Consensus and synchronization in discrete-time networks of
multi-agents with stochastically switching topologies and time
delays

Wenlian Lu; Fatihcan M. Atay; Jürgen Jost; Wenlian Lu; Fatihcan M. Atay; Jürgen Jost

doi:10.3934/nhm.2011.6.329

Networks and Heterogeneous Media

2011, Volume 6, Issue 2: 329-349. doi: 10.3934/nhm.2011.6.329

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Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays

1.
Center for Computational Systems Biology, Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433
2.
Max Planck Institute for Mathematics in theSciences, Inselstr. 22, 04103 Leipzig

Received: 01 August 2010 Revised: 01 February 2011
Primary: 93C05, 37H10, 15A51, 40A20; Secondary: 05C50, 60J10.

We analyze stability of consensus algorithms in networks of multi-agents with time-varying topologies and delays. The topology and delays are modeled as induced by an adapted process and are rather general, including i.i.d. topology processes, asynchronous consensus algorithms, and Markovian jumping switching. In case the self-links are instantaneous, we prove that the network reaches consensus for all bounded delays if the graph corresponding to the conditional expectation of the coupling matrix sum across a finite time interval has a spanning tree almost surely. Moreover, when self-links are also delayed and when the delays satisfy certain integer patterns, we observe and prove that the algorithm may not reach consensus but instead synchronize at a periodic trajectory, whose period depends on the delay pattern. We also give a brief discussion on the dynamics in the absence of self-links.
- Consensus,
- synchronization,
- delay,
- network of multi-agents,
- adapted process,
- switching topology
Citation: Wenlian Lu, Fatihcan M. Atay, Jürgen Jost. Consensus and synchronization in discrete-time networks ofmulti-agents with stochastically switching topologies and timedelays[J]. Networks and Heterogeneous Media, 2011, 6(2): 329-349. doi: 10.3934/nhm.2011.6.329

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Abstract

We analyze stability of consensus algorithms in networks of multi-agents with time-varying topologies and delays. The topology and delays are modeled as induced by an adapted process and are rather general, including i.i.d. topology processes, asynchronous consensus algorithms, and Markovian jumping switching. In case the self-links are instantaneous, we prove that the network reaches consensus for all bounded delays if the graph corresponding to the conditional expectation of the coupling matrix sum across a finite time interval has a spanning tree almost surely. Moreover, when self-links are also delayed and when the delays satisfy certain integer patterns, we observe and prove that the algorithm may not reach consensus but instead synchronize at a periodic trajectory, whose period depends on the delay pattern. We also give a brief discussion on the dynamics in the absence of self-links.

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