# Mean Square Consensus of Nonlinear Multi-Agent Systems under Markovian Impulsive Attacks
## 1 Introduction  
With the development of hardware equipment and communication technology, the collaborative control problem has recently become the focus of studies of complex systems [1, 2, 3]. The consensus of multi-agent systems (MASs) plays an important role in the field of collective behaviors, owing to its excellent model assumptions and extensive application scenarios such as control of the formation of unmanned aerial vehicles [4], restoration of a power system [5], intelligent and sustainable supplier selection of supply chains [6], etc. In real-world application scenarios, communication among agents may be affected by the environment or various attacks, especially human-made malicious attacks, which is likely to cause the system to become unstable. For attack scenarios, generally speaking, they may be classified into two main categories: denial-of-service (DoS) attacks [7, 8, 9, 10] and deception attacks [11, 12, 13] (for more details, refer to [14]). DoS attacks are attacks that make the data of sensors and controllers unavailable, resulting in packet time delays or packet drops in the signal transmission. In contrast, deception attacks are attacks that have the ability to obtain and tamper with the transmitting data and the commands from controllers. DoS attacks can destroy communication topologies and can damage the stability of the systems. Compared with DoS attacks, deception attacks are more moderate and difficult to detect, which quietly makes the system unstable. Thus, deception attacks have attracted a lot of interest and has become one of our main concerns. Much research on deception attacks has been conducted in recent years [15, 16, 17, 18, 19]. For instant, consider the existence of false data injection attacks; the authors in [16] investigated the security issue in the state estimation problem of a networked control system. Similarly, in the presence of deception attacks, the authors [19] studied secure synchronization of MASs by means of impulsive control. However, in the above research, the results were based on a linear or an additive assumption and assumptions on the attack model were made, resulting in narrow scopes of applications. Taking this restriction into account, models with more simulation significance and practical application value were established by means of stochastic processes, especially the Bernoulli distribution [20; 21; 22; 23; 24], in which the gains of attacks were assumed to be a stochastic Bernoulli variable. For example, the authors in [21] investigated the consensus of linear MASs under deception attacks with stochastic Bernoulli variables. It is worth noting that the attacks mentioned above were described as discrete events. When the state of the system suffers instantaneous disturbances and experiences sudden changes at certain moments, the system may be attacked. This is a so-called impulsive attack. An impulsive attack is a form of attack that causes instantaneous changes to the state of the system. Models of impulsive attacks with stochastic gains seem to be sufficiently good at neglecting positions. However, in actuality, such a problem is also vital to the stability of systems as we do not know a priori when attacks are made. Suppose that both the gains and instants of the impulsive attacks are stochastic. The author in [25] first considered these two characteristics of an attack, and then, some sufficient conditions for almost stable general Lipschitz-type nonlinear systems were derived. However, the work in [25] still used restricted gains under the Bernoulli distribution and the influence of attacks was assumed to monotonically decrease, which become restrictive assumptions. On the other hand, focused on the variability in an environment, a model relying on a Markovian chain is appropriate in general. Taking this into account, the consensus with Markovian switching topologies has become a hot issue regarding the performance of multi-agent systems with external disturbances.