Topological Considersations in Stability Analysis 17 November 2020

Figure 1: The manifold is contractible, but and are not. 
In this section we highlight a few of the most elegant topological results in stability theory. It can be argued that this started by Sontag his search for Control Lyapunov Functions (CLFs), although many related results can be found in, for example, the monograph by Bhatia and SzegĂ¶.
Theorem [The domain of attraction is contractible, Theorem 21] Let the flow be continuous in and let be an equilibrium point, that is, . Then, the domain of attraction is contractible. Moreover, if is not contractible, there does not exist a globally asymptotically stabilizable continuoustime system on .
This theorem by Sontag states that the underlying topology of a dynamical system 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. The next example shows that one cannot think to easily about linear systems.
Example [Dynamical Systems on ] Recall that is a smooth dimensional manifold (Lie group). Then, consider for some , the (rightinvariant) system
Indeed, the solution to (1) is given by . Since this group consists out of two disjoint components there does not exist a matrix , 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 , . Even for the path connected component , The theorem by Sontag obstructs the existence of a continuous global asymptotically stable vector field. This because the group is not simply connected for (This follows most easily from establishing the homotopy equivalence between and via (matrix) Polar Decomposition), hence the fundamental group is nontrivial and contractibility is out of the picture. See that if one would pick to be stable (Hurwitz), then for we have , however .
Following up on Sontag, Bhat and Bernstein revolutionized the field by figuring out some very important implications. The key observation is the next lemma.
Lemma [Proposition 1] Compact manifolds are never contractible.
Important examples of this lemma are the sphere and the rotation group as they make a frequent appearance in mechanical control systems, like a robotic arm.
Now the main generalization by Bhat and Bernstein with respect to the work of Sontag is to use a vector bundle construction of . Formally, given a base and total manifold , is a vector bundle when for the surjective map and each we have that the fiber is a finitedimensional 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 . 
Intuitively, vector bundles can be thought of as manifolds with vector spaces attached at each point, think of a cylinder living in , where the base manifold is and each fiber is a line through extending in the third direction, as exactly what the figure shows.
A concrete example is a rigid body with , 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 be a compact manifold giving rise to the vector bundle with , then, there is no continuous vector field on with a global asymptotically stable equilibrium point.
Indeed, compactness of can also be relaxed to 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 , is the set of all dimensional subspaces . One can identify with the Stiefel manifold , the manifold of all orthogonal frames, in the bundle sense of before, that is such that the fiber represent all frames generating the subspace . In its turn, can be indentified with the compact Lie group , such that indeed is compact. This manifold shows up in several optimization problems and from our continuoustime point of view we see that one can never find, for example, a globally converging gradientflow like algorithm.
Example [Tangent Bundles] Due to the Lagrangian nature, a lot of mechanical systems are of a secondorder nature, this means they are defined on the tangent bundle of some , that is, . However, then, if the configuration manifold 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 is defined.
One might wonder, given a vector field over a manifold , since we need contractibility of for to have a global asymptotically stable equilibrium points, how interesting are these manifolds? As it turns out, for , which some might call the highdimensional topological regime, contractible manifolds have a particular structure.
A topological space being simply connected is equivalent to the corresponding fundamental group being trivial. However, to highlight the work by Stallings, we need a slightly, less wellknown notion.
Definition [Simply connected at infinity] The topological space is simply connected at infinity, when for any compact subset there exists a where such that is simply connected.
This definition is rather complicated to parse, however, we can give a more tangible description. Let be a noncompact topological space and, then, is said to have one end when for any compact there is a where such that is connected. So, , as expected, fails to have one end, while does have one end. Now, Proposition 2.1 shows that if and are simply connected and both have one end, then is simply connected at infinity. This statement somewhat clarifies why dimension and above are somehow easier to parse in the context of topology. A similar statement can be made about the cylindrical space .
Lemma [Product representation Proposition 2.4] Let and be manifolds of dimension greater or equal than . If is contractible and , then, 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 be a contractible , , dimensional manifold which is simply connected at infinity. If , then, is diffeomorphic to .
Figure 3: The manifold being diffeomorphic to . Then, the benefit of this theorem is that in this setting, dynamical systems can be easily studied without appealing to local charts, describing curve is usually much easier than dealing with directly. 
You might say that this is all obvious, as in Figure 3, we can always do it, also for lowerdimensions. However, there is famous counterexample by Whitehead, which provided lots of motivation for the community:
‘‘There is an open, dimensional manifold which is contractible, but not homeomorphic to .’’
In fact, this example was part of the now notorious program to solve the PoincarĂ© 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.
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 nontrivial set . It seems like the shape of these objects, with respect to the underlying topology of , must indicate if this is possible again? Let denote an open neighbourhood of some . In what follows, the emphasis is on and having the same topology. We will be brief and follow Moulay and Bhat.
Lemma [Theorem 5] Consider a closedloop dynamical system given rise to a continuous flow. Suppose that the compact set is asymptotically stable with domain of attraction . Then, is a weak deformation retract of .
In loose terms, being a weak deformation retract of means that one can continuously ‘‘retract’’ points from and morph them into the set . So, this works for and , due to 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 , the set is not a deformation retract of the such that can never be a region of attraction, this should be contrasted with the setting on the Torus .
Figure 4: The topological relation between and dictates the possibility of being a domain of attraction for . 
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 is usually a lot more clear. As we saw, this determines what can, or cannot be done.
This post came about during a highly recommended course at EPFL.