## 3 Pull Based Event-Triggered Strategy
Then, we have:_  
\[\dot{\delta}_{i}(t)=\sum_{j=1}^{N}L_{d}^{T}(i,j)\delta_{j}(T_{k(T)}^{i}),t>T,i= 1,2,...,N, \tag{33}\]  
_which implies \(\delta_{i}(t)-\delta_{i}(T)=(t-T)\sum_{j=1}^{N}L_{d}^{T}(i,j)\delta_{j}(T_{k(T )}^{i})\)._  
_From Theorem 1, we observe \(\delta_{i}(t)-\delta_{j}(t)\to 0\). Thus, for all \(i,j=1,2,...,N\), we get:_  
\[\sum_{j=1}^{N}L_{d}^{T}(i,j)\delta_{j}(T_{k(T)}^{i})=\sum_{i=1}^{N}L_{d}^{T}(i,i)\delta_{i}(T_{k(T)}^{i}), \tag{34}\]  
\[\delta_{i}(T)=\delta_{j}(T). \tag{35}\]_From (35), we can easily obtain that for all \(t>T\) and \(i,j=1,2,...,N\), \(\delta_{i}(t)=\delta_{j}(t)\). Namely, there is no trigger time for all \(t>T\) such that the circle formation is achieved at time \(T\)._  
_Second, in any finite interval \([t_{1},t_{2}]\), there are only finite triggers. However, it is still an open question to prove that [PERSON] behavior can be avoided under Theorem 1._  
_To address this issue, we propose another triggering strategy as Theorem 2._  
**Theorem 2**. : _Given any admissible circle formation characterized by \(d\), considering system (4) and the 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:_  
\[t_{k+1}^{i}=\max_{\tau>t_{k}^{i}}\left\{\tau:\|\tilde{e}(t)\|^{2} \geq s^{-1}(t)\right\} \tag{36}\]  
_where weight function \(s(t)\) satisfies:_  
\[\left\{ \begin{aligned} s(t)>0,s(t)>0,\frac{s(t)}{s(t)} \leq\phi,\\ \lim_{t\to\infty}s(t)=+\infty,\end{aligned} \right. \tag{37}\]  
_and the condition_  
\[a+\phi<\frac{2\gamma_{2}}{\
u_{N}} \tag{38}\]  
_holds simultaneously, where \(a>0\), \(\phi>0\) are positive real number. That is \(\lim_{t\to\infty}y(t)=d\). Moreover, system (9) does not have Zeno behavior under event-trigger condition (36)._  
Proof. : From (24) and (26), the former part of time derivative of the Lyapunov function along the trajectory of system (11) is turned into:  
\[ \begin{aligned} s(t)&\sum_{i=1}^{N} \tilde{s}_{i}(\delta_{i}(t)-\delta(t))(\tilde{e}_{i}(t)+\tilde{\delta}_{i}(t)) \\ &=s(t)\sum_{i=1}^{N}\tilde{s}_{i}\Delta_{i}(t)\tilde{e}_{i}(t)-s( t)\sum_{i=1}^{N}\sum_{j=1}^{N}\tilde{s}_{i}\Delta_{i}(t)L_{d}^{T}(i,j)\Delta_{j}(t) \\ &\leq s(t)(-\frac{\gamma_{2}}{\
u_{N}}+\frac{a}{2})\sum_{j=1}^{N} \tilde{s}_{i}(\Delta_{i}(t))^{2}+s(t)\frac{1}{2a}\sum_{i=1}^{N}\tilde{s}_{i}( \tilde{e}_{i}(t))^{2},\end{aligned}  \tag{39}\]  
where \(\Delta_{i}(t)=\delta_{i}(t)-\tilde{\delta}_{i}(t)\) and the last inequality is deduced from Lemma 1 and inequality (19). By using (39) and (29), we can conclude:  
\[\dot{V}(t)\leq 2s(t)(-\frac{\gamma_{2}}{\
u_{N}}+\frac{a}{2}+\frac{s(t)}{2 s(t)})V(t)+s(t)\frac{1}{2a}\sum_{i=1}^{N}\tilde{s}_{i}(\tilde{e}_{i}(t))^{2}. \tag{40}\]  
To avoid Zeno behaviour, we further provide an estimate of the positive lower bound on the inter-event times. From conditions (36) and (40), it implies:  
\[\dot{V}(t)\leq 2s(t)(-\frac{\gamma_{2}}{\
u_{N}}+\frac{a}{2}+\frac{ \phi}{2})V(t)+\frac{1}{2a}. \tag{41}\]  
where \(s(t)V(t)\) are bounded, i.e., there exists \(M>0\), such that \(0<s(t)V(t)<M\) for all \(t>0\). Therefore, we have:  
\[V(t)<\frac{M}{s(t)} \tag{42}\]  
and  
\[\|\Delta_{i}(t)\|\leq\sqrt{\frac{2 Ms^{-1}(t)}{\xi_{i}}},i=1,2,...,N. \tag{43}\]  
Furthermore, for any \(t\in[t_{k}^{i},t_{k+1}^{i}]\), it yields to:  
\[\|\dot{\delta}_{i}(t)\|=\left\|\sum_{j=1}^{N}L_{d}^{T}(i,j)\Delta_{j}(t_{k}^{i })\right\|\leq 2L_{d}^{T}(i,i)\sqrt{\frac{2 Ms^{-1}(t_{k}^{i})}{\min_{j}\xi_{j}}}. \tag{44}\]  
As a results, for any \(t\in[t_{k}^{i},t_{k+1}^{i}]\), we have:  
\[\|\dot{\epsilon}_{i}(t)\|^{2}\leq M_{1}s^{-1}(t_{k}^{i}), \tag{45}\]  
where \(M_{1}\) is a positive constant, which is related to the network topology.