## 3 Theory
### Event-Triggered Consistency Theory of Multi-Microwave Sources under DoS Attacks
Define \(\Pi_{i}(t)=(\delta e_{i}\mathrm{P}(t))^{T}\times\delta e_{i}\mathrm{P}(t)\), and the derivative and inspired by [25], \(\dot{\mathrm{\Pi}}_{i}(t)=(\delta e_{i}\mathrm{P})^{T}\times\delta e_{i} \mathrm{P}+\mathrm{P}^{T}(t)h_{i}^{T}\otimes\mathrm{A}^{T}\mathrm{A}\otimes h_ {i}\mathrm{P}(t)\leq(1+2\|\mathrm{A}\|^{2})\Pi_{i}(t)+2\|\mathrm{A}\|^{2}( \delta_{i}p(t_{k}^{i}))^{T}\). Combining Equation (8), we then have:  
\[\|\delta e_{i}\mathrm{P}(t)\|^{2}\leq\frac{2\|\mathrm{A}\|^{2}\|\delta_{i} \mathrm{P}(t)\|^{2}}{1+2\|\mathrm{A}\|^{2}}\Big{(}\mathrm{e}^{((1+2\|\mathrm{ A}\|^{2})(\gamma_{1}))-1}\Big{)} \tag{15}\]  
Consequently, for any agent \(i\), \(\eta>0\) and \(\tau_{i}=t_{k}^{i}-t_{k-1}^{i}\geq\tau_{\min}\frac{\eta}{\tau_{i}}\leq 2( \mathrm{e}^{((1+2\|\mathrm{A}\|^{2})(\tau_{i}))-1})\)  
\(\big{(}\|\mathrm{A}\|^{2}\big{\|}\mathrm{Y}_{-}e_{i}p(t_{k}^{i})\big{\|}^{2} \big{)}(1+2\|\mathrm{A}\|^{2})^{-1}\) which means that the switching systems can avoid Zeno behavior [19, 25]. For system consistency, first consider the absence of DoS attacks. let \(\delta\mathrm{P}_{i}(t)=\mathrm{P}_{i}(t)-\mathrm{P}_{0}(t)\), \(\delta\mathrm{P}(t)=[\delta\mathrm{P}_{1}(t),\delta\mathrm{P}_{2}(t),\cdots, \delta\mathrm{P}_{i}(t)]\). We have  
\[\left\{ \begin{array}{l}\delta\mathrm{P}(t)=(I_{N}\otimes\mathrm{A})\delta \mathrm{P}(t),t\in(t_{k},t_{k+1}]\\ \delta\mathrm{P}(t_{k}^{+})=[I_{N}-H\sigma_{K}]\otimes I_{N}\delta\mathrm{P}(t )\end{array} \right. \tag{16}\]  
let \(\varsigma(t)=\delta\mathrm{P}(t)\) constructing Lyapunov function: \(\dot{\mathrm{N}}_{k}(\varsigma)=\varsigma^{T}(t)[(I_{N}\otimes\mathrm{A})^{T} \bar{p}+\bar{p}(I_{N}\otimes\mathrm{A})]\varsigma(t)\leq-a\|\varsigma(t)\|^{2}\). Inspired by [25] and Equation (7), Equation (9):  
\[\sum_{j=1}^{k}\left(\ln\bigg{(}\sqrt{\frac{\lambda_{\max}(\bar{p})}{\lambda_{ \min}(\bar{p})}}\|I_{N}-H\sigma(j)\|+\frac{a}{2\lambda_{\min}(\bar{p})}(t_{j}-t_ {j-1})\Big{)}\right)\leq\kappa(t_{0},t) \tag{17}\]where \(\kappa(t)\) be nonnegative and continuous function on \(t\in(t_{0},+\infty)\), constant \(a>0\), \(\bar{p}\) denotes the positive definite matrix and then \(\lambda_{\min}(\bar{p})\|\zeta(t)\|^{2}\leq\lambda_{\max}(\bar{p})\|\zeta(t_{k}^ {+})\|^{2}+\int_{t_{k}}^{t}-a\|\zeta(s)\|^{2}ds\). According to Equation (16), Equation (17) and following the proof of result:  
\[ \begin{split}\|\delta\mathrm{P}(t)\|&\leq\|I_{N}- H\sigma_{k}\|\sqrt{\frac{\lambda_{\max}(\bar{p})}{\lambda_{\min}(\bar{p})}} \times\|\delta\mathrm{P}(t_{k})\|\times e^{\left(\int_{t_{k}}^{t}-\frac{a}{ \lambda_{\min}(\bar{p})}ds\right)}\\ \cdot\cdot\cdot\cdot&\leq\|\delta\mathrm{P}(t_{0}) \|e^{(\kappa(t_{0},t))},t\in t_{k^{\prime}}t_{k+1}\|\end{split}  \tag{18}\]  
The existence of \(c>0\) makes \(\kappa(t_{0},t)\leq-c(t-t_{0}),(t\geq t_{0})\) hold, Equation (18) means that the leader-following consensus can be exponentially implemented globally. When their DoS attacks, for a single jump in the switching system, assuming \(t_{q}\) is the switching moment of \(\mathrm{P}(t)\), and that \(T_{n-1}\leq T_{n}^{on}\leq t_{q}\leq t<T_{n}\) exists, redesign Lyapunov function: \(\mathrm{N}(t)=\mathrm{N}(\zeta(t))=\varsigma^{T}(t)\bar{p}\zeta(t)\), next we have:  
\[\mathrm{N}(t)=\varsigma^{T}(t)(\mathrm{A}^{T}\bar{p}+\bar{p}\mathrm{A}^{T}) \varsigma(t)+(\delta\mathrm{P}^{T}(t))B^{T}\bar{p}\varsigma(t)+\varsigma^{T}( t)\bar{p}B(\delta\mathrm{P}(t)) \tag{19}\]  
Let \(N_{p}(0,t)\) denote the number of times the system state m switches, then there is as follows:  
\[N_{p}(0,t)=k\leq e^{-\bar{q}t}\frac{(\bar{q}t)^{k}}{k!},\forall k\geq 0 \tag{20}\]  
where \(\bar{q}=\max\bigl{\{}q_{i_{1}j_{1}}:i_{1},j_{1}\in\{1,2\},i_{1}\
eq j_{1}\bigr{\}}\), \(\bar{q}=\max\bigl{\{}|q_{i_{1}i_{1}}|\bigr{\}}\). \(i_{1},j_{1}\) denote the two states of the switching system, respectively, while \(\bar{q},\bar{q}\) denote the state switching rate parameter.