## Appendix A Proof of the Theorem 2  
For any transmission interval \([t_{k},t_{k+1})\) determined by the general event-triggered rule (3), the following two cases are divided to explain the relationship between the jump signal \(r(t)\) and the triggering sequence \(\{t_{k}\}_{k\in\mathbb{N}_{0}}\), where \(t\in\{t_{\bar{k}}\}_{\bar{k}\in\mathbb{N}_{0}}\) represents the jump instant sequence. **Case I**: No jump occurs during the interval \([t_{k},t_{k+1})\), that is, \(t_{\bar{k}}<t_{k}<t_{k+1}<t_{\bar{k}+1}\) is assumed to be established. Then, it follows from (7) that  
\[ \begin{split}\frac{d\|e_{a}(t)\|}{\text{dt}}&=\frac {1}{2}\Big{[}e_{a}^{T}(t)e_{a}(t)\Big{]}^{-\frac{1}{2}}\Big{[}2\dot{e}_{a}^{T} (t)e_{a}(t)\Big{]}\\ &\leq\Big{\|}A_{r(t_{\bar{k}})}x(t)+B_{r(t_{\bar{k}})}^{u}u(t)+ B_{r(t_{\bar{k}})}^{\omega}\omega(t)\Big{\|}.\end{split}  \tag{10}\]  
As can be seen from the dual-mode security control protocol \(u(t)\) in (10), due to the intermittent DoS attacks, two additional crucial points need to be considered in the following analysis. * DoS attacks are not activated by malicious attackers. This indicates that \(\alpha(t)=1\) and \(x(t_{k})=x(t_{n}^{\ell})\). By virtue of (10), it can be derived from (10) that \[ \begin{split}\frac{d\|e_{a}(t)\|}{\text{dt}}&\leq \Big{\|}A_{r(t_{\bar{k}})}e_{a}(t)+\hat{A}_{r(t_{\bar{k}})}x(t_{\bar{k}})+B_{r( t_{\bar{k}})}^{\omega}\omega(t)\Big{\|}\\ &\leq\theta_{1}\|e_{a}(t)\|+\theta_{2}\|x(t_{k})\|+\vartheta_{3} \|\omega(t)\|,\end{split} \] (11) where \(\hat{A}_{r(t_{\bar{k}})}=A_{r(t_{\bar{k}})}+B_{r(t_{\bar{k}})}^{u}K_{r(t_{ \bar{k}})}^{a}\), \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) are given in Theorem 2. Then, a virtual auxiliary variable \(f(t)\) satisfying \(\dot{f}(t)=\phi_{1}f(t)+\theta_{2}\|x(t_{k})\|+\vartheta_{3}\|\omega(t)\|\) needs to be introduced, aiming to constrain the change of \(\|e_{a}(t)\|\). Thus, the analytical solution of \(f(t)\) can be governed by \[f(t)=\frac{[\theta_{2}\|x(t_{k})\|+\vartheta_{3}\|\omega(t)\|]}{\theta_{1}} \Big{(}e^{\phi_{1}(t-t_{\bar{k}})}-1\Big{)}.\] (12) Combined with the actual state error \(e_{a}(t)\) mentioned in (7), one can yield that \[ \begin{split}\|e_{a}(t)\|&\leq\theta_{1}^{-1}[ \theta_{2}(\|e_{a}(t)\|+\|x(t)\|)+\vartheta_{3}\|\omega(t)\|]\\ &\quad\times\Big{(}e^{\phi_{1}(t-t_{\bar{k}})}-1\Big{)}.\end{split} \] (13)  
Based on the resilient event-triggered rule (8), it is concluded that \(e_{a}^{T}(t)\Theta_{a}e_{a}(t)\leq\varrho_{1}^{2}\|e_{a}(t)\|^{2}\) and \(x^{T}(t)\Theta_{b}x(t)\geq\varrho_{2}^{2}\|x(t)\|^{2}\) are established without DoS attacks, where \(\varrho_{1}\) and \(\varrho_{2}\) are given in Theorem 2. Then, we obtain \(e_{a}^{T}(t)\Theta_{a}e_{a}(t)\leq\sigma x^{T}(t)\Theta_{b}x(t)+\sigma^{2}\| \omega(t)\|^{2}\), which is guaranteed by \(\varrho_{1}^{2}\|e_{a}(t)\|^{2}\leq\sigma\varrho_{2}^{2}\|x(t)\|^{2}+\sigma^{2 }\|\omega(t)\|^{2}\), where \(\sigma\in\mathbb{R}_{>0}\) is a given constant. To calculate the minimum inter-execution interval \(\Delta_{\text{min}}^{1}\), we need to discuss the inherent relationship between \(\|e_{a}(t)\|\), \(\|x(t)\|\), and \(\|\omega(t)\|\). With the help of [PERSON]'s inequality, it can be further guaranteed by \(\|e_{a}(t)\|\leq\varsigma_{1}\|x(t)\|+\varsigma_{2}\|\omega(t)\|\), where the forms of \(\varsigma_{1}\) and \(\varsigma_{2}\) consist of the parameters \(\varrho_{1}\), \(\varrho_{2}\), and \(\sigma\). Combined with inequality (13), one can yield\[\frac{1}{\theta_{1}}\text{In}\bigg{[}\frac{\theta_{151}\|x(t)\|+\theta_{152}\|\omega( t)\|}{\theta_{2}(\varepsilon_{1}+1)\|x(t)\|+(\theta_{2}\varepsilon_{2}+\theta_{3})\| \omega(t)\|}+1\bigg{]}\leq\Delta_{k}, \tag{100}\]  
which is greater than or equal to \(\Delta_{\text{min}}^{1}\) in condition (29). * DoS attacks are injected remotely by malicious attackers. This means that \(\alpha(t)=0\) and \(x(t_{k})\
eq x(t_{n}^{\ell})\). Due to DoS attacks, condition (101) needs to be rewritten as \[