Citation: Jan Haskovec, Ioannis Markou. Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5651-5671. doi: 10.3934/mbe.2020304
[1] | S. Camazine, J. L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz, E. Bonabeau, Self-Organization in Biological Systems, Princeton University Press, Princeton, NJ, 2001. |
[2] | T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. |
[3] | P. Krugman, The Self Organizing Economy, Blackwell Publishers, 1995. |
[4] | G. Naldi, L. Pareschi, G. Toscani, Mathematical Modeling of Collective behaviour in Socio-Economic and Life Sciences, in Series: Modelling and Simulation in Science and Technology, Birkhäuser, 2010. |
[5] | H. Hamman, Swarm Robotics: A Formal Approach, Springer, 2018. |
[6] | A. Jadbabaie, J. Lin, A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001. |
[7] | N. Bellomo, P. Degond, E. Tamdor, Active Particles, Volume I. Advances in Theory, Models, and Applications, Series: Modelling and Simulation in Science, Engineering and Technology, Birkhäuser, 2017. |
[8] | Y.-P. Choi, S.-Y. Ha, Z. Li., Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles, Volume 1. Modeling and Simulation in Science, Engineering and Technology (eds N. Bellomo, P. Degond, E. Tadmor), Birkhäuser, 2017. |
[9] | J.-G. Dong, S.-Y. Ha, D. Kim, Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5569-5596. |
[10] | S.-Y. Ha, J. Kim, J. Park, X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365. |
[11] | D. Kalise, J. Peszek, A. Peters, Y.-P. Choi, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981. |
[12] | Z. Liu, X. Li, Y. Liu, X. Wang, Asymptotic flocking behavior of the general finite-dimensional Cucker-Smale model with distributed time delays, Bull. Malays. Math. Sci. Soc., 2020. Available from: https://doi.org/10.1007/s40840-020-00917-8. |
[13] | I. Markou, Collision avoiding in the singular Cucker-Smale model with nonlinear velocity couplings, Discrete Contin. Dyn. Syst., 38 (2018), 5245-5260. |
[14] | L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods, Oxford University Press, 2014. |
[15] | C. Pignotti, E. Trelat, Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays, Commun. Math. Sci., 16 (2018), 2053-2076. |
[16] | C. Pignotti, I. Reche Vallejo, Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, J. Math. Anal. Appl., 464 (2018), 1313-1332. |
[17] | C. Pignotti, I. Reche Vallejo, Asymptotic analysis of a Cucker-Smale system with leadership and distributed delay, in Trends in Control Theory and Partial Differential Equations (eds. F. Alabau-Boussouira, F. Ancona, A. Porretta, C. Sinestrari), Springer INdAM Series, vol. 32 (2019). |
[18] | F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. |
[19] | F. Cucker, S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227. |
[20] | S.-Y. Ha, E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435. |
[21] | S.-Y. Ha, J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. |
[22] | J. A. Carrilo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. |
[23] | I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Science Publications, Clarendon Press, Oxford, 1991. |
[24] | J. Haskovec, I. Markou, Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime, Kinet. Relat. Models, 13 (2020), 795-813. |
[25] | J. Haskovec, Exponential decay for negative feedback loop with distributed delay, Appl. Math. Lett., 107 (2020), 106419. |
[26] | Y. Liu, J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, J. Math. Anal. Appl., 415 (2014), 53-61. |
[27] | Y.-P. Choi, J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033. |
[28] | Y.-P. Choi, J. Haskovec, Hydrodynamic Cucker-Smale model with normalized communication weights and time delay, SIAM J. Math. Anal., 51 (2019), 2660-2685. |
[29] | R. Erban, J. Haskovec, Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557. |
[30] | I. Barbalat, Systèmes d'équations différentielles d'oscillations nonlinéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. |
[31] | Y.-P. Choi, Z. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56. |
[32] | H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer New York Dordrecht Heidelberg London, 2011. |