Counterexamples in synchronization: pathologies of consensus seeking gradient descent flows on surfaces

Certain consensus seeking multi-agent systems can be formulated as gradient descent flows of a disagreement function. We study how known pathologies of gradient descent flows in Euclidean spaces carry over to consensus seeking systems that evolve on nonlinear manifolds. In particular, we show that the norms of agent states can diverge to infinity, but this will not happen if the manifold is the boundary of a convex set. Moreover, the system can be initialized arbitrarily close to consensus without converging to it, but this will not happen if the manifold is analytic. For analytic manifolds, consensus is asymptotically stable. This last result summarizes a number of previous findings in the literature on generalizations of the well-known Kuramoto model to high-dimensional manifolds.