1.1 Impetus

From climate models to walking robots and from black holes to the economy; many objects of science are studied by means of dynamical systems evolving on manifolds [40, 105, 151, 156]. As models are not perfect and explicit solutions are rare, ever since the time of Poincaré the interest shifted from studying the quantitative- to studying the qualitative behaviour of a dynamical systems at hand. Not only the description of a system, but in particular, the prescription of the dynamics of a system became of increasing importance. Naturally, one must ask if the desirable dynamics are admissible in the first place. Topology provides for a rich set of answers relying on a minimal set of assumptions, as surveyed in this work.

Given some space \(\textsf{M}\) and some subset A of \(\textsf{M}\). We will be mostly concerned with studying if \(\textsf{M}\) admits a dynamical system such that the set A, e.g., some configuration of a robot, is (uniformly) globally asymptotically stable. This stability notion is captured by: (i) Lyapunov stability: that is, for each neighbourhood U of A there is another neighbourhood \(V\subseteq U\) of A such that when the system starts from a state within V, the state of the system will stay in U; and (ii) attractivity: that is, there is a neighbourhood W of A such that when the system is started from a state within W, the state of the system converges asymptotically to A. When \(W=\textsf{M}\), we speak of global asymptotic stability. We also remark that Lyapunov stability is sometimes referred to as simply stability, consequently, a set that fails to be stable, with respect to some dynamical system, is said to be unstable. For formal definitions, see Chap. 5.

Example 1

(Admissible flows on the circle) Let one be tasked with finding a continuous flow, i.e., a map that defines state propagation as a continuous function of the time to propagate and the instantaneous state, such that some point \(p^{\star }\) on the circle \(\mathbb {S}^1\) is globally asymptotically stable, e.g., see Fig. 1.1(i). The flow in any small neighbourhood around \(p^{\star }\) is well-defined, but one eventually runs into problems, see Fig. 1.1(ii), and cutting the circle (allowing for discontinuities) seems the only solution, see Fig. 1.1(iii). Although, relaxing the task, e.g., by allowing for almost surely global asymptotic stability, local asymptotic stability or merely global attractivitiy (no Lyapunov stability), also belongs to the possibilities, see Fig. 1.1(iv)–(vi). In fact, by studying Fig. 1.1 one might observe some patterns, e.g., stable and unstable equilibrium points necessarily come in pairs. The key observation, however, is that for compact nonlinear spaces, the study of global behaviour should take the global topology into account.

Fig. 1.1
figure 1

Example 1, (asymptotically) stabilizing \(p^{\star }\in \mathbb {S}^1\)

Example 1 illustrates the fact that merely the underlying topology of a space can obstruct the existence of certain qualitative behaviour.

Next, going one step beyond the circle, we consider the mathematical pendulum, cf. [142, 145], which displays a myriad of topological phenomena [97, 126] and captures the dynamics integral to the study of robotics, aerial vehicles and more.

Example 2

(The mathematical pendulum) The single-link pendulum displays intricate nonlinear behaviour by having the circle \(\mathbb {S}^1\) as its configuration space. As a pendulum is a second-order system, the state space, however, becomes the cylinder \(\mathbb {S}^1\times \mathbb {R}\), parametrizing the angle and rotational velocity, or as will be discussed, the trivial vector bundle \(\pi :\mathbb {S}^1\times \mathbb {R}\rightarrow \mathbb {S}^1\). We assume that one can control the pendulum by means of a torque applied to its axis and that this torque is chosen as a continuous function of the state. Moreover, we assume that this feedback, i.e., the torque as a function of the state, gives rise to a continuous flow on \(\mathbb {S}^1\times \mathbb {R}\) and that there are isolated fixed points of the flow, e.g., we assume there is some form of friction. Now we ask a similar question as before, can the pendulum be globally asymptotically stabilized in the upright (\(\theta =0\)) position by such a feedback? This is a stabilization problem. For example, Fig. 1.2 displays trajectories analogous to Fig. 1.1(iv)–(v). The reader is invited to construct the phase portrait akin to Fig. 1.1(vi) and recover what is called the unwinding phenomenon. One observes that generalizing the circle to the non-compact cylinder did not improve the situation, topological obstructions prevail. Simultaneously, this example shows the power of this line of study in that the results are general; we did not yet make any explicit modelling assumptions, e.g., regarding friction and inertia.

Example 2 sketches the practical value of studying topological obstructions; when explicit models are unknown or too uncertain, the underlying topology can already provide insights in admissible qualitative behaviour. In fact, one can argue that topology is the natural language to study these kind of dynamical problems [83].

The previous examples focus on equilibrium points, however, in many applications one might be interested in stabilizing non-trivial periodic orbits or other sets. The intuition from before can be generalized, for example, consider globally asymptotically stabilizing the circle \(\mathbb {S}^1\) in the plane \(\mathbb {R}^2\). Again, obstructions of this kind are purely topological.

Fig. 1.2
figure 2

Example 2, asymptotically stabilizing the inverted pendulum

Although results of this nature go back to the 1800s, we feel there is a need to survey existing material: the ever-growing field of applied dynamical systems theory, including, but not limited to, motion planning, numerical optimal control, system identification and reinforcement learning. The aforementioned observations have particularly important ramifications in those areas, as frequently, one needs to specify a space of models or policies to optimize over, a priori.

As stressed in a recent article by Schoukens and Ljung [136], nonlinear system identification poses challenges beyond linear system identification, in particular, the nonlinear structure prohibits straight-forward extrapolation, that is, inferring global information from local data is inherently difficult in the nonlinear regime. However, knowledge of the underlying topological space is occasionally present, as such it is worthwhile to study ramifications of just the topological data at hand.

Regarding policies, it is important to highlight that controllability merely implies the existence of some admissible input “steering point A to point B”. As stressed by Sussmann [146, p. 41], only when the admissible inputs are precisely detailed, one can study if some control objective can be satisfied by selecting the input as some kind of feedback controller. In practice, when numerically optimizing over policies, one is for example drawn to employing some form of function approximators [29]. It might be tempting to believe that the space of continuous functions is sufficiently rich, however, as we already saw for the most elementary nonlinear manifolds, when stability is desired, this is not necessarily true. A similar argument can be made when searching for control-Lyapunov functions (CLF) or control barrier functions [88]. This work sets out to bring results of this kind further to the attention and spur more future work towards understanding and overcoming these topological obstructions.

We will focus on continuous asymptotic stabilization of nonlinear systems. Here, nonlinear should be read as not necessarily linear. In particular, we look at dynamical systems defined on nonlinear spaces, e.g., as in Fig. 1.1, local linearizations can fail to capture topological impossibilities. The consideration of dynamical systems is largely an attractive mathematical assumption, but one that is believed to be quintessential to better understand larger classes of physical systems. The focus on continuity is historically based on implementation and robustness considerations and seems at first a general assumption. Enforcing continuity allows for a better understanding of how general this assumption really is. The focus on continuity also puts work on neural networks in perspective as common architectures result in maps which are at least continuous. The desire of (uniform) asymptotic stabilization is a natural one from the classical mechanics point of view and enviable with respect to robustness, but also here one will observe that this demand can be too strong.

1.2 Historical Remarks

The study of topological obstructions in the context of dynamical control systems is at its core the investigation of what kind of—qualitative speaking—dynamical systems a space admits. Philosophically, this is in line with the early work on topology and dynamical systems as pioneered by Poincaré. To put the material in perspective, the next section briefly covers this history at large.

1.2.1 Topology

The fourth axiom of Euclid states “Things which coincide with one another are equal to one another” [4, p. 6]. Although Euclid was a geometer and no topologist, this axiom is broadly stating what topology would be all about. Yet, “things” and “equivalence” had to come a long way since the time of Euclid.

In the early 1900s, Cantor started the development of set theory and contributed to the initial work on topology [31]. When Cantor his one-to-one map from the interval to the hypercube revealed the intricacies of defining dimension,Footnote 1 it was Dedekind to point out that perhaps something is missing: continuity [112]. Peano’s space-filling curve showed that even when continuity is satisfied (but injectivity is lost), counter-intuitive phenomena can still be observed [115]. These counterexamples revealed a lack of understanding when it comes to classifying objects as being “equivalent”.

Poincaré was amongst the first to define, in for example his 1895 “Analysis situs”, what this equivalence could be: a continuous one-to-one transformation, called a homeomorphism [121]. It took, however, a while before a homeomorphism meant what it does today. Poincaré described analysis situs (the predecessor of topology, attributed to Listing) as “This geometry is purely qualitative; its theorems would remain true if the figures, instead of being exact, were roughly imitated by a child.” [123]. A weaker invariant that plays a substantial role in this work is that of homotopy. It can be argued that homotopies, albeit with fixed endpoints, originated in the work of Lagrange on the calculus of variations [41]. Although homotopies (described as continuous deformations), the fundamental group and initial homology theory appeared in the work of Poincaré [121], the formal description of what this “continuous deformation” is supposed to be, was missing. By building upon Schönflies, the concept of dimension that bothered Cantor, was eventually put on a theoretical footing by Urysohn [152] and most notably by Brouwer [25, 26] around 1910, see also [72, Chap. 1]. This work by Brouwer also formalized homotopies and their equivalence classes as we know them today [27]. Besides, it brought forward the concept of degree, a notion of importance in this work, yet, a notion that was to some extent already known to Cauchy, Picard and in particular Kronecker [43, 114, 137]. Hurewicz added to this line of work by defining when spaces are homotopic [68] and a related theory, that of retractions was pioneered by Borsuk [16] in the late 1920s. A concept intimately related to the degree of a map is that of an index of a vector field, as arguably introduced by Poincaré and further developed by Hopf [66, 67]. An important elaboration and formalization is due to Brouwer [42, p. 168], Poincaré often assumed differentiability or even analyticity of objects under consideration [72, p. 57], while Brouwer relaxed this to mere continuity and was able to formally apply index theory to continuous vector fields on the sphere [26].

In the meantime topology branched out. Although the “Euler characteristic” was known, it can be argued that Riemann founded algebraic topology [17, pp. 162–164], as, amongst other things, he evoked, what would be called Betti numbers, in the late 1800s with his study into connectedness [130], [72, Chap. 2]. Generalizations required work from Betti [9], Poincaré [121] and most importantly, formalizations by NoetherFootnote 2 [111] and later Eilenberg [45].

Formalizing the abstract study of sets relied on early work by Hilbert [61], Fréchet [46], Riesz [131] and in particular Hausdorff, whom in 1914 published Grundzüge der Mengenlehre [56]. This was the first axiomatic work on abstract topological spaces and can be seen as the start of general topology known today. Hausdorff laid down the foundation of (general) topology and provided the neighbourhood generalization of the Bolzano–Cauchy \(\varepsilon -\delta \) continuity definition, although the neighbourhood concept was already known to some extent.

Concurrently, the notion of a (linear) manifold was already known to Gauss, but, amongst others, it were Möbius [106], Jordan [74] and in particular Riemann [129] and Poincaré [121] that initiated the classification of manifolds. The theory of differentiable manifolds, however, and thereby differential topology was largely developed by Weyl [72, Chap. 2], Veblen and Whitehead (J.H.C.) [154] and Whitney [157, 158], e.g., this includes formalizations of coordinate charts, tangent spaces and embeddings. The initial work by Cartan (Élie) on fiber bundles was further developed by Seifert, Whitney and Ehresmann [72, Chap. 22]. A later, but instrumental contribution for this work is the notion of transversality as developed largely by Thom [149]. The most important development for this work, and perhaps one of the most important series of results in the intersection of topology and dynamical systems in general, is the Poincaré–Hopf theorem, with contributions by Gauss, Kronecker, Bonnet, Dyck, Brouwer, Poincaré and in particular Lefschetz [93, 94] and Hopf [67], see also [48].

For these and more examples, see [42, 72, 137, 148, 155], the historical notes in [17, 43, 114, 163] and [124] for the English translation of [121]. Also, as highlighted, topology thrived on counterexamples, which remained an active area [143].

1.2.2 Dynamical Systems

Starting with Newton, the study of dynamical systems was dominated in the early days by the quantitative study of stability in the context of celestial mechanics. The complications that arise when trying to explicitly solve differential equations were early understood and amongst others, Laplace, Lagrange, Poisson and Dirichlet all claimed to have proven that the solar system was stable by means of analyzing series expansions, e.g., see the introduction in [1]. Motivated in part by a competition in the late 1880s hosted by Oscar II, the king of Sweden & Norway, it was Poincaré who pointed out that the series expansion approach of that time was flawed. As he puts it, “There is a sort of misunderstanding between geometers [mathematicians] and astronomers about the meaning of the word convergence.” “...take a simple example, consider the two series,

$$\begin{aligned} {\textstyle {\sum _n}} \frac{1000^n}{n!}\quad \text {and}\quad {\textstyle {\sum _n}} \frac{n!}{1000^n}, \end{aligned}$$

geometers would say the first series converges, ...but they will regard the second as divergent.Astronomers, on the contrary, will regard the first series as divergent as the first 1000 terms are increasing and the second as convergent as the first 1000 terms are decreasing” [120, pp. 1–2]. Poincaré emphasized throughout his work that divergent series could have practical value, but to prove anything meaningful about the stability of the solar system required a rigorous mathematical justification  [120, p. 2]. It was during this time that Poincaré developed his qualitative methods for mechanics and dynamical systems, e.g., [118,119,120, 122]. Amongst other things, in this work he started the classification of qualitative dynamical systems, introduced the notion of the vector field index and advocated the use of transversality (Poincaré sections), topics we return to below. Ironically, the original version of the award winningFootnote 3 1890 paper [118] did not contain the most celebrated parts of the work, e.g., work on homoclinic solutions (initially called doubly asymptotic solutions), only the corrected version did. Upon fixing the errors in the initial version of [118], Poincaré was puzzled by the chaos he created (found) and wrote to Mittag-Leffler “...I can do no more than to confess my confusion to a friend as loyal as you. I will write to you at length when I can see things more clearly.” [5, Sect. 5.8]. Almost ten years later, the situation was far from clarified, regarding homoclinic solutions to the three-body problem, Poincaré writes “One is struck by the complexity of this figure that I am not even attempting to draw.” [122, p. 389]. It would take more than half a century before Smale would clarify the situation.

As remarked previously, one of the main topics of study has always been that of stability. Around the same time as the early work of Poincaré, Lyapunov (Liapunov)—who was inspired by Poincaré [102, pp. 531–532]—published his thesis on qualitative stability theory in the early 1890s [99]. Although Lyapunov was the first to lay down the foundations, similar notions appeared in the work of Lagrange [90] and Dirichlet [44]. In this work, Lyapunov devised two lines of attack to reason about (local) stability. The first method (indirect) relies on linearizing the dynamics, whereas the second method (direct) is in the spirit of the work by Lagrange and looks for an “energy” function (Lyapunov function), which is strictly positive, yet decreasing along the dynamics. Attributed to Poission, Poincaré, however, spoke of stability of a point when a trajectory returned infinitely often to points arbitrarily close to where the point started from. Lyapunov spoke of stability of a point when for each open neighbourhood U around a point there is an open set \(V\subseteq U\) such that each trajectory starting in V remains in U. Lyapunov’s approach to stability (the second, direct method) had an intuitive and almost direct link to modelling paradigms in mechanics (energy) and it grew out to be one of the most celebrated tools in the study of dynamical systems cf. [10, 81, 89].

Concurrently, after proving Poincaré’s “last geometric theorem” in 1913 [11], it was in particular Birkhoff who propelled the qualitative study of dynamical systems [12]. This qualitative viewpoint brought the general notion of stability more to the forefront, not only stability of a system, but also the stability of its description, called structural stability. Structural stability for 2-dimensional systems was introduced by Andronov and Pontryagin [2] and extended to 2-dimensional manifolds by Peixoto [116]. Smale is largely responsible for further abstractions and continuation of this line of work [139]. In particular, his creation of the horseshoe-map clarified what Poincaré was having trouble with in his work on the three-body problem: the intricate dynamics close to a homoclinic equilibrium point. Interestingly, Smale also made a significant contribution to another, yet topological, open problem by Poincaré, he proved the (generalized) Poincaré conjectureFootnote 4 for \(n\ge 5\) [138].

In passing, we highlight a few other influential works, contributing to (the early development of) the qualitative theory of dynamical systems. The work by Hadamard [53, 54]—who was, interestingly, in close contact with Brouwer, Lefschetz [95], LaSalle (La Salle) [91] and Hartman [55]. Early contributions to stability theory by Hermite, Routh and Hurwitz, e.g., see [69, 77]. The initial work on center-manifold theory by Pliss [117] and Kelly [80]. Catastrophe theory by Thom [150]. Converse Lyapunov theorems and its topological ramifications due to Kurzweil [87], Bhatia and Szegö [10] and Wilson [159, 160]. The concept of a region of attraction due to Aĭzermann, Barbašin, Krasovskiĭ, Nemyckiĭ and Stepanov, e.g., see [10, 86]. The work by Kolmogorov, Arnol’d and Moser, e.g., see [24]. The work by Hirsch et al. [63]. The topological critical-point theory by Morse [107] and the work on chaos theory by Takens and Ruelle [132].

For more information, see [34, 48, 62, 65], or the historical notes in [1, 10, 50]. In particular, see [5] for an exposition of the competition organized by Mittag-Leffler, the initial error in the work submitted by Poincaré, how this was resolved and how the mathematical community responded. For more on the history of stability, see [77, 96]. See [125], for the English translation of [118].

1.2.3 Modern Control Theory

In the late 1700s, the field of control theory emerged due to a growing practical interest in improving the performance of mechanical systems. As discussed above, the 1800s gave rise to a lot of theoretical work on describing the dynamics of a system and in particular studying its stability. Nevertheless, motivated by the needs of war and after original work on telecommunication, filtering and circuit design in the frequency-domain, modern control theory, however, was only born in the mid 1900s out of the pioneering work by KalmanFootnote 5 [78, 79], Bellman [6], Pontryagin [47, 127] and their coworkers. This line of work emphasized some benefits of the state space approach (the time-domain) and essentially reconnected control theory to the early work of Poincaré and Lyapunov. The state space approach to linear control theory also brought linear algebra more to the forefront, which opened the door for a rigorous approach to nonlinear control, not merely by approximation, but also by appealing to differential geometric tools, cf. [21, 70, 110, 161]. Perhaps the central topic of study in (deterministic) control theory in the late 1900s was that of controllability, i.e., all questions related to the possibility of steering a system “from A to B”. Naturally, these questions relate to the aforementioned work on dynamical systems, e.g., if a space does not admit a dynamical system with a certain property, then clearly no input exists that can enforce it. Building upon the work of ChowFootnote 6 in 1940 [33], it can be argued that work on nonlinear controllability started in the 1960s—just after Kalman published his work on linear controllability—with influential contributions by Hermann [58], Lobry [100, 101], Haynes and Hermes [57], Sussmann and Jurdjevic [147], Brockett [20] and most notably Hermann and Krener [59].

In 1978, Jurdjevic and Quinn constructed a controllable system on \(\mathbb {R}^2\) that cannot be stabilized via differentiable feedback [75]. Then, against to what was a common belief at the time, by constructing an example on \(\mathbb {R}^2\), Sussmann showed in 1979 that controllability does, however, also not implies that a stabilizing continuous feedback exists [146]. A year later, Sontag and Sussmann developed theory underpinning scalar examples along these lines [140]. These examples were not unparalleled as in 1983 Brockett gave an explicit necessary (topological) condition for stabilizing differentiable feedback laws to exist [23]. Brockett’s condition gave rise to many examples, as a lot of controllable systems failed to adhere to this condition. This, and earlier work by Kurzweil [87], Wilson [160] and Bhatia and Szegö [10] can be seen as a start of the work on topological obstructions to stability and stabilization.

For more on the development of nonlinear controllability and related tools see [32, 71, 98, 153]. For more on the history of control theory, see [13] and for a historical account by Brockett himself, see [19]. See also [22, 60] for early works by Brockett and Hermann & Martin, respectively, highlighting the use of a topological and geometrical viewpoint in the context of system and control theory.

At last, we emphasize two additional schools. First, in the East, Krasnosel’skiĭ and coworkers elaborated during the second half of the 20th century on a blend of most of the aforementioned material in their study of topological methods in nonlinear analysis [84, 85]. As will be discussed below, the monograph by Krasnosel’skiĭ and Zabreĭko contains a variety of results related to arguably the most influential control-theoretic topological result produced in the West—known as “Brockett’s condition”, as discussed in detail in Chap. 6 cf. [84, Chaps. 7–8], [23]. As also pointed out in [113], although the translated version of their monograph appeared in 1984, the original Russian version appeared in 1975, well before that particular work by Brockett. Moreover, Krasnosel’skiĭ’s earlier monograph from 1968 [85] and a 1974 paper by Bobylev and Krasnosel’skiĭ [15] contain work instrumental to [84, Chaps. 7–8]. See [103] for more on the work of Krasnosel’skiĭ and [163] for an historical account by Zabreĭko. Secondly, Conley and coworkers developed their generalization of Morse- and Lyapunov theory in the late 1970s [35, 36], a topic we will only briefly touch on as it has been covered before.

What these works have in common is that they look for (algebraic) topological invariants that capture certain qualitative properties of spaces, maps, dynamical systems, and so forth. This viewpoint is at the core of this work.

Summarizing, the study into dimension and equivalences resulted in the development of a host of topological tools. Building on these tools and in part due to the inherent difficulty of solving differential equations brought about the qualitative theory of dynamical systems and control.

This brief historical overview leaves us in the 1980s. The upcoming chapters will discuss how the control-, topology- and dynamical systems communities responded over the last 40 years and what can be learned from that body of work.

1.3 Case Study: Optimal Control on Lie Groups

To illustrate the developments we consider a problem simple enough to do explicit computations, but rich enough to be of importance. Specifically, we work with Lie groups, objects ubiquitous in engineering and physics [3, 14, 28, 109, 135].

A pair \((\textsf{G},\cdot )\), with \(\textsf{G}\) a set and \(\cdot \) a binary operation, is a Lie group when

  1. (i)

    the set \(\textsf{G}\) is a group under \(\cdot \), that is, \(g\cdot h\in \textsf{G}\) for all \(g,h\in \textsf{G}\), there is an identity element \(e\in \textsf{G}\) such that \(e\cdot g= g\cdot e = e\) for any \(g\in \textsf{G}\), for all \(g\in \textsf{G}\) there is an inverse element \(g^{-1}\in \textsf{G}\) such that \(g\cdot g^{-1}=g^{-1}\cdot g = e\) and the \(\cdot \) operation is associative;

  2. (ii)

    the set \(\textsf{G}\) is a smooth manifold (informally, a set that is locally Euclidean and possesses a structure to make sense of derivations, see Chap. 2-3 for the details) and both multiplication and inversion are smooth maps.

When it clear from the context, the operator \(\cdot \) is dropped, i.e., one writes gh instead of \(g\cdot h\). For example, a Lie group of importance is the special orthogonal group \(\textsf{SO}(n,\mathbb {R})=\{A\in \mathbb {R}^{n\times n}:A^{\textsf{T}}A=I_n,\,\textrm{det}(A)=1\}\). Here the group operation \(\cdot \) is matrix multiplication and for any \(Q\in \textsf{SO}(n,\mathbb {R})\), the corresponding inverse element is \(Q^{\textsf{T}}\) with the identity element being \(e=I_n\), for \(I_n\) the identity matrix in \(\mathbb {R}^{n\times n}\).

To every Lie group \(\textsf{G}\) corresponds a Lie algebra, denoted \(\mathfrak {g}\), being a vector space identified with the tangent space of \(\textsf{G}\) at e (a vector space to be made precise in Chap. 3), denoted \(\mathfrak {g}=T_e\textsf{G}\). For example, \(\mathfrak {so}(n,\mathbb {R})=T_{I_n}\textsf{SO}(n,\mathbb {R})=\{X\in \mathbb {R}^{n\times n}:X^{\textsf{T}}+X=0\}\). Lie algebras are powerful for us in that they parametrize the tangent space of \(\textsf{G}\) at any \(g\in \textsf{G}\). To see this, pick any differentiable curve \(t\mapsto \gamma (t)\in \textsf{G}\) such that \(\gamma (0)=e\). As we work with an abstract binary operator on \(\textsf{G}\), it is convenient to define the left-translation map \(L_g\) by \(h\mapsto L_g(h)=gh\) for any \(g,h\in \textsf{G}\). Now, define the curve \(t\mapsto c(t)=L_{g}(\gamma (t))\in \textsf{G}\). Then, as \(c(0)=g\), the derivative of c with respect to t satisfies \(\dot{c}(t)|_{t=0}=DL_{g}(h)|_{h=e}\dot{\gamma }(t)|_{t=0}\in T_g\textsf{G}\) such that we have the (tangent space) isomorphism \(D(L_{g})_e:T_e\textsf{G}\rightarrow T_g\textsf{G}\) for all \(g\in \textsf{G}\).

Now, let \(\textsf{G}\) be a compact connected Lie group and let \(\langle \cdot , \cdot \rangle \) denote an \(\textrm{Ad}\)-invariant inner-product on \(\mathfrak {g}\) (the adjective “\(\textrm{Ad}\)-invariant” can be ignored if unrecognized), which always exists as \(\textsf{G}\) is compact [82, Proposition 4.24]. A vector field X on \(\textsf{G}\) is said to be left-invariant when \(D(L_g)_e X(e) = X(g) \in T_g\textsf{G}\) for all \(g\in \textsf{G}\), differently put, the evaluation of the vector field at \(e\in \textsf{G}\), defines the vector field on all of \(\textsf{G}\). The set of left-invariant vector fields is denoted by \(\textsf{Lie}(\textsf{G})\) and is isomorphic to \(\mathfrak {g}\). For a visualization of the aforementioned concepts, see Fig. 1.3.

Fig. 1.3
figure 3

Section 1.3 (i) curves and tangent spaces on the Lie group \(\textsf{G}\); (ii) a vector field \(X\in \textsf{Lie}(\mathbb {S}^1)\)

Consider for a set \(\{X_0,\dots ,X_m\}\subset \textsf{Lie}(\textsf{G})\) the input-affine control system on \(\textsf{G}\)

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}{g}(t) = X_0\left( g(t) \right) + {\textstyle {\sum \limits ^m_{i=1}}}X_i\left( g(t) \right) u_i, \end{aligned}$$

with \(\textrm{span}\{X_1,\dots ,X_m\}=\textsf{Lie}(\textsf{G})\) and the input vector \(u\in \mathbb {R}^m\). As such, (1.1a) is controllable for controls \(t\mapsto \mu (t)\in \mathbb {R}^m\) that are locally bounded and measurable [76, Theorem 7.1] (informally, one speaks of controllability of (1.1a) when for any \(g_0,g_1\in \textsf{G}\), there is a \(T\ge 0\) and a map \(\mu :[0,T]\rightarrow \mathbb {R}^m\) such that a solution \(\varphi :[0,T]\times \textsf{G}\rightarrow \textsf{G}\) to (1.1a) under \(\mu \) satisfies \(\varphi (0,g_0)=g_0\) and \(\varphi (T,g_0)=g_1\), for the precise definition see Chap. 5). Under these assumptions, one can study without loss of generality, the left-invariant control system

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}{g}(t) = D(L_{g(t)})_e ( \mu (t) ), \end{aligned}$$

where \(\mu (t)=\sum \nolimits ^m_{i=1}E_i \mu _i(t)\) and \(\textrm{span}\{E_1,\dots ,E_m\}=\mathfrak {g}\). Then, given a discount factor \(\beta > 0\), and with abuse of notation the exponential function \(e^{-\beta t}\), define the infinite-horizon optimal control problem

$$\begin{aligned} \left\{ \begin{aligned} \mathop {\mathrm {\text {minimize}}}\limits _{\mu (\cdot )}\quad&\int \limits ^{\infty }_0 e^{-\beta t} \left( d(e,g(t))^2 + \langle \mu (t),\mu (t)\rangle \right) \textrm{d}t\\ \mathop {\mathrm {\text {subject to}}}\limits \quad&(1.1b),\quad g(0)=g_0, \end{aligned}\right. \end{aligned}$$

where the distance function \(d(e,g)^2=\langle \log (g),\log (g)\rangle \) between e and any \(g\in \textsf{G}\) is defined with the aim of finding a feedback via (1.2) that stabilizes e in some sense. This construction is intended to generalize Linear-quadratic regulation (LQR) to nonlinear- systems and spaces, cf. [7, 8]. Note that \(\log :\textsf{G}\rightarrow \mathfrak {g}\) is only well-defined over the subset of \(\textsf{G}\) where \(\textrm{exp}:\mathfrak {g}\rightarrow \textsf{G}\) is injective. Now, one can show that by construction of (1.2), one can appeal to Hamilton–Jacobi–Bellman (HJB) theory which provides necessary optimality conditions for (1.2), e.g., see [18, Theorem 10.2]. Then, it can be shown that the optimal controller in (1.2) is given by \(\mu ^{\star }(g(t))=-p \log (g(t))\) for \(p> 0\) satisfying \(-\beta p -p^2 +1 =0\), under the assumption that the controlled trajectory does not pass through the singularity of \(\textrm{exp}:\mathfrak {g}\rightarrow \textsf{G}\), see [52, Theorem 4]. In fact, as \(\mu ^{\star }(e)=0\), under the aforementioned assumption, the feedback \(\mu ^{\star }\) renders e (locally) asymptotically stable [52, Theorem 5].

On the basis of this example we will further illustrate various concepts, including, but not limited to: (i) the relation, or lack thereof, between controllability and the existence of continuous feedback; (ii) the source and (in)surmountability of discontinuous controllers; and (iii) the relation between the shape of the attractor and the domain of the dynamical system.

1.4 Content and Structure

This work surveys the inception, development and future of topological obstructions in the context of dynamical control systems. The aim is to present a unified and general treatment. As such, highly specialized results, as are known for surfaces, do not belong to the core of this work. Also, we largely focus on manifolds but indicate when results hold for more general topological spaces. Besides providing a review, a secondary goal of this work is to function as an invitation to the non-specialist.

In the past, a small number of reviews appeared, for example, on low dimensional systems by Dayawansa [39]. Close to us is the work by Sánchez-Gabites [133] and Sanjurjo [134], albeit mostly focused on shape and Conley theory. The work by Byrnes [30] and later by Kvalheim and Koditschek [88] also contain overviews, but mostly focused on generalizations of Brockett’s condition. Topological obstructions are also briefly discussed in for example [141, Sect. 6], [19, Sect. 8], the monograph by Coron [37, Part 3], the monograph by Sontag [142, Sect. 5.9], the monograph by Zabczyk [162, Sect. 7.6] and the extensive survey by Vakhrameev [153] on the development of geometric methods in the study of controllability and optimal control. See also the introductions to [38, 108] and the voluminous work by Jonckheere [73] on algebraic topology and robust control. At last we highlight the thesis by Mayhew [104], containing hybrid- obstructions and solutions.

Regarding the exposition, we follow the philosophy as set forth in [51] and provide mostly arguments from differential topology with the aim of having an audience as large as possible that can follow and appreciate the complete development. Wherever insightful we do indicate how results can also be shown using arguments from algebraic topology. To further help the reader we provide ample examples, illustrations and references. Most of the results presented in this survey are already published and we systematically add a reference to which the reader can refer for more details. Some new results are added to complete those published and in this case we add a complete proof. Known proofs are occasionally presented to precisely show where assumptions are used and how to possibly relax them.

Although we will impose a smooth structure on our objects we stay in the topological realm and rarely assume knowledge of a metric on our spaces. The price to pay for this generality is that few things can be quantified.

We start by introducing a substantial amount of preliminary concepts from topology, dynamical systems and control theory. The benefit being that the core of this work can be described without technical clutter and in a somewhat self-contained manner. After presenting the topological obstructions, we also highlight how one might deal with these obstructions and what is considered future work. In particular, Chap. 2 introduces notions from general topology, e.g., homotopies and retraction theory whereas Chap. 3 introduces the prerequisite machinery for the Poincaré–Hopf theorem and the Bobylev–Krasnosel’skiĭ index theorem, that is, notions from differential topology like transversality, tubular neighbourhoods and index theory are discussed in detail. Then, Chap. 4 briefly presents material from algebraic topology and states how the Euler characteristic can be seen through different lenses, e.g., via self-intersections, combinatorially, or via homology theory. Chapter 5 introduces notions from dynamical systems theory like flows, vector fields and Lyapunov stability. Moreover, the dynamical control systems under consideration are defined. Chapter 6—the core of this work—is devoted to discussing topological obstructions to stability and stabilization. First, for equilibrium points, then for submanifolds and subsequently for generic sets. In particular, this section aims to show that just a few viewpoints allow for generalizing a wealth of results. Chapter 7 presents an overview of how to work with these obstructions, e.g., by allowing for singularities, time-dependent feedback or by employing techniques from hybrid control theory. Elaborating on some of the aforementioned tools, Chap. 8 offers a few generalizations and concludes with a list of future work.

Notation: We largely follow standard textbook notation, e.g., [51, 92, 142], but we state already that \(p\in \textsf{M}^n\) denotes an element of a n-dimensional manifold \(\textsf{M}^n\) with the variable x being reserved for the state of a dynamical system. The symbols f and F are reserved to describe those dynamical systems, whereas g and G are used for general maps. When working with differential equations we use \(\textrm{d}\xi (t)/\textrm{d}t\), \(\dot{\xi }(t)\) or simply \(\dot{\xi }\) to denote the “time”-derivative. Also, \(F_{*}\) will denote the pushforward of a map, whereas \(G_{\star }\) denotes the induced homomorphism between groups. Any subtle difference in notation will always be accompanied by clarifying text.