## 5 Conclusions  
In this paper, we mainly solve the group consensus of a second-order multi-agent system under an asynchronous DoS attack. An asynchronous DoS attack leads to continuous changes in the communication topologies; thus, we can change the multi-agent system into a switching system; then, we can use switching theory to solve the group consensus problem. Moreover, we use the persistent dwell time (PDT) to describe a DoS attack in this paper, which is greatly different from the average dwell time (ADT). For an ADT, there exists the chatter bound to constrain the number of switching intervals that are less than the average time interval length. Furthermore, an ADT requires that the length between two time intervals cannot be too short. But for the PDT, we use the \(T\)-portion to describe these short time intervals, which means that all of the time intervals will be considered equally no matter the number and length of short time intervals, greatly liberalizing the limit of the ADT. Thus, for general consensus of a multi-agent system under a DoS attack, we can try to combine the switching theory and the concept of the PDT to solve the consensus for multi-agents.  
Figure 21: The event-triggered time instants for agents.  
Figure 20: The switching signal.  
In the real world, group consensus of multi-agent systems is general, such as express deliveries with unmanned aerial vehicles. By the method in this paper, the unmanned aerial vehicles can be divided into different groups to deliver different freight to different destinations. Further, because of the vulnerability of the wireless communication, the formation of unmanned aerial vehicles may suffer cyber-attacks easily. Thus, by the method in this paper, we can embed the event-triggered controller in each agent and set the event-triggered threshold of our methods; then, if the intensity of the cyber-attack is strong enough to violate the event-triggered threshold, the controller will process the control signal to the agent. Then, the formation of unmanned aerial vehicles will be maintained.  
However, there are also some limitations existing in our study. Firstly, there are limits to the multi-agent system model. While a second-order multi-agent system model effectively captures many complex scenarios, it cannot encompass all real-world complexities. For instance, helicopter systems may involve numerous variables, necessitating a high-order multi-agent system model. In this situation, it is hard to construct a Lyapunov function for an error system like in this paper. Thus, our method may be less effective, but one can try to combine a backstepping method and the event-triggered control of this paper to explore the group consensus of a high-order multi-agent system, which will be our future work. Secondly, our analysis is limited to DoS attacks. However, various other types of cyber-attacks exist in real-world applications. For example, deception attacks can also exist in the wireless communication of agents. But the PDT is not suitable to describe a deception attack. Thus, how to describe a deception attack is a problem that we will solve in the future.  
Conceptualization, Y.P.; methodology, Y.P.; validation, Y.P.; formal analysis, Y.P.; writing--original draft preparation, Y.P.; software, Y.P.; writing--review and editing, Y.P.; visualization, Y.P.; supervision, Y.Y. and C.Y. All authors have read and agreed to the published version of the manuscript. This research received no external funding  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.