Topological Obstructions

Abstract

Given the previous chapters on topology and dynamical system theory, we can now provide a consistent treatment of topological obstructions to stability and stabilization. In particular, we will see how the retraction theory from Sect. 2.2 and the index results due to Bobylev, Krasnosel’skiĭ and Zabreĭko from Sect. 3.5 cover most of the existing results, conceptually. The current chapter focuses on the obstructions themselves and the next chapter highlights potential solutions.

Given a continuous control system $\Sigma =(\textsf{M},\mathcal {U},F)$, or merely the manifold $\textsf{M}$, in this chapter we consider stability and stabilization on three levels.

• (i) What is the topology of an admissible attractor $A\subseteq \textsf{M}$?

• (ii) Given an attractor A, what is the topology of the domain of attraction $\mathcal {D}\subseteq \textsf{M}$?

• (iii) Given the pair $(A,\mathcal {D})$, what kind of dynamics are imposed on $\textsf{M}\setminus \mathcal {D}$?

Regarding the notions of stability and stabilization, we primarily focus on (uniform) asymptotic stability and continuous static feedback, but note in passing if these obstructions prevail under other stabilization paradigms. To summarize the list of above, see Fig. 6.1, we want to understand how the topology of $\textsf{M}$ influences stability and stabilization possibilities.

We will address Questions (i)–(iii) above, by looking at the stabilization of points, submanifolds and generic sets. The obstructions differ in being of a local or global nature.¹ Moreover, some require knowledge of the vector field (dynamical system) at hand, whereas other obstruction hold for example for any continuous flow on $\textsf{M}$. For instance, with respect to the stabilization of some equilibrium point, an elementary necessary condition on $\Sigma$ is that $F(\mathcal {U})\cap Z_{\pi _p}(\textsf{M})\ne \emptyset$, or stronger, for a particular point $p^{\star }\in \textsf{M}$ it must be true that $(p^{\star },0)\in F(\pi _u^{-1}(p^{\star }))$. Similarly, for the stabilization of some invariant submanifold $\textsf{A}\hookrightarrow \textsf{M}$ one needs $F(\pi _u^{-1}(a))\cap (a,T_a\textsf{M}|_{\textsf{A}})\ne \emptyset$ for all $a\in \textsf{A}$. For a generic compact attractor $A\subseteq \textsf{M}$, however, one needs that for all $p\in \partial A$ the set $F(\pi _u^{-1}(p))$ does not only contain pairs (p, v) with v pointing outward of A, which is asserted in local coordinates. Similar conditions can be stated in the language of involutive distributions² on $\textsf{M}$ [94]. It turns out that incorporating topological aspects of the control system $\Sigma$ results in significantly deeper insights than the mere existence of equilibrium points and so forth.

Fig. 6.1

Given $\textsf{M}$, (i) what is the topology (to be understood as the homotopy type) of the attractor; (ii) what is the topology of the domain of attraction; (iii) how does this fit into $\textsf{M}$, what kind of qualitative dynamics are imposed outside of $\mathcal {D}$?

One of the first principled overviews towards answering Question (ii) appeared in the book by Bhatia and Szegö [10, Sect. V.3]. Early comments on how the global topology in combination with the Poincaré–Hopf theorem imposes an obstruction for stabilization appeared in 1988 by Koditschek [71]. Similar comments appeared in [72] with respect to navigation problems and obstacle avoidance. As the focus is on generic results, we omit early specialized (low-dimensional) results, e.g., early work by Dayawansa,³ see [38] and references therein. We also omit explicit discussions about obstructions due to quantization and the like.

With respect to Question (i), we mostly focus on compact attractors, the main reason being that asymptotic stability of a compact attractor can be characterized solely by topological means, cf. Sect. 5.1. Besides, a set of equilibrium points is closed [10, Theorem II 2.7], so that if $\textsf{M}$ is compact, any subset $A\subseteq \textsf{M}$ on which some vector field vanishes is necessarily compact.⁴ See also [79] for how compactness of A turned out to be a necessity in correcting a topological result due to Wilson.

6.1 Obstructions to the Stabilization of Points

Equipped with the tools from Chaps. 3–5, we start by considering the stabilization of an (equilibrium) point $p\in \textsf{M}$, either locally, or globally. We emphasize that locally asymptotically stable equilibrium points are isolated, yet, to aid the reader we do occasionally keep the adjective isolated.

6.1.1 Local Obstructions

In this section we mostly focus on local nonlinear models of the form $\Sigma ^{\textrm{loc}}_{n,m}$ (5.7) with $f(0,0)=0$ and seek control strategies such that $x^{\star }=0$ is an isolated and locally asymptotically stabilized equilibrium point.

We start with a few local necessary conditions for local asymptotic stabilization by means of continuous control laws. In particular, we look at Brockett’s (degree) condition, Zabczyk’s (index) condition and Coron’s (homological) condition.

Consider some non-surjective map $g:\textsf{X}\rightarrow \textsf{Y}$, as there must be a point $y\in \textsf{Y}$ such that $g^{-1}(y)=\emptyset$, y is a regular value and $\textrm{deg}(g)=0$ cf. Sect. 3.5. Now, by Example 3.4 we know that isolated asymptotically stable equilibrium points have non-zero index $(-1)^n$. Then, it follows from Lemma 3.3 and Example 3.2 that (in the Euclidean case) $\textrm{ind}_0(X)\ne 0$ implies that the vector field X is locally surjective around 0. The reason being that one can homotope $X(p)/\Vert X(p)\Vert _2$ to $(X(p)-p')/\Vert X(p)-p'\Vert _2$ for sufficiently small $p'$, preserving the degree and hence $X(p)=p'$ must have a solution for any sufficiently small $p'$. See also [131, Lemma 3]. Therefore, considering the control system $\Sigma ^{\textrm{loc}}_{n,m}$, the map $f:\mathbb {R}^n\times \mathbb {R}^m\rightarrow \mathbb {R}^n$ must at least be locally surjective from a neighbourhood of (0, 0) onto a neighbourhood of 0 to allow for a non-zero vector field index. Note that this is a necessary but not sufficient condition for the degree to be non-zero.

Given these observations, Brockett’s celebrated necessary condition follows.⁵

Theorem 6.1

(Brockett’s condition [18, Theorem 1.(iii)]) Let $\Sigma ^{\textrm{loc}}_{n,m}$ be a local continuous control system. Then, there is a continuous feedback $x\mapsto \mu (x)\in \mathbb {R}^m$ with $\mu (0)=0$ rendering $0\in \mathbb {R}^n$ locally asymptotically stable only if $(x,u)\mapsto f(x,u)$ is a surjective map from a neighbourhood of (0, 0) onto a neighbourhood of 0.

Proof

(Sketch) When the control system is smooth, assume to have knowledge of a smooth controller $\mu$ with $\mu (0)=0$ such that $f(x,\mu (x))$ is asymptotically stable. For $\textrm{ind}_0(f)\ne 0$ we clearly need $x\mapsto f(x,\mu (x))$ to be surjective.

See the proof of Theorem 6.2 for more on the purely continuous case, or see, [18, 116, Theorem 22]. In particular, after Zabczyk [131], Orsi et al. generalized Theorem 6.1 to the setting where the continuous vector field does not necessarily give rise to a (local) flow [95]. See also [12, Theorem 4] for a generalization to nonholonomic control systems. A composition operator theoretic study of feedback stabilization is undertaken by Christopherson, Mordukhovich and Jafari in [31]. Importantly, by building upon Hautus, Brockett and Zabczyk, their work goes beyond necessary conditions and towards sufficient conditions for asymptotic stabilization. See also that a bijective reparametrization of the input does not change the theorem qualitatively. As indicated in [18], the surprising element of Theorem 6.1 is that controllability does not seem to play a key role when it comes to sufficiency. Indeed, one can find examples of systems that are controllable, yet, not asymptotically stabilizable by a continuous feedback law.

Example 6.1

(The nonholonomic integrator) Consider the control system on $\mathbb {R}^3$ defined by

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}_1 =&u_1\\ \dot{x}_2 =&u_2\\ \dot{x}_3 =&x_2 u_1 - x_1 u_2, \end{aligned}\right. \end{aligned}$$

(6.1)

also referred to as the “Heisenberg system” [11, Sect. 1.8]. Regarding the equilibrium $x^{\star }=0\in \mathbb {R}^3$ under $u^{\star }=0\in \mathbb {R}^2$, although the linearization of (6.1) around $(0,0)\in \mathbb {R}^3\times \mathbb {R}^2$ does not provide a controllable linear system, the system—conveniently written as $\dot{x}=g_1(x)u_1+g_2(x)u_2$—is (globally) controllable as $\textrm{span}\left\{ g_1,g_2,[g_1,g_2]\right\} =\mathbb {R}^3$ and (6.1) is drift-free [94, Chap. 3]). Here, $[\cdot ,\cdot ]$ denotes the Lie bracket [78]. Nevertheless, $(x,u)\mapsto g_1(x)u_1+g_2(x)u_2$ is not surjective as one cannot map to $(0,0,\varepsilon )$ for any $\varepsilon \ne 0$. See [2, Sect. 4] for a class of controllable systems that cannot be stabilized by smooth feedback and see [11, 19] for further generalizations of this example.

Observe that in Example 6.1 the topological obstruction is implicit in the dynamics, one cannot move freely in $\mathbb {R}^3$. Or as put by for example Sontag in [117], the nonholonomic system imposes “virtual obstacles”. The fact that the linearization of (6.1) is not informative is inherent to nonholonomic systems. See [65, 123] for a principled methodology to obtain informative approximations using sub-Riemannian geometry. In fact, this viewpoint was exploited in the early work by Brockett on (6.1), cf. [17].

Moreover, note that Brockett’s condition prevails when one allows for dynamic feedback. Also note that Brockett’s condition essentially provides a necessary (but not sufficient) condition for $\textrm{ind}_0(f)\ne 0$, while we know more. This leads to Zabczyk’s index condition, where effectively one adapts Example 3.4 to the $C^0$ setting.

Theorem 6.2

(Zabczyk’s index condition [131]) Let $\Sigma ^{\textrm{loc}}_{n,m}$ be a local continuous control system. If a continuous feedback $x\mapsto \mu (x)\in \mathbb {R}^m$ with $\mu (0)=0$ renders $0\in \mathbb {R}^n$ locally asymptotically stable, then, the vector field index of $x\mapsto f(x,\mu (x))$ equals $(-1)^n$.

Proof

For flows this follows directly from Example 3.4. Momentarily ignoring completeness, due to asymptotic stability a smooth Lyapunov function must exist [49, 74, 126]. Following [34, p. 291], let V be a smooth Lyapunov function defined on some open ball $\mathbb {B}^n_{\varepsilon }(0)$ around 0. By construction one has $\langle \partial _x V(x),f(x,\mu (x))\rangle <0$ for all $x\in \mathbb {B}^n_{\varepsilon }(0)\setminus \{0\}$. Hence, $\partial _x V(x)\ne 0$ for all $x\in \mathbb {B}^n_{\varepsilon }(0)\setminus \{0\}$. In fact, V is Lyapunov function for $\dot{x}=-\partial _x V(x)$, hence $\textrm{ind}_0(-\partial _x V)=(-1)^n$. Now we construct the map $H(s,x)=-s\partial _x V(x) + (1-s)f(x,\mu (x))$, which is a vector field homotopy since $\langle H(s,x),\partial _x V(x)\rangle < 0$ for all $s\in [0,1]$ and $x\in \mathbb {B}^n_{\varepsilon }(0)\setminus \{0\}$. Hence, $\textrm{ind}_0(f)=\textrm{ind}_0(-\partial _x V)=(-1)^n$.

Observe that the homotopy constructed in the proof above preserves stability along the homotopy since $\langle H(s,x),\partial _x V(x)\rangle < 0$ for all $s\in [0,1]$. In other words, f can be transformed into the negated gradient flow $-\partial _x V = -\nabla V$ through qualitatively equivalent dynamical systems. Understanding when this can be done is of independent interest and has been studied in the context of gradient flows [104], locally [32, Chap. 9] and under convexity assumptions [67]. See Sect. 8 for open questions.

Although Zabczyk’s condition is stronger than Brockett’s condition, it is not necessary for dynamic feedback in the sense that the “non-lifted” dynamical control system $\dot{x}=f(x,u)$ might fail to satisfy the index condition while a stabilizing dynamic feedback exists [35]. The intuition is that the auxiliary state representing the input does not need to converge to 0 in an asymptotically stabilized manner. A secondary contribution of Zabczyk’s work was to streamline arguments via bringing the work due Krasnosel’skiĭ and Zabreĭko to the attention, e.g., [73, Theorem 52.1]. An extension to constrained (unilateral) dynamical systems is presented in [51]. Now recall Example 3.2, then geometrically, Theorem 6.2 states that one needs to be able to find a continuous map $\mu$ such that the closed-loop vector field f “points in the direction” of a vector field $f'$ with $\textrm{ind}_0(f')=(-1)^n$. This is visualized in Fig. 6.2. Note, outward pointing cones would also work here as $(1)^{n=2}=(-1)^{n=2}$.

Fig. 6.2

Given a nominal vector field $f'$ with $\textrm{ind}_0(f')=(-1)^n$. Then all vector fields f that are point-wise contained in blunt, (salient) convex cones, containing $f'$, have the same index as $f'$

Zabczyk’s condition is unfortunately not necessarily easy to check. Then, Coron’s homological index condition captures the fact that the index should be $\pm 1$ via a condition on the vector field f, whose proof follows from the homological interpretation of the degree as sketched in Sect. 4.1, see also [59, Sect. 2.2]. Here we recall that $f_{\star }$ denotes the induced homomorphism.

Theorem 6.3

(Coron’s condition [33, Theorem 2]) Let $\Sigma ^{\textrm{loc}}_{n,m}$ be a local continuous control system with $n\ge 2$ and $m\ge 1$. Assume that f is continuous over some neighbourhood $\Omega$ of $(0,0)\in \mathbb {R}^n\times \mathbb {R}^m$ and define $\Omega _{\varepsilon }=\{(x,u)\in \Omega : f(x,u)\ne 0,\, \Vert x\Vert<\varepsilon ,\,\Vert u\Vert <\varepsilon \}$ with $\varepsilon \in \mathbb {R}_{>0}\cup \{+\infty \}$. If there is a continuous feedback $x\mapsto \mu (x)\in \mathbb {R}^m$ with $\mu (0)=0$ rendering $0\in \mathbb {R}^n$ locally asymptotically stable, then, for any $\varepsilon \in \mathbb {R}_{>0}\cup \{+\infty \}$

$$\begin{aligned} f_{\star }(H_{n-1}(\Omega _{\varepsilon };\mathbb {Z})) = H_{n-1}(\mathbb {R}^n\setminus \{0\};\mathbb {Z}) \quad (\simeq \mathbb {Z}). \end{aligned}$$

(6.2)

Proof

(Sketch) Assume that there is a continuous feedback rendering 0 locally asymptotically stable, say $\mu$, with $\mu (0)=0$. By continuity, there must be a sufficiently small $\delta >0$ such that for all $x\in \textrm{cl}\,\mathbb {B}^n_{\delta }(0)\setminus \{0\}$ one has $(x,\mu (x))\in \Omega _{\varepsilon }$.

Let $g(x)=f(x,\mu (x))$ and $v(x)=(x,\mu (x))$, then, the diagram above commutes. The map g represents the closed-loop vector field and as such, the vector field index of g, with respect to 0, over a sufficiently small open neighbourhood U of 0, must be $(-1)^n$. Differently put, we know that for sufficiently small $\delta$, g is vector field homotopic to $-\textrm{id}$ on $\textrm{cl}\,\mathbb {B}^n_{\delta }(0)\setminus \{0\}$, see either Example 3.4, [34, Proof of Theorem 11.4] or [116, Proof of Theorem 22]. Indeed, one can also first homotope to a smooth vector field if desired, e.g., to $-\nabla V$. Now, recall (4.2), as $H_{n-1}(\textrm{cl}\,\mathbb {B}^n_{\delta }(0)\setminus \{0\};\mathbb {Z})\simeq \mathbb {Z}$, the induced homomorphism $g_{\star }$ simply becomes

$$\begin{aligned} g_{\star }(H_{n-1}(\textrm{cl}\,\mathbb {B}^n_{\delta }(0)\setminus \{0\};\mathbb {Z})) = H_{n-1}(\mathbb {R}^n\setminus \{0\};\mathbb {Z}) \quad (\simeq \mathbb {Z}). \end{aligned}$$

Then the result follows by commutativity of the diagram, in particular the compositional property of the induced homomorphism(s).

As highlighted in the proof sketch above, the condition $n\ge 2$ relates to the same condition in (4.2) ($H_0(\mathbb {S}^0;\mathbb {Z})\not \simeq \mathbb {Z}$). Indeed, it suffices again to find an admissible continuous feedback law such that $x\mapsto f(x,\mu (x))$ has a $(-1)^n$ vector field index, cf. Theorem 6.2. Here, one must also check controllability or better yet stabilizability, otherwise see that $\dot{x}=x$ and $\dot{x}=-x$ both satisfy the aforementioned conditions (index $\pm 1$), yet, the two systems have opposite qualitative properties. On $\mathbb {R}^{2n}$ both systems even have index $1=(-1)^{2n}$. See [34, Exercise 11.7] for a typical example that satisfies Brockett’s condition but fails to satisfy Coron’s condition. As will be discussed in Sect. 7.2, time-varying feedback is frequently a solution. Generalizations of the index condition for homogeneous stabilization are presented in [114].

Surprisingly, Brockett’s condition even prevails when certain forms of discontinuous feedback are allowed, as shown by Ryan [107]. Coron and Rosier showed a similar obstruction by concurrently linking the existence of a stabilizing discontinuous feedback to that of a stabilizing time-varying feedback [37], [36, Remark 7]. Discussing these obstructions in detail is outside the scope of this work as one needs to specify what a solution to a discontinuous differential equation (inclusion) means. Moreover, Ceragioli showed that this specification itself influences the obstructions [28].

Remark 6.1

(Robustness/fragility) Consider for some $\epsilon >0$ the following perturbation of (6.1)

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}_1 =&u_1\\ \dot{x}_2 =&u_2\\ \dot{x}_3 =&\epsilon x_1 + x_2 u_1 - x_1 u_2. \end{aligned}\right. \end{aligned}$$

(6.3)

In contrast to (6.1), the linearization of this system, around $(0,0)\in \mathbb {R}^3\times \mathbb {R}^2$, is controllable from the origin for any $\epsilon > 0$. Yet, after linearizing (6.3) in a neighbourhood of (0, 0), one finds that the smallest singular value of

$$\begin{aligned} \left[ \begin{array}{c c c | c c} 0 &{} 0 &{} 0\,\, &{}\,\, 1 &{} 0\\ 0 &{} 0 &{} 0\,\, &{}\,\, 0 &{} 1\\ \epsilon -u_2 &{} u_1 &{} 0 \,\,&{}\,\, x_2 &{} -x_1 \end{array} \right] , \end{aligned}$$

subject to $u_1=u_2=0$, is proportional to $\epsilon$. Similarly, consider the Artstein circles

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}_1=&x_1^2- x_2^2\\ \dot{x}_2 =&2x_1x_2 \end{aligned}\right. . \end{aligned}$$

(6.4a)

Here, $0\in \mathbb {R}^2$ is globally attractive (if considered on $\mathbb {S}^2$), but not globally asymptotically stable and indeed, the corresponding vector field index is 2, as in Fig. 3.4(iv). Again, we perturb the system, this time such that 0 is locally asymptotically stable for any $\epsilon >0$:

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}_1=&-\epsilon x_1+x_1^2- x_2^2\\ \dot{x}_2 =&-\epsilon x_2 + 2x_1x_2, \end{aligned}\right. \end{aligned}$$

(6.4b)

see Fig. 6.3. Computing the linearization for (6.4b), one obtains the following matrix representation of the linear map

$$\begin{aligned} \begin{bmatrix} 2x_1-\epsilon &{} -2x_2\\ 2x_2 &{} 2x_1-\epsilon \end{bmatrix}. \end{aligned}$$

Therefore, around $0\in \mathbb {R}^2$, the $\epsilon$-perturbation turned 0 into a hyperbolic equilibrium point. Yet, in a neighbourhood around 0, the smallest singular (eigen) values are proportional to $\epsilon$.

Fig. 6.3

Remark 6.1, (i): (6.4a) versus (ii): (6.4b) for $\epsilon =1$

These examples relate to the fact that hyperbolic equilibrium points are generic and have a vector field index of $\pm 1$. Similarly, recall Thom’s transversality theorem (Theorem 3.1) and consider the genericity of smooth maps $f:\mathbb {R}^n\times \mathbb {R}^m\rightarrow \mathbb {R}^n$ such that $f\pitchfork \{0\}$. As such, applying the aforementioned conditions to models obtained, for example, numerically, requires great care.

Also, the original version of Theorem 6.1 [18, Theorem 1.(iii)] by Brockett was stated under a $C^1$ assumption. Although we only present the third condition, the first condition of [18, Theorem 1] further illuminates the interest in continuous feedback laws. Specifically, this condition states that the linearized system must not have unstable uncontrollable modes, not to obstruct the existence of a differentiable feedback cf.  the Hartman–Grobman theorem [106, Chap. 5]. This condition, however, does not rule out the existence of continuous feedback laws, e.g., in [103] Qian and Lin exploit homogeneity and a cascadic structure to construct continuous controllers for systems that might not admit a smooth controller. See also the earlier work by Aeyels [2], Kawski [70] and Dayawansa, Martin and Knowles [40].

Example 6.2

(Smooth versus continuous feedback) Consider [69, Example 2], that is, the control system on $\mathbb {R}^2$ defined by

$$\begin{aligned} \left\{ \begin{aligned} \dot{x}_1 =&4x_1 + u x_2^2\\ \dot{x}_2 =&x_2 + u, \end{aligned}\right. \end{aligned}$$

(6.5)

with $u\in \mathbb {R}$. The control system can be shown to be controllable, yet, no smooth, asymptotically stabilizing feedback $x\mapsto \mu (x)\in \mathbb {R}$ exists as the differential of (6.5) around 0 will have an unstable mode. However, (6.5) does satisfy Brockett’s condition and also Coron’s condition. After a feedback transformation, a non-differentiable change of coordinates and a reparametrization of time, (6.5) becomes

$$\begin{aligned} \left\{ \begin{aligned} \dot{z}_1 =&z_1 - z_2^3\\ \dot{z}_2 =&v, \end{aligned}\right. \end{aligned}$$

(6.6)

for the new input term $v\in \mathbb {R}$. Then, the results due to Kawski provide for an explicit continuous stabilizing feedback [70] for the control system (6.6).

6.1.2 Global Obstructions

Now we have the machinery to present the first illustrative global topological obstruction with respect to the domain of attraction.

Theorem 6.4

(Sontag’s condition [116, Theorem 21]) Let $p^{\star }$ be a locally asymptotically stable equilibrium point of $X\in \mathfrak {X}^0(\textsf{M})$. Then, the set $\mathcal {D}(\varphi _X,p^{\star })$ as given by (5.3) is open and contractible to $p^{\star }$.

Proof

(Sketch) Openness follows from continuity of $\varphi _X^t$. Regarding the contractibility, the natural candidate for the homotopy would be the flow with time being rescaled from $\mathbb {R}_{\ge 0}$ to [0, 1], i.e., $H(t,p)=\varphi _X(t/(1-t),p)$ with $t\in [0,1)$ and a well-defined limit. For the full proof see [116, Theorem 21] or the remarks below Theorem 6.12.

Theorem 6.4 indicates that when $\textsf{M}$ is not contractible, there is no $p^{\star }\in \textsf{M}$ that happens to be globally asymptotically stable under the flow of some complete vector field $X\in \mathfrak {X}^r(\textsf{M})$. Indeed, $\mathbb {R}^n$ is contractible and global asymptotic stabilization of an equilibrium is not immediately obstructed on such a space. It is instrumental to remark that asymptotic stability is exploited in Theorem 6.4. Moreover, we like to recall an example due to Takens [119, p. 231], responding to a question of Thom, showing that there is a (polynomial) gradient vector field X such that the topology of the set $\Gamma =\{p\in \textsf{M}:\lim _{t\rightarrow \infty }\varphi _{X}^t(x)=p^{\star }\}$, for some equilibrium point $p^{\star }\in \textsf{M}$, is not necessarily invariant under a change of Riemannian metric (intuitively, a change of coordinates). Indeed, the equilibrium point considered is not asymptotically stable.

Akin to (5.3) one defines $\mathcal {D}(\varphi _X,A)$ for A a compact attractor, cf. Chapter 5. Early documented results on the topology of attractors and their domain of attraction can be found in [10]. For example, in the Euclidean setting, $A\subset \mathbb {R}^n$ can only be a globally asymptotically stable compact set if $\mathbb {R}^n\setminus A$ is homeomorphic to $\mathbb {R}^n\setminus \{0\}$ [10, Theorem V 3.6]. Moreover, if $p^{\star }\in \mathbb {R}^n$ is globally asymptotically stable under some flow $\varphi$, then $\mathcal {D}(\varphi ,p^{\star })\setminus \{p^{\star }\}$ is homeomorphic to $\mathbb {R}^n\setminus \{0\}$ [10, Corollary V 3.5]. Also, when A is a compact attractor, we have that A necessarily consists out of finite components [10, Theorem V 1.22]. Now, as homotopies preserve path-connected components, when $A\subset \textsf{M}$ consists out of more path-connected components than $\textsf{M}$, there is no continuous vector field on $\textsf{M}$ with A globally asymptotically stable. Interestingly, globally asymptotically stabilizing a disconnected set cannot be achieved using robust hybrid feedback either [111]. Although outside the scope of this work, see [50] for related results in the infinite-dimensional setting.

Theorem 6.4 immediately implies that vector fields over non-contractible spaces do not admit globally asymptotically stable equilibrium points. This observation clarifies the obstructions we recovered for the circle $\mathbb {S}^1$ as shown in Figure 1.1. Moreover, as contractible spaces $\textsf{M}$ are homotopy equivalent to a point they have (singular homological) Euler characteristic $\chi (\textsf{M})=1$, e.g., see (4.4).

Example 6.3

(Case study Sect. 1.3: global stabilization is obstructed) As was shown in Example 3.5, all compact Lie groups $\textsf{G}$ have $\chi (\textsf{G})=0$, hence they are not contractible and global asymptotic stabilization by means of continuous feedback is prohibited. See Example 6.10 for non-compact examples.

As was the motivation for the case study, Example 6.3 is of importance in many dynamical systems grounded in mechanics as they can be frequently identified with Lie groups [5, 11, 22, 93, 113].

An immediate but appealing manifestation of this line of reasoning is the result by Bhat and Bernstein [9]. As with Brockett’s condition, their result contradicted what was thought to be true at the time (second half of the twentieth century), cf. [90, 108].

Theorem 6.5

(Bhat–Bernstein condition [9, Theorem 1]) Let $\pi :\textsf{M}\rightarrow \textsf{B}^b$ be a vector bundle for some smooth, compact, boundaryless, base manifold $\textsf{B}^b$ with $b\ge 1$, then, there is no continuous vector field on $\textsf{M}$ with an isolated globally asymptotically stable equilibrium point.

Proof

We know from Lemma 3.1 that compact manifolds of the form $\textsf{B}^b$ are never contractible. From Example 3.1 we know that $\textsf{B}^b$ is a deformation retract of $\textsf{M}$ so that $\textsf{M}$ itself can also not be contractible by homotopy equivalence. Then, an application of Theorem 6.4 concludes the proof.

The compactness of the base manifold can be relaxed to not being contractible and the vector bundle can be generalized to a fiber bundle.⁶ Theorem 6.5 is especially of use in second order dynamical systems, e.g., Lagrangian systems in robotics are frequently defined over compact manifolds [93]. Theorem 6.5 clearly obstructs dynamic feedback,⁷ and as was shown by Bernuau, Perruquetti and Moulay, also uniform stabilization by means of time-varying feedback [8].

Example 6.4

(Obstruction to time-varying stabilization [8]) Theorem 6.5 obstructs the existence of continuous dynamic feedback to globally asymptotically stabilize an isolated equilibrium point over a fiber bundle. The reason being that in the case of dynamic feedback, the augmented system is usually rendered asymptotically stable as a whole, e.g., when employing an observer. Now if we consider time-varying feedback e.g., $\dot{s}=1$, $\dot{x}=f(x,\mu (s,x))$, then, any solution will have its last component diverge to $+\infty$ for $t\rightarrow +\infty$, hence the previous argument breaks down. To study the situation we need to define partial stability. Let $\textsf{N}=\textsf{S}\times \textsf{M}$ be a smooth product manifold and let $X=(X_s,X_p)$ be a forward complete continuous vector field on $\textsf{N}$, giving rise to the semiflow $\phi _X$, that is, the domain of $\phi _X$ is $\mathbb {R}_{\ge 0}\times \textsf{N}$ instead of $\mathbb {R}\times \textsf{N}$, and let $\pi _2$ be the projection from $\textsf{N}$ onto $\textsf{M}$. Note that $X_s:\textsf{S}\times \textsf{M}\rightarrow T\textsf{S}$ captures more time-varying schemes than simply $\dot{s}=1$. Now, the point $x_{\infty }\in \textsf{M}$ is said to be a partial equilibrium of X when $X_p(s,x_{\infty })=0$ for all $s\in \textsf{S}$. Then, $x_{\infty }$ is partially stable uniformly in s when for any neighbourhood $U_{\epsilon }\subseteq \textsf{M}$ of $x_{\infty }$ there is a neighbourhood $U_{\delta }\subseteq \textsf{M}$ of $x_{\infty }$ such that for all $x\in U_{\delta }$ and for all $s\in \textsf{S}$ one has $\pi _2\circ \phi _X(t,(s,x))\in U_{\epsilon }$ for all $t\ge 0$. Now $x_{\infty }$ is partially globally asymptotically stable uniformly in s when it is partially stable uniformly in s and for all $(s,x)\in \textsf{N}$ one has $\pi _2 \circ \phi _X(t,(s,x))\rightarrow x_{\infty }$ for $t\rightarrow +\infty$. Regarding partial global asymptotic stability uniform in s, consider for example $\dot{x}=-xs$, $\dot{s}=1$. To continue, let $\pi :\textsf{M}\rightarrow \textsf{Q}$ be a vector bundle with $\textsf{Q}$ a $C^{\infty }$ compact base manifold without boundary and assume that a stabilizing time-varying feedback does exist, that is, there is a continuous vector field $X=(X_s,X_p)$ over $\textsf{N}=\textsf{S}\times \textsf{M}$ with some point $x_{\infty }\in \textsf{M}$ being partially globally asymptotically stable uniformly in s. Let $q_{\infty }=\pi (x_{\infty })$ and define $\sigma (q)=(s',\sigma '(q))$ for some section $\sigma '\in \Gamma ^0(\textsf{M})$ and some $s'\in \textsf{S}$. It follows that $\sigma$ is a section of $\pi \circ \pi _2:\textsf{N}\rightarrow \textsf{Q}$. By assumption, X gives rise to a semiflow $\phi _X$, then consider the map $H:[0,1]\times \textsf{Q}\rightarrow \textsf{Q}$ defined by

$$\begin{aligned} (\lambda ,q) \mapsto {\left\{ \begin{array}{ll} \pi \circ \pi _2 \circ \phi _X\left( \log \left( 1/(1-\lambda ) \right) ,\sigma (q)\right) &{}\text { if }\lambda \ne 1\\ q_{\infty } &{}\text { if }\lambda = 1. \end{array}\right. } \end{aligned}$$

Indeed, $H(0,q)=q$ and $H(1,q)=q_{\infty }$. However, as one can show that H is continuous, this homotopy contradicts Q being closed. Therefore, such a time-varying feedback cannot exist. Again, one can generalize the construction to fiber bundles under the assumption that a continuous section exists. All details can be found in [8] and see for instance [34, Sect. 11.2] for more on time-varying stabilization.

Furthermore, as mentioned in [9], Theorem 6.5 can be applicable to non-compact configuration spaces, as long as $\textsf{M}$ can be written as a (vector) bundle, e.g., let $\textsf{Q}^q$ be compact with a trivial tangent bundle, like $\mathbb {S}^1$, then the configuration space $\textsf{Q}^q\times \mathbb {R}^p$ leads to $\textsf{M}=T(\textsf{Q}^q\times \mathbb {R}^p)\simeq \textsf{Q}^q\times \mathbb {R}^{q+2p}$.

One of the objectives of this work is to clarify to what extent Theorem 6.5 prevails when the global stability condition is relaxed to local multistability conditions, or when we look beyond the stabilization of points. As such, we first consider a variety of necessary conditions which indicate if a (compact) manifold admits a continuous vector field with all of its zeroes being isolated locally asymptotically stable equilibrium points.

Theorem 6.6

(A global necessary condition for local stability) Let $\textsf{M}^n$ be a smooth, compact, boundaryless, finite-dimensional manifold. Then, $\textsf{M}^n$ admits a continuous vector field $X\in \mathfrak {X}^{r\ge 0}(\textsf{M}^n)$ with $z\in \mathbb {N}_{\ge 0}$ zeroes, all of which are isolated and locally asymptotically stable, if and only if $\chi (\textsf{M}^n)=z$.

Proof

See that the special case of $z=0$ is handled by Proposition 3.6. Then, first for the “only if” direction. Consider the case $z>0$, as the index of these locally asymptotically stable equilibrium points is $(-1)^n$ (see Example 3.4), we see that $\chi (\textsf{M}^n)$ must equal $z(-1)^n$. By Theorem 6.8, this cannot be true for odd-dimensional manifolds. Therefore, a necessary condition for local asymptotic multistability is that $\chi (\textsf{M}^n)=z$. For the “if” direction one can follow the same line of arguments as used to show Proposition 3.6, e.g., see [54, pp. 141–148]. The difference being that now one starts, in local coordinates, from a set of locally asymptotically stable equilibrium points such that sum of their indices equals $\chi (\textsf{M}^n)$. The equilibrium points that emerge when these local vector fields are patched together on $\textsf{M}^n$ are removed precisely as in a general proof of Proposition 3.6.

Theorem 6.6 generalizes [91, Theorem 25] and the necessary direction can also be shown along the lines of [72] or [87, Chap. 15], see also [109, Example 17].

The next example clarifies how compact manifolds $\textsf{M}$ with $\chi (\textsf{M})=1$ do not contradict the theory, global asymptotic stability remains obstructed.

Example 6.5

(Grassmannian manifolds) Let $\textsf{Gr}(k,n)$ denote the set of k-dimensional subspaces of $\mathbb {R}^n$, which is a smooth, compact, boundaryless, $k(n-k)$-dimensional, non-orientable manifold [1, Sect. 3.4.4]. These Grassmannian manifolds appear frequently in the context of manifold optimization [1, 14]. Interestingly, one can compute that $\chi (\textsf{Gr}(2,3))=1$, e.g., by considering the normal to a subspace one can identify ($\simeq _t$) $\textsf{Gr}(2,3)$ with the real projective plane $\mathbb {R}\mathbb {P}^2$, defined as $\mathbb {R}\mathbb {P}^2=(\mathbb {R}^3\setminus \{0\})/\sim$ with $p\sim \lambda p$ for any $\lambda \in \mathbb {R}\setminus \{0\}$, and consider a 2-sheeted covering of $\mathbb {R}\mathbb {P}^2$ by $\mathbb {S}^2$.⁸ Indeed, this implies that there is a vector field on $\textsf{Gr}(2,3)$ with a single locally asymptotically stable equilibrium point. However, as $\textsf{Gr}(2,3)$ is compact, it is not contractible, hence the point cannot be globally asymptotically stable or most of Chap. 3 would be contradicted. Indeed, again identify $\textsf{Gr}(2,3)$ with $\mathbb {R}\mathbb {P}^2=\mathbb {S}^2/\sim$, for $p\sim q$ when $p=-q$. Then consider the vector field as displayed in Fig. 6.4 and observe the periodic orbit on the equator. The resulting vector field on $\mathbb {R}\mathbb {P}^2$, or $\textsf{Gr}(2,3)$ for that matter, will have a non-trivial limit set.⁹ This observation is further discussed in Sect. 8.4.

The previous example is of mathematical interest, but interestingly, the human eye (gazing) can be modelled as a control system on a space diffeomorphic to $\mathbb {R}\mathbb {P}^2$  [101]. Then, with respect to Example 6.5, we leave it to the reader to infer potential physical ramifications, e.g., does the non-trivial ($\alpha$)-limit set relate to nystagmus (unintentional oscillatory movement of the eye)?

In line with the work on almost global stability, Theorem 6.6 provides motivation for abandoning local asymptotic multistability as many spaces have their Euler characteristic being equal to 0. A straightforward necessary conditions follows.

Fig. 6.4

In line with $\chi (\mathbb {S}^2)=2$ and Theorem 6.6, a vector field on $\mathbb {S}^2$ with two isolated locally asymptotically stable equilibrium points $(p_1^{\star },p_2^{\star })$ and an unstable periodic orbit $\mathcal {O}$. This vector field also exemplifies the Poincaré–Bendixson theorem, locally, e.g., see [98, Theorem 1.8]

Corollary 6.1

(A necessary condition for local asymptotic multistability) Any, smooth, compact, boundaryless, finite-dimensional manifold $\textsf{M}$ admits a continuous vector field $X\in \mathfrak {X}^{r\ge 0}(\textsf{M})$ with all of its $z\in \mathbb {N}_{> 0}$ zeroes being locally asymptotically stable isolated equilibrium points only if $\chi (\textsf{M})>0$.

As for manifolds like any odd-sphere $\mathbb {S}^{2n+1}$, the n-torus and any Lie group, $\chi (\textsf{M})=0$. Hence, for many spaces, local asymptotic multistability is impossible. Regarding a negative Euler characteristic, recall that closed orientable surfaces $\textsf{M}^2$ with genus g have $\chi (\textsf{M}^2)=2-2g$.

Remark 6.2

(Theorem 6.4 compared with Theorem 6.6) Consider any even n-sphere like $\mathbb {S}^2$, using Sontag’s condition one concludes that continuous global asymptotic stability is impossible on such a topological manifold. Relaxing the adjective global, Fig. 6.4 shows a locally asymptotically multistable vector field on precisely $\mathbb {S}^2$. Indeed, the necessary condition from Theorem 6.6 holds true as for this example $\chi (\mathbb {S}^2)=2$, which equals the number of isolated equilibrium points. Note that in the case of Fig. 6.4, the equator functions as an unstable equilibrium point, but by means of an unstable periodic orbit, not a point. Nevertheless, as most relevant compact manifolds $\textsf{M}$ have $\chi (\textsf{M})=0$, Theorem 6.4 extends to multistability in the sense that for most compact manifolds local asymptotic multistability is prohibited.

Example 6.6

(Opinion dynamics) In [6], the authors study opinion dynamics on the compact manifolds $\mathbb {S}^1$, $\mathbb {S}^2$ and $\mathbb {T}^2$ with the purpose of understanding how the underlying state space influences the formation of opinions. Omitting a few details, the authors consider

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}{x}_i(t) = \textstyle \sum ^N_{j=1} a_{ij}\Psi \left( d(x_i(t),x_j(t)) \right) \nu _{ij} \end{aligned}$$

(6.7)

for $x=(x_1,x_2,\dots ,x_N)\in \times ^N_{i=1} \textsf{M}=(\textsf{M})^N$ denoting the set of opinions, $a_{ij}\in \mathbb {R}$ the interaction coefficients, $\Psi :\mathbb {R}\rightarrow \mathbb {R}$ the interaction potential with $\Psi (0)=0$, $d:\textsf{M}\times \textsf{M}\rightarrow \mathbb {R}_{\ge 0}$ a (Riemannian) distance on $\textsf{M}$ and $\nu _{ij}\in T_{x_i}\textsf{M}$ the direction of influence. Assume that $\Psi$ is selected such that the right-hand-side of (6.7) is continuous. As expected, the consensus setting $x_1=x_2=\cdots =x_N$ is a set of equilibrium points: the consensus manifold, denoted $\textsf{C}=\{x\in (\textsf{M})^N:x_1=x_2=\cdots =x^N\}$. Is consensus the the only equilibrium and is such an opinion stable? As $\chi ((\textsf{M})^N)=N\cdot \chi (\textsf{M})$, manifolds like $\mathbb {S}^2$ cannot only have consensus as equilibria. In fact, the upcoming obstructions to submanifold stabiliztion will indicate that on compact manifolds consensus is never a globally asymptotically stable stationary opinion when $\textsf{M}$ is not contractible.

Example 6.6 exemplifies the benefit of the topological approach, we can arrive at strong insights without any explicit study of (6.7).

Summarizing the above, global asymptotically stable equilibrium points are rare to encounter on nonlinear spaces. Indeed, in practice, global notions of stability rely on for instance exploiting model structure, optimality conditions and the existence of Lyapunov-like functions, e.g., in [124], the authors consider Jurdjevic–Quinn type systems. Chap. 7 presents more methods towards (almost) global stability.

6.1.3 A Local Odd-Number Obstruction to Multistabilization

In this section multistability is considered locally. In particular, the results are in the spirit of the work by Ortega [96] and closely related to the odd-number limitation in delayed feedback, which we briefly introduce below. To aid the exposition, we momentarily deviate from the continuous-time system (5.1) and start with a linear discrete-time system.

Following [129] we introduce the so-called odd-number limitation, which naturally appears in the context of delayed feedback control, as pioneered by Pyragas [102]. Consider a deterministic linear time-invariant system

$$\begin{aligned} \left\{ \begin{aligned} x_{k+1} =&Ax_k + Bu_k,\\ y_k =&Cx_k, \end{aligned}\right. \end{aligned}$$

(6.8)

with $x\in \mathbb {R}^n$ representing again the state, $u\in \mathbb {R}^m$ the input and $y\in \mathbb {R}^p$ the output (the observables), for some real matrices A, B and C of appropriate size. Now assume we want to control (6.8) using the delayed linear feedback $u_k = K(y_k-y_{k-1})$ for some matrix K, as is one of the key methods in the control of periodic orbits. The resulting closed-loop system can be written as

$$\begin{aligned} z_{k+1}=(A'+B'KC')z_k \end{aligned}$$

(6.9)

for

$$\begin{aligned} A' = \begin{bmatrix} A &{} 0\\ I_n &{} 0 \end{bmatrix},\quad B'=\begin{bmatrix} B\\ 0 \end{bmatrix},\quad C'=\begin{bmatrix} C&- C \end{bmatrix},\quad z_k = \begin{bmatrix} x_k \\ x_{k-1}\end{bmatrix}. \end{aligned}$$

Let $P:\mathbb {C}\rightarrow \mathbb {C}$ be the characteristic polynomial of the closed-loop system defined by $P(\lambda )=\textrm{det}(\lambda I_n - A'-B'KC')$ with $\lambda =a+ib$, then, for (6.9) to be asymptotically stable, we need¹⁰ $P(\lambda )\ne 0$ for all $\vert \lambda \vert \ge 1$. As $A'$ is even-dimensional $\lim _{a \rightarrow +\infty }P(\vert a \vert +i0)=+\infty$ irrespective of our definition of $P(\lambda )$ i.e., $\textrm{det}(\lambda I_n-(\cdot ))$ versus $\textrm{det}((\cdot )-\lambda I_n)$. Then, asymptotic stability of (6.9) together with continuity of the determinant implies that $P(1+i0)>0$. This implies, by exploiting the block structure¹¹ of the problem, that $\textrm{det}(I_n-A)>0$ must hold for (6.9) to be asymptotically stable. Hence we observe—independent of the choice of K—a manifestation of the odd-number limitation, that is, unstable eigenvalues with $\lambda _i(A)>1$ must come in pairs to allow for stabilization.

For more on this phenomenon, see [3, 62]. The odd-number limitation extends to more involved settings, and as recently shown, to hyperbolic equilibrium points [42]. Next, we go back to the continuous-time setting and highlight an obstruction to Pyragas control, e.g., stabilization of $\tau$-periodic orbits by means of $\tau$-delayed feedback.

Theorem 6.7

(De Wolff and Schneider [42, Corollary 3]) Consider the differentiable control system $\dot{x}(t)=f(x(t))+u$ over $\mathbb {R}^n$ and suppose that $p^{\star }\in \mathbb {R}^n$ is a non-degenerate equilibrium of f. Moreover, assume that $Df_{p^{\star }}$ has an odd number of eigenvalues (counting algebraic multiplicities) in the strict right half (complex) plane. Then, for all $K\in \mathbb {R}^{n\times n}$ and all $\tau >0$, $p^{\star }$ is unstable as a solution of the controlled system $\dot{x}(t) = f(x(t)) +K[x(t)-x(t-\tau )]$.

In the setting of compact manifolds, Theorem 6.7 extends, under more general types of feedback. This condition is useful in situations where $\chi (\textsf{M})$ is unavailable and/or one has no knowledge of the number of equilibrium points of $X\in \mathfrak {X}^r(\textsf{M})$, but one does have some local information.

Theorem 6.8

(Odd-number obstruction to local asymptotic multistabilization) Let $X\in \mathfrak {X}^{r\ge 0}(\textsf{M}^{n})$ be an uncontrolled vector field on a smooth, compact, boundaryless manifold $\textsf{M}^n$ with all its zeros being isolated hyperbolic equilibrium points $\{p_1^{\star },\dots ,p_{|I|}^{\star }\}\ne \emptyset$. Given any control system $\Sigma = (\textsf{M}^n,\mathcal {U},F)$ in the sense of Definition 5.5, then, the set $\{p^{\star }_i\}_{i\in {I}}$ can be locally asymptotically stabilized by continuous feedback $\mu :\textsf{M}^{n}\rightarrow \mathcal {U}$, without introducing new equilibrium points, only if $\textrm{dim}(W^u(\varphi _X,p^{\star }_i))$ is even for all $i\in {I}$.

Proof

Without loss of generality, we will consider some $I\ne 0$ and an even-dimensional manifold as the odd case cannot be handled regardless, e.g., recall Corollary 4.1. If $\textrm{dim}(W^u(\varphi _X,p^{\star }_i))$ is odd, then $\textrm{ind}_{p_i^{\star }}(X)=-1$ and as such, by the Poincaré–Hopf theorem (Corollary 3.1) and the hyperbolic index results from [73, Secti. 6], $\vert {I}\vert \ne \chi (\textsf{M}^n)$. Hence, by Theorem 6.6, this set cannot be locally asymptotically stabilized by means of continuous state-feedback.

Instead of demanding that $\textrm{dim}(W^u(\varphi _X,p^{\star }_i))$ is not odd, one could equivalently demand that, in coordinates, the differential of the uncontrolled vector field X satisfies $DX_{p^{\star }_i}\in \textsf{GL}^+(n,\mathbb {R})$ for all $i\in {I}$. As the orientation of a map is a topological invariant [77, Chap. 6], this implies that the statement of Theorem 6.8 is topologically invariant, as it should be. For example, let $a^{\star }$ and $b^{\star }$ correspond to two hyperbolic equilibrium points of some vector fields X and Y, respectively, with the dynamics around those equilibria locally captured by $\dot{x}=Ax$ and $\dot{y}=By$ for $a^{\star }$ and $b^{\star }$, respectively. Then, since $\textsf{GL}(n,\mathbb {R})=\textsf{GL}^+(n,\mathbb {R})\sqcup \textsf{GL}^-(n,\mathbb {R})$ only if the maps $f(x)=Ax$ and $g(y)=By$ have the same orientation, there is a homotopy $H(s,z)=C(s)z$ such that $H(0,x)=Ax$ and $H(1,y)=By$ with $\dot{z}=C(s)z$ hyperbolic for all $s \in [0,1]$ [106]. Better yet, let $A\in \mathbb {R}^{n\times n}$ correspond to an asymptotically stable linear system, i.e., $\textrm{ind}_0(Ax)=(-1)^n$ then for $\textrm{ind}_0(By)$, with $B\in \mathbb {R}^{n\times n}$, to be $(-1)^n$, one needs the unstable eigenspace of B to be of even dimension. In other words, the even dimension is necessary for local analysis. This is not an overly strong condition as spirals and second order systems are naturally of even dimension, see also the discussions in [42]. Note that although $\textsf{GL}^+(n,\mathbb {R})$ is a subgroup of $\textsf{GL}(n,\mathbb {R})$, it is not necessary in general to preserve the group structure as (locally) asymptotically stable systems are not necessarily hyperbolic, e.g., consider $\dot{x}=-x^3$.

Example 6.7

(Rayleigh quotient on the sphere) Consider a smooth function $\ell :\mathbb {S}^{n-1}\subset \mathbb {R}^n\rightarrow \mathbb {R}$ defined by $\ell :p\mapsto \tfrac{1}{2}\langle Ap, p \rangle$ for some symmetric matrix $A\in \textsf{Sym}(n,\mathbb {R})$ and $\langle \cdot ,\cdot \rangle$ the Euclidean inner-product. One might be interested in either minimizing or maximizing $\ell$ over $\mathbb {S}^{n-1}$. By means of this function we will study gradient flows on $\mathbb {S}^{n-1}\subset \mathbb {R}^n$ and exemplify previously discussed material. When one views $\mathbb {S}^{n-1}$ as a Riemannian submanifold of $\mathbb {R}^n$, then $\textrm{grad}\,\ell (p)= (A- I_n \langle Ap, p\rangle )p$ for all $p\in \mathbb {S}^{n-1}$ [1, Table 4.1].

• (i) Regarding Theorem 6.6, $v_i\in \mathbb {S}^{n-1}$ is a unit eigenvector of A if and only if $v_i$ is a critical point of the function $\ell$ [1, Proposition 4.6.1]. As $\chi (\mathbb {S}^{n-1})=0$ for n being even and $\chi (\mathbb {S}^{n-1})=2$ for n being odd, $\chi (\mathbb {S}^{n-1})\ne n$ for all $n\in \mathbb {N}_{\ge 0}$. Hence, by Theorem 6.6 there is no continuous vector field $X\in \mathfrak {X}^{r\ge 0}(\mathbb {S}^{n-1})$ such that every equilibrium point of X is isolated and locally asymptotically stable but also an eigenvector of $A\in \textsf{Sym}(n,\mathbb {R})$.

• (ii) To exemplify Theorem 6.8, let $n=3$ and assume to be an observer of the gradient flow (under the Euclidean metric) $\textrm{grad}\, \ell$ (maximization) on $\mathbb {S}^2\subset \mathbb {R}^3$, without having knowledge of $\ell$. One can show that this flow is equivalent to the projection of the solution to $\dot{x}(t)=Ax(t)$ onto the sphere. Let A have only simple eigenpairs $(v_i,\lambda _i)$ with $\lambda _1<\lambda _2<\lambda _3$, it follows from [1, Proposition 4.6.2] that for a curve $t\mapsto \gamma (t) (v_j+tv_i)/\Vert v_j+tv_i\Vert _2$ with $i\ne j$ one has

  $$\begin{aligned} \frac{\textrm{d}^2}{\textrm{d}t^2}\ell (\gamma (t))\vert _{t=0}= \lambda _i-\lambda _j. \end{aligned}$$

  This implies that only the points $\pm v_3\in \mathbb {S}^2$ are locally asymptotically stable equilibrium points while $\pm v_1$ and $\pm v_2$ are unstable. Indeed, the unstable manifold of $\pm v_2$ is odd-dimensional, obstructing the possibility of global continuous multistabilization as set forth by Theorem 6.8. See also [60, Sect. 1.3].

• (iii) At last, recall Brockett’s necessary condition, i.e., Theorem 6.1, and consider a Rayleigh-like control system on $\mathbb {S}^{2}\subset \mathbb {R}^3$ defined by

  $$\begin{aligned} \dot{x} = f(x,u) = (A- I_3 \langle Ax, x\rangle )x + (I_3 - xx^{\textsf{T}})u \end{aligned}$$

  (6.10)

  for some symmetric matrix $A\in \textsf{Sym}(3,\mathbb {R})$ and $u\in \mathbb {R}^3$. As $T_{x}\mathbb {S}^{2}=\{(I_3-xx^{\textsf{T}})v:v\in \mathbb {R}^3\}$, Brockett’s condition holds, locally, but we know from above that (continuous) local asymptotic multistability cannot hold for (6.10). Hence, the local condition proposed in [18] is too weak to generalize to multistable problems on compact manifolds (the global topology is not taken into account). In contrast, Theorem 6.8 catches the impossibility correctly.

See [120, Corollary 5] for an odd-number obstruction in the context of network control and [30, Theorem 1] for odd-number results in the context of optimization. See also [115, Proposition 4.11] for a result technically in the spirit of Theorem 6.8, but with ramifications in the study of Anosov diffeomorphisms, i.e., diffeomorphisms $G:\textsf{M}\rightarrow \textsf{M}$ such that $T\textsf{M}$ has a hyperbolic structure under G.

6.2 Obstructions to the Stabilization of Submanifolds

Consider again the nonholomic integrator (6.1). We know that the origin cannot be locally asymptotically stabilized by continuous feedback, but what about another set? If one ignores the $x_3$ coordinate we expect to be able to stabilize sets in some $x_1-x_2$ plane. Indeed, simply pick $u_1=-x_1$ and $u_2=-x_2$ to locally asymptotically stabilize $(0,0,x_3(t_0))\in \mathbb {R}^3$. This intuition extends and one can for example find controllers to locally asymptotically stabilize the cylinder $\{x\in \mathbb {R}^3:x_1^2+x_2^2=1\}$ [82, p. 1], see also [89, Example 5]. As stated by Mansouri, when stabilization to a point fails, stabilization to a submanifold seems the next best. Submanifold stabilization occurs for example naturally in the context of nonholonomic control systems [12] and feedback linearization [64, 94]. In particular, if stabilization to a point fails, can one stabilize the system to a small Euclidean sphere enclosing this point?

To start, let $\textsf{A}$ be a compact, connected, oriented, codimension-1 embedded submanifold of $\mathbb {R}^n$, i.e., $\textrm{dim}(\textsf{A})=n-1$, and let d denote the Euclidean metric.¹² Using ideas similar to Coron [33], Mansouri proves the following.¹³ Again, we recall that $f_{\star }$ denotes the induced homomorphism from Chap. 4.

Theorem 6.9

(Mansouri’s condition [82, Proposition 1]) Let $\Sigma ^{\textrm{loc}}_{n,m}$ be a local continuous control system. Assume that f is continuous over some open neighbourhood $\Omega$ of $\subset \mathbb {R}^n\times \mathbb {R}^m$ and define $\Omega _{f,\varepsilon }=\{(x,u)\in \Omega : f(x,u)\ne 0,\, d(x,\textsf{A})<\varepsilon \}$. Let $x \mapsto \mu (x)$ be a continuous feedback rendering $\textsf{A}$ an attractor with $g(x)=f(x,\mu (x))$ denoting the closed-loop. Then, one has for all $\varepsilon >0$

$$\begin{aligned} f_{\star }(H_{n-1}(\Omega _{f,\varepsilon };\mathbb {Z})) \supseteq \textrm{deg}(g)\cdot \mathbb {Z}. \end{aligned}$$

(6.11)

The proof is similar to that of Theorem 6.3, we only provide intuition regarding the appearance of the degree.

Proof

(Sketch) Recall Chap. 5, by construction, there is smooth Lyapunov function V for $\textsf{A}$ [49]. As $\textsf{A}$ is of codimension-1, then by the Jordan–Brouwer seperation theorem e.g., see [61, p. 107], the levelset $V^{-1}(c)$, for sufficiently small $c>0$, consists out of two components, think of $\textsf{A}=\mathbb {S}^1\subset \mathbb {R}^2$. Let $V_c$ denote the “outer” component of $V^{-1}(c)$ and let $A_{\delta }$ be a tubular neighbourhood of $\textsf{A}$, which exists [78, Theorem 6.24]. Due to asymptotic stability one can pick $\delta >0$ such that for all $x\in A_{\delta }\setminus \textsf{A}$ one has $(x,\mu (x))\in \Omega _{f,\varepsilon }$, moreover, by V being proper, there is $c>0$ such that $V_c\subseteq A_{\delta }$. Now one can essentially copy the commutative diagram from the proof of Theorem 6.3, using the same functions, that is

Therefore we have that $f_{\star }(H_{n-1}(\Omega _{f,\varepsilon };\mathbb {Z})) \supseteq g_{\star }(H_{n-1}(V_c;\mathbb {Z})$. Then, as $V_c$ is one of the two components of $V^{-1}(c)$, $H_{n-1}(V_c;\mathbb {Z})=H_{0}(V_c;\mathbb {Z})\simeq \mathbb {Z}$ by Poincaré duality. Better yet, $V_c$ is a closed, oriented manifold itself, such that $\textrm{deg}(g)$ via $g_{\star }$ is well-defined, e.g., recall the rationale of Lemma 4.1 and consider the following diagram or see [82] for the full explanation.

In contrast to Theorem 6.3, one does not assume that $\mu (a)=0$ for all $a\in \textsf{A}$. Indeed, assumptions of that form are not necessary, they rather (over)simplify the exposition.

Condition (6.11) is not particularly transparent, one rarely has access to $\textrm{deg}(g)$. However, by the assumptions on $\textsf{A}$, one can appeal to the the Gauss–Bonnet–Hopf (Dyck) theorem [53, 88], stating that for a closed manifold $\textsf{M}^{n-1}\subset \mathbb {R}^n$ the Gauss map $\gamma :\textsf{M}^{n-1}\rightarrow \mathbb {S}^{n-1}$, defined by $\gamma (p)$ being the unit normal at $p\in \textsf{M}$, one has $\textrm{deg}(\gamma )=\chi (\textsf{N})$ for $\textsf{N}$ the bounded component of $\mathbb {R}^n\setminus \textsf{M}^{n-1}$. In fact, this theorem relates directly to a manifestation of the Poincaré–Hopf theorem for manifolds with boundary and vector fields pointing outward cf. Sect. 8.2. Then, using the Gauss–Bonnet–Hopf theorem, one can relate the vector field g on the components of the levelset $V^{-1}(c)$ to inward- and outward pointing vector fields, normal to those manifolds. As such, this allows for linking $\textrm{deg}(g)$ to the topology of $\textsf{A}$, e.g., due to a result by Hopf, when n is odd, $\textrm{deg}(g)=\tfrac{1}{2}\chi (\textsf{A})$ [82, Theorem 4].¹⁴ Now one can show that the nonholonomic integrator cannot be stabilized to $\mathbb {S}^2$ either since $\chi (\mathbb {S}^2)=2$ while the left-hand-side of (6.11) evaluates to 0 [82, Corollary 2]. A similar, but not identical, result can be shown for $\textsf{A}$ being of codimension strictly larger than 1 [83].

The aforementioned results on submanifolds are of a local nature and generalize the condition due to Coron. Using retraction theory one can provide obstructions of a global nature and without explicit knowledge of vector fields.

Proposition 6.1

(Moulay and Bhat [92, Proposition 10]) Let $\textsf{M}$ be a smooth manifold and $\textsf{A}$ a compact, embedded submanifold of $\textsf{M}$. Then $\textsf{A}$ is a strong neighbourhood deformation retract of $\textsf{M}$.

Ultimately, Theorem 6.12, as discussed in the next section, states that if a set $A\subseteq \textsf{M}$ is an attractor under a flow $\varphi$, then A is a weak deformation retract of $\mathcal {D}(\varphi ,A)$. Therefore, in case $\textsf{A}$ is an attractor, combining Proposition 6.1 with Lemma 2.2 and Theorem 6.12 leads to $\textsf{A}$ being a strong deformation retract of $\mathcal {D}(\varphi ,\textsf{A})$. This implies in particular that $\textsf{A}$ can only be a global attractor if it is a strong deformation retract of $\textsf{M}$. This result also appeared, with a different proof, in [130, Lemma 4]. As highlighted in [130], indeed, for $\textsf{A}$ to be a global attractor of $\textsf{M}$, the spaces must be homotopy equivalent, e.g., stabilizing $\mathbb {S}^1$ in $\mathbb {S}^2$ fails as $\chi (\mathbb {S}^1)=0$ while $\chi (\mathbb {S}^2)=2$.

Returning to Example 6.6, consider two agents with $\textsf{M}=\mathbb {S}^1$ such that $(\textsf{M})^2=\mathbb {T}^2$ and $\textsf{C}=\Delta _{\mathbb {S}^1}\simeq _t \textsf{M}$, although $\chi (\mathbb {S}^1)=0$ and $\chi (\mathbb {T}^2)=0$, $\textsf{C}$ and $(\textsf{M})^2$ are not homotopic as they have for instance different homology groups and different fundamental groups,¹⁵ i.e., $\pi _1(\mathbb {S}^1)\simeq \mathbb {Z}$ and $\pi _1(\mathbb {T}^2)\simeq \mathbb {Z}^2$.

Proposition 6.1, however, says more, the deformation retract is of the strong type. For example, when $\textsf{M}$ is an absolute neighbourhood retract (ANR),¹⁶ then, A being a deformation retract of $\textsf{M}$ is equivalent to A being a strong deformation retract of $\textsf{M}$ [63, p. 199], see also [118, p. 31]. As before, we immediately observe some form of an odd-number limitation.

Corollary 6.2

(Odd-number limitation for attractors) Let $\textsf{M}$ be a smooth, closed manifold with $\chi (\textsf{M})\ne 0$, then $\textsf{M}$ does not admit any continuous vector field X such that any odd-dimensional, compact, embedded submanifold $\textsf{A}$ of $\textsf{M}$ is a global attractor under X.

Using Propsition 6.1 and the mod 2 intersection theory from Section 3.4, Theorem 6.5 can be generalized beyond stabilization of a point. Again, this result is stated for simplicity using vector bundles. Generalizations to fiber bundles are possible.

Theorem 6.10

(Obstruction to submanifold stabilization: bundles) Let $\pi :\textsf{M}\rightarrow \textsf{B}$ be a vector bundle over some smooth, boundaryless, compact, connected manifold $\textsf{B}$ and let $\textsf{A}\subseteq Z_{\pi }(\textsf{B})$ be a compact, embedded submanifold of $\textsf{M}$. If there is a continuous flow $\varphi$ on $\textsf{M}$ such that $\textsf{A}$ is a global attractor under $\varphi$, then, $\textsf{A}=Z_{\pi }(\textsf{B})$.

Proof

The case where $\textsf{A}$ is a point follows from Theorem  6.5 (Lemma 3.1 and Theorem 6.4). Regarding the general case, as demonstrated in Example 3.1, the zero section $Z_{\pi }(\textsf{B})$ is a deformation retract of $\textsf{M}$. If $\textsf{A}$ would be a global attractor, then, by Lemma 2.2, Proposition 6.1 and Theorem 6.12, $\textsf{A}$ would be a deformation retract of $\textsf{M}$ as well. In its turn, this implies by Lemma 2.1 that $\textsf{A}$ is a deformation retract of $Z_{\pi }(\textsf{B})$, which, by Lemma 3.2, cannot be true when $\textsf{A}\ne Z_{\pi }(\textsf{B})$.

Fig. 6.5

Example 6.8, (i): robotic arm in $\mathbb {R}^3$ with $\gamma$ the trajectory that keeps $e^{\star }$ fixed; (ii): the configuration space $\mathbb {S}^2\times \mathbb {S}^2$ of the robot arm with the curves corresponding to $\gamma$, which are not independent as indicated by the marker on the curves

Theorem 6.10 implies in particular that if $\textsf{M}^m$ is a smooth, boundaryless, compact, connected manifold. Then, given any compact, embedded submanifold $\textsf{A}^a\hookrightarrow \textsf{M}^m$ with $0\le a<m$, there is no complete vector field $X\in \mathfrak {X}^{r\ge 0}(\textsf{M}^m)$ such that $\textsf{A}^a$ is a global attractor. Regarding Section 1.3 (compact Lie groups), not only global stabilization of a point is obstructed, but effectively of any non-trivial submanifold, as Lie groups are boundaryless. Then, as was the motivation for [9], Theorem 6.10 is of use in the context of mechanical systems. Recalling Example 2, a periodic orbit of the pendulum, as seen as a curve in $T\mathbb {S}^1$ is clearly homotopic to $\mathbb {S}^1$.

Example 6.8

(Kinematic robot control) Consider a two-link robotic arm with spherical joints. The goal is to globally stabilize the end-effector position at $e^{\star }\in \mathbb {R}^3$, see Fig. 6.5(i). Here we assume to work with a dynamical control system on $\textsf{M}=T(\mathbb {S}^2\times \mathbb {S}^2)$, i.e., a second-order system. As shown in Fig. 6.5(i), the configuration of the arm is not uniquely defined by $e^{\star }$, instead, one can freely move the elbow joint over the curve $\gamma$ without changing the position of $e^{\star }$. One might want to exclude this ambiguity and render the dynamical system stationary on $\gamma$, while still globally stabilizing $e^{\star }$. As the motion of the two joints is not independent, the curve $\gamma$ represents a 1-dimensional set in the configuration space $\mathbb {S}^2\times \mathbb {S}^2$, e.g., see Fig. 6.5(ii). Regardless, one must be able to globally continuously stabilize a curve homeomorphic to $\mathbb {S}^1$ in the zero section of $\pi _p:T\mathbb {S}^2\rightarrow \mathbb {S}^2$, for an individual joint, say $j_1$. This is prohibited by Theorem 6.10.

Example 6.9

(Satellite control) Consider a free rigid body model of a satellite on $\textsf{M}^n=T\textsf{SO}(3,\mathbb {R})\simeq \textsf{SO}(3,\mathbb {R})\times \mathbb {R}^3$, where the last identification follows as Lie groups are parallelizable. Let $(R,\Omega )\in \textsf{SO}(3,\mathbb {R})\times \mathbb {R}^3$ denote the orientation and angular velocity, respectively, with $\Omega ^{\wedge }\in \mathfrak {so}(3,\mathbb {R})$ cf. Example 5.2, then, for $\times$ the cross-product, an input-affine control system is given by

$$\begin{aligned} \Sigma _{n,m}^{\textrm{aff}}:\left\{ \begin{aligned} \dot{R}(t)=&R(t)\Omega (t)^{\wedge }\\ I\dot{\Omega }(t)=&I\Omega (t) \times \Omega (t) + \textstyle \sum \limits ^m_{i=1}g_i u_i \end{aligned}\right. \end{aligned}$$

(6.12)

for I some inertia tensor, $g_i\in \mathbb {R}^3$ and $u_i\in \mathbb {R}$ available inputs for $i=1,2,\dots ,m$, see [25], [11, p. 37]. Fix any $v\in \mathbb {S}^2\subset \mathbb {R}^3$, we want to asymptotically stabilize the satellite, pointing along v, see Fig. 6.6. However, we do not specify a fixed rotation along this axis, only a constant velocity, that is, given the map $h:\textsf{SO}(3,\mathbb {R})\rightarrow \mathbb {R}^3$, $h:R\mapsto (R-I_n)v$ we want to render $\textsf{A}=\{(R,v)\in \textsf{M}:h(R)=0\}$ globally asymptotically stable. Now, the set $\textsf{A}$ is a 1-dimensional compact, embedded, submanifold of $\textsf{M}$ as the set $\{R:h(R)=0\}$ is a stabilizer subgroup of $\textsf{SO}(3,\mathbb {R})$ with respect to v [45, p. 94], in particular, $\{R:h(R)=0\}$ is isomorphic to $\textsf{SO}(2,\mathbb {R})\simeq \mathbb {S}^1$. As in [25], see that we do not demand the closed-loop vector field to vanish on $\textsf{A}$. Such a task might be locally feasible [89], however, for any m, continuous global asymptotic stabilization is obstructed by Theorem 6.10 (possibly, after a shift $\widetilde{\Omega }=\Omega -v$).

Fig. 6.6

Example 6.9: controlling a satellite (free rigid body) towards pointing along v, with angular velocity $\Omega$, that is, the orientation is not fully fixed

Similar topological obstructions occur for example in surgical robotics [125].

In [86], Mayhew and Teel extend Theorem 6.5 to the context of submanifold stabilization under set-valued maps. In particular, it is shown that for differential inclusions, i.e., vector fields of the form $\dot{x}\in X(x)$, $X:\textsf{M}\rightrightarrows T\textsf{M}$, satisfying the so-called “basic conditions” [86, Definition 5] the answer to Question (ii) is effectively the same as in the smooth case. The reason being that if a submanifold $\textsf{A}\subseteq \textsf{M}$ is an attractor under such a—possibly discontinuous—vector field, then there must exist a smooth, complete vector field $X'$, defined on the domain of attraction of $\textsf{A}$, such that $\textsf{A}$ is an attractor under $X'$ [86, Theorem 14]. Although this framework captures some discontinuities, their conditions, however, do not capture for example Fig. 1.1(iii).

We end with a remark on compactness. In [127, Theorem 3.4] it is claimed that the domain of attraction, for any submanifold $\textsf{A}\subseteq \textsf{M}$, compact or non-compact, is homeomorphic to its tubular neighbourhood. This claim has been disproven and corrected by Lin et al. [79]. See in particular [79, Sect. 4] for counterexamples.

6.3 Obstructions to the Stabilization of Sets

Regarding Question (i), in 1993, Günther and Segal showed that a finite-dimensional compact set A can be an attractor of a continuous flow on a manifold if and only if A has the shape¹⁷ of a polyhedron [55, Corollary 4]. Although the Warsaw circle is not homeomorphic to $\mathbb {S}^1$ it does have the shape of $\mathbb {S}^1$, see [58, Example 3.3] for an example by Hastings, rendering the Warsaw cirlce an attractor. Regarding realizable compact attractors, see also the article by Ortega and Sánchez–Gabites [97] and references therein. To add regarding Question (iii), given a compact attractor $A\subseteq \textsf{M}$ with domain of attraction $\mathcal {D}$, already the boundary of $\mathcal {D}$ can be arbitrarily complicated, cf. [110].

Hence, we pass to Question (ii). As mentioned before and indicated in [11, 82], if stabilization of a point is prohibited, stabilization of a set might be the next thing to consider. Concurrently, stabilization of a point might be to simplistic. In contrast to the previous section we will not assume this set to have a manifold structure, see that a variety of results in that section exploited this structure by appealing to the existence of a tubular neighbourhood.

Kvalheim and Koditschek recently generalized Brockett’s condition to stabilization of any compact subset $A\subseteq \textsf{M}$ with $\chi (A)\ne 0$ [76]. To make sure $\chi (A)$ is well-defined, the authors appeal to C̆ech–Alexander–Spanier cohomology theory [76, Sect. 2], which is outside the scope of this exposition, but when discussing their result we assume to work with this homology theory.

Theorem 6.11

(Kvalheim–Koditschek condition [76, Theorem 3.2]) Let $\Sigma =(\textsf{M},\mathcal {U},F)$ be a control system and let $A\subseteq \textsf{M}$ be a compact attractor. Assume that $\chi (A)\ne 0$, then, for any neighbourhood $W\subseteq \textsf{M}$ of A there exists a neighbourhood $V\subseteq T\textsf{M}$ of the zero section of $T\textsf{M}$ such that for any continuous vector field $X:W\rightarrow V$

$$\begin{aligned} F(\pi _u^{-1}(W))\cap X(W)\ne \emptyset . \end{aligned}$$

(6.13)

The motivation for Theorem 6.11 came from the development of repelling vector fields, i.e., to render some unsafe set $U\subset \textsf{M}$ repelling. Equivalently, one could render the safe set $S=\textsf{M}\setminus U$ attractive. After proving when $\chi (S)$ is well-defined, that result is summarized in [76, Theorem 3.6]. This result provides a variety of answers with respect to [130, Conjecture 2] stating that it is impossible for vehicles to smoothly converge to a desired configuration from every initial configuration in an environment scattered with obstacles, see [76, Example 2]. See also the work by Byrnes [24] for earlier generalizations of Brockett’s condition with respect to the global stabilization of $A\subseteq \mathbb {R}^n$.

Now, recall Example 6.1. Mansouri showed that the cylinder can be rendered an attractor [82, p. 1]. Then, using Theorem 6.11 one can show that the topology of the cylinder is crucial here as no set with non-zero Euler characteristic can be stabilized. Interestingly, when A is a point, Theorem 6.11 can be shown to be strictly stronger than Brockett’s condition, while the condition turns out to be weaker than Coron’s condition [76, Sect. 6]. This trade-off is however a recurring one, conditions based on homology theory are frequently stronger, but also significantly harder to check.

Recall the definition of a dynamical system $(\textsf{M},\mathbb {R},\varphi )$ in the sense of Sect. 5.1. Then, using the retraction theory as set forth in Chap. 2, we can provide a generic result, a generalization of the work by Sontag [116, Theorem 21] (Theorem 6.4) and Bhatia and Szegö [10, Lemma V 3.2], due to Moulay and Bhat.

Theorem 6.12

(Moulay–Bhat condition [92, Theorem 5]) Let $(\textsf{M},\mathbb {R},\varphi )$ be a dynamical system over a topological manifold $\textsf{M}$. Let $A\subseteq \textsf{M}$ be a compact attractor, with domain of attraction $\mathcal {D}(\varphi ,A)$. Then A is a weak deformation retract of $\mathcal {D}(\varphi ,A)$.

Proof

(Sketch) As a topological manifold is in particular a locally compact Hausdorff space, Theorem 5.1 applies. Now let $U_c=\{x\in \mathcal {D}(\varphi ,A):V(x)\le c\}$. Note that $U_c$ is positively invariant, i.e., $\varphi ^t(U_c)\subseteq U_c$ for all $t\ge 0$ and $\cap _{c>0}U_c=A$. Hence, for each open neighbourhood $W\subseteq \textsf{M}$ of A there is a $c>0$ such that $U_c\subset W$. Now, define $T_c(x)=\inf \{t\ge 0: \varphi ^t(x)\in U_c\}$, which can be shown to be continuous. Then, pick any W with $c>0$ such that $U_c\subset W$ and consider the map $H:I\times \mathcal {D}(\varphi ,A)\rightarrow \mathcal {D}(\varphi ,A)$ defined by $H(t,x)=\varphi (t T_c(x),x)$, which is continuous and satisfies $H(0,x)=x$ for all $x\in \mathcal {D}(\varphi ,x)$, $H(1,x)\in U_c$ for all $x\in \mathcal {D}(\varphi ,A)$ and $H(t,x)=x$ for all $x\in U_c$. Hence, H parametrizes a strong deformation retract from $\mathcal {D}(\varphi ,A)$ onto $U_c$. As W was an arbitrary open neighbourhood of A, this shows that A is weak deformation retract of $\mathcal {D}(\varphi ,A)$. See [92] for the details.

In combination with Lemma 2.2 one recovers for example from Theorem 6.12 that if $A\subseteq \textsf{M}$ is a compact attractor of a dynamical system $(\textsf{M},\mathbb {R},\varphi )$, then, if A is a strong neighbourhood retract of $\textsf{M}$, A is a strong deformation retract of its domain of attraction $\mathcal {D}(\varphi ,A)$ [92, Corollary 7]. We emphasize that Theorem 6.12 remains true for $\textsf{M}$ being a locally compact Hausdorff space.

Example 6.10

(The rotation group as a potential attractor) For any $A\in \textsf{GL}^+(n,\mathbb {R})$ let $A=UP$ be its Polar decomposition, for some $U\in \textsf{SO}(n,\mathbb {R})$ and symmetric positive definite $P\in \mathcal {S}^n_{\succ 0}$. Then consider the map $H(t,A)=U(tI_n+(1-t)P)$ for $t\in [0,1]$. See that $H(0,A)=A$ for all $A\in \textsf{GL}^+(n,\mathbb {R})$, $H(1,A)=U$ for $A=UP$ and $H(t,U)=U$ for all $U\in \textsf{SO}(n,\mathbb {R})$ and all t. Hence, as $A\mapsto U$ is continuous, $\textsf{SO}(n,\mathbb {R})$ is a strong deformation retract of $\textsf{GL}^+(n,\mathbb {R})$.¹⁸ Now for some $B\in \textsf{SL}(n,\mathbb {R})$, let $B=QR$ be its QR decomposition for $Q\in \textsf{SO}(n,\mathbb {R})$ and $R\in \mathbb {R}^{n\times n}$ being upper-triangular with strictly positive elements on its diagonal. Now define the map $H(t,B)=QZ(t)$ with $t\in [0,1]$ for Z(t) such that $z_{ii}(t)=r_{ii}^t$ and $z_{ij}(t)=t r_{ij}$ for $i\ne j$. Then we see that $H(0,B)=Q$ for $B=QR$, $H(1,B)=B$ and $H(t,Q)=Q$ for all $Q\in \textsf{SO}(n,\mathbb {R})$. Again, one can show that this decomposition is continuous such that $\textsf{SO}(n,\mathbb {R})$ is also a strong deformation retract of $\textsf{SL}(n,\mathbb {R})$. Note that for any $n>1$, $\textsf{SO}(n,\mathbb {R})$ cannot be a global attractor of a flow on $\mathbb {R}^{n\times n}\simeq \mathbb {R}^{n^2}$ by Lemma 3.1. Similarly, although we can construct a trivial embedding $\iota :\textsf{SO}(n,\mathbb {R})\hookrightarrow \textsf{SO}(n+1,\mathbb {R})$, by Lemma 3.2 $\textsf{SO}(n,\mathbb {R})$ cannot be a global attractor in such an ambient space either. For further comments, see [21].

Remark 6.3

(On a proof of Theorem6.4) Consider now A being a point, denoted $p^{\star }$, and assume that $\textsf{M}$ is locally contractible, as any topological manifold. Clearly, $p^{\star }$ is a strong neighbourhood retract of $\textsf{M}$. Hence, Theorem 6.4 follows as a corollary to Lemma 2.2 and Theorem 6.12. When $\textsf{M}=\mathbb {R}^n$, Theorem 6.12 can be strengthened to A being a strong deformation retract of $\mathcal {D}(\varphi ,A)$ [7].

Remark 6.4

(Stabilization of A with $\chi (A)=0$) Whereas works like [76, 82, 83] can in general not address the stabilization of compact sets $A\subseteq \textsf{M}$ with $\chi (A)=0$, the retraction-based results provide some necessary conditions, see Theorem 6.12 and Example 6.10. A necessary condition for global stabilization of A of the form $\chi (A)=\chi (\textsf{M})$ is clearly weaker than the deformation retract formulation, e.g., the preservation of the Euler characteristic is necessary for homotopy equivalence, but not sufficient. For instance, compare the homology groups of $\mathbb {S}^1\hookrightarrow \mathbb {T}^2$ to that of the ambient space.

Remark 6.5

(More on robustness) The retraction-based results are robust in the sense that they are true for any continuous control system. However, consider stabilizing the unit disk $\mathbb {D}_1^n$ and the punctured unit disk $\mathbb {D}_1^n\setminus \{0\}$ in $\mathbb {R}^n$. Again, arbitrarily small perturbations can potentially invalidate a necessary condition for continuous stabilization. This is important to take into account with numerical methods in mind.

Coming back to where we started, Kvalheim recently generalized the homological results due to Coron and Mansouri via appealing to homotopical arguments in the spirit of those by Bobylev, Krasnosel’skiĭ and Zabreĭko indeed, cf. Example 3.2, Example 3.4 and [75]. Now, we recall Assumption 5.1 and state the first result.

Theorem 6.13

(Kvalheim’s homotopy [75, Theorem 1]) Let $\textsf{M}$ be a smooth manifold and let $X,Y\in \mathfrak {X}^{r\ge 0}(\textsf{M})$ be such that the compact set $A\subseteq \textsf{M}$ is asymptotically stable under both. Then, there is an open neighbourhood U of A such that $X\vert _{U\setminus A}$ and $Y\vert _{U\setminus A}$ are homotopic through non-vanishing vector fields.

As in Example 3.4, using the flows corresponding to X and Y, locally, a homotopy is constructed to prove Theorem 6.13. Note that the tools from Chapter 4 now imply that for U as in Theorem 6.13 and any $W\subset U\setminus A$ we have that the following induced homomorphisms agree $(X\vert _W)_{\star }, (Y\vert _W)_{\star }:H_{(\cdot )}(W)\rightarrow H_{(\cdot )}(TW\setminus \{0\})$. This observation is exploited below.

The aforementioned results can now be unified as follows. One constructs a “canonical” vector field Y on an open neighbourhood U of A such that A is locally asymptotically stable. For this canonical vector field Y one computes some homotopy invariant of interest and by Theorem 6.13 this extends to any vector field locally asymptotically stabilizing A. The importance of having the vector fields to be nonvanishing on $U\setminus A$ is displayed in for instance Example 3.4 and the proof (sketch) of Theorem 6.3, without this requirement we are not capturing meaningful information, e.g., any two continuous vector fields on $\mathbb {R}^n$ are straight-line homotopic.

In the setting of an equilibrium point the canonical vector field (with the origin being asymptotically stable) is locally given by $\dot{x}=-x$. Indeed, from there one computes the corresponding index cf. Example 3.4. More general, for an embedded submanifold $\textsf{A}$ in $\mathbb {R}^n$, the canonical stabilizing vector field is the negated normal vector field. Indeed, Mansouri exploits this and the existence of a tubular neighbourhood in [82, Theorem 4] to relate $\textrm{deg}(g)$ from Theorem 6.9 to the underlying topology of $\textsf{A}$. In the general case, one cannot appeal to tubular neighbourhoods and the canonical vector field Y is less obvious to select. One usually passes to the dynamical system generated by the negative gradient flow of a Lyapunov function.

It is important to stress that all of these results provide necessary, but by no means sufficient, conditions, e.g., consider vector fields of the form $X_1=X$ and $X_2=-X$.

The same is true for the final Theorem of this chapter. Nevertheless, recall that $Y_{\star }$ and $F_{\star }$ denote induced homomorphisms and recall the notion of a control system $\Sigma$ as given by Definition 5.5, then, one can use Theorem 6.13 to derive rather generic homological necessary conditions.

Theorem 6.14

(Kvalheim’s condition [75, Theorem 2]) Let   $\textsf{M}$   be a smooth manifold and let $\Sigma =(\textsf{M}, \mathcal {U}, F)$ be a continuous control system. Assume there is a $Y\in \mathfrak {X}^{r\ge 0}(\textsf{M})$ such that the compact subset A of $\textsf{M}$ is asymptotically stable under Y. Moreover, assume there is a feedback $\mu$ that asymptotically stabilizes A. Then, for a sufficiently small open neighbourhood U of A one has

$$\begin{aligned} Y_{\star }H_{(\cdot )}(U\setminus A)\subseteq F_{\star }H_{(\cdot )}(\pi _u^{-1}(U\setminus A)\setminus F^{-1}(0))\subseteq H_{(\cdot )}(T(U\setminus A)\setminus \{0\}). \end{aligned}$$

(6.14)

In the proof of Theorem 6.14, one exploits the decomposition as $(F\circ \mu )_{\star }=F_{\star }\circ \mu _{\star }$. As pointed out in [75, Remark 1], the results due to Coron and Mansouri follow indeed from (6.14). Again, one can find control systems that do satisfy the conditions of Theorem 6.14, yet, continuous asymptotic stabilization is impossible. The reader is invited to find non-trivial examples. See also that Theorem 6.14 does not assume $\chi (A)\ne 0$. In conclusion,

• (i) we see that retraction theory allows for the construction of necessary conditions independent of the precise continuous control $\Sigma$ system, cf. Theorem 6.12;

• (ii) we also see that the methodology as put forth in the monograph by Krasnosel’skiĭ and Zabreĭko allows for generalizations far beyond characterizing the continuous stabilization of equilibrium points in $\mathbb {R}^n$, cf. Theorem 6.14.

6.4 Other Obstructions

Besides the aforementioned topological obstructions, a topological viewpoint can be seen to be fruitful in other modern branches of system identification and control theory [16, 27, 66, 68, 81]. In particular, in [20] a topological obstruction to the reach control problem is presented. Obstructions to simultaneous stabilization (robust control), are considered in [41]. Extensions of Brockett’s condition in the context of exponential stabilization are discussed in [56]. With respect to adaptive linear control, topological obstructions to self-tuning are presented by Polderman [100] and van Schuppen [122]. In the context of hybrid systems, Ames and Sastry present topological obstructions to zeno behaviour in [4]. In [84, 85], Mansouri presents topological obstruction to the existence of distributed controllers, that is, controllers where each input variable can only depend on a subset of the state variables.

Necessary and sufficient conditions for global, smooth, feedback linearization of a smooth, input-affine system (5.8) with $m=1$ are presented in [39], for example, $\textsf{M}$ must be simply connected, ruling out $\textsf{SO}(n,\mathbb {R})$ for $n\ge 2$.

Topological obstructions also appear in the context of motion planning algorithms. In line with Theorem 6.4, a globally defined continuous motion planning algorithm exists only if the underlying configuration space is contractible. See [47, 48], for more work by Farber on topological obstructions to motion planning algorithms and [52] for early remarks by Gottlieb. Somewhat related, see [80] for obstructions to certain tracking problems. In particular, see [80, Example 4.1] which considers Brockett’s nonholonomic integrator (6.1).

There is also a line of work by, amongst others, Byrnes, Delchamps and Hazewinkel on the geometry and topology of linear systems, providing for further obstructions to for example global system identification, e.g., see [23, 26, 43].

See [29, Sect. 18.5] or [105, pp. 35–36] for related phenomena in the calculus of variations, the so-called “Lavrentiev gap”.

In the context of physics, in particular particle physics, topological curiosities manifest themselves mathematically for example via Poincaré’s lemma, e.g., the Ehrenberg–Siday–Aharonov–Bohm effect, and by means of the Atiyah–Singer index theorem, e.g., to understand spectral flows [44]. Here, the topological obstruction oftentimes relates to not being able to apply Stokes’ theorem, e.g., a differential form fails to be exact. Earlier, topological obstructions were studied in the context of Hamiltonian mechanics [46].

Topological obstructions have also been reported in chemistry, although more related to data analysis [57]. For infinite-dimensional problems, topological obstructions frequently pertain to reachable sets being empty due to the fact that a compact set in an infinite-dimensional Banach space has empty interior, e.g., see [15]. Topological obstructions in the context of neural networks are alluded to in [128].

The topological viewpoint also provided to be useful early on in the context of Bellman equations [121], e.g., see the initial work by Petrov on regularity in the context of time-optimal processes [99].