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Research article

The critical delay of the consensus for a class of multi-agent system involving task strategies

  • Received: 26 August 2022 Revised: 25 November 2022 Accepted: 10 January 2023 Published: 18 January 2023
  • The time delay may induce oscillatory behaviour in multi-agent systems, which may destroy the consensus of the system. Therefore, the critical delay that is the maximum value of the delay to guarantee the consensus of the system, is an important performance index of multi-agent systems. This paper studies the influence of the processing delay on the consensus for a class of multi-agent system involving task strategies. The first-order system with a single delay and the second-order system with two different delays are investigated respectively. A critical delay independent of strategies and a critical region of the 2-D plane that depends on strategies is obtained for the first-order and the second-order system respectively. Specifically, a geometric method was used for the case of two different delays. Several numerical simulations are presented to explain the results.

    Citation: Yipeng Chen, Yicheng Liu, Xiao Wang. The critical delay of the consensus for a class of multi-agent system involving task strategies[J]. Networks and Heterogeneous Media, 2023, 18(2): 513-531. doi: 10.3934/nhm.2023021

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  • The time delay may induce oscillatory behaviour in multi-agent systems, which may destroy the consensus of the system. Therefore, the critical delay that is the maximum value of the delay to guarantee the consensus of the system, is an important performance index of multi-agent systems. This paper studies the influence of the processing delay on the consensus for a class of multi-agent system involving task strategies. The first-order system with a single delay and the second-order system with two different delays are investigated respectively. A critical delay independent of strategies and a critical region of the 2-D plane that depends on strategies is obtained for the first-order and the second-order system respectively. Specifically, a geometric method was used for the case of two different delays. Several numerical simulations are presented to explain the results.



    With the development of artificial intelligence, multi-agent systems have attracted extensive attention of researchers in computer science, physics, biology, social science and control engineering. The design of multi-agent systems is greatly influenced by the collective behaviour of animals in nature, such as ant colony gathering, birds flocking and fish swarming. Generally, a multi-agent system consists of multiple independent autonomous or semi-autonomous agents interconnected through a communication network, with research focuses including consensus [1], flocking [2], swarming [3], collision avoiding [4,5], formation control [6,7] and event-triggered control [8,9] etc. Consensus describes the process of agents coordination which has important applications in opinion dynamics and engineering control, and has been thoroughly analysed yielding a number of conditions guaranteeing that the agents reach consensus in the past decade, see [10,11,12,13,14].

    The early literature on consensus has mainly focused on the analysis of autonomous systems whose final state after reaching consensus depends only on the initial configuration of the system. However, the application of autonomous system is limited because it is inconvenient and inflexible to use the target state to design the initial configuration of the system when the system is required to reach a specified state. One type of intervention, adding external forces to the system to reach the desired target state, is applicable to a variety of real-world contexts, such as financial markets, public opinion and a team of UAVs, and has been prevalent in multi-agent systems. Further, such interventions usually only affect a subset of individuals in the system, resulting in the leader-follower structure of multi-agent systems [15,16,17].

    In a multi-agent system with leader-follower structure, the leaders refer to the part of agents who master information about the target state (i.e., those affected by the intervention), and the rest of the agents in the system is called followers. Leaders should not only carry out the information communication among the agents in the system, but also track the target state, while followers only are required to join in the information communication. In most previous researches, the information communication and the target tracking were separated, and the strength of target tracking was assumed to be finite based on the actual situation that the external force is limited by energy, equipment and technology.

    Different from the above studies, this paper focuses more on the agent performance than on the external force. Considering the limited ability of agents, we assume that each agent has a task strategy to properly allocate its limited energy for the information communication and the target tracking. The detail of the first-order multi-agent system involving task strategies αi[0,m](i=1,2,,N), is as follows:

    ˙xi(t)=f(t)+αi(x0(t)xi(t))+(mαi)1NNj=1(xj(t)xi(t)),i=1,2,,N, (1.1)

    where xi(t) is the position of the ith agent, x0(t) is the target state satisfying ˙x0(t)=f(t), t0, f(t)C([0,),R), mR+ represents the maximal strength of the information communication. For the ith agent, the strength of the information communication is mαi and the strength of the target tracking is αi. If αi>0, the ith agent is a leader, otherwise a follower. α=iαi is called the total strength of the target tracking of the system. The concept of the above strategy was proposed by Piccoli et al. [18] in 2016. They considered strategies {αi}Ni=1 as controls and focused on finding optimal control strategies {αi}Ni=1 to minimize the cost 1NNi=1xi(T)x0(T), where T>0 was the final time. The Eq (1.1) with the non-linear information communication was investigated in [19], and the sufficient conditions were proposed to guarantee that the system achieves consensus. In addition to the first-order system (1.1), this paper also focuses on the second-order multi-agent system involving task strategies, written as

    ¨xi(t)=g(t)+αi[γ(v0(t)˙xi(t))+(1γ)(x0(t)xi(t))]+(mαi)[γNNj=1(˙xj(t)˙xi(t))+1γNNj=1(xj(t)xi(t))], (1.2)

    where γ[0,1] is the weight coefficient of velocities information, which measures the proportion of the velocities information in the control. Then the weight coefficient of positions information is 1γ. x0(t) is the target state satisfying ˙x0(t)=v0(t), ˙v0(t)=g(t), t0, g(t)C([0,),R). Then, we present the mathematical definition of the consensus of the system for Eqs (1.1) and (1.2).

    Definition 1.1. Suppose {xi(t)}Ni=1 is a solution of the Eq (1.1), x0(t) is the target state satisfying ˙x0(t)=f(t), t0, f(t)C([0,),R). The Eq (1.1) is said to achieve consensus and reach the target state if and only if

    limt|xi(t)x0(t)|=0,i=1,2,,N.

    Suppose {xi(t)}Ni=1 is a solution of the Eq (1.2), x0(t) is the target state satisfying ˙x0(t)=v0(t), ˙v0(t)=g(t), t0, g(t)C([0,),R). The Eq (1.2) is said to achieve consensus and reach the target state if and only if

    limt|xi(t)x0(t)|=0andlimt|˙xi(t)v0(t)|=0,i=1,2,,N.

    The time delay is an important topic in the research of multi-agent systems and has been widely studied. The causes of the time delay can be divided into two types: information transmission delay and information processing delay. The transmission delay means that it takes time for agents to receive information from others limited by the speed of communication, see [20,21,22]. The processing delay, also known as the reaction delay, refers to the time required for devices to process information, see [23,24,25]. The effect of time delay on consensus formation of multi-agent systems is an issue that cannot be ignored. Olfsti-Saber and Murray [26] gave a sufficient condition of consensus for a first-order system with time delay on balanced graphs and showed that there exists a trade-off between the control gain and the critical delay. Yu et al. [27] obtained some necessary and sufficient conditions for second-order consensus on directed graphs. Ma et al. studied a second-order consensus system with unstable elements over undirected graphs and maximized the critical delay by optimizing parameters [28]. A first-order consensus system with unstable elements over directed graphs was investigated in [29] and maximal critical delay was achieved through solving a nonsmooth max-min problem. For other relevant literature, see [30,31,32].

    In this paper, we study the effect of the processing delay on the consensus of the first-order system in Eq (1.1) and the second-order system in Eq (1.2), and analyse the relationship between the processing delay and the strategies. According to the stability of linear systems in the theory of functional differential equations, the system would achieve consensus by ensuring that roots of the characteristic equation of the system have negative real parts. The specific content of the conclusion of stability is written as:

    Lemma 1.2. [33] For a linear functional equation ˙u(t)=Au(tr)+Bu(ts), where u(t)RN, A,BRN×N and r,sR+. Its characteristic equation is h(λ)=Det(λIAeλrBeλs)=0. Define a=sup{Reλ:h(λ)=0}, if a<0, the zero solution of the equation is globally asymptotically stable.

    The rest of this paper is organized as follows. In Section 2, the Eq (1.1) with a single delay is investigated. Using the continuous dependence of the equation on the processing delay, we obtain the critical delay τ that ensures that the Eq (1.1) achieves consensus, show that the critical delay τ of the Eq (1.1) is independent of the strategies αi. In Section 3, the Eq (1.2) with two different delays is investigated. Inspired by [34,35,36] and using the properties of plane geometry, we identify the critical region D in R2 that guarantees the system to achieve consensus, find that the shape of the critical region D is affected by the strategies αi. In Section 4, several numerical simulations are presented to explain our results. Finally, we give the conclusion and discussion in Section 5.

    Adding the processing delay in the Eq (1.1) yields

    ˙xi(t)=f(t)+αi(x0(tτ)xi(tτ))+(mαi)1NNj=1(xj(tτ)xi(tτ)). (2.1)

    where τR+ represents the time required for the system to process the information of positions. Next we will transform the consensus problem of the Eq (2.1) into the stability problem of a linear autonomous system with a single delay, which further becomes the problem of judging whether the roots of the characteristic equation have negative real parts. Set yi(t)=xi(t)x0(t), then the above model reduces to

    ˙yi(t)=αiyi(tτ)+(mαi)1NNj=1(yj(tτ)yi(tτ)).

    Define

    Y(t)=(y1(t),y2(t),,yN(t))TRNandΓ=[mα1Nmα1Nmα1Nmα2Nmα2Nmα2NmαNNmαNNmαNN], (2.2)

    then rewrite the above equations in matrix form

    ˙Y(t)=mY(tτ)+ΓY(tτ).

    Compute the eigenvalues of the matrix Γ

    |μIΓ|=|μmα1Nmα1Nmα1Nmα2Nμmα2Nmα2NmαNNmαNNμmαNN|=|μ(mα1)mα1Nmα1Nμ(mα2)μmα2Nmα2Nμ(mαN)mαNNμmαNN|=|μ(mα1)μNμNμ(mα2)μμNμNμ(mαN)μNμμN|=|μ(mα1)μNμNα2α1μ0αNα10μ|=|μ(mNi=1αiN)00α2α1μ0αNα10μ|=μN1[μ(mαN)]=0,

    where α=Ni=1αi. Note that rank(Γ)=1, hence the matrix Γ is similarly diagonalized, i.e., there exists a non-singular matrix P such that PΓP1=J, where J=diag(mαN,0,,0). Let Z(t)=PY(t), then the equation becomes

    ˙Z(t)=mZ(tτ)+JZ(tτ).

    The characteristic equation of the above system is

    h(λ)=(λ+meλτ)N1(λ+αNeλτ)=0. (2.3)

    Hence, applying Lemma 1.2 we know that the Eq (2.1) achieves consensus if all roots of the Eq (2.3) have negative real parts. Using the continuous dependence of the Eq (2.3) on the processing delay τ, we obtain the following result.

    Theorem 2.1. Assume α>0. Let Λ=sup{Reλ:h(λ)=0}, then there exists a critical delay

    τ=π2m

    such that Λ<0, 0τ<τ.

    Proof. Owing to m,αN,τR, ¯h(iω)=h(iω), ωR, which means that if λ=iω is a root of Eq (2.3), then also λ=iω is a root. Without loss of generality let h(iω)=0, ωR+, then we obtain

    iω+meiωτ=0oriω+αNeiωτ=0.

    From iω+meiωτ=0 we have mcos(ωτ)=0 and ω=msin(ωτ). Adding the squares of the two equations yields ω=m, then we have τkm=π2m+kπm, kZ. Similarly, from iω+αNeiωτ=0 we could obtain τkα=πN2α+kπNα, kZ. Define

    τ=min{τkm,τkα,k=0,1,}=π2m.

    When τ=0,

    h(λ)=(λ+m)N1(λ+αN)=0.

    By α>0, Λ<0 which indicates that all roots of the Eq (2.3) lie on the left half complex plane when τ=0. Since h(λ) is continuously dependent on τ, Λ is continuously dependent on τ. Therefore, as the increase of τ from 0 to , some roots of the Eq (2.3) touch the imaginary axis of the complex plane for the first time when τ=τ. Then, we conclude that Λ<0 if 0τ<τ and Λ=0 if τ=τ.

    Remark 1. Theorem 2.1 shows that the critical delay of the Eq (2.1) has nothing to do with strategies αi, but only with the maximal strength m.

    Remark 2. If Λ<0 when τ=0, then the existence of the critical delay τ is equivalent to the existence of some roots of the Eq (2.3) crossing the imaginary axis of the complex plane as the increase of τ from 0 to . If the critical delay τ exists, then it is the value of the delay when these roots first touch the imaginary axis.

    Different from the Eq (1.1), the Eq (1.2) has to process not only the information of positions but also the information of velocities, so there are two different processing delays in the system. Adding the processing delay in the Eq (1.2) yields

    ¨xi(t)=g(t)+αi[γ(v0(tτ1)˙xi(tτ1))+(1γ)(x0(tτ2)xi(tτ2))]+(mαi)[γNNj=1(˙xj(tτ1)˙xi(tτ1))+1γNNj=1(xj(tτ2)xi(tτ2))], (3.1)

    where τ1 and τ2 represent times required for the system to process the information of positions and velocities, respectively. Similarly, we will transform the consensus problem of the Eq (3.1) into the stability problem of a linear autonomous system with two different delay. Set yi(t)=xi(t)x0(t), the above model is simplified as

    ¨yi(t)=αi[γ˙yi(tτ1)+(1γ)yi(tτ2)]+(mαi)[γNNj=1(˙yj(tτ1)˙yi(tτ1))+1γNNj=1(yj(tτ2)yi(tτ2))],

    Define Y(t) and Γ same as Eq (2.2), then rewrite the equations in matrix form

    ¨Y(t)=mγ˙Y(tτ1)+γΓ˙Y(tτ1)m(1γ)Y(tτ2)+(1γ)ΓY(tτ2).

    Let Z(t)=PY(t), then the above equation becomes

    ¨Z(t)=mγ˙Z(tτ1)+γJ˙Z(tτ1)m(1γ)Z(tτ2)+(1γ)JZ(tτ2).

    The characteristic equation of the above system is

    h(λ)=[λ2+γmλeλτ1+(1γ)meλτ2]N1[λ2+γαNλeλτ1+(1γ)αNeλτ2]=0. (3.2)

    Let Λ=sup{Reλ:h(λ)=0}. When τ1=τ2=0, we have

    h(λ)=[λ2+γmλ+(1γ)m]N1[λ2+γαNλ+(1γ)αN]=0.

    If α>0, it's easy to verify that Λ<0. Next, we firstly consider three simple cases: (1) τ1>0 and τ2=0; (2) τ1=0 and τ2>0; (3) τ1=τ2>0.

    Theorem 3.1. Assume α>0, 0<γ<1. If one of the following three cases holds:

    (1)

    0=τ2<τ1<τ1=π2(γm2+(1γ)m+γ2m24),

    (2)

    0=τ1<τ2<τ2=mins{m,αN}arctan(γsγ2s22+(1γ)2s2+γ4s44)γ2s22+(1γ)2s2+γ4s44,

    (3)

    0<τ1=τ2<τ=arctan(γ1γγ2m22+(1γ)2m2+γ4m44)γ2m22+(1γ)2m2+γ4m44.

    Then Λ<0.

    Proof. Because the proofs of (1) and (2) are very similar to the proof of (3), we will just prove the case τ1=τ2=τ>0 here. Without loss of generality let h(iω)=0, ωR+, then we have

    (iω)2+iγmωeiωτ+(1γ)meiωτ=0 (3.3)

    or

    (iω)2+iγαNωeiωτ+(1γ)αNeiωτ=0. (3.4)

    According to the Eq (3.3) we obtain

    {ω2=γmωsin(ωτ)+(1γ)mcos(ωτ),0=γmωcos(ωτ)(1γ)msin(ωτ). (3.5)

    Adding the squares of the above two equations yields

    ω4=γ2m2ω2+(1γ)2m2,

    then we have

    ω=γ2m22+(1γ)2m2+γ4m44.

    On the other hand, the second equation of Eq (3.5) gives

    τ=arctan(γω1γ)ω=arctan(γ1γγ2m22+(1γ)2m2+γ4m44)γ2m22+(1γ)2m2+γ4m44τm.

    In the same way, according to the Eq (3.4) we could obtain

    τ=arctan(γ1γγ2α22N2+(1γ)2α2N2+γ4α44N4)γ2α22N2+(1γ)2α2N2+γ4α44N4τα.

    The critical delay τ=min{τm,τα}, so we need to determine the monotonicity of the function

    J(s)=arctan(γ1γγ2s22+(1γ)2s2+γ4s44)γ2s22+(1γ)2s2+γ4s44.

    Because γ2s22+(1γ)2s2+γ4s44 is monotonically increasing with respect to s and arctanξξ is monotonically decreasing with respect to ξ, J(s) is monotonically decreasing with respect to s. Hence, we conclude τ=τm.

    Remark 3. For the case (2), we cannot determine the monotonicity of the function

    J(s)=arctan(γsγ2s22+(1γ)2s2+γ4s44)γ2s22+(1γ)2s2+γ4s44

    with respect to s. Actually, the numerical curve of J(s) shows that J(s) is monotonically increasing with respect to s. In addition, we assume 0<γ<1 in Theorem 3.1. By simple calculation we could obtain that if γ=1, the critical delay of τ1 is π2m; If γ=0, the critical delay of τ2 is πm.

    In the following we consider the case τ1τ2. In this case, the method applied in Theorems 2.1 and 3.1 is no longer feasible because we cannot obtain an explicit expression of τ by solving for ω. Now we use a geometric method to analyse the Eq (3.2). From Eq (3.2) we have

    λ2+γmλeλτ1+(1γ)meλτ2=0

    or

    λ2+γαNλeλτ1+(1γ)αNeλτ2=0.

    Since λ=0 is not a root of the above equations, the deformations of the equations are

    1+γmλeλτ1+(1γ)mλ2eλτ2=0 (3.6)

    and

    1+γαNλeλτ1+(1γ)αNλ2eλτ2=0. (3.7)

    For the Eq (3.6), let a1(λ)=γmλ and a2(λ)=(1γ)mλ2. Think of 1, a1(λ)eλτ1 and a2(λ)eλτ2 as three vectors in the complex plane C, then λ is a root of the Eq (3.6) if and only if the three vectors are connected head to tail to form a triangle in the complex plane (See Figure 1).

    Figure 1.  1, a1(λ)eλτ1 and a2(λ)eλτ2 form a triangle in the complex plane.

    Therefore, we could use the triangle property to get the relationship between λ, τ1 and τ2, and try to obtain the critical delay.

    Theorem 3.2. Assume α>0, 0<γ<1. Then there exists a connected region DR+×R+ such that Λ<0, (τ1,τ2)D. In addition, the connected region D and its boundary D satisfy

    D1D{(τ1,τ2)|τ1R+,τ2=0}=[0,τ1)×{0},
    D2D{(τ1,τ2)|τ1=0,τ2R+}={0}×[0,τ2),
    D1D2{(τ,τ)}DandD(D1D2)ΦΨ,

    where

    Φ={(τ1,τ2)R+×R+|τ1=3π2+(2u1)π±θ1ω,τ2=π+(2v1)πθ2ω,u=u±0,u±0+1,,v=v0,v0+1,,ωΩ},
    Ψ={(τ1,τ2)R+×R+|τ1=3π2+(2p1)π±ϑ1ω,τ2=π+(2q1)πϑ2ω,p=p±0,p±0+1,,q=q0,q0+1,,ωΥ},

    τ1, τ2 and τ are defined in Theorem 3.1, θ1, θ2, u±0, v0, Ω, ϑ1, ϑ2, p±0, q0 and Υ are defined in the proof.

    Proof. The proof is divided into three steps.

    The first step: use the triangle inequality to obtain the range of ω such that h(iω)=0. Without loss of generality let λ=iω, ωR+, then from Eq (3.6) we have

    1+a1(iω)eiωτ1+a2(iω)eiωτ2=0,

    where a1(iω)=γmωi and a2(iω)=(1γ)mω2. According to the triangle inequality that the length of any one side does not exceed the sum of the other two sides, we obtain

    |a1(iω)|+|a2(iω)|1,1|a1(iω)||a2(iω)|1,

    i.e.

    ω2γmω(1γ)m0,ω2γmω+(1γ)m0,ω2+γmω(1γ)m0.

    Denoting the range of ω by Ω and solving the inequalities yields

    Ω={[γm2+γ2m24+(1γ)m,γm2+γ2m24+(1γ)m],4(1γ)γ2m,[γm2+γ2m24+(1γ)m,γm2γ2m24(1γ)m][γm2+γ2m24(1γ)m,γm2+γ2m24+(1γ)m],4(1γ)γ2<m.

    Ω contains all the values of ω that make the roots of the Eq (3.6) lie on the imaginary axis of the complex plane at iω.

    The second step: use Ω to calculate all the values of (τ1,τ2)R+×R+ that make h(iω)=0. Define a1(iω)[0,2π) is the angle between a1(iω) with the positive direction of the real axis, θ1[0,π] is the inner angle of the triangle formed by a1(iω)eiωτ1 and 1. Similar definitions apply to a2(iω)[0,2π) and θ2[0,π]. Because a1(iω)=γmωi and a2(iω)=(1γ)mω2, a1(iω)=3π2 and a2(iω)=π. By the law of cosine we have

    θ1=arccos(1+|a1(iω)|2|a2(iω)|22|a1(iω)|)=arccos(ω2γm+γm2ω(1γ)2m2γω3),
    θ2=arccos(1+|a2(iω)|2|a1(iω)|22|a2(iω)|)=arccos(ω22(1γ)m+(1γ)m2ω2γ2m2(1γ)).

    Using properties of plane geometry we know that the angle between a1(iω)eiωτ1 with the positive direction of the real axis plus or minus θ1 is equal to π, where plus or minus depends on whether the triangle is above or below the real axis. Similar results apply to a2(iω)eiωτ2 and θ2. Then, we could establish the relations between τ1 with ω and τ2 with ω respectively:

    ωτ1+2uπ+a1(iω)±θ1=π,uZ,
    ωτ2+2vπ+a2(iω)θ2=π,vZ,

    where ωτ1+2uπ[0,2π) is the angle between eiωτ1 with the positive direction of the real axis, ωτ2+2vπ[0,2π) is the angle between eiωτ2 with the positive direction of the real axis. Define u±0 and v0 are the smallest positive integers to guarantee τ1>0 and τ2>0 respectively, then we obtain

    τ1=a1(iω)+(2u1)π±θ1ω=3π2+(2u1)π±θ1ω,u=u±0,u±0+1,,
    τ2=a2(iω)+(2v1)πθ2ω=π+(2v1)πθ2ω,v=v0,v0+1,.

    Define

    Φ={(τ1,τ2)R+×R+|τ1=3π2+(2u1)π±θ1ω,τ2=π+(2v1)πθ2ω,u=u±0,u±0+1,,v=v0,v0+1,,ωΩ},

    then ΦR+×R+ contains all the values of (τ1,τ2)R+×R+ that make the roots of the Eq (3.6) lie on the imaginary axis.

    Repeating the above steps for the Eq (3.7) yields that the roots of the Eq (3.7) lie on the imaginary axis if and only if (τ1,τ2)Ψ, where

    Ψ={(τ1,τ2)R+×R+|τ1=3π2+(2p1)π±ϑ1ω,τ2=π+(2q1)πϑ2ω,p=p±0,p±0+1,,q=q0,q0+1,,ωΥ},
    ϑ1=arccos(ωN2γα+γα2ωN(1γ)2α2γω3N),
    ϑ2=arccos(ω2N2(1γ)α+(1γ)α2ω2Nγ2α2(1γ)N),
    Υ={[γα2N+γ2α24N2+(1γ)αN,γα2N+γ2α24N2+(1γ)αN],4(1γ)γ2αN,[γα2N+γ2α24N2+(1γ)αN,γα2Nγ2α24N2(1γ)αN][γα2N+γ2α24N2(1γ)αN,γα2N+γ2α24N2+(1γ)αN],4(1γ)γ2<αN.

    Hence, we proved that some roots of the Eq (3.2) lie on the imaginary axis if and only if (τ1,τ2)ΦΨ.

    The third step: combine Theorem 3.1 and ΦΨ to determine the connected region D. By the results in [35,37], ΦΨ characterizes a series of continuous curves on R+×R+ and divides R+×R+ into a series of connected regions. Let D represent the connected region containing the origin (0,0), then Theorem 3.1 indicates that the connected region D and its boundary D satisfy

    D1D{(τ1,τ2)|τ1R+,τ2=0}=[0,τ1)×{0},
    D2D{(τ1,τ2)|τ1=0,τ2R+}={0}×[0,τ2),
    D1D2{(τ,τ)}DandD(D1D2)ΦΨ.

    Because Λ<0 when (τ1,τ2)=(0,0)D, Λ=0 when (τ1,τ2)ΦΨ and Λ is continuously dependent on (τ1,τ2), we conclude that Λ<0, (τ1,τ2)D.

    The the connected region D in Theorem 3.2 satisfies D1D2{(τ,τ)}D and D(D1D2)ΦΨ. Define the closure of D as ¯D, then we have Λ<0, (τ1,τ2)D and Λ=0, (τ1,τ2)¯DD. Hence, we call the connected region D the critical region. However, ΦΨ contains so many curves that we cannot imagine the approximate range of D. By further analysing ΦΨ, we will show that the critical region D is contained in a bounded region.

    Theorem 3.3. Assume D is the critical region in Theorem 3.2. There exists a bounded region DR+×R+ satisfying DD1D2C1C2 and ¯DDC1C2 such that DD, where

    C1={(τ1,τ2)R+×R+|τ1=π2θ1ω,τ2=θ2ω,ωΩ}Φ,
    C2={(τ1,τ2)R+×R+|τ1=π2ϑ1ω,τ2=ϑ2ω,ωΥ}Ψ.

    D1, D2, θ1, θ2, Ω, ϑ1, ϑ2, and Υ are defined in Theorem 3.2.

    Proof. Set

    b1a1(iω)eiωτ1=iγmωeiωτ1andb2a2(iω)eiωτ2=(1γ)mω2eiωτ2.

    If (τ1,τ2)=(0,0), then b1=3π2, b2=π and the positions of 1, b1 and b2 on the complex plane are shown in Figure 2 Case 1. In this case, the Eq (3.6) do not have imaginary roots so that 1, b1 and b2 do not form a triangle in the complex plane. From Theorem 3.2 we know that 1, b1 and b2 could form a triangle for any (τ1,τ2)ΦΨ. Select curves C1 and C2 from ΦΨ, where

    C1={(τ1,τ2)R+×R+|τ1=π2θ1ω,τ2=θ2ω,ωΩ}Φ,
    C2={(τ1,τ2)R+×R+|τ1=π2ϑ1ω,τ2=ϑ2ω,ωΥ}Ψ.
    Figure 2.  The positions of 1, a1(iω)eiωτ1 and a2(iω)eiωτ2 on the complex plane. When (τ1,τ2) moves clockwise from (0,0) along the curve D, i.e., (0,0)(0,τ2)(τ,τ)(τ1,0)(0,0), the positions of 1, a1(iω)eiωτ1 and a2(iω)eiωτ2 change by Case 1 Case 2 Case 3 Case 4 Case 1.

    For the curve C1, let τ1=0 yields θ1=π2, then applying the properties of right triangle we obtain 1+|a1(iω)|2=|a2(iω)|2 and tan(θ2)=|a1(iω)|, (See Figure 2 Case 2) i.e.,

    1+γ2m2ω2=(1γ)2m2ω4andtan(θ2)=γmω.

    Solve the above equations yields

    ω=γ2m22+(1γ)2m2+γ4m44Ω,τ2=arctan(γmω)ωτ2.

    Let τ2=0 yields θ2=0, which means that the triangle degenerates into a line segment (See Figure 2 Case 4), so we have 1=|a1(iω)|+|a2(iω)| and θ1=0, and then

    ω=γm2+γ2m24+(1γ)mΩ,τ1=π2ω=τ1.

    Let τ1=τ2 yields θ1+θ2=π2, then we have 1=|a1(iω)|2+|a2(iω)|2 and tan(θ2)=|a1(iω)||a2(iω)| (See Figure 2 Case 3). By solving the above equation we obtain

    ω=γ2m22+(1γ)2m2+γ4m44Ω,τ1=τ2=arctan(γω1γ)ω=τ.

    Define a bounded region E1 formed by the curve C1 with the positive half of the horizontal and vertical axes, then the above statement indicates that DE1. Similarly, For the curve C2, define a bounded region E2 formed by the curve C2 with the positive half of the horizontal and vertical axes, then we could verify that DE2.

    Let D=(E1E2)(C1C2), then DD and D is a bounded region. In addition, D satisfies DD1D2C1C2 and ¯DDC1C2.

    Remark 4. Theorem 3.3 indicates that (τ1,0),(τ,τ)C1 and (0,τ2)C1C2. If curves in ΦC1 do not intersect the curve C1 and curves in ΨC2 do not intersect the curve C2 (except at its endpoints), then D=D. In addition, unlike the Eq (2.1), Theorem 3.3 shows that the critical region D of the Eq (3.1) is related to both the total strength α and the maximal strength m.

    In this section, a series of simulation examples are presented to illustrate Theorems 2.1, 3.1 and 3.3.

    Set the target x0(t)=sin(t)+5 and f(t)=cos(t). Let N=10 and m=2, the initial positions xi(0) and strategies αi of the Eq (2.1) are listed in Table 1.

    Table 1.  The initial positions xi(0) and strategies αi of the system (2.1).
    x1(0)=8.1472 x2(0)=9.0579 x3(0)=1.2699 x4(0)=9.1338 x5(0)=6.3236
    α1=1 α2=1 α3=0 α4=0 α5=0
    x6(0)=0.9754 x7(0)=2.7850 x8(0)=5.4688 x9(0)=9.5751 x10(0)=9.6489
    α6=0 α7=0 α8=0 α9=0 α10=0

     | Show Table
    DownLoad: CSV

    According to Theorem 2.1, the critical delay τ=π2m=π4. Take τ=0, τ=π5 and τ=π4 respectively to obtain simulations of the Eq (2.1) as shown in Figure 3.

    Figure 3.  The figures from left to right are simulations of the system (2.1) in τ=0, τ=π5 and τ=π4 respectively. In the figures, curves represent trajectories of agents xi(t)(1iN) in the system (2.1), and the inverted triangle describes the trajectory of the target x0(t). Oscillatory behaviour occurs in the system (2.1) due to the existence of delay τ and intensifies with the increase of τ. When τ<τ=π4, the system (2.1) achieves consensus and matches the dynamic behaviour of the target x0(t)=sin(t)+5. When τ=τ, periodic oscillation behaviour occurs in the system (2.1), and the system neither matches the dynamic behaviour of the target x0(t) nor achieves consensus.

    Set the target x0(t)=3sin(t)+4cos(t)+t2, v0(t)=3cos(t)4sin(t)+12 and g(t)=3sin(t)4cos(t). Let N=10, m=2 and γ=0.5, the initial positions xi(0), velocities vi(t) and strategies αi of the Eq (3.1) are listed in Table 2.

    Table 2.  The initial positions xi(0), velocities vi(t) and strategies αi of the system (3.1).
    x1(0)=7.5127 x2(0)=2.5510 x3(0)=5.0596 x4(0)=6.9908 x5(0)=8.9090
    v1(0)=2.7603 v2(0)=6.7970 v3(0)=6.5510 v4(0)=1.6261 v5(0)=1.1900
    α1=1 α2=1 α3=0 α4=0 α5=0
    x6(0)=9.5929 x7(0)=5.4722 x8(0)=1.3862 x9(0)=1.4929 x10(0)=2.5751
    v6(0)=4.9836 v7(0)=9.5976 v8(0)=3.4039 v9(0)=5.8527 v10(0)=2.2381
    α6=0 α7=0 α8=0 α9=0 α10=0

     | Show Table
    DownLoad: CSV

    According to Theorem 3.1 we have τ1=π1+5, τ2=min{1.1506,1.0166}=1.0166, τ=0.7111. Take τ1=τ2=0, τ1=τ2=0.5 and τ1=τ2=0.7111 respectively to obtain simulations of the Eq (3.1) as shown in Figure 4.

    Figure 4.  The figures from left to right are simulations of xi(t)(0iN) (bottom) and vi(t)(0iN) (top) for the system (3.1) in τ1=τ2=0, τ1=τ2=0.5 and τ1=τ2=0.7111 respectively. When τ1=τ2<τ=0.7111, the velocities vi(t)(1iN) of the system (3.1) achieve consensus and match the target velocity v0(t)=3cos(t)4sin(t)+12. And at the same time, the positions xi(t)(1iN) of the system (3.1) achieve consensus and match the target position x0(t)=3sin(t)+4cos(t)+t2. When τ1=τ2=τ=0.7111, agents in the system (3.1) move around the target, but the system (3.1) neither matches the dynamic behaviour of the target and nor achieves consensus.

    Fix m=2 and α/N=0.2, take γ=0.2, γ=0.7321 (i.e., 4(1γ)γ2=m) and γ=0.8 respectively to obtain simulations of the ΦΨ by Theorem 3.2, see Figure 5.

    Figure 5.  Fix m=2 and α/N=0.2. The figures from left to right are simulations of ΦΨ for the system (3.1) in γ=0.2, γ=0.7321 and γ=0.8 respectively. D is the critical region of the system (3.1), Denote the bounded region, containing the origin, formed by the curves C1, C2 and the positive half of the horizontal and vertical axes by D. For the one on the left, curves in ΦC1 do not intersect the curve C1 and curves in ΨC2 do not intersect the curve C2 (except at its endpoints), then D=D. For the middle one, even though curves in ΦC1 intersect the curve C1, D=D. For the one on the right, DD. In addition, when 4(1γ)γ2m, the curve C1 is smooth because of the connectedness of the interval Ω. When 4(1γ)γ2>m, the interval Ω consists of two disconnected closed intervals, so the curve C1 is made up of two smooth curves, such as the figure on the right.

    Fix m=2 and γ=0.7321, take α/N=0.2, α/N=1 and α/N=1.8 respectively to obtain simulations of the ΦΨ by Theorem 3.2, see Figure 6.

    Figure 6.  Fix m=2 and γ=0.7321. The figures from left to right are simulations of ΦΨ for the system (3.1) in α/N=0.2, α/N=1 and α/N=1.8 respectively. The critical region D of the system (3.1) changes as α changes. In particular, as α/N approaches m, curves Φ gradually converges to curves Ψ.

    In this paper, we analysed the influence of the processing delay on the consensus for the first-order system in Eq (1.1) and the second-order system in Eq (1.2). For the first-order system in Eq (2.1), by the continuous dependence of the equation on τ, we obtained the critical delay τ that ensures the Eq (2.1) to achieve consensus and showed that the critical delay τ is independent of the strategies αi. For the second-order system in Eq (3.1), from the properties of plane geometry, we identified the critical region D in R2 that guarantees the Eq (3.1) to achieve consensus and found that the shape of the critical region D is affected by the strategies αi.

    The concept of strategies αi was firstly proposed by Piccoli et al. [18] to study the problem of optimal strategy. Using Pontryagin's minimum principle in optimal control theory, they found optimal strategies {αi}Ni=1 to minimize the cost 1NNi=1xi(T)x0(T), where T>0 was the final time. More importantly, they showed that optimal strategies are sparse. From the present work, we want to find optimal strategies αi for the Eq (2.1) and analyse the influence of delays on the selection of optimal strategies αi.

    We would like to thank the editors and the reviewers for their careful reading of the paper and their constructive comments.

    This work was partially supported by the National Natural Science Foundation of China (11671011) and Hunan Provincial Innovation Foundation for Postgraduate (CN) (CX20220085).

    The authors declare there is no conflict of interest.



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