# Mean Square Consensus of Nonlinear Multi-Agent Systems under Markovian Impulsive Attacks
## 3 Main Results
 \tag{12}\]  
where \(\theta=\sum_{j=1}^{p}\max_{i\in\Gamma}\{p_{j}\}\frac{\lambda_{i}v_{j}}{ \lambda_{j}-\gamma}<1\). Substituting (12) into (11), we get that  
\[\mathbb{E}[V(t_{k},e(t_{k}))]\leq\theta^{k}\mathbb{E}[V(t_{0},e(t_{0}))]. \tag{13}\]  
from which we obtain that, with condition (6)  
\[\lim_{k\to+\infty}\mathbb{E}[V(t_{k},e(t_{k}))]=0. \tag{14}\]  
which indicates that \(\lim_{k\to+\infty}\mathbb{E}[|e(t_{k})|^{2}]=0\). Hence, the consensus of system (1) can be achieved, and the proof of Theorem 1 is completed. **Remark 2**. : _Note that the parameters \(\gamma\) and \(\lambda_{j}\) must satisfy \(\gamma<\min_{j\in\Gamma}\{\lambda_{j}\}\). If we choose \(\gamma\geq\min_{j\in\Gamma}\{\lambda_{j}\}\), we can easily know that \(\int_{0}^{+\infty}e^{\gamma\lambda_{j}}\lambda_{j}e^{-\lambda_{j}z}ds=+\infty\), so the condition that \(\gamma<\min_{j\in\Gamma}\{\lambda_{j}\}\) is necessary. Moreover, from (6), we can find a fact that the nonlinear functions has a negative effect on the consensus of system (1), which cannot deal with a high-density impulsive attack. If the system considered is linear, then the condition (6) can be released to adapt more general cases._  
**Corollary 1**. : _Under Assumptions 1 and 2, if \(\kappa\equiv 0\), then the consensus of system (1) with controller (2) under Markovian impulsive attacks can be reached if there exist symmetric positive definite matrix \(P\) and feedback gain \(K\) such that_  
\[\sum_{j=1}^{p}\max_{i\in\Gamma}\{p_{ij}\}\frac{\lambda_{j}v_{j}}{\lambda_{j}- \gamma}<1. \tag{15}\]  
_where \(\gamma=\rho\beta<\min_{j\in\Gamma}\{\lambda_{j}\}\), \(\rho=\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}\), \(\beta=\lambda_{\max}(\Pi)\), and \(\Pi=H^{\mathbb{T}}+H\)._  
**Remark 3**. : _The impulsive attacks in this paper are assumed to be stochastic, and since we do not further restrict the distribution of impulsive gains, the explicit relations between the consensus (or system stability) and the impulsive gains are ambiguous, where \(\mathbb{E}[\eta_{k}]\) is replaced by \(\mathbb{E}[\eta_{k}^{2}]\). For the above reason, it may be hard to obtain \(\mathbb{E}[\eta_{k}^{2}]\) as we do not know the distribution to which \(\eta_{k}\) subjected; we need a prior known \(\mathbb{E}[\eta_{k}^{2}]\) when the clear distribution is unknown. On the other hand, it is common and suitable to assume that the gains from attacks satisfy the Bernoulli distribution such as that in the work of [25], representing the occurrence of attacks. Moreover, the condition (6) shows that a large feedback gain is needed if the strength of impulsive attacks is strong. If the gains are deterministic (i.e., \(\eta_{k}=\eta>1,k\in\mathbb{N}\)), then we can get the following corollary, which is the direct result of model simplification, so the proof is also omitted here._  
**Corollary 2**. : _Under Assumptions 1 and 2, then the consensus of system (1) with controller (2) under Markovian impulsive attacks can be reached if there exist symmetric positive definite matrix \(P\) and feedback gain \(K\) such that_  
\[\sum_{j=1}^{p}\max_{i\in\Gamma}\{p_{ij}\}\frac{\lambda_{j}\eta^{2}}{\lambda_{ j}-\gamma}<1. \tag{16}\]  
_where \(\gamma=\rho\beta<0\), \(\rho=\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}\), \(\beta=\lambda_{\max}(\Pi)\), and \(\Pi=H^{\mathbb{T}}+H+\kappa\bar{B}^{\mathbb{T}}+\kappa\bar{B}\)._  
**Remark 4**. : _From the condition (16), one can observe that the dynamics of considered NMASS are stable without impulsive attacks. The reason for this restrictive condition is that the impulsive effects taken into account in this paper are destabilizing, which means that the parameter \(\eta\) always bigger than one. As a result, though the condition \(\gamma<\min_{j\in\Gamma}\{\lambda_{j}\}\) is satisfied, one can find that the inequality (16) never holds when \(\gamma\geq 0\).