Flocking under hierarchical leadership with a free-will leader

Recently, a model for flocking was introduced by Cucker and Smale together with a proof of convergence. This proof established unconditional convergence to flocking (i.e., to a common velocity), provided the interaction between agents was strong enough and conditional convergence otherwise. The strength of the interaction is measured by a parameter , and the critical value at which unconditional convergence stops holding is =1/2. This 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, convergence result was only for the flock with an overall leader moving with a constant velocity. In this note, we establish convergence result for the flock with a free-will leader. Copyright (c) 2012 John Wiley & Sons, Ltd.