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This work is concerned with stochastic consensus conditions of multi-agentsystems with both time-delays and measurement noises. For the case of additivenoises, we develop some necessary conditions and sufficient conditions forstochastic weak consensus by estimating the differential resolvent function fordelay equations. By the martingale convergence theorem, we obtain necessaryconditions and sufficient conditions for stochastic strong consensus. For thecase of multiplicative noises, we consider two kinds of time-delays, appearedin the measurement term and the noise term, respectively. We first show thatstochastic weak consensus with the exponential convergence rate impliesstochastic strong consensus. Then by constructing degenerate Lyapunovfunctional, we find the sufficient consensus conditions and show thatstochastic consensus can be achieved by carefully choosing the control gainaccording to the noise intensities and the time-delay in the measurement term.

Consensus conditions of continuous-time multi-agent systems with time-delays and measurement noises