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Finite-time flocking of an infinite set of Cucker-Smale particles with sublinear velocity couplings

Seung-Yeal Ha, Xinyu Wang, Fanqin Zeng

TL;DR

This work analyzes finite-time flocking for an infinite-particle Cucker-Smale system with sublinear velocity coupling under directed sender networks and switching topologies. By combining a componentwise velocity-diameter framework with a contraction-dissipation analysis, it yields explicit alignment-time estimates that remain valid as the number of agents grows and extends to switching network structures. A key finding is that non-integrable communication weights guarantee unconditional finite-time flocking, while the derived bounds depend primarily on the total effective mass rather than the population size. The results are complemented by numerical simulations confirming rapid alignment in both fixed and switching-network scenarios and by a global existence theory enabled via Schauder-Tychonoff fixed point arguments.

Abstract

We study finite-time flocking for an infinite set of Cucker-Smale particles with sublinear velocity coupling under fixed and switching sender networks. For this, we use a component-wise diameter framework and exploit sub-linear dissipation mechanisms, and derive sufficient conditions for finite-time flocking equipped with explicit alignment-time estimate. For a fixed sender network, we establish component-wise finite-time flocking results under both integrable and non-integrable communication weights. When communication weight function is non-integrable, finite-time flocking is guaranteed for any bounded initial configuration. We further extend the flocking analysis to switching sender networks and show that finite-time flocking persists under mild assumptions on the cumulative influence of time-varying sender weights. The proposed framework is also applicable to both finite and infinite systems, and it yields alignment-time estimates that do not depend on the number of agents.

Finite-time flocking of an infinite set of Cucker-Smale particles with sublinear velocity couplings

TL;DR

This work analyzes finite-time flocking for an infinite-particle Cucker-Smale system with sublinear velocity coupling under directed sender networks and switching topologies. By combining a componentwise velocity-diameter framework with a contraction-dissipation analysis, it yields explicit alignment-time estimates that remain valid as the number of agents grows and extends to switching network structures. A key finding is that non-integrable communication weights guarantee unconditional finite-time flocking, while the derived bounds depend primarily on the total effective mass rather than the population size. The results are complemented by numerical simulations confirming rapid alignment in both fixed and switching-network scenarios and by a global existence theory enabled via Schauder-Tychonoff fixed point arguments.

Abstract

We study finite-time flocking for an infinite set of Cucker-Smale particles with sublinear velocity coupling under fixed and switching sender networks. For this, we use a component-wise diameter framework and exploit sub-linear dissipation mechanisms, and derive sufficient conditions for finite-time flocking equipped with explicit alignment-time estimate. For a fixed sender network, we establish component-wise finite-time flocking results under both integrable and non-integrable communication weights. When communication weight function is non-integrable, finite-time flocking is guaranteed for any bounded initial configuration. We further extend the flocking analysis to switching sender networks and show that finite-time flocking persists under mild assumptions on the cumulative influence of time-varying sender weights. The proposed framework is also applicable to both finite and infinite systems, and it yields alignment-time estimates that do not depend on the number of agents.
Paper Structure (15 sections, 14 theorems, 203 equations, 2 figures)

This paper contains 15 sections, 14 theorems, 203 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preparatory estimates
  4. Contraction of component velocity diameters
  5. Finite-time flocking I: fixed sender network
  6. Finite-time flocking II: switching sender networks
  7. Switching sender networks
  8. Finite-time flocking
  9. Numerical simulations
  10. Conclusion
  11. Global existence of solutions to infinite-particle systems
  12. A global existence of \ref{['ICS']}
  13. Preparatory lemmas
  14. Proof of Theorem \ref{['T2.1']}
  15. Compact embedding

Key Result

Theorem 2.1

Suppose that $\psi:[0,\infty)\to[0,\infty)$ and $\Gamma$ satisfy the framework $({\mathcal{A}}_1) - ({\mathcal{A}}_2)$. Then, the Cauchy problem ICS admits a global solution $(x,v)$ such that

Figures (2)

  • Figure 1: Evolution of $\mathcal{D}_v$ for $\alpha\in\{0.1,0.4,0.8\}$
  • Figure 2: Velocity trajectories $v_i(t)$ with the weighted mean $v_c(t)$ ($N=50$, $\alpha=0.6$, $\beta=0.25$, $T=15$). Switching occurs at integer times.

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 20 more