Towards Almost Global Synchronization on the Stiefel Manifold

Abstract

A graph $\mathcal{G}$ is referred to as ${\mathsf{S}}^1$-synchronizing if, roughly speaking, the Kuramoto-like model whose interaction topology is given by $\mathcal{G}$ synchronizes almost globally. The Kuramoto model evolves on the unit circle, \ie the $1$-sphere ${\mathsf{S}}^1$. This paper concerns generalizations of the Kuramoto-like model and the concept of synchronizing graphs on the Stiefel manifold $\mathsf{St}(p,n)$. Previous work on state-space oscillators have largely been influenced by results and techniques that pertain to the ${\mathsf{S}}^1$-case. It has recently been shown that all connected graphs are ${\mathsf{S}}^n$-synchronizing for all $n\ge 2$. The previous point of departure may thus have been overly conservative. The $n$-sphere is a special case of the Stiefel manifold, namely $\mathsf{St}(1,n+1)$. As such, it is natural to ask for the extent to which the results on ${\mathsf{S}}^n$ can be extended to the Stiefel manifold. This paper shows that all connected graphs are $\mathsf{St}(p,n)$-synchronizing provided the pair $(p,n)$ satisfies $p\le \frac{2n}{3}-1$.