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

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.

💡 Research Summary

The paper investigates finite‑time flocking for an infinite‑particle Cucker‑Smale (CS) system equipped with a sub‑linear velocity coupling and directed “sender” interaction graphs, both fixed and switching. The authors first establish global existence of classical solutions for initial data in the space B = ℓ^∞(ℝ^d)×ℓ^∞(ℝ^d). By exploiting the fact that the mass profile {m_i} satisfies Σ_i m_i = 1, they obtain a compact embedding B↪H where H = ℓ^2_m(ℝ^d)×ℓ^2_m(ℝ^d). This compactness allows the use of the Schauder‑Tychonoff fixed‑point theorem to construct a global solution (x(t), v(t))∈C(