TY - JOUR
T1 - Nonexistence of the Asymptotic Flocking in the Cucker−Smale Model With Short Range Communication Weights
AU - Yin, Xiuxia
AU - Gao, Zhiwei
AU - Chen, Zili
AU - Fu, Yichuan
N1 - Funding Information: The work of Xiuxia Yin was supported by NSFC under Grant 61963028 and the Natural Science Foundation of Jiangxi Province under Grant 20192BAB207023. The work of Zhiwei Gao was supported by the Alexander von Humboldt Foundation under Grant GRO/1117303 STP. The work of Zili Chen was supported by NSFC under Grant 11961046.
PY - 2022/2
Y1 - 2022/2
N2 - For the long range communicated Cucker-Smale model, the asymptotic flocking exists for any initialcondition. It is noted that, for the short range communicated Cucker-Smale model, the asymptotic flocking only holds for very restricted initial conditions. In this case, the nonexistence of the asymptotic flocking has been frequently observed in numerical simulations, however, the theoretical results are far from perfect. In this note, we first point out that the nonexistence of the asymptotic flocking is equivalent to the unboundedness of the second order space moment, i.e., t|x i(t)-x j(t)|2=. Furthermore, by taking the second derivative and then integrating, we establish a new and key equality about this moment. At last, we use this equality and relevant technical lemmas to deduce a general sufficient condition to the nonexistence of the asymptotic flocking.
AB - For the long range communicated Cucker-Smale model, the asymptotic flocking exists for any initialcondition. It is noted that, for the short range communicated Cucker-Smale model, the asymptotic flocking only holds for very restricted initial conditions. In this case, the nonexistence of the asymptotic flocking has been frequently observed in numerical simulations, however, the theoretical results are far from perfect. In this note, we first point out that the nonexistence of the asymptotic flocking is equivalent to the unboundedness of the second order space moment, i.e., t|x i(t)-x j(t)|2=. Furthermore, by taking the second derivative and then integrating, we establish a new and key equality about this moment. At last, we use this equality and relevant technical lemmas to deduce a general sufficient condition to the nonexistence of the asymptotic flocking.
KW - Asymptotic flocking
KW - Communication weights
KW - Cucker-Smale (C-S) model
KW - Multiagent system
UR - http://www.scopus.com/inward/record.url?scp=85102306904&partnerID=8YFLogxK
U2 - 10.1109/TAC.2021.3063951
DO - 10.1109/TAC.2021.3063951
M3 - Article
AN - SCOPUS:85102306904
SN - 0018-9286
VL - 67
SP - 1067
EP - 1072
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 2
ER -