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Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays. (English) Zbl 1258.93008

Summary: We analyze the 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.

MSC:

93A14 Decentralized systems
93E03 Stochastic systems in control theory (general)
93C55 Discrete-time control/observation systems
93C05 Linear systems in control theory
37H10 Generation, random and stochastic difference and differential equations
15B51 Stochastic matrices
40A20 Convergence and divergence of infinite products
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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