# American Institute of Mathematical Sciences

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

 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.
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 and 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, Seville, Spain, 2005, 2332-2337. doi: 10.1109/CDC.2005.1582510. [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), 968-969. doi: 10.1109/TAC.2007.895885. [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), 575-600. doi: 10.1137/060657005. [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), 601-623. doi: 10.1137/060657029. [5] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. [6] R. Diestel, Graph Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1997. [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), 1913-1916. doi: 10.1109/TAC.2005.858635. [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. [9] J. M. Hendrickx, Graphs and Networks for the Analysis of Autonomous Agent Systems, Ph.D Thesis, Université Catholique de Louvain, Department of Mathematical Engineering, February 2008. [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), 988-1001. doi: 10.1109/TAC.2003.812781. [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), 1389-1419. doi: 10.1142/S0218202514500043. [12] Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774. [13] Z.-X. Liu and L. Guo, Connectivity and synchronization of Vicsek model, Sci. China Ser. F-Inf. Sci., 51 (2008), 848-858. doi: 10.1007/s11432-008-0077-2. [14] L. Moreau, Stability of multiagent systems with time-dependent communication links, IEEE Trans. Autom. Control, 50 (2005), 169-182. doi: 10.1109/TAC.2004.841888. [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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. [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), 1113-1121. doi: 10.1016/j.jmaa.2006.09.053. [17] X. Xue and Z. Li, Asymptotic stability analysis of a kind of switched positive linear discrete systems, IEEE Trans. Autom. Control, 55 (2010), 2198-2203. doi: 10.1109/TAC.2010.2052144.

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##### References:
 [1] N. Barabanov, Lyapunov exponent and joint spectral radius: Some known and new results, in Proc. 44th Conf. Decision and Control, Seville, Spain, 2005, 2332-2337. doi: 10.1109/CDC.2005.1582510. [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), 968-969. doi: 10.1109/TAC.2007.895885. [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), 575-600. doi: 10.1137/060657005. [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), 601-623. doi: 10.1137/060657029. [5] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. [6] R. Diestel, Graph Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1997. [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), 1913-1916. doi: 10.1109/TAC.2005.858635. [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. [9] J. M. Hendrickx, Graphs and Networks for the Analysis of Autonomous Agent Systems, Ph.D Thesis, Université Catholique de Louvain, Department of Mathematical Engineering, February 2008. [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), 988-1001. doi: 10.1109/TAC.2003.812781. [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), 1389-1419. doi: 10.1142/S0218202514500043. [12] Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174. doi: 10.1137/100791774. [13] Z.-X. Liu and L. Guo, Connectivity and synchronization of Vicsek model, Sci. China Ser. F-Inf. Sci., 51 (2008), 848-858. doi: 10.1007/s11432-008-0077-2. [14] L. Moreau, Stability of multiagent systems with time-dependent communication links, IEEE Trans. Autom. Control, 50 (2005), 169-182. doi: 10.1109/TAC.2004.841888. [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), 1226-1229. doi: 10.1103/PhysRevLett.75.1226. [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), 1113-1121. doi: 10.1016/j.jmaa.2006.09.053. [17] X. Xue and Z. Li, Asymptotic stability analysis of a kind of switched positive linear discrete systems, IEEE Trans. Autom. Control, 55 (2010), 2198-2203. doi: 10.1109/TAC.2010.2052144.
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