Consensus stabilization in stochastic multi-agent systems with Markovian switching topology, noises and delay

This paper investigates the consensus stabilization problem of stochastic multi-agent systems with noises, Markovian switching topology, and communication delays. To solve the consensus problem of the group of interacting agents, it is supposed to use the stochastic approximation type algorithm with the step-size non-decreasing to zero. For stochastic approximation based consensus algorithms with switching topologies, the existing convergence analysis methods may be difficult to guarantee for switching digraphs because these techniques heavily relies on quadratic Lyapunov functions. To overcome the inherent limitations of the existing methods, we develop a new ergodicity approach for backward products of degenerating stochastic matrices via a discrete time dynamical system approach to analyze the consensus stabilization problem of stochastic multi-agent systems with noises, Markovian switching topology, and communication delays; and we attain necessary and sufficient condition of the consensus stabilization. Our approach does not require the double stochasticity condition typically assumed for the existence of a quadratic Lyapunov function, and provides an effective tool for analyzing consensus problems via the existing theory of stochastic matrices and ergodicity of backward products. Finally, numerical simulation is shown to demonstrate theoretical results.