A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds

Abstract

Multi-agent systems on nonlinear spaces sometimes fail to synchronize. This is usually attributed to the initial configuration of the agents being too spread out, the graph topology having certain undesired symmetries, or both. Besides nonlinearity, the role played by the geometry and topology of the nonlinear space is often overlooked. This paper concerns two gradient descent flows of quadratic disagreement functions on general Riemannian manifolds. One system is intrinsic while the other is extrinsic. We derive necessary conditions for the agents to synchronize from almost all initial conditions when the graph used to model the network is connected. If a Riemannian manifold contains a closed curve of locally minimum length, then there is a connected graph and a dense set of initial conditions from which the intrinsic system fails to synchronize. The extrinsic system fails to synchronize if the manifold is multiply connected. The extrinsic system appears in the Kuramoto model on S 1 , rigid-body attitude synchronization on SO(3), the Lohe model of quantum synchronization on the n-sphere, and the Lohe model on U(n). Except for the Lohe model on the n-sphere where n ∈ N\{1}, there are dense sets of initial conditions on which these systems fail to synchronize. The reason for this difference is that the n-sphere is simply connected for all n ∈ N\{1} whereas the other manifolds are multiply connected.

Exponential synchronization of the high-dimensional Kuramoto model with identical oscillators under digraphs

“…In [17], a lifting method is proposed to analyze the almost globle synchronization on the unit sphere for digraphs. New exciting development on highdimensional Kuramoto model can be seen in [18,19].…”

On the Local Linear Rate of Consensus on the Stiefel Manifold

“…), where M is the constant in Lemma 2 and L t is the Lipschitz constant in Lemma 4, the sequence {x k } generated by Algorithm 1 converges to a critical point of problem (C-St) sub-linearly. Furthermore, if some critical point is a limit point of {x k } and has exponent θ = 1/2 in (Ł), {ϕ t (x k )} converges to 0 Q-linearly and the sequence {x k } converges to the critical point R-linearly 3 .…”

“…For any d, r, when local agents are close enough to each other, any first-order critical point is global optimum. 3 A sequence {a k } is said to converge R-linear to a if there exists a sequence {ε k } such that |a k −a| ≤ ε k and {ε k } converges Q-linearly to 0. Proposition 2.…”

An intrinsic aggregation model on the special orthogonal group $\mathrm{SO}(3)$: well-posedness and collective behaviours

“…The former type corresponds to a configuration where all agents occupy the same location (delta aggregation at a single point). Both extrinsic and intrinsic algorithms have been proposed and studied for achieving consensus on SO(3) [49,54]. The latter type of configurations corresponds to a group of robots well-distributed over a region/area, so that it achieves an optimal coverage needed for surveillance/tracking (the coverage problem) [53].…”

“…In literature, achieving such a state is also referred to as synchronization or rendezvous. As noted above, this represents an important problem in robotic control [49,53,54]. Consensus states(or phase-locked states) have also been investigated for the Kuramoto oscillator and related models in [19,20,22,31,34,35,37,47,48].…”