# Mean Square Consensus of Nonlinear Multi-Agent Systems under Markovian Impulsive Attacks
## 2 Preliminaries  
Before presenting the system models and the problem statement, we first give some necessary knowledge about graph theory, which is used to describe the communication topology among MASs. In this paper, we consider the MASs with \(N\) agents and the topology can be represented by a digraph \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\), where \(\mathcal{V}\) is the set of vertices, \(\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}\) is the set of edges, and \(\mathcal{A}=[a_{ij}]_{N\times N}\) is called the adjacency matrix. When the \(j\)th agent sends information to the \(i\)th agent, there exists an edge between them and \(a_{ij}=1\); otherwise, \(a_{ij}=0\). Let the diagonal matrix \(D=diag\{d_{i}\}\) be the in-degree matrix and \(\hat{d}_{i}=\sum_{j=1}^{N}a_{ij},i=1,2,\ldots,N\). Then, the Laplacian matrix \(L=[l_{ij}]_{N\times N}\) can be defined as \(L=D-\mathcal{A}\), where \(l_{ij}=-a_{ij},i\
eq j\) and \(l_{ii}=\sum_{j\
eq i}^{N}a_{ij}\). Let us consider that generally nonlinear multi-agent systems (NMASs) consist of \(N\) agents and that each can be described as follows:  
\[\dot{x}_{i}(t)=Ax_{i}(t)+Bf(x_{i}(t))+Cu_{i}(t),\quad t\geq t_{0}. \tag{1}\]  
where \(x_{i}(t)\in\mathbb{R}^{n}\) and \(u_{i}(t)\in\mathbb{R}^{n}\) are the system state and external input control, respectively. Function \(f:\mathbb{N}^{n}\rightarrow\mathbb{N}^{n}\) denotes the nonlinear items satisfying the well-known Lipschitz condition. \(A\), \(B\), and \(C\) are constant matrices with approximate dimensions. Before presenting the controller considered in this paper, we first provide some explanations about the impulsive reception attacks, so that readers can understand the design of the controller more clearly. The configuration of NMASs under impulsive reception attacks is shown in Figure 1. Generally, controllers generate control signals according to states measured by sensors and then send them to the actuators. For MASs, data transmissions are heavily dependent on topologies and wireless transmission technology. If attackers launch data modification or false data attacks to topologies in discrete-time manners or calls impulsive attacks, they result in instantaneous jumps in the transmitted signal. Figure 1: Configuration of NMASs under impulsive reception attacks (where \(Ai\) and \(Si\) denote the actuator and sensor of the \(i\)th agent, respectively, and \(l_{i}\) denotes the \(l_{i}\)th agent, \(i=1,2,\ldots,N\)). Under the configuration reflected in Figure 1, in this paper, we consider the following common linear feedback controller under Markovian impulsive attacks:  
\[u_{i}(t)=K\sum_{j=1}^{N}a_{ij}(x_{j}(t)-x_{i}(t))+\sum_{k=1}^{+\infty}d_{k}x_{i}(t )\delta(t-t_{k}). \tag{2}\]  
where \(K\in\mathbb{R}\) is the feedback gain for designed controller. Let \(\mathbb{I}:=\{t_{k}\}_{k=1}^{+\infty}\) denote the impulsive sequence in Dirac form, i.e., the function \(\delta\), and \(d_{k}\) denotes the impulsive gains. In addition, we assume that \(x_{i}(t)\) is right-hand continuous at \(t=t_{k}\), meaning that \(x_{i}(t_{k}^{+})=x_{i}(t_{k})\). Since the destabilizing impulse is adopted to model the impulsive attacks here, one can obtain the fact that \(d_{k}>0\) or \(d_{k}<-2\). Note that \(\sum_{k=1}^{+\infty}d_{k}x_{i}(t)\delta(t-t_{k})\) here denotes impulsive attacks. **Remark 1**. : _Let \(T_{i}(t,t_{0})\) and \(\bar{N}_{i}(t,t_{0})\) denote the total impulsive intervals and the occurrence number for the \(i\)th modes of impulses over the interval \([t_{0},t]\), respectively. Hence, we have \(\bar{N}(t,t_{0})=\sum_{i=1}^{p}\bar{N}_{i}(t,t_{0})\), which denotes the total number of impulses over the interval \([t_{0},t]\)._  
_Considering that the stabilization problem of an error system with ideal dynamics is a common method to tackle the consensus of multi-agent systems.