## 3 Main Results
: _In this study, the term \"fixed-time control\" refers to the use of TBG control protocols to make sure the system's status changes to the intended level within a given time interval \(t_{f}\). Consensus can be achieved through the \(-K\bar{x}_{i}(t_{k}^{i})\) term in the control protocol (12). A key advantage of our approach, compared to many existing fixed-time containment control studies, is the relatively small initial control input, which effectively reduces system costs._  
**Remark 3**. : _The time delay may cause oscillation, instability, or slow convergence. This is a challenge for the design of control systems. It is necessary to consider the time delay influence on the stability of the system. The shortcoming of this paper is that the communication delay is not considered. We plan to further explore this topic in future work to improve the control performance and propose better solutions to deal with communication delays._  
**Remark 4**. : _Event-triggered containment control can effectively reduce communication and computing loads and improve the energy efficiency of the system. The containment control based on the time reference generator can ensure that each agent in the system operates and communicates according to a unified time axis, so as to achieve coordinated and synchronous control. The choice of the appropriate method for the two control schemes depends on the specific application scenarios and system requirements, as well as the trade-off considerations for the communication load, energy efficiency, and synchronization requirements._  
**Theorem 3**. : _When neither party can reach an agreement, Zeno behavior is circumvented in control protocol (12) and under the event-triggering condition (10)._Proof of Theorem 3.: For the time interval \(0\leq t\leq t_{f}\), where  
\[\frac{d||Ke_{i}(t)||^{2}}{dt}=-2e_{i}^{T}(t)K^{T}K\,\dot{\gamma}_{i}(t) \tag{37}\]  
The prior study led to the conclusion that \(e_{i}(t)\) has an upper bound \(\vartheta=\max(|e_{i}(t)|)\). By using Equation (18), we can further derive  
\[\dot{\alpha}_{i}(t)_{\max}=\lambda_{\max}(A)\varpi-(\varphi_{\max}+1)\psi( \vartheta+\varpi) \tag{38}\]  
The following expression can be used to obtain the greatest value of \(\varpi\), of \(\dot{\alpha}_{i}(t)\), in Equation (29):  
\[\varpi=\sqrt{\frac{2\delta V(0)}{\lambda_{\min}(P)\lambda_{\min}(L1)}} \tag{39}\]  
Furthermore, \(\psi\) represents the maximum value of \(L1\otimes BK\), and \(\varphi_{\max}\) represents the maximum value of \(\varphi(t)\), which can be determined using Theorem 2. Therefore, we can conclude that  
\[\frac{d||Ke_{i}(t)||^{2}}{dt}\leq-2\vartheta||K^{T}K||(\lambda_{\max}(A)\varpi -(\varphi(t)_{\max}+1)\psi(\vartheta+\varpi)) \tag{40}\]  
Define \(\Phi_{i}=-2\vartheta||K^{T}K||(\lambda_{\max}(A)\varpi-(\varphi(t)_{\max}+1) \psi(\vartheta+\varpi))\). Thus, we have \(\frac{d||Ke_{i}(t)||^{2}}{dt}\leq\Phi_{i}\leq\infty\). Therefore, we obtain  
\[||Ke_{i}(t)||^{2}\leq\Phi_{i}(t-t_{k_{i}^{i}}^{i}) \tag{41}\]  
where \(t_{k_{i}^{i}}^{i}<t\) represents the most recent event-triggered moment of follower \(i\) at time \(t\). Thus, \(k_{t}^{i}=\arg_{i}\min_{i\in N\geq t_{i}^{i}}\{t-t_{i}^{i}\}\). Furthermore, we can deduce that  
\[||Ke_{i}(t_{k_{i}^{i}+1}^{i})||^{2}\leq\Phi_{i}(t_{k_{i}^{i}+1}^{i}-t_{k_{i}^{ i}}^{i}) \tag{42}\]  
where \(t_{k_{i}^{i}+1}^{i}\) represents the next event-triggered instant. By Equation (10), we can deduce that  
\[ \begin{split}||Ke_{i}(t_{k_{i}^{i}+1}^{i})||^{2}&= \frac{\theta}{\varphi(t_{k_{i}^{i}+1}^{i})+1}||K\dot{x}_{i}(t_{k_{i}^{i}+1}^{i })||^{2}\\ &\geq\frac{\theta}{\varphi_{\max}+1}||K\dot{x}_{i}(t_{k_{i}^{i}+ 1}^{i})||^{2}\\ &>0\end{split}  \tag{43}\]  
Therefore, we can conclude that \(t_{k_{i}^{i}+1}^{i}-t_{k_{i}^{i}}^{i}\geq\Phi_{i}^{-1}||Ke_{i}(t_{k_{i}^{i}+ 1}^{i})||>0\). When \(t>t_{f}\), similar to the example \(0\leq t\leq t_{f}\), the study of inter-event timings and the conclusion \(t_{k_{i}^{i}+1}^{i}-t_{k_{i}^{i}}^{i}>0\) can be directly obtained. Hence, [PERSON] behavior can be avoided.