Generalizations and Open Problems

Abstract

This chapter concludes the monograph and provides a brief overview of generalizations, ongoing work and open problems. In particular, by means of periodic orbits we highlight analogous discrete-time theory. Moreover, we remark how to generalize the classical Poincaré–Hopf theorem and we highlight Morse–, Lusternik–Schnirelmann– and Conley (index) theory.

8.1 Comments on Discrete-Time Systems and Periodic Orbits

We largely focused on vector fields, yet, in Chap. 33 one observes that a variety of results are shown via passing to the corresponding flow. In particular, recall the Lefschetz theory, i.e., Theorem 3.5. Akin to Example 3.4, one can compute the Lefschetz number for an asymptotically stable fixed point, e.g., recall (3.7). Consider the (time-one) map $x\mapsto Ax$ with $A\in \mathbb {R}^{n\times n}$ being Schur stable, then $\textrm{sgn}\,\textrm{det}(A-I_n)=(-1)^n$. However, instead of continuous-time dynamical systems of the form $(\textsf{M},\mathbb {R},\varphi )$ one might be interested in discrete-time dynamical systems of the form $(\textsf{M},\mathbb {Z},\psi )$, e.g., one works with discrete measurements of the state variables. Note, here the time-one map $\psi ^1:\textsf{M}\rightarrow \textsf{M}$ is enough to describe the dynamical system. Now, following [4, Sect. 1.11], we remark that such a discrete-time system corresponds directly to a continuous-time system. Define the ceiling function $c:p\in \textsf{M}\mapsto 1$ and let $\textsf{M}_1=\{(p,s)\in \textsf{M}\times [0,c(p)]\}/\sim$ for the equivalence relation $(p,c(p))\sim (\psi ^1(p),0)$. Indeed, $\textsf{M}_1$ can be a Klein bottle (see Fig. 8.1i) when $\textsf{M}=\mathbb {S}^1$. Then, one can define the semiflow $\phi ^t:\textsf{M}_1\rightarrow \textsf{M}_1$, said to be a suspension of $\psi ^1$ under c, by $\phi ^t(p,s)=(\psi ^n(p),s')$ where the free variables n and $s'$ need to satisfy $n+s'=t+s$ and $0\le s'\le 1$. The manifold $\textsf{M}_1$ is also called the mapping torus of $\psi ^1$. One should observe that although $\textsf{M}_1$ and $\textsf{M}$ are not homotopic, by identifying a set $A\subset \textsf{M}$ with a set $A'\subset \textsf{M}_1$, one can study topological obstructions in discrete-time via passing to the continuous-time. We note, however, that practically, hybrid models might be preferred over purely discrete-time models cf. [20, Chap. 1]. The other direction, from continuous-time dynamical systems to discrete-time dynamical systems finds applications in the study of periodic orbits, e.g., via Poincaré return maps [40]. Although we covered submanifold stabilization in Sect. 6.2, one can say more about the special case of periodic orbits, i.e., closed one-dimensional submanifolds of $\textsf{M}$. Here, we briefly highlight work due to Fuller. By exploiting the relation between the Euler characteristic, the Lefschetz number and homology, Fuller proved in 1952 the following.¹

Theorem 8.1

(Fuller’s condition for periodic orbits [19, Theorem 2]) Let T be a homeomorphism of a combinatorial, compact, orientable manifold $\textsf{M}$. If $\chi (\textsf{M})\ne 0$ then, T has a periodic point.

Subsequently, Fuller provides a bound on the period [19, Theorem 3]. Theorem 8.1 has direct applications with respect to Poincaré return maps. If the domain of such a map has non-trivial Euler characteristic, a periodic orbit exists. Recall that for manifolds with zero Euler characteristic, the situation is less transparent, e.g., consider an irrational map on $\mathbb {S}^1$. See [7, 8] for more details and generalizations due to Byrnes.

8.2 Comments on Generalized Poincaré–Hopf Theory

The classical Poincaré–Hopf theorem is usually presented in its smooth form (Theorem 3.6), but as we already remarked, the same is true in the continuous (topological) setting (Corollary 3.1). Note that these results assumed $\textsf{M}$ is boundaryless, this can be relaxed. Assume $\partial \textsf{M}^m\ne \emptyset$, when $X\in \mathfrak {X}^r(\textsf{M}^m)$ points inward along $\partial \textsf{M}$, then the vector field indices sum up to $(-1)^m\chi (\textsf{M}^m)$. When the vector field points outward along the boundary, the Poincaré–Hopf theorem is unchanged [34, p. 35]. This has been recently exploited to assess uniqueness of equilibria [56, Theorem 1]. Generalizations to general vector fields being merely nonsingular on $\partial \textsf{M}$ have been carried out by Morse [37] and Pugh [43]. Relaxing the compactness assumption on $\textsf{M}$ has been studied in [11]. For a generalization applicable to hybrid systems, see [30].

8.3 A Decomposition Through Morse Theory

A smooth function $g:\textsf{M}^m\rightarrow \mathbb {R}$ is called a Morse function when all its critical points are nondegenerate. The Morse index of such a critical point $p\in \textsf{M}^m$ is the dimension of the subspace on which the Hessian of p is negative definite.² Then, the Morse Lemma [6, Lemma 8.2.4] states that around a critical point with Morse index k, there is a coordinate chart $(U,\varphi )$ with $\varphi (p)=0$ such that in coordinates $g(\varphi ^{-1}(x)) = g(p) - x_1^2-\cdots -x_k^2+x_{k+1}^2+\cdots +x_m^2$. Now, looking at a gradient vector field $\textrm{grad}\,g$, an equilibrium point with Morse index k has clearly vector field index $(-1)^k$. A powerful application is that one can extract a cell decomposition.

Theorem 8.2

(Morse indices and CW complex structures [6, Theorem 8.5.3]) Let $g:\textsf{M}^m\rightarrow \mathbb {R}$ be a Morse function with $m_k$ critical points of index k. Then, $\textsf{M}^m$ is homotopy equivalent to a CW complex with $m_k$ cells of dimension k, for $k=1,\dots , m$.

As exploited in Theorem 4.2, this cell decomposition is a link to homology and $\chi (\textsf{M}^m)$. However, Theorem 8.2 is a manifestation of more general results, leading to the so-called Morse inequalities [28, Corollary 8.10.1]. In Sect. 8.5 we discuss a generalization of the work by Morse, largely due to Conley and Zehnder. Besides the original work by Morse [36] and generalizations due to Bott [2] and Smale [47], see also [33] and in particular [3] for more on Morse and his work.

8.4 An Application of Lusternik–Schnirelmann Theory

Lyapunov functions $V:\textsf{M}\rightarrow \mathbb {R}_{\ge 0}$ can be used to capture qualitative behaviour of a vector field X on $\textsf{M}$. For example, the critical points of V can be related to equilibrium points of X. Based on the topology of $\textsf{M}$, the Lusternik–Schnirelmann category can be used to bound the minimal number of critical points any V can possibly have from below, thereby, bounding the number of equilibrium points of X.

Definition 8.1

(Lusternik–Schnirelmann category [14, Definition 1.1]) The category of a topological space $\textsf{X}$ is the smallest $n\in \mathbb {N}_{\ge 0}$ such that there is an open covering $U_1,\dots ,U_{n+1}$ of $\textsf{X}$ with each set $U_i$ being contractible to a point in $\textsf{X}$. This number is denoted by $\textsf{cat}(\textsf{X})=n$ with the cover $\{U_i\}_{i\in [n+1]}$ being called categorical. If such a cover does not exist, $\textsf{cat}(\textsf{X})=+\infty$.

For the setting we consider, that of smooth or topological manifolds, one can consider a cover by open sets without loss of generality [14, Proposition 1.10].

To illustrate Definition 8.1, one can show that $\textsf{cat}(\mathbb {R}^n)=0$, $\textsf{cat}(\mathbb {S}^n)=1$ and $\textsf{cat}(\mathbb {T}^n)=n$, see also Fig. 8.1 (ii). In contrast to $\chi (\cdot )$, $\textsf{cat}(\cdot )$ is not trivial for compact Lie groups, e.g., $\textsf{cat}(\textsf{SO}(3,\mathbb {R}))=3$ and $\textsf{cat}(\textsf{SO}(5,\mathbb {R}))=8$.

Fig. 8.1

Flat representations: (i) the Klein bottle ($\textsf{M}_1$ in Sect. 8.1); (ii) For $\mathbb {T}^2$ one sees via flat quotient representation $\mathbb {T}^2=\mathbb {R}^2/\mathbb {Z}^2$ that $\textsf{cat}(\mathbb {T}^2)\le 2$

For a more control-theoretic example, we return to the switching controllers from Sect. 7.3. If $\textsf{M}$ is “covered” by local controllers all intended to asymptotically stabilize some point, their respective domains of attraction will be contractible by Theorem 6.4, so that one needs at least $\textsf{cat}(\textsf{M})+1$ controllers.

Next we provide a corollary to the Lusternik–Schnirelmann critical point theorem [14] (for any smooth function over the compact smooth manifold $\textsf{M}^n$ it holds that $\textsf{cat}(\textsf{M}^n)+1\le \textrm{crit}(\textsf{M}^n)$ [14, 48]). For the precise definition of the chain recurrent set $\mathcal {R}(\varphi _X)$ of the flow $\varphi _X$ we refer to [13, 26], but simply put, $\mathcal {R}(\varphi _X)$ is the intersection of attractor and repeller pairs under $\varphi _X$. Let A be an attractor, then, the corresponding repeller is $A^{\circ }=\textsf{M}\setminus \mathcal {D}(\varphi _X,A)$. Now the chain recurrent set is defined by $\textsf{M}\setminus \mathcal {R}(\varphi _X) = \cup _{\alpha }\mathcal {D}(\varphi _X,A_{\alpha })\setminus A_{\alpha }$, or $\mathcal {R}(\varphi _X)=\cap _{\alpha }(A_{\alpha }\cup A_{\alpha }^{\circ })$.

Corollary 8.1

(Critical points and limits sets) Let $\textsf{M}^n$ be a smooth, compact, boundaryless manifold. Then, if

$$\begin{aligned} \textsf{cat}(\textsf{M}^n)+1 > \chi (\textsf{M}^n), \end{aligned}$$

(8.1)

there is no continuous vector field $X\in \mathfrak {X}^{r\ge 0}(\textsf{M}^n)$ with all of its equilibrium points $\{p_i^{\star }\}_{i\in \mathcal {I}}$ isolated and with $\textrm{ind}_{p_i^{\star }}(X)=(-1)^n$ such that $\mathcal {R}(\varphi _X)=\{p_i^{\star }\}_{i\in \mathcal {I}}$.

Proof

For the sake of contradiction, there must be a smooth Lyapunov function $V:\textsf{M}^n\rightarrow \mathbb {R}$ that vanishes on $\mathcal {R}(\varphi _X)=\{p_i^{\star }\}_{i\in \mathcal {I}}$. Then, impose some metric $\langle \cdot , \cdot \rangle _p$ on $T_p\textsf{M}^n$, by assumption $\textsf{M}^n$ admits at least one Riemannian metric. By bilinearity of the inner-product $\langle \cdot ,\cdot \rangle _p$ on $T_p\textsf{M}^n$ the set $\{p\in \textsf{M}^n:\textrm{grad}\,V(p)=0\}$ is invariant under the choice of metric (in contrast to neighbouring points, recall the example due to Takens [49, p. 231]). Then the claim follows directly from the Lusternik–Schnirelmann critical point theorem, Theorem 6.6 and [17, Corollary 2.3].

Although Corollary 8.1 is not surprising, one can study the gap $\textsf{cat}(\textsf{M}^n)- \chi (\textsf{M}^n)$ in order to get a better understanding of admissible multistable behaviour on $\textsf{M}^n$.

Example 8.1

(Example 6.5 continued) As $\chi (\textsf{Gr}(2,3))=1$ and $\textsf{cat}(\textsf{Gr}(2,3))=2$ [1, Theorem 1.2], Corollary 8.1 applies. Hence as in Example 6.5, we recover that although Theorem 6.6 applied, stability cannot be global indeed. Yet, as also mentioned in Example 6.5, if one assumes to have a asymptotically stable equilibrium point, then, there must be a non-trivial limit set as well.

8.5 Introduction to Conley Index Theory

The index theory as developed by Conley and coworkers enlarges the scope of Morse theory beyond non-degenerate critical points, in particular, beyond points.

Let us be given a dynamical system $(\textsf{M},\mathbb {R},\varphi )$. A Morse decomposition of a compact invariant set $A\subseteq \textsf{M}$ is a finite collection of compact disjoint invariant subsets $N_1,\dots ,N_n$ of A such that when $p\in A\setminus \sqcup _{j=1}^nN_j$ there are indices $i<j$ such that $\alpha (\varphi ,p)\subset N_i$ and $\omega (\varphi ,p)\subset N_j$. Such an order on $N_1,\dots , N_n$ will be called admissible. Then, given a set $S\subseteq \textsf{M}$, let $I(S)=\{p\in S:\varphi (\mathbb {R},p)\subseteq S\}$ denote the subset of S that is invariant under the flow $\varphi$. A compact set $N\subseteq \textsf{M}$ is called an isolating neighbourhood when $I(N)\subseteq \textrm{int}\, N$. Then, a set $S\subseteq \textsf{M}$ is an isolated invariant set if $S=I(N')$ for some isolated neighbourhood $N'$ containing S.

Definition 8.2

(Index pair) Let S be an isolated invariant set. Then, the pair of compact sets (N, L) with $L\subseteq N$ is an index pair for S when

• (i) $\textrm{cl}(N\setminus L)$ is an isolating neighbourhood of S with $L\cap S = \emptyset$;

• (ii) L is positively invariant in N, that is, if $p\in L$, then $\varphi ([t_0,t_1],p)\subseteq N$ implies $\varphi ([t_0,t_1],p)\subseteq L$;

• (iii) L is an exit set on N for $\varphi$, that is, if $p\in N$ and there is a $t'>t_0$ such that $\varphi (t',p)\notin N$, then there is a $s\in [t_0,t')$ such that $\varphi (s,p)\in L$.

Now, let S be an isolated invariant set, with $N_1,\dots , N_n$ an admissible Morse decomposition, then, there is a collection of subsets $M_0,M_1,\dots ,M_n$, called a Morse filtration such that $M_n\subseteq \cdots M_1\subseteq M_0$ and whenever $i\le j$, then $(M_{i-1},M_j)$ is an index pair for $N_{ij}=\{p\in S:\alpha (\varphi ,p),\omega (\varphi ,p)\subseteq N_i\cup N_{i+1}\cup \cdots \cup N_j\}$ [27, pp. 76–77]. Evidently, $(M_0,M_n)$ is an index pair for S.

Then, the cohomological Conley index for an isolated invariant set S with index pair (N, L) is defined as $CH^{(\cdot )}(S)=H^{(\cdot )}(N,L)$. Similar to what was discussed in Chap. 6, the (co)homological Conley index can be used to provide for example necessary conditions for an equilibrium point to be asymptotically stable [13, Sect. I.4.3], [25, 38]. In this sense, $CH^{(\cdot )}(S)$ is analogous to the vector field index.

More generally, the Conley index can be defined as $h(S)=[(N/L,[L])]$, that is, as the homotopy type of the pointed space (N/L, [L]) [27, 46]. Although effectively intractable, in contrast to $CH^{(\cdot )}(S)$, the Conley index h(S) allows for a result analogous to the Poincaré–Hopf theorem. To that end, define the relative Betti numbers as $b_k (B,C) = \textrm{rank}\, H^k(B,C;\mathbb {Z})=\textrm{rank}\,H_k(B,C;\mathbb {Z})$. Additionally, set $p(t,B,C)=\sum _{k\ge 0}b_k(B,C) t^k$. If (B, C) is an index pair for some isolated invariant set A, with some abuse of notation put $p(t,h(A))=p(t,B,C)$. Under this notation, the following is a generalization of earlier work due to Morse.

Theorem 8.3

(Conley index theorem [27, Corollary 2]) Let $A\subseteq \textsf{M}$ be a compact, isolated, invariant set and $(N_1,\dots , N_n)$ an admissible ordering of a Morse decomposition of A. Then,

$$\begin{aligned} \textstyle \sum \limits ^n_{j=1}p(t,h(N_j)) = p(t,h(A)) + (1+t)Q(t), \end{aligned}$$

(8.2)

noindent for Q(t) a polynomial with non-negative integer coefficients.

We remark that Q(t) depends on the chosen decomposition. Theorem 8.3 implies in particular that for a dynamical system $(\textsf{M},\mathbb {R},\varphi )$ over a compact manifold $\textsf{M}$ one has

$$\begin{aligned} \chi (\textsf{M}) = \textstyle \sum \limits ^n_{j=1}\sum \limits _{k\ge 0}(-1)^k b_k(M_{j-1},M_j) \end{aligned}$$

(8.3)

noindent for some Morse filtration $M_0,M_1,\dots ,M_n$. See that (8.3) is inherently coarser than (8.2), i.e., the excess term Q(t) cancelled out. Also, when $\textsf{M}$ is compact, the trivial decomposition $N_1=\textsf{M}$ and the trivial filtration $(\textsf{M},\emptyset )$ are always admissible.

Example 8.2

(Admissible flows on $\mathbb {S}^2$) Recall Fig. 6.4, we will apply Theorem 8.3. First, a Morse decomposition is given by $(N_1,N_2)$ with orbit $N_1=\mathcal {O}$ and equilibrium points $N_2=\{p_1^{\star }\}\cup \{p_2^{\star }\}$. Then a Morse filtration $M_2\subseteq M_1\subseteq M_0$ is given by $M_0=\mathbb {S}^2$, $M_1$ the disjoint union of sufficiently small compact spherical caps around $p_1^{\star }$ and $p^{\star }_2$ and $M_2=\emptyset$. see Fig. 8.2i.

Fig. 8.2

(i) Example 8.2 with Morse decomposition $(N_1,N_2)$ and Morse filtration $(M_0,M_1,M_2)$; (ii) Example 8.2 with a coarse decomposition, see the equivalence through the lens of this decomposition. Also see that the rightmost figure is a typical instance where (8.1) holds with equality, that is, the vector field under consideration does exist here

Only indicating non-trivial homology groups, as $H_k(M_0,\emptyset ;\mathbb {Z})\simeq \mathbb {Z}$ for $k\in \{0,2\}$, $H_k(M_0,M_1;\mathbb {Z})\simeq \mathbb {Z}$ for $k\in \{1,2\}$ and $H_0(M_1,M_2;\mathbb {Z})\simeq \mathbb {Z}^2$ we find that $Q(t)=0$. Note, here we used the wedge sum, i.e., $M_0/M_1\simeq \mathbb {S}^2\vee \mathbb {S}^1$ [23, p. 10]. One could consider a coarser decomposition and ignore $N_1$, i.e., group $p_1^{\star }$, $p_2^{\star }$ and $N'$ as Fig. 8.2ii. In that case, $Q(t)=0$ and one effectively employs the Poincaré–Hopf theorem. Consult also the discussion in [27, Sect. 3.5].

Besides the original work by Conley, Zehnder and others [12, 13, 46], see [27, 35] for introductory works and for example [45] for a discrete-time analogue. Also, the work by Conley et al. builds upon that of Ważewski [54], [22, p. 280], which has been recently exploited in [41, 42] to provide further local obstruction theory.

8.6 Conclusion and Open Problems

This work aimed at providing an overview of how topology can provide for unique insights in control theoretic problems. This approach has a long and rich history and we believe this work shows it has more to offer for the future. In particular, Borsuk’s retraction theory and the application of homotopy- and index theory akin to the framework by Krasnosel’skiĭ and Zabreĭko provide for a fruitful and unifying approach towards a mathematical control theory. We end with potential future work.

• (i) Sufficient conditions: Controllability assumptions, the existence of a control-Lyapunov functions or structural properties like homogeneity can be exploited to provide sufficient conditions for certain feedback laws to exist. It would be interesting to see to what extent these results can be generalized to work for general control systems, e.g., $\Sigma =(\textsf{M},\mathcal {U},F)$, aiming at locally asymptotically stabilizing some closed set $A\subseteq \textsf{M}$. To that end, one needs to assert that A can be rendered invariant, there exists a dynamical system such that A is an attractor, locally, and so forth. Hence, sufficient conditions, especially conditions that lead to the construction of controllers are of great importance, yet, with the remarks of Casti [9] in mind, universal conditions might be too optimistic. Hence, a sub-question entails finding (more) appropriate classes of canonical control systems. An operator-theoretic start of this program is presented in [10].

• (ii) Numerical tools: This line of work aims to provide necessary and sufficient conditions so that one can assess—a priori—if a stabilization problem has a solution. Clearly, these conditions must be easier to check than a brute force numerical experiment for this to be of practical value. In this regard, applying tools from computational homology [29] seems interesting. However, one might obtain corrupted information regarding the control system, e.g., F is based on experimental data. Recalling Remarks 6.1 and 6.5, computational tools must be able to provide certificates of accuracy, e.g., see [53] and references therein. Here, the field of topological data analysis (TDA) is also ought to play an increasingly important role. For instance, if one can sample points from $\Omega _{\varepsilon }$ as in Theorem 6.3, TDA tools can be used to estimate if (6.2) is true.

• (iii) Other invariants: Although finding the “perfect” invariant that allows for classifying dynamical systems up to conjugacy (Smale program) is too ambitious,³ one might look for new invariants that capture (some) qualitative behaviour of interest. Related is the question, largely due to Conley, as posed in [31], can the homotopy from Theorem 6.13 be chosen to be asymptotically stabilizing throughout? Here, relaxing the notion of (uniform) asymptotic stabilization provides for many counterexamples cf. Fig. 3.4. We also believe that imposing restrictions on the class of Lyapunov functions to assert stability might be insightful. Moreover, it is not clear how to meaningfully extend this type of theory to generic input-output systems. In other words, can a similar program be outlined for general systems, going beyond dynamical systems cf. [21]. In particular, a program that allows for compositional thinking. Here we note that especially homology appears suitable, i.e., $(G_2\circ G_1)_{\star }=G_{2\star }\circ G_{1\star }$.

• (iv) Nonlinear system identification: Our understanding of statistical finite-trajectory nonlinear system identification is improving, e.g., see [57], yet it is unclear how to go about identifying a nonlinear system (or equivalence class) globally. In particular, unstable equilibria are hard to identify. From Theorem 6.6 it follows that if one assumes that all unknown equilibria are asymptotically stable and the system is noiseless, then, one needs at least $\chi (\textsf{M})$ trajectories to “identify”⁴ these equilibria and their respective dynamics, locally. Combining the topological viewpoint with TDA and modern high-dimensional probability theory, e.g., [52], could be a fruitful combination. Other popular methods like neural networks, Gaussian processes, and so forth, all require further development of the theory. A promising direction exploits Koopman operator theory [5].

• (v) Feedback invariance: As set forth in the monograph by Lewis [32], the lack of feedback-invariance present in most of the literature obstructs a principled control theory. In particular, the key necessary condition for control, that of controllability, is not invariant under feedback reparametrizations, see also early comments by Willems [55]. We also put emphasis on results that do not rely on the control system at hand, but merely on the underlying manifold $\textsf{M}$ and the set A that is deemed to be stabilized, cf. Theorem 6.12. However, a variety of results, mostly contained in Sect. 6.1.1 do rely on the control system parametrization.

• (vi) Stochastic systems: One can generalize Definition 5.5 to include disturbances [39, Definition 13.26]. Obstruction theoretic results that do include some form of noise often focus on moments [15, 16], however, how to extend the theory in the most meaningful way to stochastic systems is not clear.

• (vii) Zero Euler characteristic: Set with zero Euler characteristic are inherently difficult to handle, cf. Theorems 6.5, 6.6,  6.9 and 6.11. As these sets are, however, abundant (odd-dimensional manifolds, Lie groups, periodic orbits), more theory is needed. Recalling Remark 6.4, retraction theory is of use, at the cost of being largely independent of the control system at hand. One could also look for other topological invariants that allow for a better classification of these sets.

More questions remain, conceptually, why are odd-dimensions seemingly inherently obstructive? What can be said about finite-time stabilization? How can we further integrate the structure of $\mathcal {U}$ and F in the analysis (beyond Theorem 6.14)? For instance, to get a better grip on Question iii. Most importantly, more topological obstructions are likely to be found. For example, one might use that for closed topological spaces $\textsf{X}$ and $\textsf{Y}$ it holds that $\chi (\textsf{X} \sqcup \textsf{Y} ) = \chi (\textsf{X}) + \chi (\textsf{Y})$, or by using that an orientable manifold $\textsf{M}^n$ has even Euler characteristic if n is not a multiple of 4 [24, Theorem 1.2]. Or less recent, a closed connected smooth manifold $\textsf{M}$ has $\chi (\textsf{M})=0$ if and only if it admits a smooth codimension-1 foliation [51, Theorem 1]. However, more impactful ought to be obstructions derived from Conley– and Morse–Bott theory or obstructions applicable in the context of robust- and hybrid control theory. In particular, theory well equipped to handle input-output systems. Although progress has been made with respect to obtaining a principled approach to stabilization on Lie groups [50] cf. Sect. 1.3, the central problem remains that of finding tractable sufficient conditions—as stressed throughout the literature [10, 31]. In particular, sufficient conditions that lead to implementable controllers. Ce n’est pas tout.