Analysis Of Cucker-Smale Flocking Under Different Topology Structues

Roughly speaking,the terminology "flocking" represents the phenomena that autonomous agents,using only limited environmental information and simple rules,organize into an ordered motion,e.g.,the flock of birds fly in a formation with almost the same velocity.These collective motions have gained increasing interest from the research communities in biology and ecology,and have been extensively studied in science and engineering communities.Among many related models,we mainly study the celebrated Cucker-Smale model(in short C-S model)in this thesis.This model was proposed to describe the mechanism of collective flocking motion without central direction.And it has significant potential applications,such as unmanned aerial vehicles and formation control of robot.This thesis contains the study of C-S flocking with randomly failed interactions,multi-cluster flocking behavior of the hierarchical C-S model and flocking of C-S model with intrinsic dynamics.The main results are listed as follows:Firstly,we consider the C-S model under influenced functions with random failures.More precisely,each pair of neighboring agents can fail to interact with a certain probability.Two cases are considered: undirected connected C-S model and C-S model under rooted leadership.Our results show that unconditional flocking for long-range communication rate is still true.For short-range communication rate,conditional flocking can occur and conditions only depend on initial data.These results suggest that the C-S flocking system under various topologies can endure random failures in influenced functions.Secondly,we investigate the multi-cluster flocking behavior of the hierarchical C-S model parameterized by a constant β measuring the strength of the interaction between agents.The previous studies showed that when 0≤β < 1/2 unconditional flocking would occur,while for β≥1/2 flocking would occur provided the initial data satisfy some given conditions.We show that unconditional flocking would occur when β = 1/2.For β > 1/2,we give a complete understanding for the asymptotic multi-cluster flocking when the initial positions and velocities have the ordering relations,which means that we can specify how many clusters emerge and which agents are in the same cluster by computing the initial data.Finally,we consider the conditional flocking of modified continuous-and discretetime C-S models that every agent has its own intrinsic dynamics with Lipschitz property.The dynamics of the models are governed by the interplay between agents’ own intrinsic dynamics and C-S coupling dynamics.Based on the explicit construction of Lyapunov functionals,we show that conditional flocking would occur.And then we study the relationship between the Lipschitz constant L of the intrinsic dynamics and exponent β when flocking occurs.At last,we give two examples to show flocking might not occur for enough large L or unconditionally for β>0.