Topological Considersations in Stability Analysis |17 November 2020|
tags: math.DS

In this post we discuss very classical, yet, highly underappreciated topological results.

We will look at dynamical control systems of the form

 dot{x}=f(x,u),quad uin mathcal{U},quad xin mathsf{M}subseteq mathbf{R}^n,

where mathsf{M} is some finite-dimensional embedded manifold.

As it turns out, topological properties of mathsf{M} encode a great deal of fundamental control possibilities, most notably, if a continuous globally asymptotically stabilizing control law exists or not (usually, one can only show the negative). In this post we will work towards two things, first, we show that a technically interesting nonlinear setting is that of a dynamical system evolving on a compact manifold, where we will discuss that the desirable continuous feedback law can never exist, yet, the boundedness implied by the compactness does allow for some computations. Secondly, if one is determined to work on systems for which a continuous globally asymptotically stable feedback law exists, then a wide variety of existential sufficient conditions can be found, yet, the setting is effectively Euclidean, which is the most basic nonlinear setting.

To start the discussion, we need one important topological notion.

Definition [Contractability]. Given a topological manifold mathsf{X}. If the map mathsf{id}_{mathsf{X}}:xmapsto x is null-homotopic, then, mathsf{X} is contractible.

To say that a space is contractible when it has no holes (genus 0) is wrong, take for example the sphere. The converse is true, any finite-dimensional space with a hole cannot be contractible. See Figure 1 below.


Figure 1: The manifold mathsf{M} is contractible, but mathbf{S}^2 and mathbf{T}^1 are not.

Topological Obstructions to Global Asymptotic Stability

In this section we highlight a few (not all) of the most elegant topological results in stability theory. We focus on results without appealing to the given or partially known vector field f. This line of work finds its roots in work by Bhatia, Szegö, Wilson, Sontag and many others.

Theorem [The domain of attraction is contractible, Theorem 21] Let the flow varphi(t,x) be continuous and let x^{star} be an asymptotically stable equilibrium point. Then, the domain of attraction {xin mathsf{M};:;lim_{tto infty}varphi (t,x)=x^{star}} is contractible.

This theorem by Sontag states that the underlying topology of a dynamical system, better yet, the space on which the dynamical system evolves, dictates what is possible. Recall that linear manifolds are simply hyperplanes, which are contractible and hence, as we all know, global asymptotic stability is possible for linear systems. However, if mathsf{M} is not contractible, there does not exist a globally asymptotically stabilizable continuous-time system on mathsf{M}, take for example any sphere.

The next example shows that one should not underestimate ‘‘linear systems’’ either.

Example [Dynamical Systems on mathsf{GL}(n,mathbf{R})] Recall that mathsf{GL}(n,mathbf{R}):={Xin mathbf{R}^{ntimes n}:mathrm{det}(X)neq 0} is a smooth n^2-dimensional manifold (Lie group). Then, consider for some Ain mathbf{R}^{ntimes n}, n>1 the (right-invariant) system

     dot{X}=AX,quad X_0in mathsf{GL}(n,mathbf{R}) quad (1)

Indeed, the solution to (1) is given by X(t)=mathrm{exp}(tcdot A)X_0 in mathsf{GL}(n,mathbf{R})=:mathsf{M}. Since this group consists out of two disjoint components there does not exist a matrix A, or continuous nonlinear map for that matter, which can make a vector field akin to (1) globally asymptotically stable. This should be contrasted with the closely related ODE dot{x}=Ax, xin mathbf{R}^n=:mathsf{M}. Even for the path connected component mathsf{GL}^+(n,mathbf{R}):={Xin mathbf{R}^{ntimes n}:mathrm{det}(X)>0}, The theorem by Sontag obstructs the existence of a continuous global asymptotically stable vector field. This because the group is not simply connected for n>1 (This follows most easily from establishing the homotopy equivalence between mathsf{GL}^+(n,mathbf{R}) and mathsf{SO}(n) via (matrix) Polar Decomposition), hence the fundamental group is non-trivial and contractibility is out of the picture. See that if one would pick A to be stable (Hurwitz), then for X_0in mathsf{GL}^+(n,mathbf{R}) we have X(t)in mathsf{GL}^+(n,mathbf{R}), however lim_{tto infty}X(t)notin mathsf{GL}^+(n,mathbf{R}) .

Following up on Sontag, Bhat and Bernstein revolutionized the field by figuring out some very important ramifications (which are easy to apply). The key observation is the next lemma (that follows from intersection theory arguments, even for non-orientable manifolds).

Lemma [Proposition 1] Compact, boundaryless, manifolds are never contractible.

Clearly, we have to ignore 0-dimensional manifolds here. Important examples of this lemma are the n-sphere mathbf{S}^{n-1} and the rotation group mathsf{SO}(3) as they make a frequent appearance in mechanical control systems, like a robotic arm. Note that the boundaryless assumption is especially critical here.

Now the main generalization by Bhat and Bernstein with respect to the work of Sontag is to use a (vector) bundle construction of mathsf{M}. Loosely speaking, given a base mathsf{Q} and total manifold mathsf{M}, mathsf{M} is a vector bundle when for the surjective map pi:mathsf{M}to mathsf{Q} and each qin mathsf{Q} we have that the fiber pi^{-1}(q) is a finite-dimensional vector space (see the great book by Abraham, Marsden and Ratiu Section 3.4 and/or the figure below).


Figure 2: A prototypical vector bundle, in this case the cylinder mathsf{M}.

Intuitively, vector bundles can be thought of as manifolds with vector spaces attached at each point, think of a cylinder living in mathbf{R}^3, where the base manifold is mathbf{S}^1subset mathbf{R}^2 and each fiber is a line through mathbf{S}^1 extending in the third direction, as exactly what the figure shows. Note, however, this triviality only needs to hold locally.

A concrete example is a rigid body with mathsf{M}=mathsf{SO}(3)times mathbf{R}^3, for which a large amount of papers claim to have designed continuous globally asymptotically stable controllers. The fact that this was so enormously overlooked makes this topological result not only elegant from a mathematical point of view, but it also shows its practical value. The motivation of this post is precisely to convey this message, topological insights can be very fruitful.

Now we can state their main result

Theorem [Theorem 1] Let mathsf{K} be a compact, boundaryless, manifold, being the base of the vector bundle pi:mathsf{M}to mathsf{K} with mathrm{dim}(mathsf{K})geq 1, then, there is no continuous vector field on mathsf{M} with a global asymptotically stable equilibrium point.

Indeed, compactness of mathsf{K} can also be relaxed to mathsf{K} not being contractible as we saw in the example above, however, compactness is in most settings more tangible to work with.

Example [The Grassmannian manifold] The Grassmannian manifold, denoted by mathsf{GR}_{k,n}, is the set of all k-dimensional subspaces mathcal{S}subseteq mathbf{R}^n. One can identify mathsf{GR}_{k,n} with the Stiefel manifold mathsf{GF}_{k,n}, the manifold of all orthogonal k-frames, in the bundle sense of before, that is pi:mathsf{GF}_{k,n}to mathsf{GR}_{k,n} such that the fiber pi^{-1}(mathcal{S}) represent all k-frames generating the subspace mathcal{S}. In its turn, mathsf{GF}_{k,n} can be indentified with the compact Lie group mathsf{O}(n) (via a quotient), such that indeed mathsf{GR}_{k,n} is compact. This manifold shows up in several optimization problems and from our continuous-time point of view we see that one can never find, for example, a globally converging gradient-flow like algorithm.

Example [Tangent Bundles] By means of a Lagrangian viewpoint, a lot of mechanical systems are of a second-order nature, this means they are defined on the tangent bundle of some mathsf{Q}, that is, pi:Tmathsf{Q}to mathsf{Q}. However, then, if the configuration manifold mathsf{Q} is compact, we can again appeal to the Theorem by Bhat and Bernstein. For example, Figure 2 can relate to the manifold over which the pair (theta,dot{theta}) is defined.

Global Isomorphisms

One might wonder, given a vector field X over a manifold mathsf{M}, since we need contractibility of mathsf{M} for X to have a global asymptotically stable equilibrium points, how interesting are these manifolds? As it turns out, for ngeq 5, which some might call the high-dimensional topological regime, contractible manifolds have a particular structure.

A topological space X being simply connected is equivalent to the corresponding fundamental group being trivial. However, to highlight the work by Stallings, we need a slightly less well-known notion.

Definition [Simply connected at infinity] The topological space mathsf{X} is simply connected at infinity, when for any compact subset mathcal{K}subseteq mathsf{X} there exists a mathcal{K}' where mathcal{K}subseteq mathcal{K}'subseteq mathsf{X} such that mathsf{X}-mathcal{K}' is simply connected.

This definition is rather complicated to parse, however, we can give a more tangible description. Let mathsf{X} be a non-compact topological space, then, mathsf{X} is said to have one end when for any compact mathcal{K} there is a mathcal{K}' where mathcal{K}subseteq mathcal{K}'subseteq mathsf{X} such that mathsf{X}-mathcal{K}' is connected. So, mathbf{R}, as expected, fails to have one end, while mathbf{R}^2 does have one end. Now, Proposition 2.1 shows that if mathsf{X} and mathsf{Y} are simply connected and both have one end, then mathsf{X}times mathsf{Y} is simply connected at infinity. This statement somewhat clarifies why dimension 4 and above are somehow easier to parse in the context of topology. A similar statement can be made about the cylindrical space mathsf{X}times mathbf{R}.

Lemma [Product representation Proposition 2.4] Let mathsf{X} and mathsf{Y} be manifolds of dimension greater or equal than 1. If mathsf{M}:=mathsf{X}times mathsf{Y} is contractible and mathrm{dim}(mathsf{M})geq 3, then, mathsf{M} is simply connected at infinity.

Now, most results are usually stated in the context of a piecewise linear (PL) topology. By appealing to celebrated results on triangulization (by Whitehead) we can state the following.

Theorem [Global diffeomorphism Theorem 5.1] Let mathsf{M} be a contractible C^r, rgeq 1, n-dimensional manifold which is simply connected at infinity. If ngeq 5, then, mathsf{M} is diffeomorphic to mathbf{R}^n.


Figure 3: The manifold mathsf{M} being diffeomorphic to mathbf{R}^2. Then, the benefit of this theorem is that in this setting, dynamical systems can be easily studied without appealing to local charts, describing curve gamma_2 is usually much easier than dealing with gamma_1 directly.

You might say that this is all obvious, as in Figure 3, we can always do it, also for lower-dimensions. However, there is famous counterexample by Whitehead, which provided lots of motivation for the community:

‘‘There is an open, 3-dimensional manifold which is contractible, but not homeomorphic to mathbf{R}^3.’’

In fact, this example was part of the now notorious program to solve the Poincaré conjecture. All in all, this shows that contractible manifolds are not that interesting from a nonlinear dynamical systems point of view, compact manifolds is a class of manifolds providing for more of a challenge.

Remark on the Stabilization of Sets

So far, the discussion has been on the stabilization of (equilibrium) points, now what if we want to stabilize a curve or any other desirable set, that is, a non-trivial set mathcal{X}. It seems like the shape of these objects, with respect to the underlying topology of mathsf{M}, must indicate if this is possible again? Let U_x denote an open neighbourhood of some xin mathcal{X}. In what follows, the emphasis is on U_{mathcal{X}}:=cup_xU_x and mathcal{X} having the same topology. We will be brief and follow Moulay and Bhat.

Lemma [Theorem 5] Consider a closed-loop dynamical system given rise to a continuous flow. Suppose that the compact set mathcal{K}subseteq mathsf{M} is asymptotically stable with domain of attraction mathcal{A}subseteq mathsf{M}. Then, mathcal{K} is a weak deformation retract of mathcal{A}.

In rather loose terms, mathcal{K} being a weak deformation retract of mathcal{A} means that one can continuously ‘‘retract’’ points from mathcal{A} and morph them into the set mathcal{K}. So, this works for mathbf{S}^{n-1} and mathbf{R}^nsetminus{0}, due to mathbf{R}^n being punctured. See p.200 for more information on retracts. In Figure 4 below we show that since this lemma is mostly concerned with the local topology, the global topology is less important. For mathsf{M}, the set mathcal{X} is not a deformation retract of the set U_{mathcal{X}} such that U_{mathcal{X}} can never be a region of attraction, this should be contrasted with the setting on the Torus mathbf{T}^1.


Figure 4: The topological relation between mathcal{X} and U_{mathcal{X}} dictates the possibility of U_{mathcal{X}} being a domain of attraction for mathcal{X}.

The beauty of all of this is that even if you are not so sure about the exact dynamical system at hand, the underlying topology can already provide for some answers. In particular, to determine, what can, or cannot be done.

This post came about during a highly recommended course at EPFL.