## 3 Main Results
\begin{cases}c\sum\limits_{j=1}^{N}a_{ij}^{\delta(t)}[x_{j}(t_{k}^{i} )-x_{i}(t_{k}^{i})+v_{j}(t_{k}^{i})-v_{i}(t_{k}^{i})]-cd_{i}^{\delta(t)}[x_{i} (t_{k}^{i})-\bar{x}_{k_{i}}(t_{k}^{i})+v_{i}(t_{k}^{i})-\bar{v}_{k_{i}}(t_{k}^ {i})]\\ \\ +c\sum\limits_{j=1}^{N}l_{ij}^{\delta(t)}(\bar{x}_{k_{j}}(t_{k}^{i})+\bar{v}_{ k_{j}}(t_{k}^{i})),\hskip 56.905512 pti\in\bar{\mathcal{V}}_{\bar{k}_{i}}\\ \\ c\sum\limits_{j=1}^{N}a_{ij}^{\delta(t)}[x_{j}(t_{k}^{i})-x_{i}(t_{k}^{i})+v_{ j}(t_{k}^{i})-v_{i}(t_{k}^{i})]-cd_{i}^{\delta(t)}[x_{i}(t_{k}^{i})-\bar{x}_{k_{i}} (t_{k}^{i})+v_{i}(t_{k}^{i})-\bar{v}_{k_{i}}(t_{k}^{i})],\\ \\ \hskip 56.905512 pti\in\mathcal{V}_{\bar{k}_{i}}\setminus\bar{\mathcal{V}}_{\bar{ k}_{i}}\,and\,\mathrm{deg}(\mathrm{i})_{\mathrm{in}}=0\\ \\ c\sum\limits_{j=1}^{N}a_{ij}^{\delta(t)}[x_{j}(t_{k}^{i})-x_{i}(t_{k}^{i})+v_{ j}(t_{k}^{i})-v_{i}(t_{k}^{i})],\hskip 14.226378 pti\in\mathcal{V}_{\bar{k}_{i}} \setminus\bar{\mathcal{V}}_{\bar{k}_{i}}\,and\,\mathrm{deg}(\mathrm{i})_{ \mathrm{in}}\
eq 0\end{cases}  \tag{3}\]  
where \(c>0\) denotes the coupling strength, \(a_{ij}^{\delta(t)}\) denotes the communication link of agent \(i\) when the communication topology changes into \(G^{\delta(t)}\), and \(d_{i}^{\delta(t)}>0\) denotes the pinning control gain of agent \(i\) when the communication topology changes into \(G^{\delta(t)}\). **Remark 2**. : _It is important to point out that when \(\delta(t)=0\), there is no DoS attack. If agent \(i\) is an intra-act agent, one has \(\sum_{j=1}^{N}l_{ij}^{0}(\bar{x}_{k_{j}}(t_{k}^{i})+\bar{v}_{k_{j}}(t_{k}^{i}))=0\). When the DoS attack occurs, we do not divide the group again; thus, \(l_{ij}^{\delta(t)}=l^{0}\) or \(l_{ij}^{\delta(t)}=0\), so we can still obtain \(\sum_{j=1}^{N}l_{ij}^{\delta(t)}(\bar{x}_{k_{j}}(t_{k}^{i})+\bar{v}_{k_{j}}(t_{ k}^{i}))=0\), \(i\in\mathcal{V}_{k_{i}}\setminus\bar{\mathcal{V}}_{k_{i}}\). Then \(u_{i}(t)\) can be simplified into \(u_{i}(t)=-c(\sum_{j=1}^{N}l_{ij}^{\delta(t)}(\bar{x}_{j}(t_{k}^{i})+\bar{v}_{j}( t_{k}^{i}))+d_{i}^{\delta(t)}(\bar{x}_{i}(t_{k}^{i})+\bar{v}_{i}(t_{k}^{i})))\)._  
Suppose \(\Phi_{i}^{\delta(t)}(t)=\sum_{j=1}^{N}l_{ij}^{\delta(t)}(\bar{x}_{j}(t)+\bar{v }_{j}(t))+d_{i}^{\delta(t)}(\bar{x}_{i}(t)+\bar{v}_{i}(t))\); define \(e_{i}^{\delta(t)}(t)=\[PERSON]_{i}^{\delta(t)}(t_{k}^{i})-\[PERSON]_{i}^{\delta(t)}(t)\); then choose the event trigger function as  
\[g_{i}^{\delta(t)}(t)=||e_{i}^{\delta(t)}(t)||^{2}-a||\[PERSON]_{i}^{\delta(t)}(t)|| ^{2} \tag{4}\]  
Figure 1: Diagram of time sequence in \(l\)th stage. where \(\alpha>0\) is a positive constant. If \(g_{i}^{\delta(t)}(t)\geq 0\), then \(x_{i}(t_{k}^{i})\) will update to \(x_{i}(t_{k+1}^{i})\), which can be expressed as follows:  
\[t_{k+1}^{i}=inf\{t>t_{k}^{i}|g_{i}^{\delta(t)}(t)\geq 0\}\]  
According to (1) and (2), by the use of the Kronecker product, we can obtain the compact form  
\[ \begin{cases}\hat{x}(t)=\vartheta(t),\\ \dot{\vartheta}(t)=f(x(t),v(t))-f(\tilde{x}(t),\vartheta(t))-c[((L^{\delta(t) }+D^{\delta(t)})\otimes I_{n})(\tilde{x}(t)+\vartheta(t))+e^{\beta(t)}(t)]. \end{cases} \]  
where  
\[ \begin{array}{ll}&f(x(t),v(t))=[f^{T}(x_{1}(t),v_{1}(t)),\ldots,f^{T}(x_{N}( t),v_{N}(t))]^{T},\\ &f(\tilde{x}(t),\tilde{v}(t))=[f^{T}(\tilde{x}_{k_{1}}(t),\tilde{v}_{k_{1}}( t)),\ldots,f^{T}(\tilde{x}_{k_{N}}(t),\tilde{v}_{k_{N}}(t))]^{T},\\ &\xi(t)=[\tilde{x}_{1}^{T}(t),\ldots,\tilde{x}_{N}^{T}(t)]^{T},\vartheta(t)=[ \tilde{v}_{1}^{T}(t),\ldots,\vartheta_{N}^{T}(t)]^{T}.\end{array} \]  
Based on the above analysis, we have the following theorem:  
Theorem 1: For given non-zero scalars \(\mu_{1}>1,\mu_{2}>1,0<\mu_{3}<1\), prescribed period of persistence \(T\), and frequency \(f\), if there exist matrices \(Q_{m}>0,Q_{n}>0,m,n\in\{1,2,3,\ldots,\beta\},m\
eq n\) such that the following conditions hold:  
(1). \((\delta+\frac{c}{2})I_{N}-c\bar{L}^{0}-cD+2c\alpha(H^{0})^{T}H^{0}<0\);  
(2). \(\Gamma_{m}\leq\mu_{1}\Omega,Q_{m}\leq\mu_{2}Q_{n},\Omega\leq\mu_{3}\Gamma_{m}\);  
(3). PDT satisfies \(\tau_{D}>\frac{(\text{T}t+1)ln\mu+\rho_{2}\text{T}}{\rho_{1}}\).