arXiv:2104.02033v1 [math.OC] 5 Apr 2021
Counterexamples in synchronization: pathologies of consensus
seeking gradient descent flows on surfaces
Johan Markdahl
a,
a
Luxembourg Centre for Systems Biomedicine, University of Luxembourg
Abstract
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.
Key words: Synchronization, agents, nonlinear control, differential geometry, optimization
1 Introduction
Consensus seeking systems (CSS) on nonlinear spaces appear in various applications of control theory and physics. This includes rigid-body attitude sync on SO(3) [Sarlette et al., 2009] and quantum bit sync on the Bloch sphere [Lohe, 2010]. Most result on the convergence of such systems concern particular manifolds, e.g., the circle [Scardovi et al., 2007], the sphere [Olfati-Saber, 2006, Lohe, 2010], the Stiefel manifolds including SO(n) [DeVille, 2018, Markdahl et al., 2020], and the unitary group [Lohe, 2010]. A typical result is local stability of the synchronization manifold. However, as local convergence to sync is shown to hold on multiple manifolds, it begs the question of when it fails to hold. A theory of CSS on Riemannian manifolds can give a meaningful answer to this question. The literature contains some results that characterize CSS in terms of the known properties of the manifolds they evolve on, e.g., giving conditions for local convergence to sync [Tron et al., 2013], global convergence [Sarlette and Sepulchre, 2009], or determining when a CSS is multistable [Markdahl, 2021]. The contribution of this paper is to provide two additional results in the same spirit: (i) a CSS on a surface that is the boundary of a convex set cannot diverge to infinity and (ii) the sync manifold of a CSS on a closed analytic Riemannian manifold is asymp-totically stable. We also give counterexamples to show that the assumptions we make on the manifolds are necessary for the results (i) and (ii) to hold.
Email address: [email protected](Johan Markdahl).
1.1 Literature review
A good introduction to synchronization on nonlinear spaces is the survey paper Sepulchre [2011]. A key problem is how to design CSS or control algorithms that guarantee almost global convergence to synchronization. For homogeneous manifolds, including the Stiefel and Grassmannian mani-folds, it is given an ad hoc solution in Sarlette and Sepulchre [2009]. The motivation for studying homogeneous mani-folds is that they have the property of looking the same every-where, which suits a distributed control network. For the
n-sphere, and more generally, for most Stiefel manifolds, alter-native solutions based on generalizations of the well-known Kuramoto model are given in Markdahl et al. [2018, 2020]. Further results on local stability of CSS on Stiefel manifolds are provided in [Ha et al., 2021, Chen et al., 2021]. Com-pared to the aforementioned works, the convergence results of this paper are not limited to just homogeneous manifolds, but concerns general Riemannian manifolds.
Synchronization on general manifolds is studied in [Tron et al., 2013, Aydogdu et al., 2017, Markdahl, 2021]. Tron et al. [2013] establish local convergence to consen-sus for discrete-time systems on manifolds with bounded curvature. A key aspect of their approach is the use of in-trinsic geometry, i.e., of not relying on the embedding the manifold in some Euclidean space. Our results differ from those of [Tron et al., 2013] since we consider continuous time system dynamics in an extrinsic setting. The work [Aydogdu et al., 2017] considers both intrinsic and extrin-sic dynamics in continuous time, but do not present any
convergence results. Recently, the work [Markdahl, 2021] establishes local convergence to a class of equilibria apart from synchronization. By contrast, this paper focuses on synchronization, or consensus, which we take to have the same meaning.
For specific manifolds, there is a plethora of papers that establishes convergence to consensus. The CSS in this paper is a generalization of the Kuramoto model on its homogeneous form with identical natural frequency pa-rameters. Other results on generalized Kuramoto models include a literature on quantum synchronization on the Bloch sphere and generalizations thereof to SO(n) and U(n), see e.g., [Lohe, 2010, DeVille, 2018]. We mention just a few of the works on Kuramoto models over mani-folds: the sphere [Crnki´c and Ja´cimovi´c, 2018], the ellip-soid [Zhu, 2014], and the hyperboloid [Ha et al., 2020]. Similar results also appear in applications involving e.g., opinion consensus [Aydogdu et al., 2017], bio-inspired models of source-seeking and learning [Al-Abri et al., 2018, Crnki´c and Ja´cimovi´c, 2018], and computer science appli-cations like time-series clustering [Crnki´c and Ja´cimovi´c, 2019] and shape matching of polytopes [Ha and Park, 2020]. Some of these results on local convergence to the consensus manifold [Lohe, 2010, Aydogdu et al., 2017, DeVille, 2018, Markdahl et al., 2018, Ha et al., 2020] are summarized by our stability result for analytic manifolds.
2 Preliminaries
2.1 Manifolds and hypersurfaces
A Riemannian manifold is a pair(M, g). The set M is a real, smooth manifold and the metric tensorgxis an inner
product on the tangent space TxM at x. A chart φ : M →
Rn is a map from the manifold to a subset of Euclidean space. The composition of a chartφ and an inverse chart ψ−1: Rn→ M, i.e., φ◦ψ−1: Rn → Rn, on the manifold is called a transition map. A manifold is analytic if all its transition maps are real analytic. Familiar spaces like the
n-sphere Sn and the special orthogonal group SO(n) can be given an analytic structure. We assume that(M, g) is embedded in an ambient Euclidean space and takeg to be
the induced metric,gx(u, v) = u⊤v for any tangent vectors
u, v ∈ TxM. We use bold font to denote points x := ι(x)
onM, where ι : M ֒−→ Rnis the inclusion map. We always make the assumption of an embedding and use the induced metric, wherefore we writeM rather than (M, g).
A hypersurfaceHnis ann-dimensional surface embedded
in ann + 1-dimensional Euclidean space. The hypersurface
H is not necessarily a manifold, rather it may e.g., consist of multiple manifolds with boundaries that are glued together. We assume thatHncan be implicitly characterized as
Hn= {y ∈ Rn+1| c(y) = 0}, (1)
where c : Hn → R is a C1 function. This form of the hypersurface also yields a normal from the Gauss map n: Hn → Sn : y 7→ ∇c(y)/k∇c(y)k where we assume that ∇c(y) , 0 for all y ∈ Hn. Although some parts of this paper require the formalism of Riemannian manifolds, for the most part we work with hypersurfaces on the form (1).
2.2 Gradient descent flows
Gradient descent flows are the continuous time equivalents of gradient descent algorithms for minizing a real valued function. LetM be embedded in an Euclidean space Rn. Given a potential functionV : M → R, the gradient descent flowofV on M is the dynamical system
˙x = − gradxV (x) = −Px(∇xV (x))
wheregrad is the intrinsic gradient on the tangent space, Px : R
n
→ TxM is an orthogonal projection on the tangent
space, and∇x is the gradient in R
n
[Absil et al., 2009]. Proposition 1 (Picard-Lindel¨of) Suppose that V : R →
M is C2 on M, then the flow ˙x = − grad V (x) has a
unique solution x(t) which exists for all t ∈ R.
PROOF. See Jost [2008].
Proposition 2 The potential functionV decreases along the solutions of ˙x = − grad V (x). For any integral curve, as
t → ±∞, either grad V (x) → 0 or |V (x)| → ∞.
PROOF. See Jost [2008].
A stronger result exists for gradient flows on analytic man-ifolds. These flows either diverge to infinity or converge to a critical point of the potential function:
Proposition 3 LetM be a real-analytic Riemannian
man-ifold andV : M → R be real-analytic. Assume that x(t) is an integral curve of− grad V (x). Then the ω-limit set of x(t) consists of at most one point.
PROOF. See Lageman [2007].
In this paper we will see examples of some undesired behav-ior of consensus seeking systems (CSS) on hypersurfaces. This includes|V (x)| → ∞ as t → ∞ on an unbounded hy-persurface. We will also see CSS on bounded hypersurfaces for which theω-limit set contains infinitely many points and
2.3 Consensus seeking systems
Consider a multi-agent system with N agents. We use a
graph G = (V, E) to model interactions between agents. Each nodei ∈ V corresponds to an agent and each edge
{i, j} ∈ E corresponds to a pair of communicating agents. Items associated with agenti carry the subindex i; we denote
the state of agenti by xi∈ M, the normal of M at xiby ni
etc.We call x:= (xi)Ni=1∈ MN a configuration of agents.
For future reference we also define the cycle graph CN := ({1, 2, . . . , N }, {{1, 2}, {2, 3}, . . ., {N, 1}}). (2)
The input model of the dynamics of agenti is either
˙xi= ui(x) (3)
where ui: R n
→ TxiM is a control signal, or,
˙xi= Pxi(ui) (4)
where Pxi : R
n
→ TxiM is a orthogonal projection
opera-tor from the ambient space onto the tangent space at xi. The
two formulations (3) and (4) result in the same dynamics. The input model (3) corresponds to a situation where the constraint xi ∈ M is adopted for a specific task whereas
the model (4) refers to the case where xi∈ M is a property
of the system. An example of (3) is a team of satellites in orbit; they could leave the orbit if so desired. An example of (4) is a camera sensor networks where each camera is mounted on a spherical joint. The orientation of camerai is
always some xi∈ S2regardless of the control input.
Restrict consideration to Lipschitz feedback signals uiand
C2 manifolds M (i.e., manifolds all of whose transitions maps areC2). Note that the right-hand sides of (3) and (4) belong to TxiM. The following result guarantees that the
solutions to (3) and (4) stays onM:
Proposition 4 (Nagumo) LetM be a C2manifold and let
f be a vector field onM which is Lipschitz continuous. The
following conditions are equilvalent: (i) any integral curve of f starting onM remains in M and (ii) hf (x), ni ≤ 0
for any exterior normal vector n at all points x∈ M.
PROOF. See Blanchini and Miani [2008].
The goal of a CSS is for the agents to asymptotically ap-proach the consensus manifold
C := {(xi) N
i=1∈ M
N
| xi = xj, ∀{i, j} ∈ E}. (5)
The set C is a manifold C M by the diffeomorphism C → M : (xi)Ni=1 7→ x1. Define the distance between a
point and a set as
d(y, S) := inf
s∈S
ky − sk2.
A CSS strives to achieved((xi)Ni=1, C) → 0 as t → ∞. As
another measure of the distance to consensus, consider the
disagreement functionV : MN → R given by
V (x) := 12 X
{i,j}∈E
aijkxj− xik22, (6)
whereaij= aji∈ [0, ∞). Clearly, V (x) = 0 if and only if
x= (xi)Ni=1∈ C, i.e., if there is no disagreement
The CSS that we study in this paper, Algortihm 5, is the gra-dient descent flow of (6) onM. The general formulation of this algorithm first appears in Sarlette and Sepulchre [2009], although their work is limited to the case when the norm of the states are constant,kxik = k, i.e., the case when M is
a subset of a sphere. This assumption is required for several of the results in Sarlette and Sepulchre [2009].
Algorithm 5 Setting ˙xi = −Pxi(ui) with ui = ∇iV
yields a CSS onM given by ˙xi= −Pxi(∇iV ) = Pxi
X
j∈Ni
aij(xj− xi), (7)
whereaij = aji∈ [0, ∞) and Ni:= {j ∈ V | {i, j} ∈ E}.
Suppose that Px
iisC
2
for all xi∈ M, then the flow of (7)
exists and is unique for all timest ∈ R by Proposition 1.
For the special case that M is a hypersurface, and, more generally, for hypersurfacesHnthat may not necessarily be manifolds but can be written on the implicit form (1), there is an explicit expression for Pxi. Using this expression, at
any point where the dynamics (7) are defined they simplify: ˙xi= −Pxi(∇iV ) = (I − nin ⊤ i) X j∈Ni aij(xj− xi), (8)
where ni = ∇ic(xi)/k∇ic(xi)k. In equation (8), the
Gram-Schmidt rule n⊥ y − hy, nin is used to cancel the normal component (along ni) of xj− xi from ˙xi.
There is another algorithm in the literature:
Algorithm 6 (Zhu [2014]) A CSS on a hypersurface Hn
that satisfieshni, xii , 0 is given by
˙xi= I− xin⊤i hxi,nii X j∈Ni aijxj. (9)
A key idea of Algorithm 6 is thathni, ˙xii = 0 which ensures
that the dynamics stays onHnby Proposition 4.
Example 7 Consider synchronization on the circle. Then
Algorithm 5 and 6 coincide. Add a drift term Ωixi, where
Ωi∈ so(2), to the dynamics whereby
˙xi= Ωixi+ (I2− xix⊤i) N
X
j=1
kijxj. (10)
Make a change of variables to polar coordinates, xi = [cos θi sin θi]⊤. Then
˙θi= ωi+ N
X
j=1
kijsin(θj− θi),
whereωi is an off-diagonal element of Ωi. The above
sys-tem is the well-known Kuramoto model. The gradient de-scent flow formulation of Algorithm 5 is obtained from the Kuramoto model on the form(10) by setting Ωi = Ω and
rotating the Cartesian coordinates to yi= exp(−Ωt)xi.
To briefly compare Algorithm 5 on hypersurfaces with Al-gorithm 6, note that AlAl-gorithm 5 requires that∇c(x) , 0 on Hnwhereas Algorithm 6 also requireshx, ∇c(x)i , 0. The two algorithms are identical when∇c(x) = kx for some
k ∈ R, i.e., when Hn is Sn. Indeed, both algorithms are conceived of as generalizations of a CSS on then-sphere
by Olfati-Saber [2006], Lohe [2010]. In general, it may be more difficult to establish convergence of Algorithm 6 since it is not a gradient descent flow.
Consider equilibria of the two algorithms. Equilibria of (8) are given by X j∈Ni xj= hni, X j∈Ni xjini. (11)
whereas equilibria of (9) are given by X j∈Ni xj= 1 hxi,niihni, X j∈Ni xjixi. (12)
In particular, we note that configurations where all agents are lie on a line through the origin are equilibria of (12) since xj= kijxifor somekij ∈ R but this does not hold for (11).
The following result relates Algorithm 6 on ellispoids to Algorithm 5 onSn:
Proposition 8 The system (9) on an ellipsoid given by M = {z ∈ Rn+1| hz, Azi = 1}
for a positive definite matrix A∈ Rn+1×n+1 is equivalent to the system(8) on the unit sphere.
PROOF. The Cholesky decomposition A = LL⊤ is de-fined for any positive definite matrix A. Set yi = L⊤xi
where xi is the variable of (9). First note that
kyik2= x⊤iLL⊤xi= x⊤iAxi= 1,
i.e., yi ∈ S n
. Algorithm 6 on ellipsoids is given by ˙xi= (In+1− xix⊤iA) X j∈Ni aijxj. It follows that ˙yi = L⊤(In+1− xix⊤iLL⊤) X j∈Ni aijxj = (In+1− yiy ⊤ i) X j∈Ni aijyj,
which is (8) on then-sphere.
In particular, this implies that a number of results about the CSS (7) on the n-sphere, e.g., [Markdahl et al., 2018]
generalize directly to (9) on ellipsoids. 3 Main results
The results of this paper are divided into two parts. First, we give an example to show that a CSS on the form (8) can diverge to infinity as suggested by Proposition 2, then we provide a theorem with a condition that excludes this possibility. Second, we give an example to show that a CSS on a compact manifold does not always converge to a point as suggested by Proposition 3. Moreover, we show that a CSS on a compact manifold can be initialized arbitrarily close toC but still not converge to it. Then we give a theorem with a condition that excludes these two possibilities.
3.1 Divergence on an unbounded hypersurface
First we establish that there is a hypersurface on which the CSS (8) diverges to infinity as it approachesC. Second, we show that ifHn is the boundary of a convex set, then (??) does not diverge in the cases Algorithm 5 and 6.
A hypersurface of revolutionH2 ⊂ R3 is obtained by ro-tating a curve around an axis. It has azimuthal (cylindrical) symmetry around the axis, which we assume to be the
z-axis. The surfaceH2can then be parametrized as
x: R2→ R3: (u, v) 7→ φ(u) cos v φ(u) sin v ψ(u) , (13)
whereφ, ψ : R → R are two continuous functions. For
ex-ample, a cylinder is obtained by settingφ(u) = 1, ψ(u) = u.
Spherical coordinates onS2are obtained by settingφ(u) =
sin u, ψ(u) = cos u.
Definition 9 For surfaces of revolution in R3, defined by two given functionsφ, ψ, and cycle networks CN given by
(2), we defineq-twisted configurations as
Tq = n (xi)Ni=1 ∈ MN| xi=
φ(u) cos(θ + 2πqi/N ) φ(u) sin(θ + 2πqi/N )
ψ(u) , θ ∈ R, u ∈ R, ∀ i ∈ Vo
The interpretation of a1-twisted configuration on S1is that the agents are distributed equidistantly over the circle, each an angle2π/N removed from its neighbors on either side. Note thatq-twisted configurations are not equilibria in
gen-eral; it depends on the manifold. They are however asymp-totically stable equilibria of the standard Kuramoto model on the circle [Wiley et al., 2006]. For spheres,Tq are
equi-libria if they lie on a great circle, i.e., the equator. However, Tq onSn forn ≥ 2 are unstable [Markdahl et al., 2018].
The next example shows that the gradient descent flow di-verges on some manifolds, even as it condi-verges to consen-sus, i.e., it is possible thatlimt→∞xi(t) − xj(t) = 0 for
all{i, j} ∈ E while limt→∞kxi(t)k = ∞ as suggested by
Proposition 2. Moreover, this example has the property that bothd((xi)Ni=1, C) → 0 and d((xi)Ni=1, T1) → 0 as n → ∞.
Example 10 Let H2 be the pseudosphere, which is a hy-persurface of constant negative curvature in R3, see Fig 1. The points that consitituteH2 are implicitly characterized as the solution set to the equation
c x y z = z2− sech−1 q x2+ y2− q 1 − x2− y2 2 = 0.
Because the above expression is somewhat complicated, we prefer to work with a parametrization instead:
y: R2→ M ⊂ R3: (u, v) 7→ sech u cos v sech u sin v u − tanh u (14)
wherev ∈ (−π, π] and u ∈ R. The pseudosphere consists of two manifolds with boundary that have been glued together,
one corresponding to the parameter valuesu ∈ (−∞, 0] and the other tou ∈ (0, ∞). For the sake of simplicity we limit consideration tou ∈ (0, ∞). For this case, an outward pointing unit normal of the pseudosphere is given by
n= tanh u cos v tanh u sin v sech u .
Fig. 1. A CSS of 10 agents on the pseudosphere diverges to infinity.
Since the pseudosphere is a surface of revolution around the z-axis, we can distribute the agents equidistantly at a constantu value to obtain a 1-twisted configuration:
xi(u) =
sech u cos 2πi/N sech u sin 2πi/N
u − tanh u ,
(xi)Ni=1∈ T1. The agent are spread out on a circle of radius
sech u (all agents have the same u value) and are connected
by the cycle graph(2).
The input uisatisfies
ui=
sech u cos 2π(i − 1)/N sech u sin 2π(i − 1)/N
u − tanh u +
sech u cos 2π(i + 1)/N sech u sin 2π(i + 1)/N
u − tanh u − 2
sech u cos 2πi/N sech u sin 2πi/N
u − tanh u = 2(cos 2π/N − 1)
sech u cos 2πi/N sech u sin 2πi/N
0 =2(cos 2π/N − 1) sinh u ni− 2(cos 2π/N − 1) sech u sinh u e3,
e3= [0 0 1]⊤. It follows that ˙xi = (I3− nin⊤i)ui= (I3− nin⊤i)f (u)e3, (15) f (u) = 2(1 − cos 2π/N ) cosh u sinh u = 4(1 − cos 2π/N ) sinh(2u) .
Introduce a Lyapunov functionU = e⊤3xi = u − tanh(u)
with derivatives
˙
U = f (u)(1 − (e⊤3ni)
2
) = f (u) tanh2(u), ˙
U = ˙u − ˙u(1 − tanh2(u)) = ˙u tanh2(u).
We thus find ˙u = f (u). This equation can be solved for
u(t) = cosh−1(8(1 − cos 2π/N )t + cosh(u(0))).
Note thatu(t) → ∞ as t → ∞, see also Fig. 1.
3.2 Boundedness on boundaries of convex sets
Our next result, Proposition 12, shows that CSS on bound-aries of convex sets cannot diverge to infinity. For the sake of simplicity, we assume that the manifoldM is a hypersur-face. This assumption is made without loss of generality: Lemma 11 A convex setK of dimension m in Rn,n ≥ m, can be translated and rotated to anm-dimensional subspace.
PROOF. The convex hull ofK is contained within its own affine span. This implies thatK is contained in an affine sub-space of dimensionm. A translation takes the affine space
to a linear subspace. Take an orthonormal basis of the sub-space. An orthogonal transformation takes the basis of the subspace to the firstm unit vectors {e1, . . . , em}.
Theorem 12 If a hypersurfaceHn is the boundary∂K of a convex setK ⊂ Rn+1, then the CSS (8) on M, where zi∈ {ni, xi}, satisfies
lim
t→∞kxi(t)k < ∞
for all initial conditions on(Hn)N and alli ∈ V.
PROOF. First we establish a property of unbounded se-quences in convex sets. Translate K so that 0 ∈ K. Let s := (si)∞i=1 be any sequenence such that si ∈ K and
limi→∞ksik = ∞. Take a subsequence s′ := (s′i)∞i=1 of
s such that ks′ik ≥ i. Form a sequence r := (ri)∞i=1 on
Snx by setting ri := s′i/ks′ik. Since r is a sequence on a
bounded set, by the Bolzano-Weierstrass theorem, there ex-ists a d∈ Sn such that r has a convergent subsequence r′ withlimi→∞r′i= d. Consider any point αd for some fixed
α ∈ (0, ∞). Because 0, s′i ∈ K and K is convex, for i ≥ α
we haveα/ks′ik ∈ [0, 1] whereby
αs′i/ks′ik = α/ks′ik s′i+ (1 − α/ks′ik) 0 ∈ K.
Moreover, sinceK is closed, αd = limi→∞αs′i/ks′ik ∈ K.
It follows that{αd | α ∈ [0, ∞)} ⊂ K.
Suppose thatkxi(t)k → ∞ as t → ∞. By the reasoning of
the previous paragraph, there exists a d∈ Snsuch that sup
i∈N
hxi(t), di = ∞.
LetPαbe a family of parallel affine planes such thatαd ∈
Pαand d⊥ p − q for all p, q ∈ Pα, i.e., d is a normal of
Pα. Note thatPαseparatesK into two subsets K0andK∞:
0∈ K0, K0∪ K∞= K, K0∩ K∞⊂ Pα,
andK∞is unbounded. Note thatkkk ≥ α for all k ∈ K∞.
Sincekxi(0)k is finite we can pick an α ∈ (0, ∞) such that
{x1(0), . . . , xN(0)} ⊂ K0.
Note that for xito diverge along the direction of d, it must
first pass throughPα. Leti be the index of the agent whose
state xi passes throughPα first, i.e., at the earliest timeτ .
At that timeτ , we note that
hd, ˙xii = hd, (I − nin⊤i) X j∈Ni aij(xj− xi)i = h(I − nin⊤i)d, X j∈Ni aij(xj− xi)i = hd, X j∈Ni aij(xj− xi)i− hni, dihni, X j∈Ni aij(xj− xi)i. (16)
The above quantities are illustrated in Fig. 2.
We claim that the last expression in (16) is negative. To see this, note thathd,P
j∈Niaij(xj− xi)i ≤ 0 since d is
nor-mal toPαand points towardsK∞whereas xj− xi points
away from K∞ due to xj beloning toK0. The inequality
follows from d and xj− xi belonging to different
half-spaces. By the Hahn-Banach separation theorem, since K is convex, the tangent plane TxiK separates R
n+1
into two halfspaces, one that containsK and one that does not. More-over, niis normal to TxiK and we may assume that it points
towards the halfspace which does not containK. As such hni, k − xii ≤ 0 for all k ∈ K, including xj∈ K. This also
yieldshni, di ≤ 0 since d = k − xifor k= xi+ d ∈ K.
In summary, xi cannot pass through Pα since hd, xii is
d
P
αK
0K
1x
jn
i0
x
iT
xiK
Fig. 2. Illustration of vector quantities for the proof of Theorem 12. Note that d is the normal of Pαand niis the normal of Tx
iK
3.3 Divergence on a compact hypersurface
Consider a gradient descent flow off : R2→ R, " ˙x ˙y # = " ∂f (x, y)/∂x ∂f (x, y)/∂y # , (17)
i.e., a continuous time version of the standard gradient de-scent algorithm for optimization. A result based on the Ło-jasiewicz gradient inequality [?] implies that each gradient flow line of (17) converges to a single equilibrium point if the functionf is analytic [Absil et al., 2005, ?]. This property,
often refered to as point-wise convergence, is generalized to gradient flows on manifolds in [Kurdyka et al., 2000], see also Proposition 3. For functions that are not analytic, it is possible for the gradient flow to approach a limit set rather than converge to a singleton. This is illustrated by a coun-terexample based on a so-called Mexican hat function in Absil et al. [2005].
Example 13 We explore the possibility that a CSS can
be-have similarly to the gradient descent flow of the Mexican hat function. To this end, we create a gradient descent flow of a functionf using a CSS on a manifold. Any pathologies of gradient descent flows in Rncan thus appear as patholo-gies of CSS on hypersurfaces.
Because the domain off is a compact set, there is no loss of generality in assuminginfx,yf (x, y) > 0. Then we can
imagine a hypersurfaceH2 in R3 that partly corresponds to the graph off , i.e., the set {(x, y, f (x, y))}. We design the hypersurface to have a part corresponding tof mirrored in thexy-plane (i.e., with negative z-coordinate). The two hypersurfaces do not intersect since we assumed f to be strictly positive. These two hypersurfaces are then joined together in a suitable fasion. This can be done outside the range of(x, y, f (x, y)) triplets that we are interested in. As
such, we skip the details on joining the two surfaces. Locally, the hypersurfaceH2is given implicitly by
|z| − f (x, y) = 0.
An outward pointing unit normal toHn is
n= p 1 1+(∂xf (x,y))2+(∂yf (x,y))2 −∂xf (x, y) −∂yf (x, y) sgn z ,
where∂xf (x, y) = ∂f (x, y)/∂x, ∂yf (x, y) = ∂f (x, y)/∂y.
Place two agents at equal x- and y-coordinates but at z-coordinates with opposite sign. The CSS onH2 is
˙x1= (I3− n1n⊤1)(x2− x1) = 0 0 −2f − n1n⊤1 0 0 −2f = 0 0 −2f + 2f 1+(∂xf )2+(∂yf )2 −∂xf −∂yf 1 ˙x2= (I3− n1n⊤1)(x1− x2) = 0 0 2f − n2n⊤2 0 0 2f = 0 0 2f + 2f 1+(∂xf )2+(∂yf )2 −∂xf −∂yf −1
Note that the time-derivatives for the xy-coordinates are the same for both agents. Moreover, the derivatives of the z-coordinates are equal up to sign. This implies that the symmetry of the two agents, i.e., equalxy-coordinates and oppositez-coordinates, is maintained at all times.
Because of the symmetry of the two agents and the rela-tion zi = f (xi, yi), we may limit consideration to the
xy-coordinates of agent 1. We reduce the system to
" ˙x ˙y # = − 2f (x,y) 1+(∂xf (x,y))2+(∂yf (x,y))2 " ∂xf (x, y) ∂yf (x, y) # (18)
This is the gradient descent flow(17), up to a scalar
multi-plicative factor. Note that by assumption,infx,yf (x, y) > 0
whereby scalar factor in(18) does not affect the trajectory
of the system, only its speed. As such, for initial conditions from which the gradient descent flow off does not converge to a single point (e.g., a Mexican hat function), neither does the consensus seeking gradient descent flow(18).
As a second counterexample, consider a telescope like sur-faceH2similar to the one in Fig. 3. We show that there is an initial state arbitrarily close to synchronization from which the system converges to a1-twisted configuration.
Fig. 3. A nonsmooth telescope shaped hypersurface.
Example 14 Construct a telescope like manifold by
glue-ing together infinitely many cylinders. Denote the cylinders byk ∈ N. The length and diameter of the kth cylinder is
2−k. Note that each pair of subsequent cylindersk, k + 1 can be glued together in an arbitrarily smooth manner (i.e., C∞) using bump functions. However, they cannot be glued together using analytic functions. The final result will hence not be an analytic manifold, as is important in Theorem 15. We derive the dynamics of a CSS on the telescope. First, we find an implicit characterization of thekth cylinder
x y z ∈ R3| x2+ y2= 2−k, k−1 X j=1 2−j ≤ z ≤ k X j=1 2−j .
Moreover, the normal of thekth cylinder is given by
n= q 1 x2+ y2 x y 0 = 2k/2 x y 0 .
Assume the agents are connected by the cycle graph(2) and
setaij = 1. It follows that the dynamics (8) are
˙xi= I3− 2k xi yi 0 h xi yi 0 i (xi−1+ xi+1− 2xi),
which decouples into
" ˙xi ˙yi # =I3− 2k " xi yi # h xi yi i · " xi−1 yi−1 # + " xi+1 yi+1 # − 2 " xi yi # ˙zi= zi+1+ zi−1− 2zi.
Since the z-dynamics are a linear CSS on R, it follows thatPk−1 j=12 −j ≤ z i ≤P k j=12 −j
for all i is an invariant
set. This means the agents will stay on a single cylinder if they are initialized on it. The dynamics of xi and yi are
the CSS (8) on the circle S1 = {y ∈ R2| kyk2 = 1}
with the cycle graph(2). This dynamics is the well-known
homogeneous Kuramoto model on the circle in cartesian coordinates. In particular, it holds that1-twisted equilibria
are asymptotically stable for this CSS [Wiley et al., 2006]. The point of this example is to show that if we initialize the agents in a 1-twisted equilibrium on the kth cylinder with
aij = 1, then the value of the disagrement function is
V (x) = 1 2 X i,j∈E aijkxj− xik 2 2= 2−k+1N sin π/N,
where we used the fact that the agents are equidistantly spread over a circle and the chord length formula. It is now clear that for any ε > 0 we can position the agents on a cylinder with a k sufficiently large that the disagrement function satisfies V (x) < ε. In particular, the agents can start out arbitrarily near C, i.e., near V (x) = 0, without
converging toC since they converge to T1instead.
3.4 Point-wise convergence on closed analytic manifolds
Finally, we give a condition such that the CSS (7) converges to a single equilibrium point (i.e., point-wise convergence) for all initial conditions. We also prove convergence to syn-chronization if x is initialized sufficiently close toC. Theorem 15 LetM ⊂ Rm be a closed analytic Rieman-nian manifold. Consider the gradient descent flow
˙x = −Px∇V,
where x= (xi)Ni=1 and
V (x) :=1 2 X {i,j}∈E kxj− xik 2 .
i.e., the system given by Algorithm 5. This system con-verges to a single equilbrium point (point-wise convergence). Moreover, the synchronization manifold C := {(xi)Ni=1 ∈
MN| xi= xj, ∀ {i, j} ∈ E} is asymptotically stable.
PROOF. Since M is closed, the system cannot diverge to infinity. Point-wise convergence follows directly from Proposition 3 on analytic gradient flows on manifolds. The potential function of a gradient descent flow decreases with time,
˙
V (x) = h∇V (x), ˙xi = −k∇V (x)k2. (19) SinceV (x) ≥ 0 with V (x) = 0 if and only if x ∈ C, we can
stable. See Markdahl [2021] for a Lyapunov theorem in this context. Essentially, it requires that the setC of equilibria is isolated for asymptotic stability to hold.
SinceM is closed, the gradient descent flow converges to a connected component of the set of critical points of V
[Helmke and Moore, 2012]. By (19), any sublevel set of
V is forward invariant. Moreover, all sublevel sets contain
C = {x ∈ MN| V (x) = 0}. Let Q denote the set of equilibria of (7) that do not belong toC. If there is an open sublevel set ofV which does not intersect Q, then there is
an open neighborhood ofC from which x converges to C. This is also the notion of isolated sets in Markdahl [2021]. SinceV (x) is analytic it satisfies the Łojasiewicz inequality
on Riemannian manifolds [Kurdyka et al., 2000]. For every x∈ C there is an open ball B(x), an α < 1, and a k > 0 such that
V (y)α≤ kk∇V (y)k
for all y ∈ B(x). If y ∈ Q, then ∇V (y) = 0 whereby
V (y) = 0. However, this implies y ∈ C, a contradiction.
HenceQ ∩ B(x) = ∅.
Consider the value of q = infx∈QV (x). If q = 0, then
there is a sequence{xk}∞k=1such thatlimk→∞V (xk) = 0.
Since M is a closed manifold embedded in Rn, by the Bolzano-Weierstrass theorem, the sequence{xk}∞k=1 has a
subsequence which converges to some y ∈ M. Moreover,
V (y) = 0 whereby y ∈ C. For each ε > 0 there must
be a z(ε) ∈ Q (an element of the convergent subsequence) such thatky − z(ε)k < ε. This contradicts Q ∩ B(y) = ∅. Henceq > 0 and all trajectories that start in the level set
{x ∈ M | V (x) < q} converges to C.
4 Conclusions
The aim of this paper is to formulate a geometric theory of consensus on nonlinear spaces. Examples show how con-sensus seeking systems (CSS) can go wrong, figuratively speaking, and theorems provide the assumptions needed to overcome these deficiencies. In particular, a general CSS can diverge to infinity but a CSS on the boundary of a con-vex set cannot. Moreover, a CSS on a bounded hypersurface can fail to converge to consensus, but this does not happen if the surface is an analytic manifold. These results should be understood in relation to the literatur on synchronization over surfaces, in particular the many results that apply to a specific manifold. We show that it is also possible to obtain interesting results by working in a more general setting. References
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