TY - JOUR
T1 - On the critical exponent for flocks under hierarchical leadership
AU - Cucker, Felipe
AU - Dong, Jiu-Gang
PY - 2009/8
Y1 - 2009/8
N2 - Very recently, a model for flocking was introduced by Cucker and Smale together with a proof of convergence. This proof established unconditional convergence 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 β <0 and the critical value at which unconditional convergence stops holding is β = 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, unconditional convergence was proved only for $\beta <\frac{1}{2k}$ (k being the number of agents). In this note we improve on this result showing that unconditional convergence holds indeed for β <1/2. © 2009 World Scientific Publishing Company.
AB - Very recently, a model for flocking was introduced by Cucker and Smale together with a proof of convergence. This proof established unconditional convergence 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 β <0 and the critical value at which unconditional convergence stops holding is β = 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, unconditional convergence was proved only for $\beta <\frac{1}{2k}$ (k being the number of agents). In this note we improve on this result showing that unconditional convergence holds indeed for β <1/2. © 2009 World Scientific Publishing Company.
KW - Hierarchical leadership
KW - Unconditional convergence.
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-69849097029&origin=recordpage
U2 - 10.1142/S0218202509003851
DO - 10.1142/S0218202509003851
M3 - RGC 21 - Publication in refereed journal
VL - 19
SP - 1391
EP - 1404
JO - Mathematical Models and Methods in Applied Sciences
JF - Mathematical Models and Methods in Applied Sciences
SN - 0218-2025
IS - SUPPL. 1
ER -