# Mean Square Consensus of Nonlinear Multi-Agent Systems under Markovian Impulsive Attacks
## 2 Preliminaries
Define the error state as \(e_{i}(t):=x_{i}(t)-\bar{x}(t)=x_{i}(t)-\frac{1}{N}\sum_{i=1}^{N}x_{i}(t)\), which also can be written as \(e(t)=[e_{1}^{\mathsf{T}}(t),e_{2}^{\mathsf{T}}(t),\ldots,e_{N}^{\mathsf{T}}(t )]^{\mathsf{T}}=Ex(t)=((I_{N}-\frac{1}{N}1_{N}1_{N}^{\mathsf{T}})\otimes I_{n} )x(t)\), and the vector of states \(x(t)=[x_{1}^{\mathsf{T}}(t),x_{2}^{\mathsf{T}}(t),\ldots,x_{N}^{\mathsf{T}}(t )]^{\mathsf{T}}\). Then, we can rewrite system (1) with controller (2) in the error system in a compact form as follows:_  
\[ \begin{cases}\dot{e}(t)=He(t)+\bar{B}F(e(t)),\quad t\
eq t_{k},t\geq t_{0}\\ e(t_{k})=\eta_{k}e(t_{k}^{-}),\quad t=t_{k}.\end{cases}  \tag{3}\]  
_where \(H=I_{N}\otimes A-KL\otimes C\). Define \(\bar{B}=I_{N}\otimes B\) and function \(F(e(t))=Ef(x(t))\), where the vector function \(f(x(t))=[f^{\mathsf{T}}(x_{1}(t)),f^{\mathsf{T}}(x_{2}(t)),\ldots,f^{\mathsf{ T}}(x_{N}(t))]^{\mathsf{T}}\). Moreover, \(\eta_{k}=1+d_{k}\), and then, one has \(|\eta_{k}|>1\) according to the values of \(d_{k}\)._  
Before ending this section, here, we present some assumptions and definitions used in this paper. **Assumption 1**. : _Suppose that there exists \(p\) types of impulses, and both the gains and instants of impulsive sequence are stochastic. The gains \(\eta_{k}\) take values from a finite set \(\mathbb{G}=\{\mu_{i}\}_{i\in\Gamma}\) satisfying \(\mathbb{E}[\mu_{i}^{2}]=v_{i}<+\infty\) and \(\vartheta=\max_{i\in\Gamma}\{v_{i}\}\), and \(\Gamma:=\{1,2,\ldots,p\}\). The impulsive sequence \(\{t_{k}\}_{k=1}^{+\infty}\) is governed by a right-continuous homogeneous Markovian chain with \(p\) modes, i.e., for \(t\in[t_{k},t_{k+1}),r_{k}:=r(t)=i\in\Gamma\) with the generator \(\Lambda=[\lambda_{ij}]_{p\times p},i,j\in\Gamma\) given by_  
\[P(r_{t+h}=j|r_{t}=i)=\left\{ \begin{aligned} &\lambda_{ij}h+o(h),& i \
eq j,\\ & 1+\lambda_{ij}h+o(h),& i=j.\end{aligned} \right.\]  
_where \(h>0,\lim_{h\to 0}\frac{o(h)}{h}=0\) and \(\lambda_{ij}\) denotes the transition rate from \(i\) to \(j\), which satisfies that \(\lambda_{ij}\geq 0\) for \(i\
eq j\) and \(\lambda_{ii}=-\sum_{j\
eq i}\lambda_{ij}\)._  
_According to properties of the Markovian chain, we know that the impulsive interval \(\tau(k)=t_{k}-t_{k-1}\) satisfies the exponential distribution, i.e., \(\tau(k)\sim Exp(\lambda_{i})\), when \(r_{k}=i\) and \(\lambda_{i}=|\lambda_{ii}|\). Moreover, we obtain the transition probability that \(p_{ij}=\frac{\lambda_{ij}}{\lambda_{i}}\) for \(i\
eq j\) and \(p_{ij}=0\) for \(i=j\)._  
_Based on Assumption 1, we further assume that \(\eta_{k}\) is a stochastic process on \((\Omega^{1},\mathcal{F}^{1},\mathcal{F}^{1}\mathbb{R}_{k\in\mathbb{N}}, \mathcal{P}^{1})\), where \(\mathcal{F}^{1}\mathbb{R}_{k\in\mathbb{N}}=\sigma(\eta_{1},\eta_{2},\ldots, \eta_{k})\) is the filtration of \(\mathcal{F}^{1}\). On the other hand, the impulsive instants satisfies the Markovian chain, meaning that \(r_{1},r_{2},\ldots,r_{k}\) are independent random variables on a probability space \((\Omega^{2},\mathcal{F}^{2},\mathcal{F}^{2}\mathbb{R}_{k\in\mathbb{N}}, \mathcal{P}^{2})\); similarly, \(\mathcal{F}^{2}\mathbb{R}_{k\in\mathbb{N}}=\sigma(r_{1},r_{2},\ldots,r_{k})\) is the filtration of \(\mathcal{F}^{2}\). Hence, denote \(\Omega:=\Omega^{1}\times\Omega^{2},\mathcal{F}:=\mathcal{F}^{1}\times\mathcal{ F}^{2}\), \(\mathcal{P}:=\mathcal{P}^{1}\times\mathcal{P}^{2}\), and \(\mathcal{F}_{k}:=\sigma(\mathcal{F}^{1}\mathbb{R}\times\mathcal{F}^{2})\). As a result, the jointly stochastic process \(\{(r_{k},\eta_{k})\}_{k\in\mathbb{N}}\) can be defined on the probability space \((\Omega,\mathcal{F},\mathcal{F}_{k},\mathcal{P})\)._  
**Assumption 2**. : _The nonlinear function \(f:\mathbb{R}\rightarrow\mathbb{R}\) is said to be satisfied the Lipschitz condition if there exists a constant \(\kappa>0\) such that_  
\[|f(x_{1})-f(x_{2})|\leq\kappa|x_{1}-x_{2}|. \tag{4}\]  
**Definition 1**.