## 3 Pull Based Event-Triggered Strategy
Further, the derivative of the Lyapunov function (23) along of the trajectories of the system yields to:  
\[\dot{V}(t)=s(t)\sum_{i=1}^{N}\bar{\xi}_{i}(\delta_{i}(t)-\bar{\delta}(t))( \dot{\delta}_{i}(t)-\dot{\bar{\delta}}(t))+\frac{1}{2}s(t)\delta^{T}(t)\Psi \delta(t). \tag{24}\]  
Owing to \(\xi^{T}L_{d}^{T}=0\), it leads to:  
\[\dot{\bar{\delta}}(t) =\sum_{i=1}^{N}\bar{\xi}_{i}\dot{\delta}_{i}(t)=\sum_{i=1}^{N} \bar{\xi}_{i}\sum_{j=1}^{N}L_{d}^{T}(i,j)\hat{\delta}_{j}(t) \tag{25}\] \[=\sum_{i=1}^{N}\bar{\delta}_{j}(t)\sum_{j=1}^{N}\bar{\xi}_{i}L_{d }^{T}(i,j)=0.\]  
For clarity, (24) can be divided into two parts, the former part of (24) is calculated by:  
\[s(t)\sum_{i=1}^{N}\bar{\xi}_{i}(\delta_{i}(t)-\bar{\delta}(t)) \dot{\delta}_{i}(t) \tag{26}\] \[=-s(t)\sum_{i=1}^{N}\bar{\xi}_{i}(\delta_{i}(t)-\bar{\delta}(t)) \sum_{j=1}^{N}L_{d}^{T}(i,j)\delta_{j}(t)\] \[=s(t)\sum_{i=1}^{N}\bar{\xi}_{i}(\delta_{i}(t)-\bar{\delta}(t)) \bar{\delta}_{i}(t_{k}^{i})\] \[=s(t)\sum_{i=1}^{N}\bar{\xi}_{i}(\delta_{i}(t)-\bar{\delta}(t))( \bar{\sigma}_{i}(t)+\bar{\delta}_{i}(t))\] \[\lim_{t\to\infty}\delta_{i}(t)=\lim_{t\to\infty}\delta_{j}(t)=\sum_{i=1}^{N} \xi_{i}\delta_{i}(0)=c, \tag{22}\]0 \[\lim_{t\to\infty}\delta_{i}(t)=\lim_{t\to\infty}\delta_{j}(t)=\sum_{i=1}^{N} \xi_{i}\delta_{i}(0)=c, \tag{22}\]1  
Together with Lemma 1 and (17), (26) is rewritten as:  
\[\lim_{t\to\infty}\delta_{i}(t)=\lim_{t\to\infty}\delta_{j}(t)=\sum_{i=1}^{N} \xi_{i}\delta_{i}(0)=c, \tag{22}\]2 \[\lim_{t\to\infty}\delta_{i}(t)=\lim_{t\to\infty}\delta_{j}(t)=\sum_{i=1}^{N} \xi_{i}\delta_{i}(0)=c, \tag{22}\]3  
where \(0<a<\frac{2\gamma_{2}}{v_{N}^{2}}\). By using (19), the former part of (24) is rearranged as:  
\[ \begin{split}&-s(t)(1-\frac{av_{N}^{2}}{2\gamma_{2}})\frac{\gamma_{ 2}}{\rho(L_{d}L_{d}^{T})}\delta^{T}(t)L_{d}L_{d}^{T}\delta(t)+s(t)\frac{1}{2a} \tilde{e}^{T}(t)\tilde{e}(t)\\ &\leq-s(t)\left((1-\frac{av_{N}^{2}}{2\gamma_{2}})\frac{\gamma_{ 2}}{\rho(L_{d}L_{d}^{T})}\tilde{\delta}^{T}(t)\tilde{\delta}(t)+\frac{1}{2a} \tilde{e}^{T}(t)\tilde{e}(t)\right).\end{split}  \tag{28}\]  
Meanwhile, from (18), the latter part of (24) is written as:  
\[\frac{\dot{s}(t)\delta^{T}(t)\Psi\Psi\delta(t)}{2\Psi}\leq\dot{s}(t)\frac{ \omega_{2}}{\
u_{N}}\delta^{T}(t)\tilde{\delta}(t). \tag{29}\]  
Together with (28) and (29), (24) can be rearranged as:  
\[\dot{V}(t)\leq-s(t)\left((1-\frac{av_{N}^{2}}{2\gamma_{2}})\frac{\gamma_{2}}{ \rho(L_{d}L_{d}^{T})}\tilde{\delta}^{T}(t)\tilde{\delta}(t)-\frac{\dot{s}(t) \omega_{2}}{s(t)\
u_{N}}\delta^{T}(t)-\frac{1}{2a}\tilde{e}^{T}(t)\tilde{e}(t )\right). \tag{30}\]  
Enforcing the event condition in (20), we can immediately obtain that \(\dot{V}(t)\leq 0\) and \(\dot{V}(t)=0\) if and only if the Circle Formation Problem is solvable. The proof is completed. As a particular case, we have the following results. **Corollary 1**. : _Given any admissible circle formation characterized by \(d\), considering system (4) and the designed control law (9) over a directed graph \(\mathcal{G}\) containing a spanning tree, the Circle Formation Problem is solvable when the event-trigger condition designs as:_  
\[t_{k+1}^{i}=\max_{\tau>t_{k}^{i}}\left\{\tau:\|\tilde{e}(t)\|\geq\frac{\| \tilde{\delta}(t)\|}{\sqrt{2a}\varphi}\right\}, \tag{31}\]  
_where \(0<a<\frac{2\gamma_{2}}{\
u_{N}^{a}}\), and the condition_  
\[ \begin{split}&\varphi=(1-\frac{av_{N}^{2}}{2\gamma_{2}})\frac{ \gamma_{2}}{\rho(L_{d}L_{d}^{T})}-\frac{\omega_{2}\phi}{\
u_{N}}>0,\\ &\phi\geq\dot{s}(t)/s(t)\end{split}  \tag{32}\]  
_holds simultaneously, where \(s(t)=e^{\phi t}\) is the weight function. That is, \(\lim_{t\to\infty}y(t)=d\)._  
_To use the event-triggering strategy, the following two issues should be addressed. First, under the proposed policy in Theorem 1, there exists at least one agent \(i\), where the event interval between \(t_{1}^{k}\) and \(t_{1}^{k+1}\) is strictly positive before the circle formation is achieved._  
_As a matter of fact, we have the assumption that there is no trigger event when \(t>T\).