Applications of non-autonomous discrete dynamical systems into nonlinear consensus problems

Historically, an idea of reaching consensus through repeated averaging was introduced by DeGroot (see [1, 3]) for a structured time-invariant and synchronous environment. Since that time, the consensus which is the most ubiquitous phenomenon of multi-agent systems becomes popular in various scientific communities, such as biology, physics, control engineering and social science. Roughly speaking, a trajectory of a row-stochastic matrix presents DeGroot’s model of the structured time-invariant synchronous environment. In [2], Chatterjee and Seneta considered a generalization of DeGroot’s model for the structured time-varying synchronous environment. A trajectory of a sequence of row-stochastic matrices (a non-homogeneous Markov chain) presents the Chatterjee- Seneta model of the structured time-varying synchronous environment. In this paper, we shall consider a nonlinear model of the structured time-varying synchronous environment which generalizes both DeGroot’s and the Chatterjee-Seneta models. Namely, by means of multidimensional stochastic hypermatrices, we present an opinion sharing dynamics of the multi-agent system as a trajectory of non-autonomous polynomial stochastic operators (nonlinear Markov operators). We show that the multiagent system eventually reaches to a consensus under suitable conditions