American Institute of Mathematical Sciences

Advanced
June  2014, 9(2): 335-351. doi: 10.3934/nhm.2014.9.335

A revisit to the consensus for linearized Vicsek model under joint rooted leadership via a special matrix

Zhuchun Li 1, , Xiaoping Xue 1, and  Seung-Yeal Ha 2,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747

Received  April 2013 Revised  March 2014 Published  July 2014

We address the exponential consensus problem for the linearized Vicsek model which was introduced by Jadbabaie et al. in [10] under a joint rooted leadership via the $(sp)$ matrices. This model deals with self-propelled particles moving in the plane with the same speed but different headings interacting with neighboring agents by a linear relaxation rule. When the time-varying switching topology of the neighbor graph satisfies some weak connectivity condition, namely, `` joint connectivity condition'' in the spatial-temporal domain, it is well known that the consensus for the linearized Vicsek model can be achieved asymptotically. In this paper, we extend the theory of $(sp)$ matrices and apply it to revisit this asymptotic consensus problem and give an explicit estimate on the maximum Lyapunov exponent, when the underlying network topology satisfies the joint rooted leadership which is directed and non-symmetric.
Keywords: The Vicsek model, $(sp)$ matrix, joint rooted leadership, maximum Lyapunov exponent., consensus.
Mathematics Subject Classification: Primary: 92D25, 39A12; Secondary: 92C4.
Citation: Zhuchun Li, Xiaoping Xue, Seung-Yeal Ha. A revisit to the consensus for linearized Vicsek model under joint rooted leadership via a special matrix. Networks & Heterogeneous Media, 2014, 9 (2) : 335-351. doi: 10.3934/nhm.2014.9.335
References:
[1]

N. Barabanov, Lyapunov exponent and joint spectral radius: Some known and new results,, in Proc. 44th Conf. Decision and Control, (2005), 2332.  doi: 10.1109/CDC.2005.1582510.  Google Scholar

[2]

D. P. Bertsekas and J. N. Tsitsiklis, Comments on "Coordination of groups of mobile autonomous agents using nearest neighbor rules'',, IEEE Trans. Automat. Control, 52 (2007), 968.  doi: 10.1109/TAC.2007.895885.  Google Scholar

[3]

M. Cao, A. S. Morse and B. D. O. Anderson, Reaching a consensus in a dynamically changing environment: A graphic approach,, SIAM J. Control Optim., 47 (2008), 575.  doi: 10.1137/060657005.  Google Scholar

[4]

M. Cao, A. S. Morse and B. D. O. Anderson, Reaching a consensus in a dynamically changing environment: Vonvergence rates, meansurement delays, and asynchronous events,, SIAM J. Control Optim., 47 (2008), 601.  doi: 10.1137/060657029.  Google Scholar

[5]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[6]

R. Diestel, Graph Theory,, Graduate Texts in Mathematics, (1997).   Google Scholar

[7]

L. X. Gao and D. Z. Cheng, Comment on ‘Coordination of groups of mobile agents using nearest neighbor rules',, IEEE Trans. Autom. Control, 50 (2005), 1913.  doi: 10.1109/TAC.2005.858635.  Google Scholar

[8]

S.-Y. Ha, Z. Li, M. Slemrod and X. Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit,, to appear in Quart. Appl. Math., ().   Google Scholar

[9]

J. M. Hendrickx, Graphs and Networks for the Analysis of Autonomous Agent Systems,, Ph.D Thesis, (2008).   Google Scholar

[10]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile agents using nearest neighbor rules,, IEEE Trans. Autom. Control, 48 (2003), 988.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[11]

Z. Li, S.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership,, Math. Mod. Meth. Appl. Sci., 24 (2014), 1389.  doi: 10.1142/S0218202514500043.  Google Scholar

[12]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies,, SIAM J. Appl. Math., 70 (2010), 3156.  doi: 10.1137/100791774.  Google Scholar

[13]

Z.-X. Liu and L. Guo, Connectivity and synchronization of Vicsek model,, Sci. China Ser. F-Inf. Sci., 51 (2008), 848.  doi: 10.1007/s11432-008-0077-2.  Google Scholar

[14]

L. Moreau, Stability of multiagent systems with time-dependent communication links,, IEEE Trans. Autom. Control, 50 (2005), 169.  doi: 10.1109/TAC.2004.841888.  Google Scholar

[15]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett. 75 (1995), 75 (1995), 1226.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[16]

X. Xue and L. Guo, A kind of nonnegative matrices and its application on the stability of discrete dynamical systems,, J. Math. Anal. Appl., 331 (2007), 1113.  doi: 10.1016/j.jmaa.2006.09.053.  Google Scholar

[17]

X. Xue and Z. Li, Asymptotic stability analysis of a kind of switched positive linear discrete systems,, IEEE Trans. Autom. Control, 55 (2010), 2198.  doi: 10.1109/TAC.2010.2052144.  Google Scholar

show all references

References:
[1]

N. Barabanov, Lyapunov exponent and joint spectral radius: Some known and new results,, in Proc. 44th Conf. Decision and Control, (2005), 2332.  doi: 10.1109/CDC.2005.1582510.  Google Scholar

[2]

D. P. Bertsekas and J. N. Tsitsiklis, Comments on "Coordination of groups of mobile autonomous agents using nearest neighbor rules'',, IEEE Trans. Automat. Control, 52 (2007), 968.  doi: 10.1109/TAC.2007.895885.  Google Scholar

[3]

M. Cao, A. S. Morse and B. D. O. Anderson, Reaching a consensus in a dynamically changing environment: A graphic approach,, SIAM J. Control Optim., 47 (2008), 575.  doi: 10.1137/060657005.  Google Scholar

[4]

M. Cao, A. S. Morse and B. D. O. Anderson, Reaching a consensus in a dynamically changing environment: Vonvergence rates, meansurement delays, and asynchronous events,, SIAM J. Control Optim., 47 (2008), 601.  doi: 10.1137/060657029.  Google Scholar

[5]

F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[6]

R. Diestel, Graph Theory,, Graduate Texts in Mathematics, (1997).   Google Scholar

[7]

L. X. Gao and D. Z. Cheng, Comment on ‘Coordination of groups of mobile agents using nearest neighbor rules',, IEEE Trans. Autom. Control, 50 (2005), 1913.  doi: 10.1109/TAC.2005.858635.  Google Scholar

[8]

S.-Y. Ha, Z. Li, M. Slemrod and X. Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit,, to appear in Quart. Appl. Math., ().   Google Scholar

[9]

J. M. Hendrickx, Graphs and Networks for the Analysis of Autonomous Agent Systems,, Ph.D Thesis, (2008).   Google Scholar

[10]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile agents using nearest neighbor rules,, IEEE Trans. Autom. Control, 48 (2003), 988.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[11]

Z. Li, S.-Y. Ha and X. Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership,, Math. Mod. Meth. Appl. Sci., 24 (2014), 1389.  doi: 10.1142/S0218202514500043.  Google Scholar

[12]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies,, SIAM J. Appl. Math., 70 (2010), 3156.  doi: 10.1137/100791774.  Google Scholar

[13]

Z.-X. Liu and L. Guo, Connectivity and synchronization of Vicsek model,, Sci. China Ser. F-Inf. Sci., 51 (2008), 848.  doi: 10.1007/s11432-008-0077-2.  Google Scholar

[14]

L. Moreau, Stability of multiagent systems with time-dependent communication links,, IEEE Trans. Autom. Control, 50 (2005), 169.  doi: 10.1109/TAC.2004.841888.  Google Scholar

[15]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles,, Phys. Rev. Lett. 75 (1995), 75 (1995), 1226.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[16]

X. Xue and L. Guo, A kind of nonnegative matrices and its application on the stability of discrete dynamical systems,, J. Math. Anal. Appl., 331 (2007), 1113.  doi: 10.1016/j.jmaa.2006.09.053.  Google Scholar

[17]

X. Xue and Z. Li, Asymptotic stability analysis of a kind of switched positive linear discrete systems,, IEEE Trans. Autom. Control, 55 (2010), 2198.  doi: 10.1109/TAC.2010.2052144.  Google Scholar

[1]

Victor Kozyakin. Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 143-158. doi: 10.3934/dcdsb.2010.14.143
[2]

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
[3]

Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048
[4]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[5]

Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249
[6]

Jairo Bochi, Michal Rams. The entropy of Lyapunov-optimizing measures of some matrix cocycles. Journal of Modern Dynamics, 2016, 10: 255-286. doi: 10.3934/jmd.2016.10.255
[7]

Ai-Guo Wu, Ying Zhang, Hui-Jie Sun. Parametric Smith iterative algorithms for discrete Lyapunov matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019093
[8]

Ewa Girejko, Luís Machado, Agnieszka B. Malinowska, Natália Martins. On consensus in the Cucker–Smale type model on isolated time scales. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 77-89. doi: 10.3934/dcdss.2018005
[9]

Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
[10]

Shichen Zhang, Jianxiong Zhang, Jiang Shen, Wansheng Tang. A joint dynamic pricing and production model with asymmetric reference price effect. Journal of Industrial & Management Optimization, 2019, 15 (2) : 667-688. doi: 10.3934/jimo.2018064
[11]

Li Deng, Wenjie Bi, Haiying Liu, Kok Lay Teo. A multi-stage method for joint pricing and inventory model with promotion constrains. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020097
[12]

Julia Amador, Mariajesus Lopez-Herrero. Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3137-3151. doi: 10.3934/dcdsb.2017211
[13]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
[14]

Hyukjin Lee, Cheng-Chew Lim, Jinho Choi. Joint backoff control in time and frequency for multichannel wireless systems and its Markov model for analysis. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1083-1099. doi: 10.3934/dcdsb.2011.16.1083
[15]

Erica M. Rutter, Yang Kuang. Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1001-1021. doi: 10.3934/dcdsb.2017050
[16]

Luciano Pandolfi. Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-11. doi: 10.3934/dcdss.2020090
[17]

Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031
[18]

Marco Scianna, Luigi Preziosi, Katarina Wolf. A Cellular Potts model simulating cell migration on and in matrix environments. Mathematical Biosciences & Engineering, 2013, 10 (1) : 235-261. doi: 10.3934/mbe.2013.10.235
[19]

Zhuchun Li. Effectual leadership in flocks with hierarchy and individual preference. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3683-3702. doi: 10.3934/dcds.2014.34.3683
[20]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences