Abstract. We present a simple proof on the formation of flocking to the Cucker-Smale system based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to flocking state will occur. For large particle systems, we give a rigorous justification for the mean-field limit from the many particle Cucker-Smale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity.Key words. Flocking, swarming, emergence, self-driven particles system, autonomous agents, Vlasov equation, Lyapunov functional, measure valued solution, Kantorovich-Rubinstein distance.AMS subject classifications. Primary 92C17; secondary 82C22, 82C40. IntroductionCollective self-driven synchronized motion of autonomous agents appears in many applications ranging from animal herding to the emergence of common languages in primitive societies [1,9,10,11,12,14,16,18,19,20,2,3]. The word flocking in this paper refers to general phenomena where autonomous agents reach a consensus based on limited environmental information and simple rules. In the seminal work of Cucker and Smale [2,3], they postulated a model for the flocking of birds, and verified the convergence to a consensus (the same velocity) depending on the spatial decay of the communication rate between autonomous agents.In this paper, we present a simple and complete analysis of the flocking to the Cucker-Smale system (in short C-S system) using the explicit Lyapunov functional approach. In particular we improve the flocking estimates of the C-S system for regular and algebraically decaying communication rates [2,3] in two ways. First, we present flocking estimates for general communication rates which can be singular when two particles are very close enough. Secondly, we remove the conditional assumption in [2,3] on the initial configuration in critical case, where the communication rate behaves like |x| −1 , as |x| → ∞. We show that the standard deviations of particle phase-space positions are dominated by the system of dissipative differential inequalities (SDDI):

We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior without closure of higher moments.

We present a strong asymptotic stochastic flocking estimate for the stochastically perturbed Cucker–Smale model. We characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states. When the strength of multiplicative noises is sufficiently large, we show that the strong stochastic flocking occurs even for negative communication weights.

We present a class of initial-configurations for the Cucker-Smale flocking type models leading no finite-time collisions between particles. For this class of initial-configurations, the global existence of smooth solutions to the Cucker-Smale model with singular communication weights is established. This also generalizes the earlier flocking estimates for the Cucker-Smale type models with regular communication weights.