## 3 Main Results
### Dynamic Event Triggering Mechanism  
Let us consider the multi-agents given in (2) with the undirected strongly connected graph \(\mathcal{G}\) and take \(\big{\{}t_{k_{i}+1}^{i}\big{\}}_{k_{i}=1}^{\infty}\) as the triggering time sequence of the \(i\)th agent. Each element \(t_{k_{i}}^{i}\) stands for a triggering time at which the \(i\)th agent can take its information \(p_{i}(t_{k_{i}}^{i})\), \(v_{i}(t_{k_{i}}^{i})\), the last information \(p_{i}(t_{k_{j}}^{i})\), \(v_{i}(t_{k_{j}}^{i})\) of its neighbors, and \(r(t_{k_{i}}^{i})\) of formation trajectory \(r(t)\). It is worth noting that the communication between two consecutive \(i\)th and \(j\)th agents only occurs at their triggering time. Hence, to reduce the number of communication times, our goal in this part is to find a mechanism for each agent that can properly determine the triggering time. Suppose that the first triggering time corresponds to initial state \(t_{1}^{i}=0\); the \(i\)th agent determines its triggering time sequence as follows:  
\[t_{1}^{i}=0,\] \[t_{k_{i}+1}^{i}=\min_{r>t_{k_{i}}^{i}}\big{\{}r\in\mathbb{R}_{+}: \sigma_{i}\chi_{i}(r)+\hat{\chi}_{i}(r)\leq 0\big{\}}, \tag{6}\]  
Figure 1: Formation shape of six agents.  
where:  
\[\hat{\chi}_{i}(t) =\gamma\theta_{p}\big{\|}z_{p,i}(t)\big{\|}_{2}^{2}+\theta_{v} \big{\|}z_{v,i}(t)\big{\|}_{2}^{2}-\big{(}\gamma z_{p,i}(t)+z_{v,i}(t)\big{)}^{T }q_{i}(t), \tag{7}\] \[q_{i}(t) =\gamma_{p}\sum_{j\in\mathcal{N}_{i}}\ell_{ij}\left(\widetilde{ \eta}_{p,i}(t)-\widetilde{\eta}_{p,j}(t)\right)+\gamma_{v}\sum_{j\in\mathcal{ N}_{i}}\ell_{ij}\left(\widetilde{\eta}_{v,i}(t)-\widetilde{\eta}_{v,j}(t) \right)-\gamma_{v}\delta_{i}\widetilde{\eta}_{v,i}(t)-\gamma_{p}\delta_{i} \widetilde{e}_{p}(t)+\widetilde{f}_{a,i}(t),\] (8) \[z_{p,i}(t) =\delta_{i}\big{(}\eta_{p,i}(t)-r(t)\big{)}-\sum_{j\in\mathcal{ N}_{i}}\ell_{ij}\left(\eta_{p,i}(t)-\eta_{p,j}(t)\right),\] (9) \[z_{v,i}(t) =\delta_{i}\big{(}\eta_{v,i}(t)-v_{0}\big{)}-\sum_{j\in\mathcal{ N}_{i}}\ell_{ij}\left(\eta_{v,i}(t)-\eta_{v,j}(t)\right),\] (10) \[\widetilde{\eta}_{p,i}(t) =\eta_{p,i}(t_{k_{i}}^{i})-\eta_{p,i}(t),\] \[\widetilde{\eta}_{v,i}(t) =\eta_{v,i}(t_{k_{i}}^{i})-\eta_{v,i}(t),\] \[\widetilde{r}(t) =r(t_{k_{i}}^{i})-r(t),\] \[\widetilde{e}_{p}(t) =\widetilde{\eta}_{p,i}(t)-\widetilde{r}(t),\] \[\widetilde{f}_{a,i}(t) =f_{a,i}(t_{k_{i}}^{i})-f_{a,i}(t),t\in\big{[}t_{k_{i}}^{i},t_{k_ {i}+1}^{i}\big{)},\]  
and the internal dynamic variable \(\chi_{i}(t)\) with \(\chi_{i}(0)>0\) is given as follows:  
\[\hat{\chi}_{i}(t)=-\beta_{i}\chi_{i}(t)+\hat{\chi}_{i}(t), \tag{11}\]  
where \(\beta_{i},\sigma_{i},\gamma,\gamma_{p},\gamma_{v},\theta_{p}\), and \(\theta_{v}\in\mathbb{R}_{+}\).  
**Remark 2**.: _In the conventional event triggering mechanism, the triggering condition was constructed from the current and latest triggered states, e.g., in this case, the system states will be triggered when \(\hat{\chi}_{i}(t)\leq 0\) occurs instead of (6). Thus, it is also known as the static event-triggered controller [16]. According to [24], the static event-triggered controller guarantees the asymptotic stability of the system, as well as excluding the Zeno behavior phenomenon. In this study, the dynamic triggering law has been enriched with the time-varying formation control of the multi-agent system._