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

Zhuchun Li; Xiaoping Xue; Seung-Yeal Ha

doi:10.3934/nhm.2014.9.335

Article Contents

2014, Volume 9, Issue 2: 335-351. Doi: 10.3934/nhm.2014.9.335

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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
2.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747

Received: March 31, 2013

Revised: February 28, 2014

Published: May 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 92D25, 39A12; Secondary: 92C42.

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References

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