Complete Cluster Predictability of the Cucker–Smale Flocking Model on the Real Line

Ha, Seung Yeal; Kim, Jeongho; Park, Jinyeong; Zhang, Xiongtao

doi:10.1007/s00205-018-1281-x

Cited by 68 publications

(54 citation statements)

References 38 publications

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“…Comparing to the regular case [26,22], the construction of the first order equation in the present paper is a little different due to the singularity of the communication function.…”

Section: Preliminarymentioning

confidence: 83%

“…Therefore, the emergence of mono-cluster flocking depends on the initial configuration and multi-cluster formation may occur. Similar as in [22], we will construct the critical value κ c so that mono-cluster formation will emerge if and only if κ > κ c and multi-cluster formation will emerge if and only if κ ≤ κ c .…”

Section: Higher Order Singularitymentioning

confidence: 99%

“…The order preservation allows us to apply similar arguments in [22] to yield the asymptotic nonoscillatory behavior of relative distance between particles governed by the Cucker-Smale system (2.4). In fact, based on the analysis in [22], we have the following lemma.…”

Section: 2mentioning

confidence: 99%

“…In [6,7], the authors proved the collision avoidance for C-S model in any finite time, but the uniform-in-time lower bound is still unknown. Moreover, in [26,22], the authors studied the C-S model with regular short range interaction and constructed a sufficient and necessary condition for the emergence of mono-cluster and multi-cluster formation, respectively. While, it is not clear wether the similar results can be obtained in the singular case.…”

mentioning

confidence: 99%

“…Therefore, may we address three natural questions as follows,• (Q1) Can we derive the uniqueness of the solution to the C-S model (1.1) with singular communication (1.2)? Moreover, can we find the sufficient and necessary condition for emergence of finite time flocking in N particle system?• (Q2) Can we construct the uniform-in-time lower bound between two particles of the C-S model with singular interaction, so that the collision avoidance occurs when time tends to infinity and asymptotical equilibrium state can be constructed?• (Q3) Can we obtain the sufficient and necessary condition for the emergence of mono-cluster and multi-cluster formation, and thus construct the critical coupling strength κ c for flocking emergence as in [22]? Moreover, can we obtain the information of cluster number and the asymptotical velocity of each cluster?On the other hand, for β ≥ 1, avoidance of collision will be guaranteed according to [7].…”

mentioning

confidence: 99%

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Complete classification of the asymptotical behavior for singular C-S model on the real line

Zhang

Zhu

2020

Journal of Differential Equations

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In this paper, we study the singular Cucker-Smale (C-S) model on the real line. For long range case, i.e. β < 1, we prove the uniqueness of the solution in the sense of Definition 2.1 and the unconditional flocking emergence. Moreover, the sufficient and necessary condition for collision and sticking phenomenon will be provided. For short range case, i.e. β > 1, we construct the uniform-in-time lower bound of the relative distance between particles and provide the sufficient and necessary condition for the emergence of multi-cluster formation. For critical case, i.e. β = 1, we show the uniform lower bound of the relative distance and unconditional flocking emergence. These results provide a complete classification of the collective behavior for C-S model on the real line.(In fact, recently, the C-S model with singular interaction (1.2) attracts a lot of attention from various area. This is mainly due to that the Coulomb type interaction will automatically generates the repulsion and leads to the avoidance of collision [7], which is more physical and very important for application in engineering such as formation of unmanned aerial vehicles. However, this singular communication weight causes a lot of difficulty in the mathematical analysis. For instance, the uniqueness of the solution to (1.1) cannot be guaranteed by the fundamental theory of ODE, because the vector field on the right hand side of (1.1) is no more Lipschitz. Therefore, comparing to the extensive study on regular communication weight, there are very few works concerning on the C-S model with singular interaction: flocking dynamics and mean-field limit [25], avoidance of collision [1,7], global existence of weak solutions in particle and kinetic level [6,35,39,40]. More precisely,in [39], the author constructed the global existence of weak solution without uniqueness. Meanwhile, the author in [39] found the finite time flocking phenomenon but only in two particles system. In [6,7], the authors proved the collision avoidance for C-S model in any finite time, but the uniform-in-time lower bound is still unknown. Moreover, in [26,22], the authors studied the C-S model with regular short range interaction and constructed a sufficient and necessary condition for the emergence of mono-cluster and multi-cluster formation, respectively. While, it is not clear wether the similar results can be obtained in the singular case. Therefore, may we address three natural questions as follows,• (Q1) Can we derive the uniqueness of the solution to the C-S model (1.1) with singular communication (1.2)? Moreover, can we find the sufficient and necessary condition for emergence of finite time flocking in N particle system?• (Q2) Can we construct the uniform-in-time lower bound between two particles of the C-S model with singular interaction, so that the collision avoidance occurs when time tends to infinity and asymptotical equilibrium state can be constructed?• (Q3) Can we obtain the sufficient and necessary condition for the emergence of mono-cluster and multi-cluster...

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“…Comparing to the regular case [26,22], the construction of the first order equation in the present paper is a little different due to the singularity of the communication function.…”

Section: Preliminarymentioning

confidence: 83%

Section: Higher Order Singularitymentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Complete classification of the asymptotical behavior for singular C-S model on the real line

Zhang

Zhu

2020

Journal of Differential Equations

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Emergence of time‐asymptotic flocking for a general Cucker–Smale‐type model with distributed time delays

Liu

Wang

2020

Math Methods in App Sciences

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In this paper, we analyze the asymptotic flocking behavior for a Cucker–Smale‐type model with a disturbed delayed coupling, where delays are information processing and reactions of individuals. By constructing a new energy functional combined with L2‐analysis, we obtain the uniform bound of particle velocities, and then by establishing a system of dissipative differential inequalities together with L∞‐analysis, we prove the existence of asymptotic flocking solutions when the maximum value of time delays is sufficiently small.

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Flocking control for Cucker–Smale model under denial‐of‐service attacks

Shi

Cheng

et al. 2022

Intl J Robust & Nonlinear

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In this article, the issue of flocking control for Cucker–Smale model under denial‐of‐service (DoS) attacks is discussed. It is assumed that there is an active stage and an asleep stage for each period of the DoS. Malicious attacks partially or completely destroy the network communication links in the active stages, and replenish energy without any attacks in the dormant stages. By constructing the agents' position and velocity errors, the C–S model under DoS attacks is equivalently transformed into error systems. By applying the approach of products convergence for infinite sub‐stochastic matrices, a sufficient condition for producing the flocking behavior is obtained. An upper bound of relative errors for agents' final positions is established, which depends on initial states, the topology structure and the weight function. Finally, simulation experiments verify the effectiveness of the theoretical results.

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Complete Cluster Predictability of the Cucker–Smale Flocking Model on the Real Line

Cited by 68 publications

References 38 publications

Complete classification of the asymptotical behavior for singular C-S model on the real line

Complete classification of the asymptotical behavior for singular C-S model on the real line

Emergence of time‐asymptotic flocking for a general Cucker–Smale‐type model with distributed time delays

Flocking control for Cucker–Smale model under denial‐of‐service attacks

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