## 3 Stability Analysis under Resilient Event-Triggered Rule and DoS Attacks
### Stability Criterion
 \tag{15}\]where  
\[V_{1}(x(t),r(t)) =x^{T}(t)P_{r(t)}x(t), \tag{16}\] \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\] \[\qquad+\int_{t-\tau_{q}}^{t-\tau_{p}}x^{T}(s)Q^{b}_{r(t)}x(s) \text{d}s,\] \[V_{3}(x(t),r(t)) =\int_{-\tau_{p}}^{0}\int_{t+\theta}^{t}x^{T}(s)S_{a}x(s)\text{d }s\] \[\qquad+\int_{-\tau_{q}}^{-\tau_{p}}\int_{t+\theta}^{t}x^{T}(s)S_{b }x(s)\text{d}s\text{d}^{\cdot},\] \[V_{4}(x(t),r(t)) =\int_{-\tau_{p}}^{0}\int_{t+\theta}^{t}\dot{x}^{T}(s)R_{a}\dot{x }(s)\text{d}s\text{d}^{\cdot}\] \[\qquad+\int_{-\tau_{q}}^{-\tau_{p}}\int_{t+\theta}^{t}\dot{x}^{T} (s)R_{b}\dot{x}(s)\text{d}s\text{d}^{\cdot}.\]  
For \(r(t)=i\in\mathbb{S}\), let \(\mathfrak{L}\) be the weak infinitesimal generator, which is computed along the state trajectory of MJCPSs (1) as  
\[\mathfrak{L}V_{1}(x(t),r(t)) \tag{17}\] \[=2x^{T}(t)P_{i}\big{[}A_{i}x(t)+B_{i}^{\omega}\omega(t)-B_{i}^{u} K_{i}e_{a}(t)\] \[\quad+\bar{B}_{i}^{u}\bar{K}_{i}x(t-\tau(t))\big{]}+x^{T}(t)\sum _{j=1}^{S}\pi_{ij}P_{j}x(t),\] \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]0 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]1 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]2 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]3 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]4 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]5 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]6 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]7 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]8 \[V_{2}(x(t),r(t)) =\int_{t-\tau_{p}}^{t}x^{T}(s)Q^{q}_{r(t)}x(s)\text{d}s\]9 (18) \[\qquad+\int_{t-\tau_{q}}^{t-\tau_{p}}x^{T}(s)Q^{b}_{r(t)}x(s) \text{d}s,\]0  
where \(\hat{\tau}\in\mathbb{R}_{>0}\) is defined in Theorem 1. For clarity, let \(\eta(t):=\text{col}\{x(t),x(t-\tau_{p}),x(t-\tau(t)),x(t-\tau_{q}),e_{a}(t), \omega(t)\}\) represent the augmented vector. Then, in view of the mathematical nature of \(\alpha(t)\), one can obtain that  
\[\qquad+\int_{t-\tau_{q}}^{t-\tau_{p}}x^{T}(s)Q^{b}_{r(t)}x(s) \text{d}s,\]1 \[\qquad+\int_{t-\tau_{q}}^{t-\tau_{p}}x^{T}(s)Q^{b}_{r(t)}x(s) \text{d}s,\]2where \(\tilde{\alpha}\in\mathbb{R}_{>0}\), \(\zeta_{1}=\left[A_{i},0,\tilde{B}_{i}^{u}\tilde{K}_{i},0,-\tilde{B}_{i}^{u}\tilde {K}_{i},B_{i}^{u}\right]\), and \(\zeta_{2}=\left[0,0,\tilde{B}_{i}^{u}\tilde{K}_{i},0,-\tilde{B}_{i}^{u}\tilde {K}_{i},0\right]\). By virtue of the Jessen's inequality, the integral terms in (19) are calculated as  
\[ \begin{split}&-\int_{t-\tau_{p}}^{t}\dot{x}^{T}(s)R_{a}\dot{x}(s) \mathrm{d}s\\ &\leq-\tilde{\tau}\left[x(t)-x(t-\tau_{p})\right]^{T}R_{a}[x(t)-x (t-\tau_{p})],\\ &-\int_{t-\tau_{q}}^{t-\tau_{p}}\dot{x}^{T}(s)R_{b}\dot{x}(s) \mathrm{d}s\\ &\leq-\tilde{\tau}\big{[}x(t-\tau_{p})-x(t-\tau(t))\big{]}^{T}R_{ b}[x(t-\tau_{p})\\ &\quad-x(t-\tau(t))\big{]}-\tilde{\tau}\big{[}x(t-\tau(t))-x(t- \tau_{q})\big{]}^{T}\\ &\quad\times R_{b}[x(t-\tau(t))-x(t-\tau_{q})]-2\tilde{\tau} \big{[}x(t-\tau_{p})\\ &\quad-x(t-\tau(t))\big{]}^{T}\tilde{R}_{b}\big{[}x(t-\tau(t))-x (t-\tau_{q})\big{]},\end{split}  \tag{22}\]  
where \(\tilde{\tau}\) and \(\tilde{\tau}\in\mathbb{R}_{>0}\) are defined in Theorem 1, respectively. Combining condition (14), it follows from (16)-(22) that  
\[ \begin{split}\mathfrak{L}V(x(t),r(t))&\leq\eta^{T}( t)\big{[}\Omega_{a}+\zeta_{1}^{T}(\tau_{p}R_{a}+\tau R_{b})\zeta_{1}\\ &\quad+\xi_{2}^{T}(\tilde{\alpha}\tau_{p}R_{a}+\tilde{\alpha} \hat{\tau}R_{b})\zeta_{2}]\eta(t),\end{split}  \tag{23}\]  
where  
\[\Omega_{a}=\left[ \begin{array}{ccccc}\varphi_{11}&\tau R_{a}&\varphi_{13} &0&\varphi_{15}&\varphi_{16}\\ *&\varphi_{22}&\varphi_{23}&\tilde{\tau}\tilde{R}_{b}&0&0\\ *&*&\varphi_{33}&\varphi_{34}&0&0\\ *&*&*&\varphi_{44}&0&0\\ *&*&*&*&0\\ *&*&*&*&*&0\\ \end{array}