Consensus of overflowing clocks via repulsive Laplacian laws

The main objective of this paper consists in imposing consensus in a network of overflowing clocks with identical speeds but potentially different initial offsets. Each (overflowing) clock is modeled as a single integrator with the state confined in a bounded set such that, whenever the state reaches its maximum allowed value, it is immediately reset to zero (overflowing phenomenon), thus exhibiting both continuous-time and discrete-time behaviours. In this framework, control techniques inspired by the classical Laplacian philosophy lead to a somewhat unexpected result. In fact, it is shown that both an attractive and a repulsive Laplacian law induce two periodic orbits of the closed-loop system, characterized by the feature that along only one of these trajectories consensus is reached. It is then proved that the error-zeroing periodic orbit is unstable with the attractive Laplacian, hence agreement of the clocks is not achieved, while an asymptotic convergence on it is guaranteed with the repulsive Laplacian, hence consensus is reached with a repulsive control law among the clocks.