{i},0,-\tilde{D}_{i}^{u}\tilde {K}_{i},0\right]\), and  
\[\Omega_{b}=\left[ \begin{array}{ccccc}\varphi_{11}&\tilde{\tau}R_{a}&\varphi_ {13}&0&\varphi_{15}&\varphi_{16}\\ *&\varphi_{22}&\varphi_{23}&\tilde{\tau}R_{b}&0&0\\ *&*&\varphi_{33}&\varphi_{34}&0&0\\ *&*&*&\varphi_{44}&0&0\\ *&*&*&*&0&0\\ *&*&*&*&*&-\gamma^{2}I\\ \end{array} \right],\]  
which implies that \(\zeta_{1}^{T}(\tau_{p}R_{a}+\hat{\tau}R_{b})\zeta_{1}+\zeta_{2}^{T}(\tilde{ \alpha}\tau_{p}R_{a}+\tilde{\alpha}\hat{\tau}R_{b})\zeta_{2}+\zeta_{3}^{T}\zeta_ {3}+\tilde{\alpha}\zeta_{4}^{T}\zeta_{4}+\Omega_{b}<0\) is established. In this way, we obtain that \(\Lambda(t)\leq-\|\eta(t)\|^{2}<-\lambda\|x(t)\|^{2}\), where \(\lambda\in\mathbb{R}_{>0}\). From this, we can conclude that MJCPs (1) are stochastically stable and have an \(H_{\infty}\) disturbance attenuation level \(\gamma\). Finally, based on the developed resilient event-triggered rule (8), it follows from condition (24) that \[\Lambda(t) \leq\bar{\Lambda}(t)+\alpha(t)\mathcal{J}_{d}\] \[=\eta^{T}(t)\left[\Omega_{c}+\xi_{1}^{T}(\tau_{p}R_{a}+\tau R_{b}) \zeta_{1}+\xi_{2}^{T}(\bar{\alpha}\tau_{p}R_{a}\right. \tag{25}\] \[+\bar{\alpha}\tau R_{b})\zeta_{2}+\xi_{3}^{T}\zeta_{3}+\bar{\alpha }\xi_{4}^{T}\zeta_{4}]\eta(t)+(1-\alpha(t))\mathcal{J}_{d},\]  
where \(\bar{\Lambda}(t)=\Lambda(t)+\sigma x^{T}(t_{k})\Theta_{b}x(t_{k})-e_{a}^{T}(t) \Theta_{a}e_{a}(t)\) and \(\Omega_{c}\) is given in condition (13). By [PERSON]'s complement lemma, it is concluded that the objective term \(\bar{\Lambda}(t)\) in (25) is equivalent to \(\Omega\) in (13). Thus, it is derived that \(\bar{\Lambda}(t)\leq-\bar{\lambda}\|x(t)\|^{2}\), where \(\bar{\lambda}\in\mathbb{R}_{>0}\). This implies that \(\mathfrak{E}V(x(t),r(t))\leq-\bar{\lambda}V(x(t),r(t))\) is established, where \(\bar{\lambda}=\bar{\lambda}/\lambda_{\text{max}}(P_{i})\) represents the decay rate and \(\lambda_{\text{max}}(P_{i})\) represents the maximum eigenvalue of \(P_{i}\). From this, we can obtain that  
\[\mathfrak{E}V(x(t),r(t))\leq-\bar{\lambda}V(x(t),r(t))+(1-\alpha(t_{n}^{\ell} ))\mathcal{J}_{d}, \tag{26}\]  
where \(t\in[t_{n}^{\ell},t_{n}^{\ell+1})\). Then, it follows from condition (26) that  
\[\mathbb{E}[V(x(t),r(t))]\leq\mathbb{E}[V(x(0),r(0))]+\frac{(1-\alpha(t_{n}^{ \ell}))\mathcal{J}_{d}}{\bar{\lambda}}, \tag{27}\]  
which implies that  
\[\mathbb{E}\bigg{\{}\int_{0}^{\infty}x^{T}(t)x(t)\mathrm{dt}\mid(x(0),r(0)) \bigg{\}} \tag{28}\] \[\leq\hbar\mathbb{E}[V(x(0),r(0))]+\frac{(1-\alpha(t_{n}^{\ell})) \hbar\mathcal{J}_{d}}{\bar{\lambda}}\] \[=\mathcal{M}(x(0),r(0)),\]  
where \(\hbar=\lambda_{\text{min}}^{-1}(P_{i})\) and \(\lambda_{\text{min}}(P_{i})\) is the minimum eigenvalue of \(P_{i}\). In this case, it can be concluded that \(\|x(t)\|^{2}\) is bounded by \(\mathcal{M}(x(0),r(0))\). Meanwhile, the maximum performance loss induced by DoS attacks is calculated as \(\mathcal{J}_{\text{max}}=\hbar\lambda_{d}\mathcal{J}_{d}\), where \(\lambda_{d}=1/\bar{\lambda}\). This completes the proof.