Sure Almost Global vs. Global Almost Sure Synchronization on the Circle: The Virtues of Stochastic Hybrid Feedback

This paper introduces a stochastic, hybrid algorithm for global almost sure synchronization of two agents evolving on the circle. Recent results on stochastic hybrid systems are exploited and a Lyapunov-based proof of global almost sure synchronization is obtained. The same arguments establish that global almost sure practical synchronization is achieved in the presence of sufficiently small disturbances. In contrast, sure almost global algorithms are more seriously affected by adversarial disturbances. Namely, there is a set of initial conditions with nonzero measure from which the agents may not approximately synchronize.