A necessary and sufficient condition for discrete-time consensus on star boundaries

It is intuitive and well known, that if agents in a multi-agent system iteratively update their states in the Euclidean space as convex combinations of neighbors’ states, all states eventually converge to the same value (consensus), provided the interaction graph is sufficiently connected. However, this seems to be also true in practice if the convex combinations of states are mapped or radially projected onto any unit $l_p$-sphere or even boundaries of star-convex sets, herein referred to as star boundaries. In this paper, we present insight into this matter by providing a necessary and sufficient condition for asymptotic consensus of the normalized states (directions) for strongly connected directed graphs, which is equivalent to asymptotic consensus of states when the star boundaries are the same for all agents. Furthermore, we show that when asymptotic consensus occurs, the states converge linearly and the point of convergence is continuous in the initial states. Assuming a directed strongly connected graph provides a more general setting than that considered, for example, in gradient-based consensus protocols, where symmetric graphs are assumed. Illustrative examples and a vast number of numerical simulations showcase the theoretical results.

💡 Research Summary

This paper investigates a class of discrete‑time consensus algorithms in which each agent’s state is constrained to lie on the boundary of a star‑convex set (referred to as a “star boundary”). The most familiar examples of such boundaries are the unit ℓₚ‑spheres, but the analysis covers any continuous directional function γ that maps the unit Euclidean sphere to a positive scaling factor, thereby defining a family of star‑shaped surfaces S^{d‑1}_γ.

The agents interact over a directed graph G that is strongly connected. The interaction matrix A=