On asymptotic properties of continuous-time stochastic approximation type consensus protocols

For the multi-agent consensus with stochastic communication noises, the stochastic approximation type protocol is an effective method. In this paper, we will analyze the asymptotic properties of continuous-time stochastic approximation type consensus protocols in both mean square and probability one. We clarify the relationship between the convergence rate of the consensus error and a representative class of consensus gains. Different from the previous literature in which stochastic Lyapunov functions and the supermartingale convergence theorem were used, basic results of stochastic analysis and algebraic graph theory are used in this paper. By the law of the iterated logarithm of stochastic integrals, more precise estimates of the convergence rate are given.