Consensus of impulsive coupled networked delay systems under edge-based attacks

Abstract

This paper investigates the consensus of impulsive coupled networked delay systems (NDSs) under edge-based attacks. First, an edge-based attack model is established, which is asynchronous and can change the states of nodes. Next, according to the number of edges allowed to be attacked at a time instant, the edge-based attack is divided into two cases. For each case, the sufficient condition is given to guarantee the system consensus against the edge-based attacks. Then, by considering a special type of edge-based attacks on the NDSs with star topology, a less conservative consensus criterion is obtained. Finally, two numerical examples are given to verify the correctness of the obtained results.

Keywords

Consensus impulsive coupling edge-based attacks time delay networked systems

Introduction

In recent years, networked systems (NSs) have attracted much attention due to their wide application in pattern recognition (Hoppensteadt and Izhikevich, 2000), image encryption (Wei et al., 2019), and so on. Consensus, as one of the basic and important properties of NSs, has attracted more and more researchers’ interests (Qi et al., 2021; Ren et al., 2022).

It is well known that nodes in NSs are usually coupled through communication channels, and appropriate coupling schemes are beneficial to achieving some good properties such as consensus (Guo et al., 2015; Sun et al., 2022; Wu and Chen, 2008; Yu et al., 2019). According to existing works, there are mainly two common ways of nodes coupling: continuous-time coupling, which often requires continuous transmission between nodes (Li et al., 2018, 2022b; Yang et al., 2015); and impulsive coupling, which is activated only at some discrete time instants. In comparison, the impulsive coupling has the feature of less energy consumption (Yang, 2001). The investigation of impulsive coupling has attracted the interest of some scholars in recent years. For example, in Chen et al. (2019), the consensus of NSs is analyzed through the appropriate choice of impulsive coupling strength. To realize the consensus of linear NSs, literature by Wang et al. (2017) introduces a stochastic impulsive coupling protocol. By designing the control gain of coupled impulsive control strategy, it is shown that the consensus of multi-agent system is achieved in Zhang et al. (2020) and Zhu et al. (2020).

In real NSs, the network communication environment is not absolutely safe. Instead, it is often subject to all kinds of cyberattacks, which may destroy the consensus of the network. There have been some investigations on cyberattacks in recent years (Beg et al., 2019; Fawzi et al., 2014). According to existing research, cyberattacks can be roughly divided into three categories: denial-of-service (DoS) attacks, which make some information transmission between nodes unavailable (Befekadu et al., 2015); replay attacks, which maliciously duplicate the information transmitted between nodes (Chen et al., 2018); and deception attacks, which infuse false data in information transmission (Xie et al., 2011). To the best of the authors’ knowledge, when investigating these attacks, most existing works tend to study cyberattacks in systems with a single node (Liu et al., 2022b; Wakaiki et al., 2020). Although there are some exceptions that consider the case of multiple nodes, they often assume that all nodes are attacked simultaneously (Wang et al., 2022). However, such results are sometimes difficult to apply since many real NSs have large number of nodes. Real examples include neural network systems, social communication networks, and traffic systems. Therefore, when the network scale is relatively large, attacking all the nodes of the network at the same time requires huge energy consumption and is difficult to achieve. As a contrast, it seems easier for the hackers to attack only part of the network. However, such attacks have not been fully studied.

The spatial distribution of nodes and the communication links between nodes make it possible for both nodes and edges to be attacked. The corresponding attacks are called node attacks and edge attacks, respectively (He et al., 2022). Because of the important role of communication channels (or edges) in NSs, edge-based attacks have attracted much interest of scholars. It is known that edge-based attacks are often asynchronous, which makes it possible for attackers to attack part of the network. To be specific, each edge in the network topology is attacked by an individual attacker, whose strategy is independent of others. In addition, the above mentioned classic cyberattacks inject false data into the network during information transmission to destroy the system, but system states are not directly changed. Different from the the above mentioned synchronous launching of cyberattacks, this paper is devoted to the study of asynchronous edge-based attacks that can directly affect the state of the system.

In real NSs such as neural network and biological network, time delay phenomenon is universal. The existence of time delay affects the performance of the system. Hence, it is necessary to take time delay into account (Li et al., 2019; Lu et al., 2022; Song et al., 2021). There exist a lot of work on investigating NSs with time delays. For example, the NSs studied in Li et al. (2022a) consider input delay and time-varying communication delays simultaneously. In Zhang et al. (2022), both stochastic deception attacks and time delay are considered in the network model. It is shown that the time delay plays a negative role for the consensus of coupled networks. In Ni et al. (2022), the consensus problem of the multi-agent systems under DoS attacks is studied. In Jiang et al. (2020), it is shown that the delay in impulses has double effects, that is, it can stabilize an unstable system or destroy a stable system under the corresponding conditions. When the time delay is within the specified upper and lower bounds, Qiu et al. (2023) provide the conditions for the system to achieve consensus against DoS attacks. Although both attacks and time delays are considered in the literature mentioned above, it should be pointed out that the attacks occured are synchronized for all the nodes or edges. Hence, the consensus of NSs under asynchronous attacks and time delays simultaneously is still an open problem.

In NSs, the neighboring nodes will interact with each other through edges. On one hand, the open communication environment makes edges vulnerable to attackers. The asynchrony and flexibility of edge-based attacks make them ubiquitous in large-scale practical systems such as traffic network systems (He et al., 2022). On the other hand, the time delay of the node itself will affect the dynamics of the node, and then affect the consensus of the whole system. Therefore, the investigation of edge-based attack strategy is important to maintain the desired performance of the system with time delay. Although there have been some interesting works about cyberattacks in NSs (Liu et al., 2022a; Persis and Tesi, 2015), few of them have been devoted to analyzing the effect of edge-based attacks on the consensus of impulsive coupled networked delay systems. As with many other attacks, the edge-based attacks usually have a negative impact on the consensus of NSs. Inspired by Xu et al. (2020) and Maity and Tsiotras (2022), this paper will focus on the influence of the edge-based attacks for the consensus of an NS. Up to now, there is not much literature to analyze edge attacks because the asynchronous nature of edge-based attacks brings challenges to the analysis process. This paper establishes a new edge-based attack mode and proposes two attack strategies. It is shown that as more information is known about the attackers, the easier and more accurately we can deduce the intensity of the attacks the system can tolerate.

Based on the above analysis, this paper studies the consensus of impulsive coupled NDSs under edge-based attacks. The contributions of this paper are as follows:

In this paper, the new edge-based attack models are established, which do not require all nodes to be attacked at the same time. It is worth mentioning that this paper divides the attacker’s strategy into two situations. In one case, only one edge is attacked at the impulsive coupling time instants, and in the other case, multiple edges are allowed to be attacked simultaneously.

The effect of edge-based attacks is analyzed, and two sufficient conditions are given for the consensus of the impulsive coupled delay system against edge-based attacks, respectively. Furthermore, the less conservative consensus criteria is derived by applying the general result to a typical star networks. Table 1 summarizes the meaning of notations used in this article.

Table 1.

Notations.

Notations	Definitions
$R^{n}$	N-dimensional Euclidean space
$R^{n \times n}$	Set of $n \times n$ real matrices
⊗	Kronecker product
$N_{+}$	Set of positive integers
$R_{+}$	Set of positive real numbers
$B^{T}$	The transpose of $B$
$δ (\cdot)$	Dirac function
$λ_{max} (B)$	Largest eigenvalue of real symmetric matrix $B$
$max (x, y)$	The maximum value of $x$ and $y$
$C ([t_{0} - τ, t_{0}], R_{+})$	Class of continuous functions from $[t_{0} - τ, t_{0}]$ to $R_{+}$

Model description

We consider the following NDSs with $N$ nodes

{\overset{\cdot}{x}}_{i} (t) = C x_{i} (t) + B x_{i} (t - τ (t)) + u_{i} (t), i = 1, \dots, N

(1)

where $t \geq t_{0}$ , $x_{i} (t) \in R^{n}$ is the state vector of the $i th$ node, and $x_{i} (t) = ϕ_{i} (t) \in C ([t_{0} - τ, t_{0}], R^{n})$ is the initial state of node $i$ ; $C, B \in R^{n \times n}$ ; $0 \leq τ (t) \leq τ$ is time delay; $u_{i} (t) \in R^{n}$ is the impulsive coupling term of node $i$ .

Let $G = (V, E, A)$ represents an undirected graph, where $V = {1, 2, \dots, N}$ is the set of nodes of $G$ , $E \in V \times V$ is the set of edges, $A = (a_{ij})_{N \times N}$ is the adjacency matrix, and $a_{ij} > 0$ if and only if $(i, j) \in E$ , $a_{ij} = 0$ otherwise. The impulsive coupling term $u_{i} (t)$ we considered in this paper is as follows

{\begin{matrix} u_{i} (t) = 0, t \in [t_{k}, t_{k + 1}), \\ u_{i} (t) = - \sum_{k = 1}^{\infty} δ (t - t_{k}) α h_{i} (t_{k}^{-}), t = t_{k} \end{matrix}

(2)

where $t_{k}, k \in N_{+}$ represent the time instants of impulsive coupling; $h_{i} (t) = \sum_{j = 1}^{N} a_{ij} (x_{i} (t) - x_{j} (t))$ is the coupling term and $α$ is the coupling strength.

Recall that coupling is activated only at discrete time instants ${t_{k}}$ . Each node transmits information to its neighbors only at these coupling time instants. As can be seen from Figure 1, on the one hand, node $i$ can always receive its own state information through its own sensor and actuator. On the other hand, the nodes of the NS exchange information at ${t_{k}}$ . At $t \neq t_{k}$ , the communication channel is closed and the transmitted $u_{i} (t)$ is $0$ . On the contrary, at time instant $t = t_{k}$ , the signal from the sensor will be transmitted to the neighboring nodes via the communication network. Meanwhile, the signals of its neighbors will also be known by node $i$ through the channels. However, open communication channels easy subject to attacks, which may cause the system to lose consensus.

Figure 1.

Block diagram of impulsive coupled NSs.

Suppose there is an attacker on each edge. Each attacker has its own attack strategy and is independent of the others. The purpose of edge-based attack is to undermine the consensus of the system. Hence, for node $i$ , the more edges connected to node $i$ are attacked, the more damage node $i$ suffers. We first consider a simple case, that is, at most one edge is attacked at each impulsive coupling time instant. Inspired by literature Zhang et al. (2022) and He et al. (2020), the attack on each edge considered in this paper is in the form of impulse. Specifically, if the edge $(i, j) \in E$ is attacked at time instant $t$ , the states of nodes $i$ and $j$ will be mutated, that is, $x_{i} (t) = θ^{ij} x_{i} (t^{-})$ and $x_{j} (t) = θ^{ji} x_{j} (t^{-})$ , where $θ^{ij} = θ^{ji}$ represents the attacked strength of the attacker on edge $(i, j)$ . In addition, due to the malicious nature of the attack, it is natural to assume $θ^{ij} = θ^{ji} > 1$ . Let $t_{k}^{ij}$ represents the $k$ th attacked time instant on edge $(i, j)$ . Then, according to the right continuity of the states, the attack on edge $(i, j) \in E$ has the following form

{\begin{matrix} x_{i} (t_{k}^{ij}) = θ^{ij} x_{i} ({(t_{k}^{ij})}^{-}), \\ x_{j} (t_{k}^{ij}) = θ^{ji} x_{j} ({(t_{k}^{ij})}^{-}) \end{matrix}

(3)

For edge $(i, j) \in E$ , $T^{ij} = {t_{k}^{ij}}_{k = 1}^{\infty}$ represents all the time instants it is attacked. Because the attack is edge-based, we have $t_{k}^{ij} \in \bar{T} = {t_{k}}_{k = 1}^{\infty}$ . In this section, without loss of generality, we assume that one and only one edge is attacked at each $t_{k}$ . Then, it can be obtained that $\bar{T} = {T^{ij} | (i, j) \in E} = {t_{k}}_{k = 1}^{\infty}$ .

Under edge-based attacks (3) and impulsive coupling (2), system (1) can be rewritten as

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = C x_{i} (t) + B x_{i} (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ x_{i} (t) = (θ^{ij} + 1) x_{i} (t^{-}) - α h_{i} (t^{-}), t \in T^{ij}, \\ x_{i} (t) = x_{i} (t^{-}) - α h_{i} (t^{-}), t \in \bar{T} \ T^{ij}, \\ x_{i} (t) = ϕ_{i} (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(4)

where $i, j = 1, 2, \dots, N$ and $(i, j) \in E$ . For node $i$ , define function $q_{i} (t)$ as follows

{\begin{matrix} q_{i} (t) = 1 & t \in T^{ij} \\ q_{i} (t) = 0 & otherwise \end{matrix}

(5)

Further, system (4) can be written in the following form

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = C x_{i} (t) + B x_{i} (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ x_{i} (t) = x_{i} (t^{-}) + q_{i} (t) θ^{ij} x_{i} (t^{-}) - α h_{i} (t^{-}), t = t_{k}, \\ x_{i} (t) = ϕ_{i} (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(6)

Let $L = (l_{ij})_{N \times N}$ , where $l_{ij} = - a_{ij}$ , if $i \neq j$ and $l_{ii} = \sum_{k = 1}^{N} a_{ik}$ , if $i = j$ . Denote $x (t) = [x_{1} (t)^{T}, x_{2} (t)^{T}, \dots, x_{N} (t)^{T}]^{T}$ . Then, one has

{\begin{matrix} \overset{\cdot}{x} (t) = (I_{N} \otimes C) x (t) + (I_{N} \otimes B) x (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ x (t_{k}) = [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}] x (t_{k}^{-}), t = t_{k}, \\ x (t) = ϕ (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(7)

where $ϕ (t) = [ϕ_{1}^{T} (t), ϕ_{2}^{T} (t), \dots, ϕ_{N}^{T} (t)]^{T}$ and $Q (t_{k}) = diag (q_{1} (t_{k}), q_{2} (t_{k}), \dots, q_{N} (t_{k}))$ . Since only one edge is attacked at any time instant $t_{k}$ , two elements of matrix $Q (t_{k})$ are equal to 1 and the rest are equal to 0. To be specific, if edge $(i, j) \in E$ is attacked at $t_{k}$ , then, we have $Q (t_{k}) = diag (0, \dots, 0, 1, 0, \dots, 0, 1, 0, \dots, 0)$ , where the two elements equal to 1 are in row $i$ and row $j$ , and corresponding $θ_{k} = diag (0, \dots, 0, θ^{ij}, 0, \dots, 0, θ^{ji}, 0, \dots, 0)$ .

Remark 1. NSs are susceptible to malicious attacks. The purpose of the attacker is to destroy or eavesdrop the signal transmission on the targeted channel, thereby undermining the consensus of the system. The idea of one-to-one correspondence between attacker and edge in this paper comes from literature Xu et al. (2020) and Maity and Tsiotras (2022). We use this idea to investigate a new kind of attack in the impulsive coupled NSs. In addition, different from the DoS attacks in Xu et al. (2020), this paper studies a new type of asynchronous edge-based attack and takes time delay into account. Compared with the work by Maity and Tsiotras (2022), which estimates the system state by eavesdropping the information on the channel, the attacker strategy in this paper can directly change the node state.

Main results

Assumption 1. For the impulsive coupling time instants $\bar{T} = {t_{k}}_{k = 1}^{\infty}$ , the following inequality holds

N (t, s) \leq \frac{t - s}{T} + N_{0}

(8)

where $t \geq s \geq t_{0}$ , $T > 0$ , $N_{0} > 0$ , and $N (t, s) \geq 0$ represent the number of impulsive coupling instants within $[s, t]$ .

Theorem 1. For the NDSs (1) with impulsive coupling (2), suppose the edge-based attacks satisfying (3) and impulsive coupling time instants satisfying (8). Then, system (1) can realize consensus, if

\begin{matrix} λ_{max} (C + C^{T}) + 1 + \frac{\ln max (η, η^{- 1})}{T} + λ_{max} (B^{T} B) \\ {(max (η, η^{- 1}))}^{N_{0}} < 0 \end{matrix}

(9)

where $η$ is defined in equation (12).

Proof. Construct the Lyapunov function $V (t) = x (t)^{T} x (t)$ . When $t \in [t_{0} - τ, t_{0}]$ , it holds that

V (t) = x (t)^{T} x (t) = ϕ (t)^{T} ϕ (t) sup_{t \in [t_{0} - τ, t_{0}]} ϕ (t)^{T} ϕ (t)

(10)

when $t \in [t_{k}, t_{k + 1})$ , take the derivative of $V (t)$ along equation (7) to get

\begin{matrix} \overset{\cdot}{V} (t) = 2 x (t)^{T} \overset{\cdot}{x} (t) \\ = 2 x (t)^{T} (I_{N} \otimes C) x (t) + 2 x (t)^{T} (I_{N} \otimes B) x (t - τ (t)) \\ = x (t)^{T} [I_{N} \otimes (C + C^{T})] x (t) + 2 x (t)^{T} (I_{N} \otimes B) x (t - τ (t)) \\ \leq (λ_{max} (C + C^{T}) + 1) x (t)^{T} x (t) + x (t - τ (t))^{T} (I_{N} \otimes B^{T}) \\ (I_{N} \otimes B) x (t - τ (t)) \\ \leq (λ_{max} (C + C^{T}) + 1) x (t)^{T} x (t) + λ_{max} (B^{T} B) x (t - τ (t))^{T} \\ x (t - τ (t)) \\ = (λ_{max} (C + C^{T}) + 1) V (t) + λ_{max} (B^{T} B) V (t - τ (t)) \end{matrix}

(11)

when $t = t_{k}$ , one has

\begin{matrix} V (t_{k}) = x (t_{k})^{T} x (t_{k}) \\ = x (t_{k}^{-})^{T} [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} \\ [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}] x (t_{k}^{-}) \\ \leq λ_{max} ([I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} \\ [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]) x (t_{k}^{-})^{T} x (t_{k}^{-}) \\ \leq max_{k \in N_{+}} {λ_{max} ([I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} \\ [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}])} x (t_{k}^{-})^{T} x (t_{k}^{-}) \\ \overset{Δ}{=} η V (t_{k}^{-}) \end{matrix}

(12)

where $η = max_{k \in N_{+}} {λ_{max} ([I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}])}$ . From the previous analysis, it is easy to know that there are at most $N (N - 1) / (2)$ possible values of $Q (t_{k}) θ_{k}, k \in N_{+}$ . Hence, $max_{k \in N_{+}} {λ_{max} ([I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}])}$ is achievable.

Combining equations (10), (11), and (12), we have

{\begin{matrix} \overset{\cdot}{V} (t) \leq (λ_{max} (C + C^{T}) + 1) V (t) \\ + λ_{max} (B^{T} B) V (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ V (t_{k}) \leq η V (t_{k}^{-}), t = t_{k}, \\ V (t) \leq {sup}_{t \in [t_{0} - τ, t_{0}]} ϕ {(t)}^{T} ϕ (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(13)

In the following, we will apply comparison principle to prove the theorem. Given $W (t)$ is the solution of the following system

{\begin{matrix} \overset{\cdot}{W} (t) = (λ_{max} (C + C^{T}) + 1) W (t) \\ + λ_{max} (B^{T} B) W (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ W (t_{k}) = η W (t_{k}^{-}), t = t_{k}, \\ W (t) = {sup}_{t \in [t_{0} - τ, t_{0}]} ϕ {(t)}^{T} ϕ (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(14)

By the comparison principle, one has

0 \leq V (t) \leq W (t), t \geq t_{0}

(15)

The solution of system (14) is as follows

\begin{matrix} W (t) = U (t, t_{0}) W (t_{0}) + \int_{t_{0}}^{t} U (t, s) λ_{max} (B^{T} B) W (s - τ (s)) ds, t \geq t_{0} \end{matrix}

(16)

where $U (t, s)$ is the Cauchy matrix of the system

{\begin{matrix} \overset{\cdot}{v} (t) = (λ_{max} (C + C^{T}) + 1) v (t), & t \in [t_{k}, t_{k + 1}) \\ v (t_{k}) = η v (t_{k}^{-}), & t = t_{k} \end{matrix},

(17)

where $v (t) \in R$ . Further, by condition (8), it can be calculated that

\begin{matrix} U (t, s) = e^{(λ_{max} (C + C^{T}) + 1) (t - s)} \underset{s < t_{k} \leq t}{Π} η \\ \leq {\begin{matrix} e^{(λ_{max} (C + C^{T}) + 1) (t - s)}, η \in (0, 1) \\ e^{(λ_{max} (C + C^{T}) + 1) (t - s)} η^{\frac{t - s}{T} + N_{0}}, η \geq 1 \end{matrix} \\ = {\begin{matrix} η^{- N_{0}} e^{(λ_{max} (C + C^{T}) + 1 + \frac{\ln η^{- 1}}{T}) (t - s)} e^{\frac{\ln η}{T} (t - s)} η^{N_{0}}, η \in (0, 1) \\ η^{N_{0}} e^{(λ_{max} (C + C^{T}) + 1 + \frac{\ln η}{T}) (t - s)}, η \geq 1 \end{matrix} \\ \leq {\begin{matrix} η^{- N_{0}} e^{(λ_{max} (C + C^{T}) + 1 + \frac{\ln η^{- 1}}{T}) (t - s)}, η \in (0, 1) \\ η^{N_{0}} e^{(λ_{max} (C + C^{T}) + 1 + \frac{\ln η}{T}) (t - s)}, η \geq 1 \end{matrix} \\ = max {(η, η^{- 1})}^{N_{0}} e^{- p (t - s)} \end{matrix}

(18)

where $p = - (λ_{max} (C + C^{T}) + 1 + \ln max (η, η^{- 1}) / (T))$ and the second inequality of equation (18) applies to $e^{\ln η / (T) (t - s)} η^{N_{0}} \leq 1$ when $η \in (0, 1)$ . By condition (9), we have $p > λ_{max} (B^{T} B) (max {η, η^{- 1}})^{N_{0}} > 0$ . Then, combining equations (16) and (18), one has

\begin{matrix} W (t) \leq max {(η, η^{- 1})}^{N_{0}} \\ [W (t_{0}) e^{- p (t - t_{0})} + \int_{t_{0}}^{t} e^{- p (t - s)} λ_{max} (B^{T} B) W (s - τ (s)) ds], t \geq t_{0} \end{matrix}

(19)

When $t \in [t_{0} - τ, t_{0}]$ , because of the boundedness of $W (t_{0})$ , for $\forall ε > 0$ , we have

W (t) \equiv W (t_{0}) < W (t_{0}) + ε \overset{Δ}{=} W_{0} max {(η, η^{- 1})}^{N_{0}} W_{0} e^{- \bar{λ} (t - t_{0})}

(20)

where $max (η, η^{- 1})^{N_{0}} \geq 1$ and $\bar{λ}$ is the root of function $f (x) = x - (p - max (η, η^{- 1})^{N_{0}} λ_{max} (B^{T} B) e^{x τ})$ . The derivative of $f (x)$ is $\overset{\cdot}{f} (x) = 1 + max (η, η^{- 1})^{N_{0}} λ_{max} (B^{T} B) e^{x τ} τ > 0$ and hence, $f (x)$ is strictly increasing. On the one hand, by condition (9), we know

\begin{matrix} f (0) = - p + max {(η, η^{- 1})}^{N_{0}} λ_{max} (B^{T} B) \\ = λ_{max} (C + C^{T}) + 1 + \frac{\ln max (η, η^{- 1})}{T} max \\ {(η, η^{- 1})}^{N_{0}} λ_{max} (B^{T} B) \\ < 0 \end{matrix}

(21)

On the other hand, $lim_{x \to + \infty} f (x) = + \infty$ . Hence, $f (x) = 0$ has a unique positive root $\bar{λ}$ . It can be obtained that equation (20) holds.

Next, we prove that the following inequality is true

W (t) < max {(η, η^{- 1})}^{N_{0}} W_{0} e^{- \bar{λ} (t - t_{0})}, \forall t \geq t_{0}

(22)

Assume equation (22) is not true. By equation (20) and $W (t) \in C ([t_{k - 1}, t_{k}), R_{+})$ , we can find a $t_{*} > t_{0}$ such that

W (t_{*}) \geq max {(η, η^{- 1})}^{N_{0}} W_{0} e^{- \bar{λ} (t_{*} - t_{0})}

(23)

and

W (t) < max {(η, η^{- 1})}^{N_{0}} W_{0} e^{- \bar{λ} (t - t_{0})}, t < t_{*}

(24)

Denote $ζ = max (η, η^{- 1})^{N_{0}}$ . Combining equations (19) and (24), one has

\begin{matrix} W (t_{*}) \leq ζ W (t_{0}) e^{- p (t_{*} - t_{0})} + ζ \int_{t_{0}}^{t_{*}} e^{- p (t_{*} - s)} λ_{max} (B^{T} B) ζ W (t_{0}) \\ e^{- \bar{λ} (s - τ (s) - t_{0})} ds \\ < ζ W_{0} e^{- p (t_{*} - t_{0})} + ζ \int_{t_{0}}^{t_{*}} e^{- p (t_{*} - s)} λ_{max} (B^{T} B) ζ W_{0} e^{- \bar{λ} (s - τ (s) - t_{0})} ds \\ \leq ζ W_{0} e^{- p (t_{*} - t_{0})} + ζ W_{0} \int_{t_{0}}^{t_{*}} ζ λ_{max} (B^{T} B) e^{\bar{λ} τ} e^{- p t_{*} + \bar{λ} t_{0}} e^{(p - \bar{λ}) s} ds \\ = ζ W_{0} [e^{- p (t_{*} - t_{0})} + ζ λ_{max} (B^{T} B) e^{\bar{λ} τ} e^{- p t_{*} + \bar{λ} t_{0}} \int_{t_{0}}^{t_{*}} e^{(p - \bar{λ}) s} ds] \\ = ζ W_{0} [e^{- p (t_{*} - t_{0})} + ζ λ_{max} (B^{T} B) e^{\bar{λ} τ} \frac{e^{- \bar{λ} (t_{*} - t_{0})} - e^{- p (t_{*} - t_{0})}}{p - \bar{λ}}] \\ = ζ W_{0} e^{- \bar{λ} (t_{*} - t_{0})} \\ = max {(η, η^{- 1})}^{N_{0}} W_{0} e^{- \bar{λ} (t_{*} - t_{0})} \end{matrix}

(25)

where $p - \bar{λ} = max (η, η^{- 1})^{N_{0}} λ_{max} (B^{T} B) e^{\bar{λ} τ}$ is used. It is obvious that equations (23) and (25) contradict with each other. Hence, by equations (15) and (22), one has

0 \leq V (t) \leq W (t) < max {(η, η^{- 1})}^{N_{0}} W_{0} e^{- \bar{λ} (t - t_{0})}, t \geq t_{0}

(26)

By $\bar{λ} < 0$ , it can be obtained that

lim_{t \to + \infty} max {(η, η^{- 1})}^{N_{0}} W_{0} e^{- \bar{λ} (t - t_{0})} = 0

Hence, one has

lim_{t \to + \infty} V (t) = 0

Further, we have $x (t) \to 0$ when $t \to + \infty$ , which implies that system (1) can achieve consensus.

Remark 2. It should be noted that the value of $η = max_{k \in N_{+}} {λ_{\max} ([I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}])}$ is achievable. From the assumption that only one edge is attacked at each impulsive coupling time instant, we know that $Q (t_{k})$ and $θ_{k}$ are diagonal matrices with only two non-zero elements and the rest are zero, and the corresponding positions of non-zero elements are the same. If edge $(i, j) \in E$ is attacked at time $t_{k}$ , then $Q (t_{k}) θ_{k} = diag (0, \dots, 0, θ^{ij}, 0, \dots, 0, θ^{ji}, 0, \dots, 0)$ , where the two non-zero elements are in row $i$ and row $j$ . For an undirected graph with $N$ nodes, there are at most $N (N - 1) / 2$ edges, and there are at most $N (N - 1) / 2$ values of $Q (t_{k}) θ_{k}$ for any $k \in N_{+}$ . Therefore, $η$ can be obtained from these finite values.

Remark 3. Note that the lower bound of convergence rate of $V (t)$ satisfies $\bar{λ} = p - max (η, η^{- 1})^{N_{0}} λ_{max} (B^{T} B) e^{\bar{λ} τ}$ . It can be observed that $\bar{λ}$ decreases with the increase of $τ$ . From this point of view, the time delay in the system has a negative effect on the network consensus.

Remark 4. Assume that $η > 1$ , it means that the impulsive coupling term only reduces the effect of the attacks, and the state of $V (t)$ has a sudden change that undermines the consensus of the system at the coupling time instant. From $\bar{λ} + λ_{max} (C + C^{T}) + 1 + \ln max (η, η^{- 1} / T) + λ_{max} (B^{T} B) (max (η, η^{- 1}))^{N_{0}} e^{\bar{λ} τ} = 0$ , it can be seen that the larger $η$ will lead to the smaller $\bar{λ}$ . That is to say, in a certain range of attack strength, the larger $θ_{k}$ will lead to the larger $η$ , which further leads to the more difficult satisfying condition (9) and the slower convergence speed of the system. When $θ_{k}$ is larger than a certain bound, the system will lose consensus. This can also be observed from two numerical examples in the paper. The case $η \leq 1$ can be discussed similarly.

Remark 5. In Theorem 1, we only know that the attacked time instants $\bar{T} = {T^{ij} | (i, j) \in E} = {t_{k}}_{k = 1}^{\infty}$ satisfy inequality (8). It is not known which edge is attacked at each impulsive coupling time instant. Therefore, when estimating $η$ , we need to compare the maximum value of $λ_{\max} ([I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} max [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}])$ at all attacked time instants, which is achievable (see Remark 2). We have to estimate this because of the limited information about the attacks. However, when more information about the attacks and the network topology is known, this condition can be relaxed, which can be seen in Corollary 2.

Corollary 1. For the NDSs (1) with $τ (t) \equiv 0$ and impulsive coupling (2), suppose the edge-based attacks satisfying (3) and impulsive coupling time instants satisfying (8). Then, system (1) can realize consensus, if

λ_{max} (C + B + C^{T} + B^{T}) + \frac{\ln max (η, η^{- 1})}{T} < 0

(27)

where $η$ is defined in Theorem 1.

Proof. When $τ (t) \equiv 0$ , system (1) can be rewritten as

{\begin{matrix} \overset{\cdot}{x} (t) = (I_{N} \otimes (C + B)) x (t), t \in [t_{k}, t_{k + 1}), \\ x (t_{k}) = [(I_{N} + Q (t_{k}) θ_{k} - α L) \otimes I_{n}] x (t_{k}^{-}), t = t_{k} \end{matrix}

(28)

where $Q (t_{k}) θ_{k}$ is defined as Theorem 1. Similarly, the Lyapunov function $V (t) = x (t)^{T} x (t)$ satisfies

{\begin{matrix} \overset{\cdot}{V} (t) \leq λ_{max} (C + B + C^{T} + B^{T}) V (t), t \in [t_{k}, t_{k + 1}), \\ V (t_{k}) \leq η V (t_{k}^{-}), t = t_{k} \end{matrix}

(29)

Hence, it can be obtained that

0 \leq V (t) \leq max {(η, η^{- 1})}^{N_{0}} V_{t_{0}} e^{- p (t - t_{0})}, t \geq t_{0}

(30)

Here, $p = - (λ_{max} (C + B + C^{T} + B^{T}) + \ln max (η, η^{- 1}) / (T)) > 0$ . Therefore, one has $lim_{t \to + \infty} max (η, η^{- 1})^{N_{0}} V_{t_{0}} e^{- p (t - t_{0})} = 0$ and $lim_{t \to + \infty} V (t) = 0$ .

In Theorem 1, it has been proved that system (1) can achieve consensus under certain conditions. However, it is required that at most one edge is attacked by an attacker at each impulsive coupling time instant. In the following, we shall consider a more general case where attackers are allowed to attack the multiple edges simultaneously. It should be addressed that the state of the node will produce greater mutations if this node suffers several attacks at the same time.

$T^{ij}$ and $\bar{T}$ are defined as equation (2). If multiple edges are allowed to be attacked at the same time, there may exist an instant coincidence of multiple attacked time instants in ${T^{ij} | (i, j) \in E}$ , and the corresponding nodes may be attacked for several times. For example, at time instant $t_{k}$ , if $m_{k}^{i}$ edges connecting $i$ , that is, $(i, j_{1}) \in E, \dots, (i, j_{m_{k}^{i}}) \in E$ , are attacked by attackers, then we have $t_{k_{1}}^{i j_{1}} = t_{k_{2}}^{i j_{2}} = \dots = t_{k_{m_{k}^{i}}}^{i j_{m_{k}^{i}}} = t_{k}$ , and the states of the corresponding nodes will undergo different mutations as follows

{\begin{matrix} x_{i} (t_{k}) = (θ^{i j_{1}} + θ^{i j_{2}} + \dots + θ^{i j_{m_{k}^{i}}} θ^{i j_{m_{k}^{i}}}) x_{i} (t_{k}^{-}), \\ x_{j_{l}} (t_{k}) = θ^{i j_{l}} x_{j_{l}} (t_{k}^{-}), l \in {1, 2, \dots, m_{k}^{i}} \end{matrix}

(31)

Similar to the previous section, we also introduce the function $q_{i} (t_{k})$ and $Q (t_{k})$ , which are defined as equations (5) and (7). Under edge-based attacks (31) and impulsive coupling (2), system (1) can be rewritten as

{\begin{matrix} \overset{\cdot}{x} (t) = (I_{N} \otimes C) x (t) + (I_{N} \otimes B) x (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ x (t_{k}) = [I_{N} \otimes I_{n} + (Q (t_{k}) β_{k} - α L) \otimes I_{n}] x (t_{k}^{-}), t = t_{k}, \\ x (t) = ϕ (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(32)

where $β_{k}$ is a diagonal matrix, whose diagonal elements represent the attacked intensity of each node at time instant $t_{k}$ . The positions of the zero elements in matrix $β_{k}$ and matrix $Q (t_{k})$ are the same. For example, at the time instant $t_{k}$ , nodes $1, 2, \dots, i_{1}$ $(i_{1} \leq N - 1)$ are attacked $m_{k}^{1}, m_{k}^{2}, \dots, m_{k}^{i_{1}}$ times, respectively. Then, we have $Q (t_{k}) = diag (1, \dots, 1, 0, \dots, 0)$ , where the first $i_{1}$ elements are 1 and the remaining elements are 0. Correspondingly, $β_{k} = diag (θ^{1 j_{1}} + θ^{1 j_{2}} + \dots + θ^{1 j_{m_{k}^{1}}}, \dots, θ^{i_{1} j_{1}} + θ^{i_{1} j_{2}} + \dots + θ^{i_{1} j_{m_{k}^{i_{1}}}}, 0, \dots, 0)$ , in which $θ^{l j_{1}} + θ^{l j_{2}} + \dots + θ^{l j_{m_{k}^{l}}}$ is related to the intensity of attack on $m_{k}^{l}$ edges $(l = 1, 2, \dots, i_{1})$ . In this case, we only know that node $l$ has been attacked $m_{k}^{l}$ times, but we do not know which edges have been attacked. For an NS with $N$ nodes, node $i$ is attacked at most $N - 1$ times at the same time. Therefore, there is only a finite number of different values of $β_{k}$ $(k = 1, 2, \dots)$ .

Based on the above analysis, the following Theorem 2 can be obtained.

Theorem 2. For the NDSs (1) with impulsive coupling (2), suppose the edge-based attacks satisfying (31) and impulsive coupling time instants satisfying (8). Then, system (1) can realize consensus, if

\begin{matrix} λ_{\max} (C + C^{T}) + 1 + \frac{\ln max (\bar{η}, {\bar{η}}^{- 1})}{T} + λ_{\max} (B^{T} B) \\ {(max (\bar{η}, {\bar{η}}^{- 1}))}^{N_{0}} < 0 \end{matrix}

(33)

where $\bar{η} = max_{k \in N_{+}} {λ_{max} ([I_{N} \otimes I_{n} + (Q (t_{k}) β_{k} - α L) \otimes I_{n}]^{T} [I_{N} \otimes I_{n} + (Q (t_{k}) β_{k} - α L) \otimes I_{n}])}$ .

The proof of this theorem is similar to the proof of Theorem 1. Hence, it is omitted here.

Remark 6. Compared with Theorem 1, Theorem 2 represents a more general case and has a wider range of applications. Due to multiple edges allowed to be attacked in Theorem 2, the attacker’s strategy is more difficult to be estimated. Hence, the estimation of parameter $\bar{η}$ is much more difficult than that of parameter $η$ in Theorem 1.

As discussed in Remark 5, the estimation of $η = max_{k \in N_{+}} {λ_{max} ([I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}]^{T} [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}])}$ would be greatly simplified if more information is known about the attacks. In the following, we shall consider the special type of edge-based attacks on the NDSs with star topology. It shows that with more knowledge of the attacker’s strategy, the estimation of $η$ can be simplified and accurate, which is less conservative than the result of Theorem 1.

Consider NDSs with $N$ nodes with the star network structure (see Figure 2). The edge-based attacks satisfy the following assumption:

Assumption 2. Each edge is attacked periodically and with equal intensity, that is

{\begin{matrix} T^{12} = T^{21} = {t_{(N - 1) l - (N - 2)}}, \\ T^{13} = T^{31} = {t_{(N - 1) l - (N - 3)}}, \\ \dots, \\ T^{1 N - 1} = T^{N - 11} = {t_{(N - 1) l - 1}}, \\ T^{1 N} = T^{N 1} = {t_{(N - 1) l}}, l \in N_{+} \end{matrix}

(34)

and

θ^{1 i} = θ^{i 1} = θ > 1 . \forall i = 2, 3, \dots, N

(35)

Figure 2.

The structure of a star network.

Corollary 2. Consider NDSs (1) with star network topology, whose impulsive coupling satisfies Assumption 1. Suppose the edge-based attacks satisfy Assumption 2 and $α = 1$ . Then, system (38) can realize consensus, if

λ_{\max} (C + C^{T}) + 1 + \frac{\ln η_{*}}{T} + λ_{\max} (B^{T} B) η_{*}^{N_{0}} < 0

(36)

where $η_{*} = max (λ_{*}, 0)$ and $λ_{*}$ is defined in equation (41).

Proof. The Laplacian matrix $L$ of the star network is

L = [\begin{matrix} N - 1 & - 1 & - 1 & - 1 & \dots & - 1 & - 1 & - 1 \\ - 1 & 1 & 0 & 0 & \dots & 0 & 0 & 0 \\ - 1 & 0 & 1 & 0 & \dots & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & \dots & ⋮ & ⋮ & ⋮ \\ - 1 & 0 & 0 & 0 & \dots & 0 & 1 & 0 \\ - 1 & 0 & 0 & 0 & \dots & 0 & 0 & 1 \end{matrix}]

(37)

According to Assumptions 1 and 2, the dynamics of the system can be written as follows

{\begin{matrix} \overset{\cdot}{x} (t) = (I_{N} \otimes C) x (t) + (I_{N} \otimes B) x (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ x (t_{k}) = [I_{N} \otimes I_{n} + (Q (t_{k}) θ_{k} - α L) \otimes I_{n}] x (t_{k}^{-}) \\ = {\begin{matrix} [I_{N} \otimes I_{n} + (Q_{2} ω_{2} - α L) \otimes I_{n}] x (t_{k}^{-}), \\ k = (N - 1) l - (N - 2), \\ [I_{N} \otimes I_{n} + (Q_{3} ω_{3} - α L) \otimes I_{n}] x (t_{k}^{-}), \\ k = (N - 1) l - (N - 3), \\ \dots, \\ [I_{N} \otimes I_{n} + (Q_{N - 1} ω_{N - 1} - α L) \otimes I_{n}] x (t_{k}^{-}), \\ k = (N - 1) l - 1, \\ [I_{N} \otimes I_{n} + (Q_{N} ω_{N} - α L) \otimes I_{n}] x (t_{k}^{-}), k = (N - 1) l, \end{matrix} \\ x (t) = ϕ (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(38)

where $Q_{i} = diag (1, 0, \dots, 0, 1, 0, \dots, 0)$ , the $i th$ entry of $Q_{i}$ is equal to 1 and the rest are equal to 0 $(i = 2, 3, \dots, N - 1)$ . Correspondingly, $ω_{i} = diag (θ, 0, \dots, 0, θ, 0, \dots, 0)$ , the $i th$ entry of $ω_{i}$ is equal to $θ$ and the rest are equal to 0 $(i = 2, \dots, N - 1)$ .

Construct the Lyapunov function $V (t) = x (t)^{T} x (t)$ . Similar to the proof of Theorem 1, one has

{\begin{matrix} \overset{\cdot}{V} (t) \leq (λ_{max} (C + C^{T}) + 1) V (t) + \\ λ_{\max} (B^{T} B) V (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ V (t_{k}) \leq λ_{max} (M^{T} M) x {(t_{k}^{-})}^{T} x (t_{k}^{-}), t = t_{k} \\ V (t) \leq {sup}_{t \in [t_{0} - τ, t_{0}]} ϕ {(t)}^{T} ϕ (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(39)

where $M = I_{N} \otimes I_{n} + (Q_{2} ω_{2} - L) \otimes I_{n}$ and $α = 1$ is used.

In the following, we will calculate $λ_{max} ([I_{N} \otimes I_{n} + (Q_{2} ω_{2} - L) \otimes I_{n}]^{T} [I_{N} \otimes I_{n} + (Q_{2} ω_{2} - L) \otimes I_{n}])$ . The characteristic polynomial of matrix $[I_{N} \otimes I_{n} + (Q_{2} ω_{2} - L) \otimes I_{n}]^{T} [I_{N} \otimes I_{n} + (Q_{2} ω_{2} - L) \otimes I_{n}]$ can be obtained as follows:

\begin{matrix} | {[I_{N} + Q_{2} (ω_{2} - L)]}^{T} [I_{N} + Q_{2} (ω_{2} - L)] - λ E | \\ = (- λ)^{N - 3} [\begin{matrix} {(N - 1 - λ)}^{2} (θ^{2} - λ) - (N - 1 - λ) λ - \\ θ^{2} (N - 1 - λ) - θ (θ - N + 2) λ + θ (θ - N + 2) λ - \\ {(θ - N + 2)}^{2} (θ^{2} - λ) λ \end{matrix}] \\ \overset{Δ}{=} f (λ) (- λ)^{N - 3} \\ = 0 \end{matrix}

(40)

where $f (λ) = - λ^{3} + (2 (N - 1) + 1 + θ^{2} + (θ - N + 2)^{2}) λ^{2} - (N - 1 + (N - 1)^{2} + (2 N - 3) θ^{2} + θ^{2} (θ - N + 2)^{2}) λ + (N - 1) (N - 2) θ^{2}$ .

According to the semidefinite property of matrix $[I_{N} + (Q_{2} ω_{2} - α L)]^{T} [I_{N} + (Q_{2} ω_{2} - α L)]$ , it can be known that all the roots of $f (λ) = 0$ are non-negative. Denote the largest root of $f (λ) = 0$ is $λ_{*}$ . By using the the famous Cardano formula (see reference Artin, 2010), it can be obtained

λ_{*} = \frac{1}{3} (- b + \sqrt{A} (\cos \frac{ϑ}{3} + \sqrt{3} \sin \frac{ϑ}{3}))

(41)

where $b = 2 (N - 1) + 1 + θ^{2} + (θ - N + 2)^{2}, A = b^{2} +$ $3 (N - 1 + (N - 1)^{2} + (2 N - 3) θ^{2} + θ^{2} (θ - N + 2)^{2}), ϑ = \arccos$ $(2 Ab - 3 B) / 2 \sqrt{A^{3}}, B = - b (N - 1 + (N - 1)^{2} +$ $(2 N - 3) θ^{2} + θ^{2} (θ - N + 2)^{2}) + 9 (N - 1) (N - 2) θ^{2}$ . Combining equations (40) and (41), we have

λ_{max} (M^{T} M) = max (λ_{*}, 0) \overset{Δ}{=} η_{*}

(42)

By equations (41) and (42), equation (39) can be rewritten as

{\begin{matrix} \overset{\cdot}{V} (t) \leq (λ_{max} (C + C^{T}) + 1) V (t) + \\ λ_{\max} (B^{T} B) V (t - τ (t)), t \in [t_{k}, t_{k + 1}), \\ V (t_{k}) = x {(t_{k})}^{T} x (t_{k}) \leq η_{*} x {(t_{k}^{-})}^{T} x (t_{k}^{-}), \\ V (t) \leq {sup}_{t \in [t_{0} - τ, t_{0}]} ϕ {(t)}^{T} ϕ (t), t \in [t_{0} - τ, t_{0}] \end{matrix}

(43)

where $η_{*} = max (λ_{*}, 0) = λ_{*}$ . According to Theorem 1, system (38) can achieve consensus.

Numerical simulation

Example 1. Consider the NDSs (1) with $N = 5$ and $d = 3$ . The network structure is shown in Figure 2. Choose $α = 1$ , $τ (t) = τ = 0.4$

\begin{matrix} C = [\begin{matrix} - 4 & 0 & 0 \\ 0 & - 4 & 0 \\ 0 & 0 & - 4 \end{matrix}], & B = [\begin{matrix} 0.08 & - 0.75 & - 0.25 \\ - 0.25 & 0.06 & - 1.0 \\ - 1.1 & 1.0 & 0.06 \end{matrix}] \end{matrix}

Select the attack sequence as shown in Figure 3, where $T = 0.7, N_{0} = 0.1$ and $θ = 2.5$ , which satisfies Assumptions 1 and 2.

Figure 3.

The attack sequence with $T = 0.7, N_{0} = 0.1$ , and $θ = 2.5$ .

By simple computation, it can be obtained that $λ_{*} = 1.9001$ , $η_{*} = 1.9001$ , and $λ_{max} (C + C^{T}) + 1 + (\ln η_{*} / (T)) + λ_{max} (B^{T} B) η_{*}^{N_{0}} = - 3.2955 < 0$ . Hence, the condition of Corollary 2 holds. In this example, $λ_{*} = 1.9001$ means that the effect of attack is to destroy system consensus. Figure 4 shows that under the given attack sequence, the system can still maintain consensus. Besides, for the attack sequence satisfying $T = 0.7, N_{0} = 0.1$ , and $θ = 3.5$ , the system trajectory is shown in Figure 5. The comparison between Figures 4 and 5 shows that a higher intensity of attack will result in a slower convergence rate. However, if the system is subjected to a more severe attack with a higher intensity shown as Figure 6, the consensus of the system will be eventually destroyed, which can be seen from Figure 7.

Figure 4.

The trajectories of all nodes under the attack sequence with $T = 0.7, N_{0} = 0.1$ , and $θ = 2.5$ .

Figure 5.

The trajectories of all nodes under the attack sequence with $T = 0.7, N_{0} = 0.1$ , and $θ = 3.5$ .

Figure 6.

The attack sequence with $T = 0.7, N_{0} = 0.1$ , and $θ = 6$ .

Figure 7.

The trajectories of all nodes under the attack sequence shown in Figure 6.

To test the effect of time delay $τ$ , let $τ = 0.9$ and other parameters remain unchanged. Figure 8 shows the evolution of trajectories of all nodes. Comparing Figures 4 and 8, convergence speed of the system will be faster with decreasing the value of time delay, which demonstrates the negative effect of time delay as discussed in Remark 3.

Figure 8.

The trajectories of all nodes under the attack sequence shown in Figure 3 and $τ = 0.9$ .

Select $τ = 0$ and other parameters unchanged. From Figure 9, it can be observed that when the time delay does not exist, the system trajectory can converge quickly and realize consensus. When $τ = 1.9$ , it can be seen from Figure 10 that the system is consensus. But the convergence time required is much longer than the cases with $τ = 0.9$ and 0. When $τ = 2$ , the system becomes divergent, which can be seen from Figure 11. Therefore, we can say that the maximum time delay that the system can tolerate in this example approximately is $1.9$ . This also illustrates the negative effect of time delay on system performance.

Figure 9.

The trajectories of all nodes under the attack sequence shown in Figure 3 and $τ = 0$ .

Figure 10.

The trajectories of all nodes under the attack sequence shown in Figure 3 and $τ = 1.9$ .

Figure 11.

The trajectories of all nodes under the attack sequence shown in Figure 3 and $τ = 2$ .

Example 2. Consider the NDSs with $N = 4$ and $d = 3$ . The network structure is shown in Figure 12. Matrices $C$ and $B$ are selected as Example $1$ . Choose $τ (t) = τ = 0.4$ and $α = 0.4$ . Select the attack sequence as shown in Figure 13, where $T = 0.9, N_{0} = 0.1$ , and $θ = 3$ , which satisfies Assumptions 1 and 2. By simple computation, it can be obtained that $η = 12.2658$ and $λ_{max} (C + C^{T}) + 1 + (\ln max (η, η^{- 1}) / (T)) + λ_{max} (B^{T} B) (max (η, η^{- 1}))^{N_{0}} = - 0.8557 < 0$ . Hence, the conditions of Theorem 1 hold. From Figure 14, it can be observed that the system under the given attack sequence can achieve consensus. $η = 12.2658$ means that the the effect of attack sequence is to destroy the consensus of the system. If the system is subjected to a more severe attack with a higher intensity, then it will become divergent. For example, when we take the attack sequence shown in Figure 15, it can be obtained $λ_{\max} (C + C^{T}) + 1 + (\ln max (η, η^{- 1}) / (T)) + λ_{max} (B^{T} B) (max (η, η^{- 1}))^{N_{0}} = 0.7113 > 0$ . Hence, Theorem 1 does not hold. Figure 16 shows the evolution of trajectories of all nodes, which demonstrate that the system cannot achieve consensus.

Figure 12.

The structure of the network in Example 2.

Figure 13.

The attack sequence with $T = 0.9, N_{0} = 0.1,$ and $θ = 3$ .

Figure 14.

The trajectories of all nodes under the attack sequence shown in Figure 13.

Figure 15.

The attack sequence with $T = 0.9, N_{0} = 0.1,$ and $θ = 5.5$ .

Figure 16.

The trajectories of all nodes under the attack sequence shown in Figure 15.

To test the effect of coupling gain $α$ , let $α = 1.0$ and other parameters remain unchanged. Also select the attack sequence in Figure 13. Then, one has $η = 9.4035$ and $λ_{max} (C + C^{T}) + 1 + (\ln max (η, η^{- 1} / T)) + λ_{max} (B^{T} B) (max (η, η^{- 1}))^{N_{0}} = - 1.2390 < 0$ . The trajectories of the nodes are obtained as shown in Figure 17. By comparing Figures 14 and 17, it is obvious that the convergence speed of the system accelerates with the increase of coupling strength $α$ .

Figure 17.

The trajectories of all nodes under the attack sequence shown in Figure 13 and $α = 1.0$ .

Example 3. Consider the NDSs with $N = 15$ and $d = 3$ . The network structure is shown in Figure 2. Select $τ (t) = τ = 0.2$ and $α = 0.15$ . The attack sequence is shown in Figure 18. By simple computation, we have $η = 1.4119$ and $λ_{max} (C + C^{T}) + 1 + (\ln max (η, η^{- 1}) / (T)) + λ_{max} (B^{T} B) (max (η, η^{- 1}))^{N_{0}} = - 3.8013 < 0$ . Hence, the conditions of Corollary 2 hold. It can be seen from Figure 19 that the system is consensus under the attack sequence.

Figure 18.

The attack sequence with $T = 0.7, N_{0} = 0.1,$ and $θ = 3$ .

Figure 19.

The trajectories of all nodes under the attack sequence shown in Figure 18.

Remark 7. In Example 2 of Zhu et al. (2020), the consensus of the system can be maintained by an impulsive coupling control strategy, but it does not take the attacks and time delay into account. In Examples 1 and 2, we have verified that the system can still maintain the consensus under the influence of attacks, and the negative effect of time delay on the system. In addition, when we take $τ = 0$ and $θ = 1$ , Examples 1 and 2 in this paper become similar to Example 2 of Zhu et al. (2020). In the example of Liu et al. (2022c), the system state can only converge to a bounded region in the presence of attacks, while the system in our examples can achieve consensus. Besides, different from the previous related works, asynchronous edge-based attacks and time delays are simultaneously considered in our model.

Conclusion

This paper has studied the consensus of impulsive coupled NDSs under edge-based attacks. The attack here can not only change the state of system nodes but also is asynchronous, which is different from most previous works. The attack strategies are divided into two cases, that is, only one edge is attacked and multiple edges are allowed to be attacked at one impulsive coupling time instant. By investigating the two cases of edge-based attacks, sufficient conditions for system consensus are given, respectively. In addition, the results have been applied to special edge-attacked NDSs with star network topology and a less conservative consensus criterion is obtained. Finally, the correctness of the results are illustrated by three numerical examples. In the future, we will take communication delays into account in the networked delayed systems. In addition, it is also interesting to consider communication delays and other types of asynchronous attacks including DoS attacks and deception attacks in NSs.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number: 61503115).

ORCID iD

Lulu Li

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