Theorems about Ergodicity and Class-Ergodicity of Chains with Applications in Known Consensus Models

摘要

In a multi-agent system, unconditional (multiple) consensus is the propertyof reaching to (multiple) consensus irrespective of the instant and values atwhich states are initialized. For linear algorithms, occurrence ofunconditional (multiple) consensus turns out to be equivalent to (class-)ergodicity of the transition chain (A_n). For a wide class of chains, chainswith so-called balanced asymmetry property, necessary and sufficient conditionsfor ergodicity and class-ergodicity are derived. The results are employed toanalyze the limiting behavior of agents' states in the JLM model, the Krausemodel, and the Cucker-Smale model. In particular, unconditional single ormultiple consensus occurs in all three models. Moreover, a necessary andsufficient condition for unconditional consensus in the JLM model and asufficient condition for consensus in the Cucker-Smale model are obtained.