Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings

Previous Article
Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation
DCDS Home
This Issue
Next Article
Dispersive effects of the incompressible viscoelastic fluids

October 2018, 38(10): 5245-5260. doi: 10.3934/dcds.2018232

Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings

Institute of Applied and Computational Mathematics, Foundation for Research and Technology - Hellas, N. Plastira 100, Vassilika Vouton, GR - 700 13, Heraklion, Crete, Greece

Received March 2018 Revised June 2018 Published July 2018

Collision avoidance is an interesting feature of the Cucker-Smale (CS) model of flocking that has been studied in many works, e.g. [2,1,4,6,7,20,21,22]. In particular, in the case of singular interactions between agents, as is the case of the CS model with communication weights of the type $ψ(s) = s^{-α}$ for $α ≥ 1$, it is important for showing global well-posedness of the underlying particle dynamics. In [4], a proof of the non-collision property for singular interactions is given in the case of the linear CS model, i.e. when the velocity coupling between agents $i,j$ is $v_{j}-v_{i}$. This paper can be seen as an extension of the analysis in [4]. We show that particles avoid collisions even when the linear coupling in the CS system has been substituted with the nonlinear term $Γ(·)$ introduced in [12] (typical examples being $Γ(v) = v|v|^{2(γ -1)}$ for $γ ∈ (\frac{1}{2},\frac{3}{2})$), and prove that no collisions can happen in finite time when $α ≥ 1$. We also show uniform estimates for the minimum inter-particle distance, for a communication weight with expanded singularity $ψ_{δ}(s) = (s-δ)^{-α}$, when $α ≥ 2γ$, $δ ≥ 0$.

Keywords: Nonlinear Cucker-Smale system, emergent behavior, collision avoidance, singular communication weight.

Mathematics Subject Classification: 82C22, 92D50.

Citation: Ioannis Markou. Collision-avoiding in the singular Cucker-Smale model with nonlinear velocity couplings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5245-5260. doi: 10.3934/dcds.2018232

References:

[1]	M. Agueh, R. Illner and A. Richardson, Analysis and simulation of a refined flocking and swarming model of Cucker-Smale type, Kinet. Relat. Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1. Google Scholar
[2]	S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10. Google Scholar
[3]	J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model, ESAIM:Proc., 47 (2014), 17-35. doi: 10.1051/proc/201447002. Google Scholar
[4]	J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar
[5]	Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033. doi: 10.3934/krm.2017040. Google Scholar
[6]	F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129. doi: 10.1109/TAC.2011.2107113. Google Scholar
[7]	F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Contin. Dyn. Syst., 34 (2014), 1009-1020. doi: 10.3934/dcds.2014.34.1009. Google Scholar
[8]	F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002. Google Scholar
[9]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. Google Scholar
[10]	F. Cucker and S. Smale, On the mathematics of emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x. Google Scholar
[11]	R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557. doi: 10.1137/15M1030467. Google Scholar
[12]	S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113. Google Scholar
[13]	S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201, 19pp. doi: 10.1088/1751-8113/43/31/315201. Google Scholar
[14]	S.-Y. Ha, K. Lee and D. Levy, Emergence of time asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9. Google Scholar
[15]	S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar
[16]	S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar
[17]	M. Hauray and P.-E. Jabin, Particles approximations of Vlasov equations with singular forces: Propagation of chaos, Ann. Sci. Éc. Norm. Supér., 48 (2015), 891-940. doi: 10.24033/asens.2261. Google Scholar
[18]	P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661. Google Scholar
[19]	S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. Google Scholar
[20]	P. B. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308. doi: 10.1007/s00205-017-1160-x. Google Scholar
[21]	J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003. Google Scholar
[22]	J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686. doi: 10.1137/15M1009299. Google Scholar
[23]	J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719. doi: 10.1137/060673254. Google Scholar
[24]	T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73. doi: 10.1142/S0219530513500255. Google Scholar
[25]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

show all references

References:

[1]	M. Agueh, R. Illner and A. Richardson, Analysis and simulation of a refined flocking and swarming model of Cucker-Smale type, Kinet. Relat. Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1. Google Scholar
[2]	S. M. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643. doi: 10.4310/CMS.2012.v10.n2.a10. Google Scholar
[3]	J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model, ESAIM:Proc., 47 (2014), 17-35. doi: 10.1051/proc/201447002. Google Scholar
[4]	J. A. Carrillo, Y.-P. Choi, P. B. Mucha and J. Peszek, Sharp conditions to avoid collisions in singular Cucker-Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328. doi: 10.1016/j.nonrwa.2017.02.017. Google Scholar
[5]	Y.-P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033. doi: 10.3934/krm.2017040. Google Scholar
[6]	F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129. doi: 10.1109/TAC.2011.2107113. Google Scholar
[7]	F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Contin. Dyn. Syst., 34 (2014), 1009-1020. doi: 10.3934/dcds.2014.34.1009. Google Scholar
[8]	F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296. doi: 10.1016/j.matpur.2007.12.002. Google Scholar
[9]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. Google Scholar
[10]	F. Cucker and S. Smale, On the mathematics of emergence, Japan J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x. Google Scholar
[11]	R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM J. Appl. Math., 76 (2016), 1535-1557. doi: 10.1137/15M1030467. Google Scholar
[12]	S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113. Google Scholar
[13]	S.-Y. Ha, T. Ha and J.-H. Kim, Asymptotic dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A, 43 (2010), 315201, 19pp. doi: 10.1088/1751-8113/43/31/315201. Google Scholar
[14]	S.-Y. Ha, K. Lee and D. Levy, Emergence of time asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469. doi: 10.4310/CMS.2009.v7.n2.a9. Google Scholar
[15]	S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2. Google Scholar
[16]	S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar
[17]	M. Hauray and P.-E. Jabin, Particles approximations of Vlasov equations with singular forces: Propagation of chaos, Ann. Sci. Éc. Norm. Supér., 48 (2015), 891-940. doi: 10.24033/asens.2261. Google Scholar
[18]	P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661. Google Scholar
[19]	S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. Google Scholar
[20]	P. B. Mucha and J. Peszek, The Cucker-Smale equation: Singular communication weight, measure-valued solutions and weak-atomic uniqueness, Arch. Ration. Mech. Anal., 227 (2018), 273-308. doi: 10.1007/s00205-017-1160-x. Google Scholar
[21]	J. Peszek, Existence of piecewise weak solutions of a discrete Cucker-Smale's flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925. doi: 10.1016/j.jde.2014.06.003. Google Scholar
[22]	J. Peszek, Discrete Cucker-Smale flocking model with a weakly singular weight, SIAM J. Math. Anal., 47 (2015), 3671-3686. doi: 10.1137/15M1009299. Google Scholar
[23]	J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719. doi: 10.1137/060673254. Google Scholar
[24]	T. V. Ton, N. T. H. Linh and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73. doi: 10.1142/S0219530513500255. Google Scholar
[25]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[1]	Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic & Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028
[2]	Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks & Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017
[3]	Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic & Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039
[4]	Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic & Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040
[5]	Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019168
[6]	Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[7]	Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[8]	Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447
[9]	Laure Pédèches. Asymptotic properties of various stochastic Cucker-Smale dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115
[10]	Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419
[11]	Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116
[12]	Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062
[13]	Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223
[14]	Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial & Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443
[15]	Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
[16]	Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045
[17]	Chiun-Chuan Chen, Seung-Yeal Ha, Xiongtao Zhang. The global well-posedness of the kinetic Cucker-Smale flocking model with chemotactic movements. Communications on Pure & Applied Analysis, 2018, 17 (2) : 505-538. doi: 10.3934/cpaa.2018028
[18]	Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[19]	Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019072
[20]	Matthias Gerdts, René Henrion, Dietmar Hömberg, Chantal Landry. Path planning and collision avoidance for robots. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 437-463. doi: 10.3934/naco.2012.2.437

American Institute of Mathematical Sciences