Hierarchical leadership and collision avoidance in flocking

Abstract

Flocking is a phenomenon in which a number of interacting agents, using only limited information and simple rules, organize into an ordered motion without a central direction. In recent years, this phenomenon has attracted more and more attention from researchers in biology, physics, robotics, control theory and computer science. Recently, a flocking model was introduced by Cucker and Smale together with a result establishing the convergence to flocking depending on initial conditions and system parameters. This thesis investigates some problems related to the Cucker-Smale model. The main contributions of the thesis can be summarized as follows. The chapter 2 of the thesis considers the Cucker-Smale model under hierarchical leadership. For the original Cucker-Smale model, Cucker and Smale established unconditional convergence to a common velocity provided the interaction between the agents was strong enough, and conditional convergence otherwise. The strength of the interaction is measured by a parameter β ≥ 0 and the critical value at which the unconditional convergence stops holding is β = 1/2. The Cucker-Smale model was extended by Shen to allow for a hierarchical leadership structure among the agents and similar convergence results were proved. But, for discrete time, unconditional convergence was proved only for β < 1/2k (k being the number of agents). In this chapter we improve this result by showing that unconditional convergence holds indeed for β < 1/2. The chapter 3 of the thesis considers the Cucker-Smale model with collision avoid ance. We extend the Cucker-Smale model by adding to it a repelling force between agents and show that, for this modified model, convergence to flocking is established while, in addition, avoidance of collisions (i.e., the respect of a minimal distance between agents) is ensured. The chapter 4 of the thesis proves a general result of collision-avoiding flocking. The underlying model allows several forms of coupling forces and the main result ensures collision-avoiding flocking provided the initial state of the population does not show simultaneously very different velocities, very spread positions or very close agents. The results of this thesis have already been published (or accepted for publication) in academic journals. Thus, the contents in Chapter 2 appear in [10], those of Chapter 3 in [11], and those of Chapter 4 are to appear in [12].

Date of Award	15 Jul 2011
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Felipe CUCKER (Supervisor)