Research article Special Issues

Flocking dynamics and pattern motion for the Cucker-Smale system with distributed delays


  • In this paper, a new class of Cucker-Smale systems with distributed delays are developed from the measurement perspective. By combining dissipative differential inequalities with a continuity argument, some new sufficient criteria for the flocking dynamics of the proposed model with general communication rate, especially the non-normalized rate, are established. In order to achieve the prescribed pattern motion, the driving force term is incorporated into the delayed collective system. Lastly, some examples and simulations are provided to illustrate the validity of the theoretical results.

    Citation: Jingyi He, Changchun Bao, Le Li, Xianhui Zhang, Chuangxia Huang. Flocking dynamics and pattern motion for the Cucker-Smale system with distributed delays[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 1505-1518. doi: 10.3934/mbe.2023068

    Related Papers:

    [1] Hyunjin Ahn, Myeongju Kang . Emergent dynamics of various Cucker–Smale type models with a fractional derivative. Mathematical Biosciences and Engineering, 2023, 20(10): 17949-17985. doi: 10.3934/mbe.2023798
    [2] Jan Haskovec, Ioannis Markou . Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays. Mathematical Biosciences and Engineering, 2020, 17(5): 5651-5671. doi: 10.3934/mbe.2020304
    [3] Fei Cao, Sebastien Motsch, Alexander Reamy, Ryan Theisen . Asymptotic flocking for the three-zone model. Mathematical Biosciences and Engineering, 2020, 17(6): 7692-7707. doi: 10.3934/mbe.2020391
    [4] Le Li, Lihong Huang, Jianhong Wu . Flocking and invariance of velocity angles. Mathematical Biosciences and Engineering, 2016, 13(2): 369-380. doi: 10.3934/mbe.2015007
    [5] Huazong Zhang, Sumin Yang, Rundong Zhao, Qiming Liu . Finite-time flocking with collision-avoiding problem of a modified Cucker-Smale model. Mathematical Biosciences and Engineering, 2022, 19(10): 10332-10343. doi: 10.3934/mbe.2022483
    [6] Pedro Aceves-Sánchez, Mihai Bostan, Jose-Antonio Carrillo, Pierre Degond . Hydrodynamic limits for kinetic flocking models of Cucker-Smale type. Mathematical Biosciences and Engineering, 2019, 16(6): 7883-7910. doi: 10.3934/mbe.2019396
    [7] Yuanpei Xia, Weisong Zhou, Zhichun Yang . Global analysis and optimal harvesting for a hybrid stochastic phytoplankton-zooplankton-fish model with distributed delays. Mathematical Biosciences and Engineering, 2020, 17(5): 6149-6180. doi: 10.3934/mbe.2020326
    [8] Max-Olivier Hongler, Roger Filliger, Olivier Gallay . Local versus nonlocal barycentric interactions in 1D agent dynamics. Mathematical Biosciences and Engineering, 2014, 11(2): 303-315. doi: 10.3934/mbe.2014.11.303
    [9] Kai Wang, Zhidong Teng, Xueliang Zhang . Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1425-1445. doi: 10.3934/mbe.2017074
    [10] Ryoto Himo, Masaki Ogura, Naoki Wakamiya . Iterative shepherding control for agents with heterogeneous responsivity. Mathematical Biosciences and Engineering, 2022, 19(4): 3509-3525. doi: 10.3934/mbe.2022162
  • In this paper, a new class of Cucker-Smale systems with distributed delays are developed from the measurement perspective. By combining dissipative differential inequalities with a continuity argument, some new sufficient criteria for the flocking dynamics of the proposed model with general communication rate, especially the non-normalized rate, are established. In order to achieve the prescribed pattern motion, the driving force term is incorporated into the delayed collective system. Lastly, some examples and simulations are provided to illustrate the validity of the theoretical results.



    In the last years, self-organization systems have been getting a great deal of attention from researchers around the world, and have conducted a lot of research in many fields such as artificial intelligence, physics, biology and social sciences. The famous Cucker-Smale model [1,2] offered a frame to describe aggregation behaviors, for instance, the flocking of birds, reaching a consensus. Considering that the interaction intensity depends on the number of agents, the Cucker-Smale model can not well describe the flocking behavior of non-uniform multi-particle swarm optimization. Motsch and Tadmor generalized the flocking system to the case of asymmetric interactions [3]. Recently, the classical Cucker-Smale model has been generalized and modified to several cases, such as various forms of stochastic noise, cone-vision constraints, the presence of leadership and more general interaction potentials [4,5,6,7,8,9,10,11,12,13,14,15,22,23,24,25,26,27,28].

    In practical applications, time-delay often leads to system instability, and its impact can not be ignored [29,30,31]. It has become a broad consensus that mathematical models with time delay always have greater practicability[23,24,25,32,33,34,35]. The authors took into account heterogeneous delays in [23]. The velocity asymptotic alignment of the delayed Cucker-Smale model was investigated in the presence or absence of noise [24]. Very recently, the authors in [25] proposed the following improved model

    {˙pi=ci,˙ci=1h(t)Nm=1ttT(t)β(ts)Φ(pm(s),pi(t))(cm(s)ci(t))ds,i=1,2,,N, (1.1)

    where pi(t) and ci(t) denote the position and velocity of agent i at time t, β:[0,T0][0,) is a weight function which requires

    ˆT0β(s)ds>0andh(k):=T(k)0β(s)ds,k0. (1.2)

    Φ(pm(s),pi(t)) is the normalized communication weights provided by

    Φ(pm(s),pi(t))={ψ(|pm(s)pi(t)|)jiψ(|pj(s)pi(t)|),ifmi,0,ifm=i, (1.3)

    with ψ:[0,)(0,), is positive, bounded, nonincremental and Lipschitz continuous on [0,), and ψ(0)=1. Besides, there has ˆT>0 and T0>0 obeying

    T(k)ˆT,T(k)0,andˆTT(k)T0fork0. (1.4)

    On the one hand, assumption (1.4) is reasonable requirement for time-varying delays in specific background [21,27]. As we all know, measurement of velocity is much more sensitive than measurement of position from the perspective of time delay and the measurement delay in this system mainly responds to velocity, not position. Furthermore, the introduction of distributed delay can effectively characterize the fact that the position and velocity of agents are not only affected by the behavior of other individuals in a certain period of time, but also affected by the behavior of other individuals at a varying time period [37]. And few researchers have considered the following distributed delay of Cucker-Smale model which governed by

    {˙pj=cj,˙cj=1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,j(k)(cm(s)cj(k))ds,j=1,2,,N, (1.5)

    where Φm,j(k)=Φ(pm(k),pj(k)), (ϕj,ψj)S2=S×S and S:=S([T,0],Rd) is the Banach space of all continuous functions. Moreover, Φ:R+R+ requires ϕ(k)1 for each k0. R+ denotes the set of positive real number.

    pj(s)=:Φj(s),cj(s)=:ψj(s),j=1,,N,s[T0,0], (1.6)

    On the other hand, the pattern motion to self-organized systems come out so unaffectedly at lots of physical and biological scenes, which is important for us to further study the mechanism of swarm intelligence systems. However, there are few more practical research on flocking formation behavior based on Cucker-Smale model.

    Illuminated by the aforementioned arguments, the main objective of this article is to establish the flocking behavior of the system (1.5). Specifically speaking, the focus of this article is as follows.

    (i) In this article, we propose a new class of Cucker-Smale model incorporating distributed delays from the measurement perspective, which is different from the existing models, see, e.g., [23,24,25].

    (ii) Under certain assumptions, by exploiting dissipative differential inequality, some new sufficient criteria for the flocking behavior of (1.5) and (1.6) with general communication rate (especially the non-normalized rate) are gained for the first time.

    (iii) Numerical simulations are arranged to verify the effectiveness of the main theoretical analysis results.

    In the rest of this paper, we give the definition and several useful lemmas in Section 2. The flocking result and motion pattern of the system (1.5) with a driving force are presented in Section 3. In Section 4, we also give the numeric calculations which are very good agreement with theoretical results. Lastly, we draw a brief conclusion in Section 5.

    In this section, for purpose of obtaining the main results of this paper, we firstly require the following definition and lemmas. Define the following quantities:

    dP(t):=max1j,iN||pj(t)pi(t)||,dC(t):=max1j,iN||cj(t)ci(t)||.

    Definition 2.1 We say that a solution {pj(t),cj(t)},j{1,...,N} of the systems (1.5) and (1.6) converges to flocking while the conditions as follow are satisfied.

    supt0dP(k)<, and limkdC(k)=0.

    Due to the fact which the functions dP and dC usually are not at C1 smooth, we do with the upper Dini derivative. For a function G(t), G's upper Dini derivative at k is defined by

    D+G(k)=lim supm0+G(k+m)G(k)m.

    In particular, the Dini derivative is the same as the usual derivative when G is differential at k. In this paper, we assume that the function Φ is bounded, positive, nonincremental and at Lipschitz continuous in R+, with Φ(0)=1.

    Remark 2.1 We notice that the theoretical analysis in [21] the assumption of the function Φ. that it have the strictly positive lower bound. However, it isn't needed in our framework.

    Lemma 2.2 Let {(pj,cj)}Nj=1 be a solution to (1.5) and (1.6) and F=maxk[T,0]max1jN|Ψj(k)|>0. Assume the premier velocity Ψj(j=1,2,...,N) is continuous on [T,0]. Then the solution satisfies

    max1jN|cj(t)|FforkT.

    Proof Choose any ϵ>0 and set

    Qϵ:={k>0:max1iN|ci(t)|<F+ε,t[0,k)}.

    According to the assumption, Qε. Denote Rε:=supQε>0. We will prove that Rε=+. For contradiction, suppose Rε<+. This gives, by continuity, max1jN|cj(Rε)|=F+ϵ.

    On another scale, from (1.5) and (1.6), for k<Rε and j=1,,N, we have

    12D+|cj(k)|2cj(k),dcj(k)dk=cj(k),1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)(cm(s)cj(k))ds=1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)cj(k),cm(s)cj(k)ds=1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)(cj(k),cm(s)|cj(k)|2)ds1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)|cj(k)|(|cm(s)|cj(k)|)ds.

    Note that 1NNm=1,mjΦm,i(k)1 and max1jN|cm(s)|<F+ϵ for k<Rϵ. Thus we receive

    12D+|cj(k)|21h(t)kkT(k)β(ks)ds|cj(k)|(F+ϵ|cj(k)|)=|cj(k)|(F+ϵ|cj(k)|),

    which yields

    D+|cj(k)|(F+ϵ)|cj(k)|.

    With the help of Gronwall inequality, we have

    |cj(k)|ek(cj(0)Fϵ)+F+ϵ<F+ϵ.

    Hence,

    limtRϵmax1jN|cj(k)|<F+ϵ,

    which is in contradiction with hypothesis. Therefore, Rϵ=+. Moreover, since ϵ is arbitrary, the lemma is proved.

    Lemma 2.3. Let {(pj,cj)}Nj=1 be the solution to (1.5) and (1.6). Afterwards, the diameters functions dP(k) and dC(k) require

    D+dP(k)dC(k),D+dC(k)Φ(dP(k))dC(k)+2ΔTN(k), (2.1)

    for all k>0, where ΔTN(k) is given by

    ΔTN(k)=1Nh(k)max1jNNm=1,mjkkT(k)β(ks)Φm,i(k)|cm(k)cm(s)|ds, (2.2)

    and satisfies

    ΔTN(k)kkT(k)[ΔTN(s)+dC(s)]ds. (2.3)

    Proof We firstly obtain from (1.5) that

    D+dP(k)dC(k).

    On account of the continuity of cj(k)(j{1,...,N}), there is a times sequence {km}mN such that

    mN[km,km+1)=[0,+).

    And for each kN, there has i,j{1,...,N} so that dC(k)=|cj(k)ci(k)| for k[km,km+1).

    Consequently, one can get

    12D+d2C(k)=12ddk|cj(k)ci(k)|2=cj(k)ci(k),˙cj(k)˙ci(k)=cj(k)ci(k),1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)[cm(s)cj(k)]dscj(k)ci(k),1Nh(k)Nm=1,mikkT(k)β(ks)Φm,i(k)[cm(s)ci(k)]ds=:W1(k)+W2(k). (2.4)

    For any j{1,...,N} and the function Φ is non-increasing, one can obtain

    Φm,i(k)Φ(dP(k)), (2.5)

    and

    cj(k)ci(k),cm(k)cj(k)0,k0,m{1,...,N}. (2.6)

    According to the definitions of h(k), and with Φ1, α(k), (2.5) and (2.6), we obtain as follows:

    W1(k)=cj(k)ci(k),1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)[cm(s)cj(k)]ds=cj(k)ci(k),1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)[cm(k)cj(k)]ds+cj(k)ci(k),1Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)[cm(s)cm(k)]dsΦ(dP(k))NNm=1,mjcj(k)ci(k),cm(k)cj(k)+dC(k)Nh(k)Nm=1,mjkkT(k)β(ks)Φm,i(k)|cm(k)cm(s)|ds. (2.7)

    Similarly

    W2(k)=cj(k)ci(k),1Nh(k)Nm=1,mikkT(k)β(ks)Φk,j(k)[cm(k)ci(k)]ds+cj(k)ci(k),1Nh(k)Nm=1,mikkT(k)β(ks)Φk,j(k)[cm(s)cm(k)]dsΦ(dP(k))NNm=1,micj(k)ci(k),cm(k)ci(k)+dC(k)Nh(k)Nm=1,mikkT(k)β(ks)Φm,i(k)|cm(k)cm(s)|ds. (2.8)

    From (2.4), (2.7) and (2.8), for k0 we obtain that

    12D+dC(k)2Φ(dP(k))dC(k)2+2dC(k)Nh(k)max1jiNNm=1,mikkT(k)β(ks)Φm,i(k)|cm(k)cm(s)|ds.

    Thereby,

    D+dC(k)Φ(dP(k))dC(k)+2ΔTN(k).

    We next estimate the term ΔTN(k). One can get

    ΔTN(k)=1Nh(k)max1iNNm=1,mikkT(k)β(ks)Φm,i(s)|cm(k)cm(s)|ds1Nh(k)Nm=1kkT(k)β(ks)ks|˙cm(θ)|dθds1NNm=1kkT(k)|˙cm(s)|dskkT(k)|˙cm(s)|ds. (2.9)

    Furthermore, it obeys from (2.7), that

    |˙cm(s)|=|1Nh(s)Nl=1,lmssT(s)β(sθ)Φm,l(s)|cl(θ)cm(s)|dθ|1Nh(s)Nl=1ssT(s)β(sθ)Φm,l(s)(|cl(s)cm(s)|+|cl(θ)cl(s)|)dθΔTN(s)+dC(s). (2.10)

    Hence, combining with (2.9) and (2.10), we obtain

    ΔTN(k)kkT(k)[ΔTN(s)+dC(s)]ds, (2.11)

    which proves the Lemma 2.3.

    Remark 2.2 In the perspective of Lemma 2.3, we achieve

    |˙cm(s)||1Nh(s)Nl=1ssT(s)β(sθ)Φm,l(s)|cl(θ)cm(s)|dθ|2F.

    Estimation are given as follow:

    ΔTN(k)2FT(k)fork0.

    Theorem 3.1.1 Make {(pi,cj)}Ni=1 as the solution to (1.5) and (1.6). Assume that there have some constants κ,δ>0 satisfying

    0<κ<Φ(δ),dP(0)+C2κ<δ, (3.1)

    where C2:=2C1Φ(δ)κ with C1> max {dC(0)2[Φ(δ)κ],2GT0}. Then, if

    Φ(δ)κ+2κ[Φ(δ)κ](eκT01)<1, (3.2)

    we have

    dP(k)<δanddC(k)C2eκk,k0.

    Aiming at proving Theorem 3.1.1, we give Lemma 3.1.2 as follow.

    Lemma 3.1.2 Make {(pi,cj)}Ni=1 as a global solution to the model (1.5) and (1.6) satisfying a priori assumption on the relative position:

    sup0k<+dP(k)δ. (3.3)

    Then, we can gain

    ΔTN(k)<C1eκkanddC(k)<C2eκkk>0, (3.4)

    where κ>0 and C1,C2>0 have been given in Theorem 3.1.1.

    Proof Firstly if ΔTN(k)<C1eκk for all k[0,H] with fixing H>0, we prove that dC(k)<C2eκk for all k[0,T]. Actually, according to Lemma 2.3, one can easily obtain

    D+dC(k)Φ(dP(k))dC(k)+2C1eκk,

    for k[0,H]. Using Growall's inequality and the fact that dC(0)<2C1Φ(δ)κ yield

    dC(k)dC(0)eΦ(δ)k+2C1Φ(δ)κ[eκkeΦ(δ)t]=[dC(0)2C1Φ(δ)κ]eΦ(δ)k+2C1Φ(δ)κeκk<2C1Φ(δ)κeκk.

    Set

    M:={H>0:ΔTN(k)<C1eκkanddC(k)<C2eκk,k[0,H]}.

    Which obeys Lemma 2.3 and 2GT0<C1 that 0M. Thus, M. We will prove supM=. Suppose that ¯H=supM<. Therefore by continuity of functions dC(k) and ΔTN(k), we have

    ΔTN(¯H)=C1eκ¯HordC(¯H)=C2eκ¯H, (3.5)

    Afterwards ¯HM. Whatsmore, with using ΔTN(k) and M yields definitions.

    ΔTN(¯H)=limk¯HΔTN(k)limk¯HkkT(k)[ΔTN(s)+dC(s)]dslimk¯H(C1+C2)tkT(k)eκsdsC1+C2κeκ¯H(eκT01)<C1eκ¯H. (3.6)

    Thus, from the assertion of the proof of the lemma, we can get dC(¯H)<C2eκ¯H. Consequently, (3.5) doesn't hold, and we have ¯H=. This completes the proof.

    Next, we prove Theorem 3.1.1.

    We prove that a priori assumption (3.3) is effective for given δ in Theorem 3.1.1. Indeed, label

    S:={H>0:dP(k)<δ,k[0,H]}.

    In addition, through (3.3) and the function dP(t) continuity, one can conclude that S. We are now ready to deduce that supS=. Suppose that ¯¯H:=supS<, then we get dP(¯¯H)<δ. On the other hand, applying Lemma 3.1.2 yields that dC(k)C2eκk for k[0,¯¯H). So, by defining dC(k), one can obtain that for i,j{1,...,N}

    |pi(¯¯H)pj(¯¯H)||pi(0)pj(0)|+¯¯H0|cj(s)ci(s)|dsdP(0)+¯¯H0|dC(s)|dsdP(0)+¯¯H0C2eκsds<dP(0)+C2κ. (3.7)

    It can deduce dP(¯¯H)<dP(0)+C2κ. And we have

    δ=dP(¯¯H)<dP(0)+C2κ<δ,

    which conflicts. Thus, the priori assumption (3.3) is valid for δ. Combining with Lemma 3.1.2, we finish the proof of Theorem 3.1.1.

    In practical application, we hope that agents can form formation, so we establish a distributed time-delay flocking model with controller. The driving force term is taken into account, precisely, a suitable F is introduced into the system (1.5), so that all the agents converge to flocking and achieve the prescribed pattern motion. The modified Cucker-Smale is given as follows:

    {˙pi=cj,˙vi=1Nh(k)Nm=1,mikkT(k)β(ks)Φm,i(k)(cm(s)cj(k))ds+F(pi),i=1,2,,N. (3.8)

    Inspired by the work of [36], the function is taken as

    F(pi(k))=γ(sin(pa,c)pa,w)(ccos(pa,v)w), (3.9)

    where a=1NNi=1pi(k), v,w are two given vectors, γ is a positive force strength measured constant.

    For the sake of brevity, we only list a framework model. Some simulations for special experiments are conducted in the next section. The range of control parameters for formation of flock, how to select the control design parameters effectively, and strict theoretical analysis will be our future study.

    In this section, we provide several numerical simulations, which confirm the delay can affect the dynamic position and velocity of system (4.1).

    {˙pj=cj,˙cj=120h(k)20m=1,mjkkT(k)ϕm,i(k)(cm(s)cj(k))ds,j=1,2,,20, (4.1)

    with the initial functions as below

    pj(s)=:Φj(s),cj(s)=:ψj(s),j=1,,N,s[T0,0],

    where ϕm,i(k)=ϕ(pm(k),pi(k)), (Φj,ψj)S2=S×S and S:=S([T,0],Rd) is the Banach space of all continuous functions. Moreover, Φ:R+R+ requires ϕ(k)1 for each k0. R+ denote the set of positive real number.

    The effectiveness and validity of the analytical results are demonstrated by the following examples.

    Example 4.1 For system (4.1), we choose

    ϕ(r)=1(1+r2)0.2,Φi(s)=(0.01,i1),ψi(s)=(0.01,i1),i=1,,20,s[T0,0]. (4.2)

    Taking δ=70, C1=0.84, T(k)T0=0.04, by some simple calculations, it is easy to see that all conditions of Theorem 3.1.1 are satisfied. Therefore, system (4.1) under condition (4.2) will asymptotically converge to a flock. This conclusion can be verified by the following numerical simulations in Figure 1.

    Figure 1.  N = 20, T=0.04, the system (4.1) asymptotically converges to a flock.

    Example 4.2 With the purpose of demonstrating the dynamic process of agents' flock convergence, for system (4.1), we choose

    ϕ(r)=1(1+r2)0.4,Φi(s)=(0.01,2i1),ψi(s)=(0.01,2i1),i=1,,20,s[T0,0]. (4.3)

    Taking δ=70, C1=0.84 and T(k)T0=0.01, by some simple calculations, it is easy to see that all conditions of Theorem 3.1.1 are satisfied. Figure 2 shows the process of agents' flock convergence, which demonstrates that the dynamic graph of agents connected and changing until system (4.1) under condition (4.3) flocking achieved.

    Figure 2.  The dynamic process of agents' flock convergence.

    Example 4.3 Aiming at demonstrating that agents starting at different initial position and initial velocity can converging to varieties of different stable structure eventually. For system (4.1), we take δ=70, C1=0.84 and T(k)T0=0.01. It is easy to see that the below three situations satisfy Theorem 3.1.1, and therefore, the system (4.1) can work normally.

    (i) ϕ(r)=er0.8, Φi(s)=(0.01,i1),ψi(s)=(0.01,i1),i=1,,20,s[T0,0]. The simulation results are shown at (a) in Figure 3

    Figure 3.  Agents start at different position with different velocity, shape various different stable formation at last.

    (ii) ϕ(r)=(sinr2)0.8, Φi(s)=(0.01,2i1),ψi(s)=(0.01,2i1),i=1,,20,s[T0,0]. The simulation results are shown at (b) in Figure 3.

    (iii) ϕ(r)=(cosr2)0.8, Φi(s)=(0.01,3i1),ψi(s)=(0.01,3i1),i=1,,20,s[T0,0]. The simulation results are shown at (c) in Figure 3.

    Figure 3 displays a variety of different stable formation of 20 agents which start at random initial positions with different velocities.

    Example 4.4 For system (4.1), we choose

    Φ(r)=er0.6,Φi(s)=(0.01,i1),ψi(s)=(0.01,i1),i=1,,20,s[T0,0]. (4.4)

    Taking T(k)T0=3.0, by some simple calculations, then the condition of Theorem 3.1.1 fails, which is perfectly verified in Figure 4.

    Figure 4.  N = 20, T=1.0, system (4.1) with (4.4) cannot asymptotically converges to a flock.

    Example 4.5 Consider the system (4.1), where ϕ(r)=1(1+r2)16, and take γ=1 as the communication rate function and T(k)T0=0.8. Besides, we let pi(k) and ci(k) (i=1,2,,N) for k[T0,0] generated randomly and diverse in the region [0,10]×[0,1]. Then our simulation verifies that the solution of the system (4.1) can converge to a flock with the prescribed motion pattern, which is shown in Figure 5.

    Figure 5.  The position distribution of population at t = 0 s, t = 20 s and t = 100 s, respectively. And the values of each parameter are given as w=(0,1) and v=(1,0).

    It is practical to understand how autonomous agents organize orderly movements based on finite information about the environment and monotonous rules. By utilizing dissipative differential inequality with a continuity argument, abundant conditions are the key insurance to the existence of flocking for the Cucker-Smale system with distributed delays from a measurement perspective. Several numerical simulations, it is a confirmation of that distributing delays can affect the flocking behavior. Meanwhile, the driving force term is added to the delay collective system to realize the specified pattern motion through numerical experiments.

    This work was supported by the National Natural Science Foundation of China (Nos. 11971076, 11801562). The authors are grateful to the editor and reviewers for their constructive comments, which led to a significant improvement of our original manuscript.

    The authors declare that they have no conflict of interest.



    [1] F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852–862. https://doi.org/10.1109/tac.2007.895842 doi: 10.1109/tac.2007.895842
    [2] F. Cucker, S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197–227. https://doi.org/10.1007/s11537-007-0647-x doi: 10.1007/s11537-007-0647-x
    [3] S. Motsch, E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923–947. https://doi.org/10.1007/s10955-011-0285-9 doi: 10.1007/s10955-011-0285-9
    [4] F. Dalmao, E. Mordecki, Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307–1316. https://doi.org/10.1137/100785910 doi: 10.1137/100785910
    [5] F. Cucker, J. G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Autom. Control, 56 (2011), 1124–1129. https://doi.org/10.1109/tac.2011.2107113 doi: 10.1109/tac.2011.2107113
    [6] S. Y. Ha, J. G. Liu, A simple proof of the cucker-smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297–325. https://doi.org/10.4310/cms.2009.v7.n2.a2 doi: 10.4310/cms.2009.v7.n2.a2
    [7] S. Y. Ha, T. Ha, J. H. Kim, Emergent behavior of a cucker-smale type particle model with nonlinear velocity couplings, IEEE Trans. Autom. Control, 55 (2010), 1679–1683. https://doi.org/10.1109/tac.2010.2046113 doi: 10.1109/tac.2010.2046113
    [8] J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Phys. D, 261 (2013), 42–51. https://doi.org/10.1016/j.physd.2013.06.006 doi: 10.1016/j.physd.2013.06.006
    [9] L. Li, L. Huang, J. Wu, Cascade flocking with free-will, Discrete Contin. Dyn. Syst. B, 21 (2015), 497–522. https://doi.org/10.3934/dcdsb.2016.21.497 doi: 10.3934/dcdsb.2016.21.497
    [10] Z. Li, X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156–3174. https://doi.org/10.1137/100791774 doi: 10.1137/100791774
    [11] Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst., 34 (2014), 3683–3702. https://doi.org/10.3934/dcds.2014.34.3683 doi: 10.3934/dcds.2014.34.3683
    [12] H. Liu, X. Wang, Y. Liu, X. Li, On non-collision flocking and line-shaped spatial configuration for a modified singular Cucker-Smale model, Commun. Nonlinear Sci. Numer. Simul., 75 (2019), 280–301. https://doi.org/10.1016/j.cnsns.2019.04.006 doi: 10.1016/j.cnsns.2019.04.006
    [13] L. Ru, Z. Li, X. Xue, Cucker-Smale flocking with randomly failed interactions, J. Franklin Inst., 352 (2015), 1099–1118. https://doi.org/10.1016/j.jfranklin.2014.12.007 doi: 10.1016/j.jfranklin.2014.12.007
    [14] L. Ru, X. Xue, Multi-cluster flocking behavior of the hierarchical Cucker-Smale model, J. Franklin Inst., 354 (2017), 2371–2392. https://doi.org/10.1016/j.jfranklin.2016.12.018 doi: 10.1016/j.jfranklin.2016.12.018
    [15] J. J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2008), 694–719. https://doi.org/10.1137/060673254 doi: 10.1137/060673254
    [16] Y. P. Choi, J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011–1033. https://doi.org/10.3934/krm.2017040 doi: 10.3934/krm.2017040
    [17] Y. Liu, J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Ana. Appli., 415 (2014), 53–61. https://doi.org/10.1016/j.jmaa.2014.01.036 doi: 10.1016/j.jmaa.2014.01.036
    [18] C. Pignotti, E. Trélat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053–2076. https://doi.org/10.4310/cms.2018.v16.n8.a1 doi: 10.4310/cms.2018.v16.n8.a1
    [19] J. G. Dong, S. Y. Ha, D. Kim, J. Kim, Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differ. Equation, 266 (2019), 2373–2407. https://doi.org/10.1016/j.jde.2018.08.034 doi: 10.1016/j.jde.2018.08.034
    [20] J. G. Dong, S. Y. Ha, D. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. B, 24 (2017), 1–28. https://doi.org/10.3934/dcdsb.2019072 doi: 10.3934/dcdsb.2019072
    [21] C. Pignotti, E. Trélat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053–2076. https://doi.org/10.4310/cms.2018.v16.n8.a1 doi: 10.4310/cms.2018.v16.n8.a1
    [22] I. D. Couzin, J. Krause, N. R. Franks, S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513–516. https://doi.org/10.1038/nature03236 doi: 10.1038/nature03236
    [23] Y. P. Choi, S. Y. Ha, Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Act. Part., 1 (2017), 299C331. https://doi.org/10.1007/978-3-319-49996-3_8 doi: 10.1007/978-3-319-49996-3_8
    [24] R. Erban, J. Haškovec, Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535–1557. https://doi.org/10.1137/15m1030467 doi: 10.1137/15m1030467
    [25] Y. P. Choi, C. Pignotti, Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays, Network Heterog. Med., 14 (2019), 789–804. https://doi.org/10.3934/nhm.2019032 doi: 10.3934/nhm.2019032
    [26] X. Wang, L. Wang, J. Wu, Impacts of time delay on flocking dynamics of a two-agent flock model, Commun. Nonlinear Sci. Numer. Simul., 70 (2019), 80–88. https://doi.org/10.1016/j.cnsns.2018.10.017 doi: 10.1016/j.cnsns.2018.10.017
    [27] E. I. Verriest, Inconsistencies in systems with time-varying delays and their resolution, IMA J. Math. Control Inf., 28 (2011), 147–162. https://doi.org/10.1093/imamci/dnr013 doi: 10.1093/imamci/dnr013
    [28] S. Wongkaew, M. Caponigro, A. Borzì, On the control through leadership of the hegselmann–krause opinion formation model, Math. Models Method Appl. Sci., 25 (2014), 565–585. https://doi.org/10.1142/s0218202515400060 doi: 10.1142/s0218202515400060
    [29] C. Huang, X. Zhao, J. Cao, F. E. Alsaadi, Global dynamics of neoclassical growth model with multiple pairs of variable delays, Nonlinearity, 33 (2020), 6819–6834. https://doi.org/10.1088/1361-6544/abab4e doi: 10.1088/1361-6544/abab4e
    [30] C. Huang, Y. Tan, Global behavior of a reaction-diffusion model with time delay and dirichlet condition, J. Differ. Equation, 271 (2021), 186–215. https://doi.org/10.1016/j.jde.2020.08.008 doi: 10.1016/j.jde.2020.08.008
    [31] C. Huang, L. Huang, J. Wu, Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays, Discrete Contin. Dyn. Syst. B, 27 (2022), 2427–2440. https://doi.org/10.3934/dcdsb.2021138 doi: 10.3934/dcdsb.2021138
    [32] C. Huang, B. Liu, Traveling wave fronts for a diffusive nicholson's blowflies equation accompanying mature delay and feedback delay, Appl. Math. Lett., 134 (2022), 108321. https://doi.org/10.1016/j.aml.2022.108321 doi: 10.1016/j.aml.2022.108321
    [33] X. Li, S. Song, J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Autom. Control, 64 (2019), 4024–4034. https://doi.org/10.1109/tac.2019.2905271 doi: 10.1109/tac.2019.2905271
    [34] X. Li, X. Yang, S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica, 103 (2019), 135–140. https://doi.org/10.1016/j.automatica.2019.01.031 doi: 10.1016/j.automatica.2019.01.031
    [35] X. Li, D. Peng, J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908–4913. https://doi.org/10.11092Ftac.2020.2964558
    [36] X. Li, Y. Liu, J. Wu, Flocking and pattern motion in a modified Cucker-Smale model, Bull. Korean Math. Soc., 53 (2016), 1327–1339. https://doi.org/10.41342Fbkms.b150629
    [37] C. M. Farza, M. M'Saad, Observer design for a class of disturbed nonlinear systems with time-varying delayed outputs using mixed time-continuous and sampled measurements, Automatica, 107 (2019), 231–240. https://doi.org/10.1016/j.automatica.2019.05.049 doi: 10.1016/j.automatica.2019.05.049
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2042) PDF downloads(103) Cited by(0)

Figures and Tables

Figures(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog