# Strong chain control sets and affine control systems

Fritz Colonius11footnotemark: 1<EMAIL_ADDRESS>Alexandre J.
Santana222Partially supported by CNPq grant n. 309409/2023-333footnotemark: 3

###### Abstract

For control-affine systems on non-compact manifolds, the notion of strong
chain control sets is introduced and related to the strong chain transitivity
of the associated control flows. Affine control systems on $\mathbb{R}^{n}$
are embedded into bilinear control systems in an extended state space and it
is shown that they are topologically conjugate to the induced system on the
northern hemisphere of the Poincaré sphere. This preserves strong chain
control sets. Further chain controllability properties on spheres are
analyzed.

###### keywords:

affine control system , strong chain control set , strong chain transitivity ,
Poincaré sphere 93B05 , 37B20

††journal: Journal of Differential Equations

[label1]organization=Institut fur Mathematik, addressline=Universitatsstraße
12, city=Augsburg, postcode=86159, country=Germany

[label2]organization=Department of Mathematics, addressline=Av. Colombo, 5790,
city=Maringa, postcode=87020-900, country=Brazil

##  1 Introduction

This paper is mainly motivated by the problem of describing generalized
controllability properties for affine control systems on $\mathbb{R}^{n}$ of
the form

$\dot{x}(t)=A_{0}x(t)+a_{0}+\sum_{i=1}^{m}u_{i}(t)[A_{i}x(t)+a_{i}],\,u\in\mathcal{U},$
(1)

where $A_{i}\in\mathbb{R}^{n\times
n},a_{i}\in\mathbb{R}^{n},i\in\\{0,1,\ldots,m\\}$. The control range
$\Omega\subset\mathbb{R}^{m}$ is a convex and compact neighborhood of
$0\in\mathbb{R}^{m}$, and the controls $u$ are in

$\mathcal{U}:=\left\\{u\in
L^{\infty}(\mathbb{R},\mathbb{R}^{m})\left|u(t)\in\Omega\text{ for almost all
}t\in\mathbb{R}\right.\right\\}.$ (2)

We denote the solution for initial condition $x(0)=x_{0}\in\mathbb{R}^{n}$ and
control $u\in\mathcal{U}$ by $\psi(t,x_{0},u),t\in\mathbb{R}$. The homogeneous
part of this system is the bilinear control system

$\dot{x}(t)=[A_{0}+\sum_{i=1}^{m}u_{i}(t)A_{i}]x(t),\,u\in\mathcal{U}.$ (3)

Controllability questions for bilinear and affine control systems are a
classical topic of control theory. Early contributions are due to Rink and
Mohler [28] who took the set of equilibria as a starting point for
establishing results on complete controllability. A theorem due to Jurdjevic
and Sallet [21, Theorem 2] (cf. also Do Rocio, Santana, and Verdi [13]) shows
that affine system (1) is controllable on $\mathbb{R}^{n}$ if it has no fixed
points and its homogeneous part (3) is controllable on
$\mathbb{R}^{n}\setminus\\{0\\}$. Cannarsa and Sigalotti [5] prove that
approximately controllable bilinear systems are controllable. As Jurdjevic
[20, p. 182] emphasizes, the controllability properties of affine systems are
substantially richer than for linear systems and may require “entirely
different geometrical considerations”, based on Lie-algebraic methods, which
have yielded important insights. There is substantial literature on
controllability properties of bilinear and affine control systems. Here we
only refer to the monographs Mohler [24], Elliott [15], and Jurdjevic [20].

In the present paper, we study chain controllability properties, which
constitute a weaker version of approximate controllability in infinite time.
Here small jumps in the trajectories are allowed. The approach we propose is
to extend the affine system to a bilinear control system on $\mathbb{R}^{n+1}$
(cf. Jurdjevic [19] for this idea in the context of the Lie-algebraic
approach). This system can be projected to the unit sphere $\mathbb{S}^{n}$ or
to projective space $\mathbb{P}^{n}$. We call $\mathbb{S}^{n}$ the Poincaré
sphere since a similar construction in the theory of ordinary differential
equation goes back to Poincaré [27] (cf. Perko [26]). The induced system on
the open northern hemisphere $\mathbb{S}^{n,+}$ is conjugate to the system on
$\mathbb{R}^{n}$, hence one can study the chain controllability properties on
the sphere. This continues our previous work, where we consider control sets
for affine system (Colonius, Santana, Setti [11]), chain control sets (i.e.,
maximal regions of chain controllability) for linear control systems
(Colonius, Santana, and Viscovini [12]) and for affine control systems
(Colonius and Santana [9]), where we show that the chain control sets in
$\mathbb{R}^{n}$ are mapped to subsets of the central chain control set of the
induced system on projective space $\mathbb{P}^{n}$.

The proposed approach for affine control systems is confronted with several
obstacles, which we can only partially surmount.

In general, chain controllability is only preserved under uniformly continuous
conjugation maps, which does not hold for the conjugation between the non-
compact spaces $\mathbb{R}^{n}$ and $\mathbb{S}^{n,+}$. We introduce a
modified version of chain controllability, which we term strong chain
controllability. Instead of bounding the jump lengths of controlled chains by
constants $\varepsilon>0$ we consider continuous jump functions defined on the
state space with values in $(0,\infty)$. This is based on an approach by
Hurley to dynamical systems on non-compact state spaces; cf. [17, 18] and also
Choi, Chu, and Park [6]. Hurley’s approach is motivated by the desire to
develop a theory, that only depends on the topology and not on the metric. We
apply it to the associated control flows. Note that a different strengthening
of chain transitivity (also called strong chain transitivity) is due to Easton
[14], who proposes to sum up the length of the jumps along an
$\varepsilon$-chain. This has e.g. been explored by Ahmadi [2], where also
further references are given. Patrão and San Martin [25] treat chain
transitivity in topological spaces based on families of coverings by
constructing associated shadowing semigroups.

Strong chain control sets are contained in chain control sets and coincide
with them when the state space is compact. For non-compact state spaces, the
use of jump functions instead of constant jump bounds entails that close to
the boundary of the state space or close to “infinity” very small jump lengths
are required. Definition 4 introduces the notion of strong chain control sets,
which allows us to show that the strong chain control sets of affine system
(1) on $\mathbb{R}^{n}$ are mapped onto the strong chain control sets of the
induced system on the open northern hemisphere $\mathbb{S}^{n,+}$ of the
Poincaré sphere (Corollary 2). The same is true for the southern hemisphere
$\mathbb{S}^{n,-}$. Hence the problem of determining the strong chain control
sets of affine system (1) is transformed into the problem of describing the
strong chain control sets on the hemispheres. For the latter problem, we have
only very partial results. We can describe the chain control sets for the
systems on the sphere induced by general bilinear control systems.
Furthermore, also for the system restricted to the closed northern hemisphere
$\overline{\mathbb{S}^{n,+}}$ some results are derived.

Section 2 cites a fundamental lemma due to Hurley [18] and uses it to describe
the behavior of strongly chain transitive sets of flows under topological
conjugation. For general control-affine systems, the relation of chain control
sets to maximal chain transitive sets of the associated control flows is
recalled. Affine systems of the form (1) are embedded into bilinear control
systems on an extended state space. The associated control flow possesses a
Selgrade decomposition, which is closely related to the Selgrade decomposition
of the homogeneous part (3). This yields that there is a unique central chain
control set ${}_{\mathbb{P}}E_{c}$ on the projective Poincaré sphere that is
not contained in the projective equator. These results are taken from Colonius
and Santana [9]. Section 3 introduces the notion of strong chain
controllability and strong chain control sets for control-affine systems on
non-compact manifolds $M$. Theorem 5 shows that the lifts to
$\mathcal{U}\times M$ of the strong chain control sets are the maximal
invariant strongly chain transitive sets of the control flow. Example 2
presents a linear control system, where the control set around the origin is
inside the strong chain control set $E^{\ast}$ and $E^{\ast}$ is a proper
subset of the chain control set $E$. Section 4 establishes that every affine
control system on $\mathbb{R}^{n}$ is conjugated to the induced system on the
northern hemisphere $\mathbb{S}^{n,+}$ of the Poincaré sphere and that the
strong chain control sets are preserved under this conjugacy. Analogous
results hold for the conjugation to the induced system on a subset of
$\mathbb{P}^{n}$; cf. Theorem 7. The ensuing sections contribute to the
analysis of chain controllability on spheres. For general bilinear control
systems, Section 5 describes the chain control sets for the induced systems on
the sphere. Theorem 8 proves that for any chain control set in projective
space, the preimage on the sphere is either a single chain control set or the
union of two chain control sets. This applies, in particular, to the central
chain control set ${}_{\mathbb{P}}E_{c}\subset\mathbb{P}^{n}$ of an affine
control system. Section 6 analyzes the chain control sets on the closed
hemispheres of the Poincaré sphere. Here additional assumptions are imposed on
the chain control sets ${}_{\mathbb{S}}E^{\hom}$ of the system on
$\mathbb{S}^{n-1}$ induced by the homogeneous part (3). They determine the
chain control sets of the system restricted to the equator $\mathbb{S}^{n,0}$
of the Poincaré sphere $\mathbb{S}^{n}$.

Notation. The origin in $\mathbb{R}^{n}$ is denoted by $0_{n}$ and $0_{1}$ is
abbreviated by $0$. The projections from $\mathbb{R}^{n}$ to the sphere
$\mathbb{S}^{n-1}$ and projective space $\mathbb{P}^{n-1}$ are denoted by
$\mathbb{S}x$ and $\mathbb{P}x$ for $0_{n}\not=x\in\mathbb{R}^{n}$.

##  2 Preliminaries

In the first subsection, we recall properties of strong chain recurrence for
flows on non-compact metric spaces. The second subsection collects notions and
results for control systems and the associated control flows. In particular,
some properties of affine control systems are reviewed.

###  2.1 Strong chain recurrence for flows

We formulate a lemma due to Hurley [18] characterizing strong chain recurrence
for continuous flows $\phi$ on non-compact metric spaces $X$ with metric $d$,
hence $\phi:\mathbb{R}\times X\rightarrow X$ is continuous with $\phi(0,x)=x$
and $\phi(t+s,x)=\phi(t,\phi(s,x))$ for $t,s\in\mathbb{R}$ and $x\in X$.
Furthermore, an example illustrates the difference between chain recurrence
and strong chain recurrence.

Let
$\mathcal{P}(X):=\\{\varepsilon:X\rightarrow(0,\infty),~{}$continuous$\\}$.
Given a function $\varepsilon\in\mathcal{P}(X)$ and $T>0$ sequences
$x_{0}=x,x_{1},\ldots,x_{k-1},x_{k}=y$ in $X$ and $t_{0},\ldots,t_{k-1}\geq T$
form an $(\varepsilon,T)$-chain from $x$ to $y$ if
$d(\phi(t_{j},x_{j}),x_{j+1})<\varepsilon(\phi(t_{j},x_{j}))$ for
$\,j=0,\ldots,k-1$. A point $x\in X$ is called strongly chain recurrent if for
all $T>0$ and $\varepsilon\in\mathcal{P}(X)$ there is an
$(\varepsilon,T)$-chain from $x$ to $x$. A set $Y\subset X$ is called strongly
chain transitive, if for all $x,y\in Y$ and all $T>0$ and
$\varepsilon\in\mathcal{P}(X)$ there is an $(\varepsilon,T)$-chain from $x$ to
$y$. The set of all strongly chain recurrent points is denoted by
$\mathcal{CR}^{\ast}(X)$. If $X$ is compact, the set $\mathcal{CR}^{\ast}(X)$
coincides with the chain recurrent set $\mathcal{CR}(X)$, i.e., the set of all
points $x$ such that for all constants $\varepsilon>0$ and all $T>0$ there is
an $(\varepsilon,T)$-chain from $x$ to $x$ (this can be seen similarly as
Proposition 1).

In the following the symbol “$\varepsilon$” often denotes a function. It will
be clear from the context when a constant $\varepsilon>0$ is meant. We will
need the following fundamental lemma due to Hurley [18, Lemma 2]; cf. also
Choi, Chu, and Park [6, Lemma 2].

######  Lemma 1.

Suppose that $(X,d)$ and $(Y,\rho)$ are metric spaces and that $g:X\rightarrow
Y$ and $\varepsilon:Y\rightarrow(0,\infty)$ are continuous. Then there is a
continuous map $\delta:X\rightarrow(0,\infty)$ such that $d(x,y)<\delta(x)$
implies $\rho(g(x),g(y))<\varepsilon(g(x))$.

This lemma has the following consequence for strong chain transitivity.

######  Theorem 1.

Let $\phi$ and $\psi$ be continuous flows on metric spaces $(X,d)$ and
$(Y,\rho)$, respectively. Suppose that $h:X\rightarrow Y$ is a homeomorphism
conjugating $\phi$ and $\psi$, i.e.,

$h(\phi(t,x))=\psi(t,h(x))\text{ for all }x\in X\text{ and
}t\in\mathbb{R}\text{.}$

Then $\mathcal{M}\subset X$ is a maximal invariant strongly chain transitive
set for $\phi$ if and only if $h(\mathcal{M})$ is a maximal invariant strongly
chain transitive set for $\psi$.

###### Proof.

Let $\varepsilon\in\mathcal{P}(Y)$ and choose $\delta\in\mathcal{P}(X)$
according to Hurley’s lemma, Lemma 1. Consider
$y=h(x),y^{\prime}=h(x^{\prime})\in h(\mathcal{M})$ and let $T>0$. Since
$\mathcal{M}$ is strongly chain transitive there is a $(\delta,T)$-chain in
$X$ given by $x_{0}=x,x_{1},\ldots,x_{k}=x^{\prime}$ in $X$ and
$T_{0},\ldots,T_{k-1}\geq T$ with

$d(\phi(T_{j},x_{j}),x_{j+1})<\delta(\phi(T_{j},x_{j}))\text{ for
}\,j=0,\ldots,k-1.$

Then the points $y_{j}:=h(x_{j}),\,j=0,\ldots,k-1$, satisfy

$\displaystyle\rho(\psi(T_{j},y_{j}),y_{j+1})$
$\displaystyle=\rho(\psi(T_{j},h(x_{j})),h(x_{j+1}))=\rho(h(\phi(T_{j},x_{j})),h(x_{j+1}))$
$\displaystyle<\varepsilon(h(\phi(T_{j},x_{j}))=\varepsilon(\psi(T_{j},h(x_{j})))=\varepsilon(\psi(T_{j},y_{j})).$

It follows that an $(\varepsilon,T)$-chain in $Y$ is given by
$y_{0}=y=h(x),y_{1}=h(x_{1}),\ldots,\allowbreak
y_{k}=y^{\prime}=h(x^{\prime})$ and $T_{0},\ldots,T_{k-1}\geq T$. Since
$\mathcal{M}$ is an invariant strongly chain transitive set for $\phi$ it
follows that $h(\mathcal{M})$ is an invariant strongly chain transitive for
$\psi$. The converse follows by considering $h^{-1}$. Thus also the maximality
property follows. ∎

In the following example, the state space is chain transitive but not strongly
chain transitive.

######  Example 1.

Consider the following two-dimensional autonomous differential equation

$\left(\begin{array}[c]{c}\dot{x}(t)\\\
\dot{y}(t)\end{array}\right)=\left(\begin{array}[c]{cc}0&1\\\
0&0\end{array}\right)\left(\begin{array}[c]{c}x(t)\\\ y(t)\end{array}\right).$

In the following, we show that the strong chain recurrent set equals the
$x$-axis $\mathbb{R}\times\\{0\\}$.

It is clear that the $x$-axis is contained in the strong chain recurrent set
since it consists of equilibria. The components of the solutions are
$\phi_{1}(t,x,y)=x+ty,\phi_{2}(t,x,y)=y$ for $t\in\mathbb{R}$. Hence in the
open upper half-plane, the trajectories move to the right on parallels to the
$x$-axis. The state space $\mathbb{R}^{2}$ is easily seen to be chain
transitive: For example, for constant $\varepsilon>0$ and $T>0$ any
$(\varepsilon,T)$-chain from an initial point $(0,y_{0}),y_{0}>0$, on the
$y$-axis to the $x$-axis follows parallels to the $x$-axis till time $T$,
jumps to a lower parallel to the $x$-axis, etc.

Claim: There are $T>0$ and $\varepsilon\in\mathcal{P}(\mathbb{R}^{2})$ such
that there is no $(\varepsilon,T)$-chain from $(x_{0},y_{0})=(0,2)$ to the
line $y=1$.

Let $T>0,\varepsilon\in\mathcal{P}(\mathbb{R}^{2})$ with
$\varepsilon(x,y)=\varepsilon^{\prime}(x)+\delta(x,y),$ where
$\varepsilon^{\prime}:\mathbb{R}\rightarrow(0,\infty)$ and
$\delta:\mathbb{R}^{2}\rightarrow(0,\infty)$ are continuous with
$\delta(0,y)>0$ for all $y\in\mathbb{R}$ and $\delta(x,y)=0$ for
$\left|x\right|\geq 1,y\in\mathbb{R}$. Furthermore, we assume that
$\varepsilon^{\prime}(x)$ strongly decreases to $0$ for $x\rightarrow\infty$.

Let the jump points be $(x_{i},y_{i})$ with $y_{i}\in(1,2]$ and
$x_{i+1}=\phi_{1}(T_{i},x_{i},y_{i})$. It suffices to show that
$Y:=\lim_{i\rightarrow\infty}y_{i}>1$, hence the line $y=1$ is not reached.

Note that $\varphi_{1}(T_{0},0,2)=2T_{0}>1$ for $T>\frac{1}{2}$, and

$\displaystyle\varepsilon(\phi(T_{0},0,2))$
$\displaystyle=\varepsilon(\phi_{1}(T_{0},0,2),\phi_{2}(T_{0},0,2))=\varepsilon^{\prime}(\phi_{1}(T_{0},0,2))+\delta(\phi_{1}(T_{0},0,2),2)$
$\displaystyle=\varepsilon^{\prime}(2T_{0})\leq\varepsilon^{\prime}(2T).$

Hence in the following only the function $\varepsilon^{\prime}$ will be
estimated. The first jump to $(x_{1},y_{1})$ occurs at
$x_{1}=\phi_{1}(T_{0},0,2)=2T_{0}$ and

$\phi_{2}(T_{0},y_{0})-y_{1}=y_{0}-y_{1}<\varepsilon^{\prime}(2T_{0})\text{,
i.e., }y_{1}>y_{0}-\varepsilon^{\prime}(2T_{0})\geq
2-\varepsilon^{\prime}(2T).$

We may assume that $y_{1}>2-\varepsilon^{\prime}(2T)>1$. In general a jump
occurs at

$\phi_{2}(T_{i},x_{i},y_{i})-y_{i+1}=y_{i}-y_{i+1}<\varepsilon^{\prime}(\phi_{1}(T_{i},x_{i},y_{i}))=\varepsilon^{\prime}(x_{i}+T_{i}y_{i}),$

hence $y_{i+1}>y_{i}-\varepsilon^{\prime}(x_{i}+T_{i}y_{i})$ and
$x_{i+1}=x_{i}+T_{i}y_{i}$. Since
$\varepsilon^{\prime}(x_{i}+T_{i}y_{i})<\varepsilon^{\prime}(x_{i})$ it
follows that $y_{i+1}>y_{i}-\varepsilon^{\prime}(x_{i})$ implying

$y_{i}>y_{1}-\sum\nolimits_{j=1}^{i-1}\varepsilon^{\prime}(x_{j})\text{ for
}i\geq 2.$

The $x$-component satisfies $x_{i+1}=x_{i}+T_{i}y_{i}\geq x_{i}+T$ and hence
$x_{i}>x_{1}+(i-1)T$ for $i\geq 2$. Since $y_{1}-1>0$ we can, additionally,
suppose that for the function $\varepsilon^{\prime}$ the following limit
exists and satisfies

$\alpha:=\sum\nolimits_{j=1}^{\infty}\varepsilon^{\prime}(x_{1}+(j-1)T)<y_{1}-1.$

This implies
$\sum_{j=1}^{i-1}\varepsilon^{\prime}(x_{j})<\sum_{j=1}^{i-1}\varepsilon^{\prime}(x_{1}+(j-1)T)\leq\alpha$.
The claim follows from

$y_{i}>y_{1}-\sum\nolimits_{j=1}^{i-1}\varepsilon^{\prime}(x_{j})>y_{1}-\alpha>1\text{
for all }i\geq 2.$

A consequence of the claim is that there is no $(\varepsilon,T)$-chain from
$(x_{0},y_{0})=(0,2)$ to the $x$-axis.

Analogous, but technically more involved arguments show for every point
$(x_{0},y_{0})$ in the open upper and the open lower half-planes that there
are $T>0$ and $\varepsilon\in\mathcal{P}(\mathbb{R}^{2})$ such that there is
no $(\varepsilon,T)$-chain from $(x_{0},y_{0})$ to the $x$-axis. One concludes
that the strong chain recurrent set coincides with the $x$-axis.

###  2.2 Control systems

Consider control-affine systems of the form

$\dot{x}(t)=X_{0}(x(t))+\sum_{i=1}^{m}u_{i}(t)X_{i}(x(t)),\,u\in\mathcal{U},$
(4)

where $X_{0},X_{1},\ldots,X_{m}$ are smooth ($C^{\infty}$-)vector fields on a
smooth manifold $M$ and $\mathcal{U}$ is defined by (2). We assume that for
every control $u\in\mathcal{U}$ and every initial state $x(0)=x_{0}\in M$
there exists a unique (Carathéodory) solution
$\varphi(t,x_{0},u),t\in\mathbb{R}$.

Recall the notion of chain controllability (cf. Colonius and Kliemann [7, p.
80], Kawan [23, Definition 1.14]), which allows for (small) jumps between
pieces of trajectories; cf. also Kawan [22] and Ayala, Da Silva, and San
Martin [1]. Here a metric $d_{M}$ compatible with the topology on $M$ is
fixed.

######  Definition 1.

Let $x,y\in M$. For $\varepsilon,T>0$ a controlled $(\varepsilon,T)$-chain
$\zeta$ from $x$ to $y$ is given by $k\in\mathbb{N},\
x_{0}=x,x_{1},\ldots,x_{k}=y\in M,\ u_{0},\ldots,u_{k-1}\in\mathcal{U}$, and
$T_{0},\ldots,T_{k-1}\geq T$ with

$d_{M}(\varphi(T_{j},x_{j},u_{j}),x_{j+1})<\varepsilon\text{
}\,\text{for\thinspace all}\,\,j=0,\ldots,k-1.$

If for every $\varepsilon,T>0$ there is a controlled $(\varepsilon,T)$-chain
from $x$ to $y$, the point $x$ is chain controllable to $y$. The chain
reachable set from $x\ $and the chain controllable set to $x$ are

$\displaystyle\mathbf{R}^{c}(x)$ $\displaystyle=\left\\{y\in M\left|x\text{ is
chain controllable to }y\right.\right\\},$ $\displaystyle\mathbf{C}^{c}(x)$
$\displaystyle=\left\\{y\in M\left|y\text{ is chain controllable to
}x\right.\right\\},$

respectively. A nonvoid set $F\subset M$ is said to be chain controllable, if
$x$ is chain controllable to $y$ for all $x,y\in F$.

Chain controllability is a generalized version of approximate controllability
in infinite time. We define chain control sets as maximal chain controllable
sets.

######  Definition 2.

A nonvoid set $E\subset M$ is a chain control set of system (4) if for all
$x,y\in E$ and $\varepsilon,T>0$ there is a controlled $(\varepsilon,T)$-chain
from $x$ to $y$, and $E$ is maximal with this property.

We cite the following result from Colonius, Santana, and Viscovini [12,
Theorem 2.14 (iii)].

######  Theorem 2.

Let $E$ be a chain control set. Then it follows for all $x\in E$ that
$E=\mathbf{R}^{c}(x)\cap\mathbf{C}^{c}(x)$ and that there exists
$u\in\mathcal{U}$ such that $\varphi(t,x,u)\in E$ for all $t\in\mathbb{R}$.

Topological conjugacies between control systems are defined as follows. For
simplicity, we suppose that the sets of control functions coincide.

######  Definition 3.

Consider two control systems of the form (4) on manifolds $M$ and $N$,
respectively, with controls $u\in\mathcal{U}$, given by

$\dot{x}(t)=X_{0}(x(t))+\sum_{i=1}^{m}u_{i}(t)X_{i}(x(t))\text{ and
}\dot{y}(t)=Y_{0}(y(t))+\sum_{i=1}^{m}u_{i}(t)Y_{i}(y(t)).$ (5)

Denote the trajectories by $\varphi(t,x_{0},u)$ and
$\psi(t,y_{0},u),t\in\mathbb{R}$, respectively. A homeomorphism
$H:M\rightarrow N$ is a topological conjugacy of these control systems, if

$H(\varphi(t,x_{0},u))=\psi(t,H(x_{0}),u)\text{ for }t\in\mathbb{R},x_{0}\in
M,u\in\mathcal{U}.$

Control systems of the form (4) come with an associated flow called the
control flow. Endow the set $\mathcal{U}$ of controls with a metric compatible
with the weak∗ topology on $L^{\infty}(\mathbb{R},\mathbb{R}^{m})$. One may
choose

$d_{\mathcal{U}}(u,v)=\sum_{i=1}^{\infty}\frac{1}{2^{i}}\frac{\left|\int_{\mathbb{R}}\left\langle
u(t)-v(t),y_{i}(t)\right\rangle
dt\right|}{1+\left|\int_{\mathbb{R}}\left\langle
u(t)-v(t),y_{i}(t)\right\rangle dt\right|},$ (6)

where $\\{y_{i}\\}$ is an arbitrary countable dense set in
$L^{1}(\mathbb{R},\mathbb{R}^{m})$. A metric on $\mathcal{U}\times M$ is
defined by $d((u,x),(v,y))=\max\\{d_{\mathcal{U}}(u,v),d_{M}(x,y)\\}$. The
control flow is defined as

$\Phi:\mathbb{R}\times\mathcal{U}\times M\rightarrow\mathcal{U}\times
M,\,(t,u,x_{0})\mapsto(u(t+\cdot),\varphi(t,x_{0},u)),$ (7)

where $u(t+\cdot)(s):=u(t+s),s\in\mathbb{R}$, is the right shift. The control
flow $\Phi$ is continuous and $\mathcal{U}$ is compact and chain transitive;
cf. Colonius and Kliemann [7, Chapter 4] or Kawan [23, Section 1.4].

Chain control sets can be characterized as maximal chain transitive sets of
the control flow. By [7, Theorem 4.3.11] or [23, Proposition 1.24(iv)] a chain
control set $E$ yields a maximal invariant chain transitive set $\mathcal{E}$
of the control flow $\Phi$ via

$\mathcal{E}:=\\{(u,x)\in\mathcal{U}\times M\left|\varphi(t,x,u)\in E\mbox{
for all }t\in\mathbb{R}\right.\\},$ (8)

and for any maximal invariant chain transitive set in $\mathcal{U}\times M$
the projection to $M$ is a chain control set.

The affine control system (1) and its homogeneous, linear part (3) generate
control flows which we denote by $\Psi$ and $\Phi^{\hom}$, respectively, on
the vector bundle $\mathcal{V}:=\mathcal{U}\times\mathbb{R}^{n}$. Here
$\Phi^{\hom}$ is a linear flow. Fundamental information on linear flows on
vector bundles is provided by Selgrade’s theorem; cf. Selgrade [31], Salamon
and Zehnder [30], Colonius and Kliemann [7, 8]. It provides the finest
decomposition of the underlying vector bundle into exponentially separated
subbundles and, equivalently, the finest Morse decomposition of the associated
flow on the projective bundle $\mathcal{U}\times\mathbb{P}^{n-1}$, which is
compact. The Morse sets in this decomposition are the maximal invariant chain
transitive subsets.

For the linear flow $\Phi^{\hom}$, Selgrade’s theorem yields the decomposition

$\mathcal{U}\times\mathbb{R}^{n}=\mathcal{V}_{1}\oplus\cdots\oplus\mathcal{V}_{\ell}\text{
with }1\leq\ell\leq n,$ (9)

where the Selgrade bundles $\mathcal{V}_{i}$ are exponentially separated and
the projections $\mathbb{P}\mathcal{V}_{i}$ to the projective bundle
$\mathbb{P}\mathcal{V}$ are the maximal invariant chain transitive sets of the
induced flow $\mathbb{P}\Phi^{\hom}$ on
$\mathbb{P}\mathcal{V}=\mathcal{U}\times\mathbb{P}^{n-1}$. The bilinear
control system (3) induces control systems on the unit sphere
$\mathbb{S}^{n-1}$ and on projective space $\mathbb{P}^{n-1}$ (cf., e.g.,
Colonius and Kliemann [7, Chapter 6]) with trajectories
$\mathbb{P}\varphi(t,\mathbb{P}x_{0},u),t\in\mathbb{R}$, for $x_{0}\not=0$.
The projective flow $\mathbb{P}\Phi^{\hom}$ is the control flow for the
induced control system on $\mathbb{P}^{n-1}$.

Affine control system (1) can be embedded into a bilinear control system on
$\mathbb{R}^{n+1}$ of the form (cf. Jurdjevic [19] or Elliott [15, Subsection
3.8.1])

$\left(\begin{array}[c]{c}\dot{x}(t)\\\
\dot{z}(t)\end{array}\right)=\left(\begin{array}[c]{cc}A_{0}&a_{0}\\\
0&0\end{array}\right)\left(\begin{array}[c]{c}x(t)\\\
z(t)\end{array}\right)+\sum_{i=1}^{m}u_{i}(t)\left(\begin{array}[c]{cc}A_{i}&a_{i}\\\
0&0\end{array}\right)\left(\begin{array}[c]{c}x(t)\\\
z(t)\end{array}\right),u\in\mathcal{U},$ (10)

with trajectories denoted by $\psi^{1}(t,x_{0},z_{0},u),t\in\mathbb{R}$. This
control system induces control systems on the unit sphere $\mathbb{S}^{n}$ and
on projective space $\mathbb{P}^{n}$ with trajectories
$\mathbb{P}\psi^{1}(t,\mathbb{P}(x_{0},z_{0}),u),t\in\mathbb{R}$, for
$(x_{0},z_{0})\not=(0,0)$. The associated control flow $\Psi^{1}$ on
$\mathcal{U}\times\mathbb{R}^{n+1}$ is linear.

The following result (cf. Colonius and Santana [9, Corollary 32]) shows that
the Selgrade decomposition of $\Psi^{1}$ is closely related to the Selgrade
decomposition (9) of the homogeneous part. The Selgrade bundles of $\Psi^{1}$
are essentially the Selgrade bundles of the homogeneous part $\Phi^{\hom}$,
with one exception.

######  Theorem 3.

Consider an affine control system of the form (1) and the associated affine
control flow $\Psi$ on $\mathcal{U}\times\mathbb{R}^{n}$. For
$i\in\\{1,\ldots,\ell\\}$ let
$\mathcal{V}_{i}\subset\mathcal{U}\times\mathbb{R}^{n}$ be the Selgrade
bundles of the linear flow $\Phi^{\hom}$ associated with control system (3),
and let $\mathcal{V}_{i}^{\infty}=\mathcal{V}_{i}\times\\{0\\}$.

(i) The Selgrade decomposition of the lifted flow $\Psi^{1}$ has the form

$\mathcal{U}\times\mathbb{R}^{n+1}=\mathcal{V}_{1}^{\infty}\oplus\cdots\oplus\mathcal{V}_{\ell^{+}}^{\infty}\oplus\mathcal{V}_{c}^{1}\oplus\mathcal{V}_{\ell^{+}+\ell^{0}+1}^{\infty}\oplus\cdots\oplus\mathcal{V}_{\ell}^{\infty},$
(11)

for some numbers $\ell^{+},\ell^{0}\geq 0$ with $\ell^{+}+\ell^{0}\leq\ell$.

(ii) The subbundle $\mathcal{V}_{c}^{1}$ called the central Selgrade bundle
satisfies

$\mathcal{V}_{c}^{1}\cap\left(\mathcal{U}\times\mathbb{R}^{n}\times\\{0\\}\right)=\bigoplus_{i=\ell^{+}+1}^{i=\ell^{+}+\ell^{0}}\mathcal{V}_{i}^{\infty}:=\mathcal{V}_{c}^{\infty}.$

(iii) The dimension of $\mathcal{V}_{c}^{1}$ is given by
$\dim\mathcal{V}_{c}^{1}=1+\dim\mathcal{V}_{c}^{\infty}$, and
$\dim\mathcal{V}_{c}^{1}=1$ holds if and only if
$\mathcal{V}_{c}^{1}\cap\left(\mathcal{U}\times\mathbb{R}^{n}\times\\{0\\}\right)=\mathcal{U}\times\\{0_{n}\\}\times\\{0\\}$.

(iv) Suppose that (3) is uniformly hyperbolic, i.e.,
$\mathcal{U}\times\mathbb{R}^{n}$ can be split into a stable and an unstable
subbundle. Then the central Selgrade bundle is the line bundle

$\mathcal{V}_{c}^{1}=\\{(u,-re(u,0),r)\in\mathcal{U}\times\mathbb{R}^{n}\times\mathbb{R}\left|u\in\mathcal{U},r\in\mathbb{R}\right.\\},$
(12)

where $e(u,t),t\in\mathbb{R}$, is the unique bounded solution of (1) for
$u\in\mathcal{U}$, and $\mathbb{P}\mathcal{V}_{c}^{1}$ is a compact subset of
$\mathcal{U}\times\left\\{\mathbb{P}(x,1)\left|x\in\mathbb{R}^{n}\right.\right\\}$.

This theorem also yields information on chain control sets on
$\mathbb{P}^{n}$, based on the relation between maximal chain transitive sets
of control flows and chain control sets; cf. (8). Projective space
$\mathbb{P}^{n}$ can be written as the disjoint union
$\mathbb{P}^{n}=\mathbb{P}^{n,1}\cup\,\mathbb{P}^{n,0}$, where
$\mathbb{P}^{n,1}:=\left\\{\mathbb{P}(x,1)\left|x\in\mathbb{R}^{n}\right.\right\\}$
and
$\mathbb{P}^{n,0}:=\left\\{\mathbb{P}(x,0)\left|0\not=x\in\mathbb{R}^{n}\right.\right\\}$.
Note that $\mathbb{P}^{n,1}$ can be identified with the northern hemisphere
$\mathbb{S}^{n,+}$ of the unit sphere $\mathbb{S}^{n}$ and $\mathbb{P}^{n,0}$
corresponds to the equator of $\mathbb{S}^{n}$. A special feature of the
system induced on $\mathbb{P}^{n}$ by the bilinear system (10) is that the
sets $\mathbb{P}^{n,1}$ and $\mathbb{P}^{n,0}$ are invariant. In this context,
we have established in [9, Theorem 35] the following relation between the
chain control sets in these spaces.

######  Theorem 4.

Consider an affine control system of the form (1).

(i) There is a unique chain control set ${}_{\mathbb{P}}E_{c}$, called the
central chain control set, of the induced control system on the projective
Poincaré space $\mathbb{P}^{n}$ such that
${}_{\mathbb{P}}E_{c}\cap\mathbb{P}^{n,1}\not=\varnothing$. It is given by

${}_{\mathbb{P}}E_{c}=\\{\mathbb{P}(x,r)\in\mathbb{P}^{n}\left|\exists
u\in\mathcal{U}:(u,\mathbb{P}(x,r))\in\mathbb{P}\mathcal{V}_{c}^{1}\right.\\}.$

(ii) If there is a chain control set $E$ in $\mathbb{R}^{n}$ of the affine
control system (1), the image $\mathbb{P}\left(E\times\\{1\\}\right)$ in the
projective Poincaré space $\mathbb{P}^{n}$ is contained in
${}_{\mathbb{P}}E_{c}$.

(iii) If (3) is uniformly hyperbolic, then there is a unique chain control set
$E$ in $\mathbb{R}^{n}$. It is compact and the central chain control set is a
compact subset of $\mathbb{P}^{n,1}$ satisfying ${}_{\mathbb{P}}E_{c}$
$=\left\\{\mathbb{P}\left(x,1\right)\left|x\in E\right.\right\\}$. For every
$u\in\mathcal{U}$ there exists a unique element $x\in E$ with $\psi(t,x,u)\in
E$ for all $t\in\mathbb{R}$.

In the nonhyperbolic case, the information given in this theorem on the chain
control sets of affine control systems is rather scarce: It is only shown that
their images on the projective Poincaré sphere are contained in the central
chain control set ${}_{\mathbb{P}}E_{c}$. A main goal of this paper is to
derive more information on chain controllability properties of affine control
systems.

##  3 Noncompact spaces and strong chain controllability

We introduce a stronger version of chain control sets for noncompact spaces
and provide a characterization in terms of the control flow.

Consider a control system of the form (4) on a manifold $M$ with trajectories
$\varphi(t,x,u),\allowbreak t\in\mathbb{R}$, and define the set of jump
functions by $\mathcal{P}(M):=\\{\varepsilon:M\rightarrow(0,\infty),$
continuous$\\}$.

Define for $\varepsilon\in\mathcal{P}(M)$ a controlled $(\varepsilon,T)$-chain
from $x$ to $y$ by $T_{0},\ldots,T_{k-1}\allowbreak\geq
T,u_{0},\ldots,u_{k-1}\in\mathcal{U}$ and $x_{0}=x,x_{1},\ldots,x_{k}=y\in M$
with

$d_{M}(\varphi(T_{j},x_{j},u_{j}),x_{j+1})<\varepsilon(\varphi(T_{j},x_{j},u_{j}))\text{
for }j=0,\ldots,k-1.$

A point $x$ is strongly chain controllable to a point $y$ if for each
$\varepsilon\in\mathcal{P}(M)$ there is a controlled $(\varepsilon,T)$-chain
from $x$ to $y$. A nonvoid subset $F$ of $M$ is called strongly chain
controllable if every point $x\in F$ is strongly chain controllable to every
point $y\in F$.

######  Definition 4.

For a control system on $M$ a nonvoid set $E^{\ast}\subset M$ is a strong
chain control set if (i) every $x\in E^{\ast}$ is strongly chain controllable
to every $y\in E^{\ast}$ (ii) for every $x\in E^{\ast}$ there is
$u\in\mathcal{U}$ with $\varphi(t,x,u)\in E^{\ast}$ for all $t\in\mathbb{R}$,
and (iii) $E^{\ast}$ is maximal with these properties.

Note that jump functions $\varepsilon\in\mathcal{P}(M)$ may have values
arbitrarily close to $0$. In particular, if the boundary $\partial M$ of $M$
is nonvoid and $\partial M\cap M=\varnothing$, there are
$\varepsilon\in\mathcal{P}(M)$ with $\lim_{d_{M}(x,\partial M)\rightarrow
0}\varepsilon(x)=0$.

The following proposition presents some basic properties of strong chain
control sets. In particular, it shows that for compact state spaces chain
control sets and strong chain control sets coincide.

######  Proposition 1.

(i) Every strong chain control set $E^{\ast}$ is closed, and if the
intersection of strong chain control sets $E_{1}^{\ast}$ and $E_{2}^{\ast}$ is
nonvoid, then $E_{1}^{\ast}=E_{2}^{\ast}$.

(ii) Every strong chain control set is contained in a chain control set.

(iii) Every compact chain control set $E$ is also a strong chain control set.

(iv) If $M$ is compact, then chain control sets and strong chain control sets
coincide.

###### Proof.

(i) Suppose that a sequence $\left(x^{i}\right)$ in $E^{\ast}$ converges to
$x\in M$. By compactness of $\mathcal{U}$ and continuity it follows that there
exists $u\in\mathcal{U}$ with $\varphi(t,x,u)\in\overline{E^{\ast}}$ for all
$t\in\mathbb{R}$.

Consider $y\in E^{\ast}$ and let $\varepsilon\in\mathcal{P}(M),T>0$. First, we
show that there exists a controlled $(\varepsilon,T)$-chain from $y$ to $x$.
There are $v^{i}\in\mathcal{U}$ with $\varphi(-T,x^{i},v^{i})\in E^{\ast}$.
Hence there are controlled $(\varepsilon,T)$-chains $\zeta^{i}$ from $y\in
E^{\ast}$ to $\varphi(-T,x^{i},v^{i})$ satisfying
$x_{0}^{i}=y,x_{k_{i}}^{i}=\varphi(-T,x^{i},v^{i})$, and for
$j=0,\ldots,k_{i}-1$

$d_{M}(\varphi(T_{j}^{i},x_{j}^{i},u_{j}^{i}),x_{j+1}^{i})<\varepsilon(\varphi(T_{j}^{i},x_{j}^{i},u_{j}^{i})).$

Extend each chain $\zeta^{i}$ by defining
$T_{k_{i}}^{i}:=T,x_{k_{i}+1}^{i}:=x,$ and $u_{k_{i}}^{i}:=v^{i}(-T+\cdot)$.
Now take $i$ large enough such that
$\varepsilon(x^{i})\geq\frac{1}{2}\varepsilon(x)$ and
$d_{M}(x^{i},x)<\frac{1}{2}\varepsilon(x)$. It follows that
$\varphi(T_{k_{i}}^{i},x_{k_{i}}^{i},u_{k_{i}}^{i})=x^{i}$ and

$\displaystyle
d_{M}(\varphi(T_{k_{i}}^{i},x_{k_{i}}^{i},u_{k_{i}}^{i}),x_{k_{i}+1}^{i})$
$\displaystyle=d_{M}(\varphi(T,\varphi(-T,x^{i},v^{i}),v^{i}(-T+\cdot)),x)$
$\displaystyle=d_{M}(x^{i},x)<\frac{1}{2}\varepsilon(x)\leq\varepsilon(x^{i})=\varepsilon(\varphi(T_{k_{i}}^{i},x_{k_{i}}^{i},u_{k_{i}}^{i})).$

Hence this defines a controlled $(\varepsilon,T)$-chain from $y$ to $x$.

Next we show that there is a controlled $(\varepsilon,T)$-chain from $x$ to
$y$. There is a controlled $(\varepsilon,2T)$-chain $\zeta^{i}$ from $x^{i}\in
E^{\ast}$ to $y$ satisfying

$d_{M}(\varphi(T_{j}^{i},x_{j}^{i},u_{j}^{i}),x_{j+1}^{i})<\varepsilon(\varphi(T_{j}^{i},x_{j}^{i},u_{j}^{i}))\text{
for }j=0,\ldots,k_{i}-1.$

Introduce an artificial jump point $z_{1}^{i}:=\varphi(T,x^{i},u_{0}^{i})$ at
time $S_{0}=T$, let $v_{0}^{i}:=u_{0}^{i}$, and define
$S_{1}^{i}:=T_{0}^{i}-T\geq T$ and $v_{1}^{i}:=u_{0}^{i}(T+\cdot)$.

Then replace $T_{0}^{i},u_{0}^{i}$ by $S_{0},v_{0}^{i}$ and
$S_{1}^{i},v_{1}^{i}$. This yields again a controlled $(\varepsilon,T)$-chain
from $x^{i}\in E^{\ast}$ to $y$: In fact,

$d_{M}(\varphi(S_{0},x^{i},v_{0}^{i}),z_{1}^{i})=d_{M}(\varphi(T,x^{i},u_{0}^{i}),\varphi(T,x^{i},u_{0}^{i}))=0<\varepsilon(\varphi(S_{0},x^{i},v_{0}^{i}))$

and
$\varphi(S_{1}^{i},z_{1}^{i},v_{1}^{i})=\varphi(T_{0}^{i}-T,\varphi(T,x^{i},u_{0}^{i}),u_{0}^{i}(T+\cdot))=\varphi(T_{0}^{i},x^{i},u_{0}^{i})$,
hence

$d_{M}(\varphi(S_{1}^{i},z_{1}^{i},v_{1}^{i}),x_{2}^{i})<\varepsilon(\varphi(S_{1}^{i},z_{1}^{i},v_{1}^{i})).$

We claim that $\inf_{j\in\mathbb{N}}\varepsilon(\varphi(T,x,u_{0}^{j}))>0$. In
fact otherwise there is a subsequence $u_{i_{j}}$ such that
$\varepsilon(\varphi(T,x,u_{0}^{i_{j}}))\rightarrow 0$. By compactness of
$\mathcal{U}$ and continuity it follows that there is a subsequence converging
to some element $u\in\mathcal{U}$ with
$\varepsilon\left(\varphi(T,x,u)\right)=0$ which contradicts the definition of
$\varepsilon$.

The claim together with continuity implies that one can take $i$ large enough
such that

$d_{M}(\varphi(T,x,u_{0}^{i}),\varphi(T,x^{i},u_{0}^{i}))<\frac{1}{2}\inf_{j\in\mathbb{N}}\varepsilon(\varphi(T,x,u_{0}^{j}))\leq\frac{1}{2}\varepsilon(\varphi(T,x,u_{0}^{i})).$

Taking, if necessary, a subsequence we may assume that $u_{0}^{i}$ converges
to an element $u\in\mathcal{U}$. Then

$\varepsilon(\varphi(T,x,u_{0}^{i}))\rightarrow\varepsilon(\varphi(T,x,u))\text{
and
}\varepsilon(\varphi(T,x^{i},u_{0}^{i}))\rightarrow\varepsilon(\varphi(T,x,u)).$

Hence for $i$ large enough
$\varepsilon(\varphi(T,x,u_{0}^{i}))<2\varepsilon(\varphi(T,x^{i},u_{0}^{i}))$.
Then we replace the starting point $x^{i}$ by $x$ and find

$\displaystyle d_{M}(\varphi(S_{0},x,v_{0}^{i}),z_{1}^{i})$
$\displaystyle=d_{M}(\varphi(T,x,v_{0}^{i}),z_{1}^{i})$ $\displaystyle\leq
d_{M}(\varphi(T,x,v_{0}^{i}),\varphi(T,x^{i},u_{0}^{i}))+d_{M}(\varphi(T,x^{i},u_{0}^{i}),z_{1}^{i})$
$\displaystyle=d_{M}(\varphi(T,x,u_{0}^{i}),\varphi(T,x^{i},u_{0}^{i}))+0$
$\displaystyle<\frac{1}{2}\varepsilon(\varphi(T,x,u_{0}^{i}))<\varepsilon(\varphi(T,x,u_{0}^{i})).$

This shows that the modified chain is a controlled $(\varepsilon,T)$-chain
from $x$ to $y$.

Since $\varepsilon\in\mathcal{P}(M)$ and $T>0$ are arbitrary it follows that
$x\in E^{\ast}$. If the intersection of two chain control sets is nonvoid,
they coincide by the maximality property since the concatenation of controlled
$(\varepsilon,T)$-chains is a controlled $(\varepsilon,T)$-chain.

(ii) This follows by considering constant jump functions
$\varepsilon(x)\equiv\varepsilon>0$.

(iii) In view of assertion (ii) it suffices to show strong chain
controllability in $E$. Let $\varepsilon\in\mathcal{P}(M)$. Then
$\varepsilon(x)>0$ for all $x\in E$ and compactness of $E$ imply that
$\varepsilon_{0}:=\min\left\\{\varepsilon(x):x\in E\right\\}>0$. Thus any
controlled $(\varepsilon_{0},T)$-chain defines a strong controlled
$(\varepsilon,T)$-chain, hence strong chain controllability holds in $E$.

(iv) Here chain control sets and strong chain control sets are compact. Hence
the assertion follows by (ii) and (iii). ∎

From Example 1 we can construct a linear control system, where a strong chain
control set is a proper subset of a chain control set and contains a control
set, see Figure 1.

######  Example 2.

Consider a linear control system given by

$\left(\begin{array}[c]{c}\dot{x}(t)\\\ \dot{y}(t)\\\
\dot{z}(t)\end{array}\right)=\left(\begin{array}[c]{ccc}0&1&0\\\ 0&0&0\\\
0&0&1\end{array}\right)\left(\begin{array}[c]{c}x(t)\\\ y(t)\\\
z(t)\end{array}\right)+\left(\begin{array}[c]{c}0\\\ 0\\\
1\end{array}\right)u(t)\text{ with }u(t)\in\Omega=[-1,1].$

By Example 1 it follows for the subspace $\mathbb{R}^{2}\times\\{0\\}$ that
the strong chain recurrent set is $\mathbb{R}\times\\{0\\}\times\\{0\\}$ (here
the controls $u$ do not act). The control set (i.e., the maximal set of
approximate controllability) containing $0\in\mathbb{R}^{3}$ is
$D_{0}=\\{(0,0)\\}\times(-1,1)$. Since in the interior of $D_{0}$ with respect
to $\\{(0,0)\\}\times\mathbb{R}$ exact controllability holds, the strong chain
control set is

$E^{\ast}=\mathbb{R}\times\\{0\\}\times[-1,1].$

On the other hand, $L^{0}=\mathbb{R}^{2}\times\\{0\\}$ is the center subspace
of the homogeneous (uncontrolled) part, and Colonius, Santana, and Viscovini
[12, Theorem 4.8] implies that the chain control set is

$E=\overline{D_{0}}+L^{0}=\left(\\{(0,0)\\}\times[-1,1]\right)+\left(\mathbb{R}^{2}\times\\{0\\}\right)=\mathbb{R}^{2}\times[-1,1]\not=E^{\ast}.$

The next theorem clarifies the relation between strong chain control sets and
maximal invariant strongly chain transitive sets of the control flow. The
proof is a modification of the one for chain control sets; cf. Colonius and
Kliemann [7, Theorem 4.3.11] or Kawan [23, Proposition 1.24(iv)].

######  Theorem 5.

Consider a control system of the form (4) on $M$ with control flow $\Phi$ on
$\mathcal{U}\times M$.

(i) If $E^{\ast}\subset M$ is a strong chain control set, then

$\mathcal{E}^{\ast}:=\\{(u,x)\in\mathcal{U}\times M\left|\varphi(t,x,u)\in
E^{\ast}\text{ for all }t\in\mathbb{R}\right.\\}$ (13)

is a maximal invariant strongly chain transitive set for the control flow
$\Phi$.

(ii) Conversely, let $\mathcal{E}^{\ast}\subset\mathcal{U}\times M$ be a
maximal invariant strongly chain transitive set for the control flow $\Phi$.
Then

$E^{\ast}:=\left\\{x\in M\left|\exists
u\in\mathcal{U}:(u,x)\in\mathcal{E}^{\ast}\right.\right\\}$ (14)

is a strong chain control set.

###### Proof.

(i) Let $E^{\ast}$ be a strong chain control set and consider
$(u,x),\;(v,y)\in\mathcal{E}^{\ast}$ defined by (13). Pick $T>0$ and
$\varepsilon\in\mathcal{P}(\mathcal{U}\times M)$.

Claim 1: Define
$\varepsilon_{M}(x):=\min\left\\{\varepsilon(u,x)\left|u\in\mathcal{U}\right.\right\\},x\in
M.$ Then $\varepsilon_{M}\in\mathcal{P}(M)$.

By compactness of $\mathcal{U}$ it is clear that $\varepsilon_{M}(x)>0$ for
all $x\in M$ and it remains to show that $\varepsilon_{M}$ is continuous.
Consider a sequence $x_{i}\rightarrow x$. It follows that
$\varepsilon(u,x_{i})\geq\varepsilon_{M}(x_{i})$ for all $u\in\mathcal{U}$.
Continuity of $\varepsilon$ implies that
$\varepsilon(u,x)\geq\lim\sup_{i\rightarrow\infty}\varepsilon_{M}(x_{i})$ for
all $u\in\mathcal{U}$ and hence
$\varepsilon_{M}(x)\geq\lim\sup_{i\rightarrow\infty}\varepsilon_{M}(x_{i})$.

By compactness of $\mathcal{U}$ there are $u_{i}\in\mathcal{U}$ with
$f(u_{i},x_{i})=\varepsilon_{M}(x_{i})$ and there is a subsequence
$(u_{i_{k}})$ converging to some $u^{\ast}\in\mathcal{U}$. It follows that
$\varepsilon_{M}(x_{i_{k}})=\varepsilon(u_{i_{k}},x_{i_{k}})\rightarrow\varepsilon(u^{\ast},x)\geq\varepsilon_{M}(x)$.
Since this applies to any subsequence of $(u_{i})$ it follows that
$\varepsilon_{M}(x)\leq\lim\inf_{i\rightarrow\infty}\varepsilon_{M}(x_{i})$
proving Claim 1.

Strong chain controllability from $\varphi(2T,x,u)\in E^{\ast}$ to
$\varphi(-T,y,v)\in E^{\ast}$ yields the existence of a controlled
$(\varepsilon_{M},T)$-chain given by
$x_{0}=\varphi(2T,x,u),x_{1},\ldots,x_{k}=\varphi(-T,y,v)$ in
$M,\,u_{0},\ldots,u_{k-1}\in\mathcal{U},\,T_{0},\ldots,T_{k-1}\geq T$ with

$d_{M}(\varphi(T_{j},x_{j},u_{j}),x_{j+1})<\varepsilon_{M}(\varphi(T_{j},x_{j},u_{j}))\text{\
for }j=0,\ldots,k-1.$ (15)

We construct an $(\varepsilon,T)$-chain from $(u,x)$ to $(v,y)$ in the
following way. Define

$\begin{array}[c]{cccc}T_{-2}=T,&x_{-2}=x,&v_{-2}=u,&\\\
T_{-1}=T,&x_{-1}=\varphi(T,x,u),&v_{-1}(t)=&\left\\{\begin{array}[c]{cc}u(T_{-2}+t)&\text{for
}t\leq T_{-1}\\\ u_{0}(t-T_{-1})&\text{for
}t>T_{-1}\end{array}\right.\end{array}$

and let the times $T_{0},\ldots,T_{k-1}$ and the points $x_{0},\ldots,x_{k}$
be as given earlier. Furthermore, set

$\begin{array}[c]{ccc}T_{k}=T,&x_{k+1}=y,&v_{k+1}=v,\end{array}$

and define for $j=0,\ldots,k-2$

$\displaystyle v_{j}(t)$
$\displaystyle=\left\\{\begin{array}[c]{lll}v_{j-1}(T_{j-1}+t)&\text{for}&t\leq
0\\\ u_{j}(t)&\text{for}&0<t\leq T_{j}\\\
u_{j+1}(t-T_{j})&\text{for}&t>T_{j},\end{array}\right.$ $\displaystyle
v_{k-1}(t)$
$\displaystyle=\left\\{\begin{array}[c]{lll}v_{k-2}(T_{k-2}+t)&\text{for}&t\leq
0\\\ u_{k-1}(t)&\text{for}&0<t\leq T_{k-1}\\\
v(t-T_{k-1}-T)&\text{for}&t>T_{k-1},\end{array}\right.$ $\displaystyle
v_{k}(t)$
$\displaystyle=\left\\{\begin{array}[c]{ll}v_{k-1}(T_{k-1}+t)&\text{for }t\leq
0\\\ v(t-T)&\text{for }t>0.\end{array}\right.$

Define a constant $\widehat{\varepsilon}$ by

$\widehat{\varepsilon}:=\frac{1}{2}\min_{j=-2,\ldots,k}\varepsilon_{M}(\varphi(T_{j},x_{j},v_{j})).$

Recall the definition of the metric $d_{\mathcal{U}}$ on $\mathcal{U}$ given
in (6) and choose $N\in\mathbb{N}$ large enough such that
$\sum_{i=N+1}^{\infty}2^{-i}<\widehat{\varepsilon}$. There exists $S>0$ such
that

$\int_{\mathbb{R}\setminus\left[-S,S\right]}\left|y_{i}(\tau)\right|\,\,d\tau<\widehat{\varepsilon}/\mathrm{\,diam}\,U\text{
for }i=1,\ldots,N.$

We can assume without loss of generality that $T\geq S$. Since
$\varepsilon\in\mathcal{P}(\mathcal{U}\times M)$ and $T\geq S$ are arbitrary,
the following claim implies that $\mathcal{E}$ is strongly chain transitive.

Claim 2: An $(\varepsilon,T)$-chain from $(u,x)$ to $(v,y)$ is given by

$(v_{-2},x_{-2}),\,(v_{-1},x_{-1}),\ldots,(v_{k+1},x_{k+1})\text{ in
}\mathcal{U}\times M\text{ and }T_{-2},\,T_{-1},\ldots,T_{k}\geq T.$

For the proof of the claim observe first that there are no jumps at
$x_{-1},x_{0}$, and $x_{k+1}$ and

$\varphi(T_{j},x_{j},u_{j})=\varphi(T_{j},x_{j},v_{j})\text{ for
}j=0,\ldots,k-1.$

Hence (15) yields for $j=-2,\ldots,k$

$\displaystyle d_{M}(\varphi(T_{j},x_{j},v_{j}),x_{j+1})$
$\displaystyle<\varepsilon_{M}(\varphi(T_{j},x_{j},v_{j}))$
$\displaystyle\leq\varepsilon(v_{j}(T_{j}+\cdot),\varphi(T_{j},x_{j},v_{j}))=\varepsilon(\Phi_{T_{j}}(v_{j},x_{j})).$

Concerning the component in $\mathcal{U}$ we have to show that

$d_{\mathcal{U}}(v_{j}(T_{j}+\cdot),v_{j+1})<\varepsilon(\Phi_{T_{j}}(v_{j},x_{j}))\text{
for }j=-2,\,-1,\ldots,k.$ (16)

By choice of $T$ and $N$ all $w_{1},\,w_{2}\in\mathcal{U}$ satisfy

$\displaystyle
d_{\mathcal{U}}(w_{1},w_{2})=\sum_{i=1}^{\infty}\frac{1}{2^{i}}\frac{\left|\int_{\mathbb{R}}\left\langle
w_{1}(t)-w_{2}(t),y_{i}(t)\right\rangle\,dt\right|}{1+\left|\int_{\mathbb{R}}\left\langle
w_{1}(t)-w_{2}(t),y_{i}(t)\right\rangle\,dt\right|}$
$\displaystyle\leq\sum_{i=1}^{N}\frac{1}{2^{i}}\left\\{\left|\int_{\mathbb{R}\setminus\left[-T,T\right]}\left\langle
w_{1}(t)-w_{2}(t),y_{i}(t)\right\rangle\,dt\right|\right.+\left.\left|\int_{-T}^{T}\left\langle
w_{1}(t)-w_{2}(t),y_{i}(t)\right\rangle\,dt\right|\right\\}+\widehat{\varepsilon}$
$\displaystyle<2\widehat{\varepsilon}+\max_{i=1,\ldots,N}\int_{-T}^{T}\left|w_{1}(t)-w_{2}(t)\right|\,\left|y_{i}(t)\right|\,dt.$

For all considered pairs of control functions in (16) the integrands vanish,
hence the distances are bounded by $2\widehat{\varepsilon}$, and for
$j=0,\ldots,k-1$ the definition of $\widehat{\varepsilon}$ shows that

$2\widehat{\varepsilon}=\min_{j=-2,\ldots,k}\varepsilon_{M}(\varphi(T_{j},x_{j},v_{j}))\leq\min_{j=-2,\ldots,k}\varepsilon(\Phi_{T_{j}}(v_{j},x_{j})).$

Thus the claim in (16) holds.

(ii) Let $\mathcal{E}^{\ast}$ be a maximal invariant strongly chain transitive
set in $\mathcal{U}\times M$. Suppose that for $x,y\in M$ there are
$u,v\in\mathcal{U}$ with $(u,x),(v,y)\allowbreak\in E^{\ast}$, hence
$\varphi(t,x,u),\varphi(t,y,v)\allowbreak\in E^{\ast}$ defined by (14) for all
$t\in\mathbb{R}$. Choose $\varepsilon_{M}\in\mathcal{P}(M)$ and $T>0$ and
define $\varepsilon\in\mathcal{P}(\mathcal{U}\times M)$ by
$\varepsilon(u,x):=\varepsilon_{M}(x),u\in\mathcal{U},x\in M$. A corresponding
$(\varepsilon,T)$-chain in $\mathcal{U}\times M$ for the control flow $\Phi$
yields $x_{j},u_{j},T_{j}$ such that the corresponding trajectories satisfy

$\displaystyle d_{M}(\varphi(T_{j},x_{j},u_{j}),x_{j+1})$ $\displaystyle\leq
d(\Phi_{T_{j}}(x_{j},u_{j}),(u_{j+1},x_{j+1}))<\varepsilon(\Phi_{T_{j}}(x_{j},u_{j}))$
$\displaystyle=\varepsilon_{M}(\varphi(T_{j},x_{j},u_{j})).$

This shows that $E^{\ast}$ defined by (14) is strongly chain controllable.

The proof of (i) and (ii) is concluded by the observation that $E^{\ast}$ is a
maximal invariant strongly chain controllable set if and only if
$\mathcal{E}^{\ast}$ is a maximal invariant strongly chain transitive set. ∎

As an easy consequence, we find that conjugacies of control systems (cf.
Definition 3) preserve strong chain control sets.

######  Corollary 1.

Suppose that two control systems of the form (5) on $M$ and $N$, respectively,
are topologically conjugate by a homeomorphism $H:M\rightarrow N$. Then
$E^{\ast}$ is a strong chain control set in $M$ if and only if $H(E^{\ast})$
is a strong chain control set in $N$.

###### Proof.

The homeomorphism $H$ can be extended to a topological conjugacy
$\mathrm{id}_{\mathcal{U}}\times H$ of the associated control flows $\Phi$ on
$\mathcal{U}\times M$ and $\Psi$ on $\mathcal{U}\times N$. By Theorem 5 the
set $E^{\ast}$ is a strong chain control set if and only if the lift
$\mathcal{E}^{\ast}$ is a maximal invariant strongly chain transitive set for
$\Phi$. Theorem 1 shows that it is mapped onto the maximal invariant strongly
chain transitive set $\left(\mathrm{id}_{\mathcal{U}}\times
H\right)(\mathcal{E}^{\ast})$, which again by Theorem 5 is the lift of a
strong chain control set which has the form $H(E^{\ast})$. The converse
follows by considering $H^{-1}$. ∎

######  Remark 1.

An alternative proof of this corollary uses Hurley’s lemma, Lemma 1, directly
for topologically conjugate control systems.

##  4 Strong chain control sets of affine systems

In this section, we consider affine control systems of the form (1) on
$\mathbb{R}^{n}$ and study the induced control systems on projective space
$\mathbb{P}^{n}$ and on the northern hemisphere $\mathbb{S}^{n,+}$ of the
Poincaré sphere,

$\mathbb{S}^{n,+}:=\\{s=(s_{1},\ldots,s_{n+1})\in\mathbb{S}^{n}\left|{}\right.\sum_{i=1}^{n+1}s_{i}^{2}=1,\left|s_{i}\right|\leq
1,i=1,\ldots,n+1,s_{n+1}>0\\}.$

Define a function $h:\mathbb{R}^{n}\rightarrow\mathbb{S}^{n,+}$ by

$h(x)=h(x_{1},\ldots,x_{n}):=\left(\frac{x_{1}}{\left\|(x,1)\right\|},\ldots,\frac{x_{n}}{\left\|(x,1)\right\|},\frac{1}{\left\|(x,1)\right\|}\right)=\frac{(x,1)}{\left\|(x,1)\right\|}.$
(17)

Note that $\left\|x\right\|\rightarrow\infty$ if and only if $h(x)$ approaches
the equator

$\mathbb{S}^{n,0}:=\\{s=(s_{1},\ldots,s_{n+1})\in\mathbb{S}^{n}:s_{n+1}=0\\}.$

Define a control system on $\mathbb{S}^{n}$ by the projection of the bilinear
control system (10); cf. Colonius and Kliemann [7, Section 6.1]. With

$A_{i}^{\prime}:=\left(\begin{array}[c]{cc}A_{i}&a_{i}\\\
0&0\end{array}\right)\text{ for }i=0,\ldots,m,$

the projected system is given by

$\dot{s}=\left[A_{0}^{\prime}-s^{T}A_{0}^{\prime}s\cdot
I\right]s+\sum_{i=1}^{m}u_{i}(t)\left[A_{i}^{\prime}-s^{T}A_{i}^{\prime}s\cdot
I\right]s,u\in\mathcal{U}.$ (18)

It is clear that the control system (18) also induces a control system on
projective space $\mathbb{P}^{n}$. The trajectories on $\mathbb{S}^{n}$ are
denoted by

$\mathbb{S}\psi^{1}(t,s,u)=\frac{\psi^{1}(t,x,1,u)}{\left\|\psi^{1}(t,x,1,u)\right\|}=\frac{\left(\psi(t,x,u),1\right)}{\left\|\left(\psi(t,x,u),1\right)\right\|}\text{
with }s=\frac{(x,1)}{\left\|(x,1)\right\|}.$

This control-affine system can be restricted to the northern hemisphere with
trajectories $\mathbb{S}\psi^{1,+}(t,s,u),t\in\mathbb{R}$. We denote the
corresponding control flow by

$\mathbb{S}\Psi^{1,+}:\mathbb{R}\times\mathcal{U}\times\mathbb{S}^{n,+}\rightarrow\mathcal{U}\times\mathbb{S}^{n,+},~{}\mathbb{S}\Psi^{1,+}(t,u,s):=(u(t+\cdot),\mathbb{S}\psi^{1,+}(t,s,u)),$

for $(t,u,s)\in\mathbb{R}\times\mathcal{U}\times\mathbb{S}^{n,+}$.

######  Remark 2.

The elements of the southern hemisphere $\mathbb{S}^{n,-}$ satisfy $s_{n+1}<0$
and one obtains a map $h_{-}:\mathbb{R}^{n}\rightarrow\mathbb{S}^{n,-}$ by

$h_{-}(x)=h_{-}(x_{1},\ldots,x_{n}):=\left(\frac{-x_{1}}{\left\|(x,1)\right\|},\ldots,\frac{-x_{n}}{\left\|(x,1)\right\|},\frac{-1}{\left\|(x,1)\right\|}\right).$

Since $\psi^{1}$ is linear there is a control system on $\mathbb{S}^{n,-}$
with trajectories

$-\mathbb{S}\psi^{1}(t,s,u)=\frac{-\psi^{1}(t,x,1,u)}{\left\|\psi^{1}(t,x,1,u)\right\|}=\frac{\psi^{1}(t,-x,-1,u)}{\left\|\psi^{1}(t,x,1,u)\right\|}=\mathbb{S}\psi^{1}(t,-s,u).$

This shows that the control systems on the northern and the southern
hemispheres are conjugate.

In the following, we analyze the relation between the original control system
on $\mathbb{R}^{n}$ and the induced control system on $\mathbb{S}^{n,+}$. A
metric on $\mathbb{S}^{n,+}$ is given by

$d(s,s^{\prime})=\max_{i=1,\ldots,n+1}\left|s_{i}-s_{i}^{\prime}\right|\text{
for
}s=(s_{1},\ldots,s_{n+1}),s^{\prime}=(s_{1}^{\prime},\ldots,s_{n+1}^{\prime})\in\mathbb{S}^{n,+}.$

On $\mathbb{R}^{n}$ we consider the maximum norm
$\left\|\cdot\right\|_{\infty}$.

######  Lemma 2.

The map $h:\mathbb{R}^{n}\rightarrow\mathbb{S}^{n,+}$ is a homeomorphism with
inverse

$h^{-1}:\mathbb{S}^{n,+}\rightarrow\mathbb{R}^{n},h^{-1}(s)=h^{-1}(s_{1},\ldots,s_{n+1})=\left(\frac{s_{1}}{s_{n+1}},\ldots,\frac{s_{n}}{s_{n+1}}\right).$

###### Proof.

The map $h$ is injective since $h(x)=h(y)$ implies
$\frac{1}{\left\|(x,1)\right\|}=\frac{1}{\left\|(y,1)\right\|}$ and

$\frac{x_{i}}{\left\|(x,1)\right\|}=\frac{y_{i}}{\left\|(y,1)\right\|}\text{,
i.e., }x_{i}=y_{i}\text{ for }i=1,\ldots,n.$

Surjectivity follows since for $s=(s_{1},\ldots,s_{n+1})\in\mathbb{S}^{n,+}$
the point $x=\left(\frac{s_{1}}{s_{n+1}},\ldots,\frac{s_{n}}{s_{n+1}}\right)$
satisfies

$\left\|(x,1)\right\|^{2}=\frac{s_{1}^{2}}{s_{n+1}^{2}}+\cdots+\frac{s_{n}^{2}}{s_{n+1}^{2}}+1=\frac{1}{s_{n+1}^{2}},$

hence

$h\left(\frac{s_{1}}{s_{n+1}},\ldots,\frac{s_{n}}{s_{n+1}}\right)=\left(s_{1},\ldots,s_{n},s_{n+1}\right).$

This also implies the formula for $h^{-1}$. The map $h$ is continuous by
Colonius and Santana [9, Proposition 9(iii)], which shows that

$d(h(x),h(y))\leq 2\left\|x-y\right\|_{\infty}\text{ for
}x,y\in\mathbb{R}^{n}.$

It remains to prove the continuity of $h^{-1}$. First note that for
$s,s^{\prime}\in\mathbb{S}^{n,+}$ we find

$\displaystyle\left\|h^{-1}(s)-h^{-1}(s^{\prime})\right\|_{\infty}$
$\displaystyle=\left\|\left(\frac{s_{1}}{s_{n+1}},\ldots,\frac{s_{n}}{s_{n+1}}\right)-\left(\frac{s_{1}^{\prime}}{s_{n+1}^{\prime}},\ldots,\frac{s_{n}^{\prime}}{s_{n+1}^{\prime}}\right)\right\|_{\infty}$
$\displaystyle=\max_{i=1,\ldots,n}\left|\frac{s_{i}}{s_{n+1}}-\frac{s_{i}^{\prime}}{s_{n+1}^{\prime}}\right|.$

Now let $s^{k}$ converge to $s$ in $\mathbb{S}^{n,+}$, hence
$d(s^{k},s)=\max_{i=1,\ldots,n+1}\left|s_{i}^{k}-s_{i}\right|\rightarrow 0$.
Then it follows that

$\displaystyle\left\|h^{-1}(s^{k})-h^{-1}(s)\right\|_{\infty}$
$\displaystyle=\max_{i=1,\ldots,n}\left|\frac{s_{i}^{k}}{s_{n+1}^{k}}-\frac{s_{i}}{s_{n+1}}\right|$
$\displaystyle\leq\max_{i=1,\ldots,n}\left|\frac{s_{i}^{k}}{s_{n+1}^{k}}-\frac{s_{i}}{s_{n+1}^{k}}\right|+\max_{i=1,\ldots,n}\left|\frac{s_{i}}{s_{n+1}^{k}}-\frac{s_{i}}{s_{n+1}}\right|$
$\displaystyle\leq\frac{1}{\left|s_{n+1}^{k}\right|}\max_{i=1,\ldots,n}\left|s_{i}^{k}-s_{i}\right|+\left|\frac{1}{s_{n+1}^{k}}-\frac{1}{s_{n+1}}\right|\max_{i=1,\ldots,n}\left|s_{i}\right|$
$\displaystyle\leq\frac{1}{\left|s_{n+1}^{k}\right|}\max_{i=1,\ldots,n}\left|s_{i}^{k}-s_{i}\right|+\left|\frac{1}{s_{n+1}^{k}}-\frac{1}{s_{n+1}}\right|.$

The right hand side converges to $0$ for $k\rightarrow\infty$, since
$s_{i}^{k}\rightarrow s_{i}$ for $i=1,\ldots,n+1$. ∎

The next result shows that the control flows on
$\mathcal{U}\times\mathbb{R}^{n}$ and on $\mathcal{U}\times\mathbb{S}^{n,+}$
are topologically conjugate.

######  Theorem 6.

Consider the affine control flow $\Psi$ on $\mathcal{U}\times\mathbb{R}^{n}$
corresponding to the affine control system (1) and the control flow
$\mathbb{S}\Psi^{1,+}$ on $\mathcal{U}\times\mathbb{S}^{n,+}$ corresponding to
the induced control system (18) on $\mathbb{S}^{n,+}$.

(i) The map $\mathrm{id}_{\mathcal{U}}\times
h:\mathcal{U}\times\mathbb{R}^{n}\rightarrow\mathcal{U}\times\mathbb{S}^{n,+}$
is a topological conjugation of the flows $\Psi$ and $\mathbb{S}\Psi^{1,+}$,
hence

$\left(\mathrm{id}_{\mathcal{U}}\times
h\right)\Psi_{t}(u,x)=\mathbb{S}\Psi_{t}^{1,+}(u,h(x))\text{ for
}t\in\mathbb{R},(u,x)\in\mathcal{U}\times\mathbb{R}^{n}.$

(ii) The maximal invariant strongly chain transitive sets for $\Psi$ are
mapped by $\mathrm{id}_{\mathcal{U}}\times h$ onto the maximal invariant
strongly chain transitive sets for $\mathbb{S}\Psi_{t}^{1,+}$.

###### Proof.

(i) By Lemma 2 the map $\mathrm{id}_{\mathcal{U}}\times h$ is a homeomorphism
and it remains to prove the conjugation equality. For the component in
$\mathbb{R}^{n}$ one computes

$\displaystyle h(\psi(t,x,u))$
$\displaystyle=\left(\frac{\psi_{1}(t,x,u)}{\left\|(\psi(t,x,u),1)\right\|},\ldots,\frac{\psi_{n}(t,x,u)}{\left\|(\psi(t,x,u),1)\right\|},\frac{1}{\left\|(\psi(t,x,u),1)\right\|}\right)$
$\displaystyle=\frac{\psi^{1}(t,x,1,u)}{\left\|\psi^{1}(t,x,1,u)\right\|}=\frac{\psi^{1}\left(t,\frac{\left(x,1\right)}{\left\|(x,1)\right\|},u\right)}{\left\|\psi^{1}\left(t,\frac{\left(x,1\right)}{\left\|(x,1)\right\|},u\right)\right\|}$
$\displaystyle=\mathbb{S}\psi^{1,+}(t,h(x),u).$

This implies the conjugation equality

$\displaystyle(\mathrm{id}_{\mathcal{U}}\times h)\Psi_{t}(u,x)$
$\displaystyle=\left(u(t+\cdot),h(\psi(t,x,u))\right)=\left(u(t+\cdot),\mathbb{S}\psi^{1,+}(t,h(x),u)\right)$
$\displaystyle=\mathbb{S}\Psi_{t}^{1,+}(u,h(x)).$

(ii) This follows by (i) and Theorem 1. ∎

For the strong chain control sets of the affine control system on
$\mathbb{R}^{n}$ and the induced control system on $\mathbb{S}^{n,+}$ we
obtain the following consequence.

######  Corollary 2.

Consider the affine control system (1) on $\mathbb{R}^{n}$ and the induced
control system (18) on the upper hemisphere $\mathbb{S}^{n,+}$ of the Poincaré
sphere. The strong chain control sets on $\mathbb{S}^{n,+}$ have the form
${}_{\mathbb{S}}E^{\ast}=h(E^{\ast})$, where the $E^{\ast}$ are the strong
chain control sets on $\mathbb{R}^{n}$.

###### Proof.

Theorem 6 shows that the maximal invariant strongly chain transitive sets on
$\mathcal{U}\times\mathbb{R}^{n}$ for $\Psi$ are mapped by
$\mathrm{id}_{\mathcal{U}}\times h$ onto the maximal invariant strongly chain
transitive sets on $\mathcal{U}\times\mathbb{S}^{n,+}$ for
$\mathbb{S}\Psi_{t}^{1,+}$. By Theorem 5 the strong chain control sets
uniquely correspond to the maximal invariant strongly chain transitive sets of
the control flows. Thus the assertion follows. ∎

Next we briefly indicate the relation between the affine control system (1) on
$\mathbb{R}^{n}$ and the induced control system on $\mathbb{P}^{n}$. Recall
from Subsection 2.2 that
$\mathbb{P}^{n,1}=\\{\mathbb{P}(x,1)\in\mathbb{P}^{n}\left|x\in\mathbb{R}^{n}\right.\\}$.

######  Theorem 7.

Consider the affine control flow $\Psi$ on $\mathcal{U}\times\mathbb{R}^{n}$
corresponding to the affine control system (1) and the control flow
$\mathbb{P}\Psi^{1}$ on $\mathcal{U}\times\mathbb{P}^{n}$ corresponding to the
control system induced by (10) on $\mathbb{P}^{n}$. Let
$\mathbb{P}h:\mathbb{R}^{n}\rightarrow\mathbb{P}^{n,1}$ be the map
$\mathbb{P}h(x):=\mathbb{P}(x,1),x\in\mathbb{R}^{n}$.

(i) The map
$\mathrm{id}_{\mathcal{U}}\times\mathbb{P}h:\mathcal{U}\times\mathbb{R}^{n}\rightarrow\mathcal{U}\times\mathbb{P}^{n,1}$
is a topological conjugation of the flows $\Psi$ and $\mathbb{P}\Psi^{1}$
restricted to $\mathbb{P}^{n,1}$, hence

$\left(\mathrm{id}_{\mathcal{U}}\times\mathbb{P}h\right)\Psi_{t}(u,x)=\mathbb{P}\Psi_{t}^{1}(u,\mathbb{P}h(x))\text{
for }t\in\mathbb{R},(u,x)\in\mathcal{U}\times\mathbb{R}^{n}.$

(ii) The maximal invariant strongly chain transitive sets for $\Psi$ are
mapped by $\mathrm{id}_{\mathcal{U}}\times\mathbb{P}h$ onto the maximal
invariant strongly chain transitive sets for $\mathbb{P}\Psi^{1}$ restricted
to $\mathcal{U}\times\mathbb{P}^{n,1}$.

(iii) The strong chain control sets on $\mathbb{P}^{n,1}$ have the form
${}_{\mathbb{P}}E^{\ast}=\mathbb{P}h(E^{\ast})$, where the $E^{\ast}$ are the
strong chain control sets on $\mathbb{R}^{n}$.

###### Proof.

We only show that the map $\mathbb{P}h$ is a homeomorphism. Then the theorem
follows along the same lines as the proofs of Theorem 6 and Corollary 2.

Recall that $\mathbb{P}^{n}=(\mathbb{R}^{n+1}\setminus\\{0\\})/\thicksim$,
where $\thicksim$ is the equivalence relation $x\thicksim y$ if $y=\lambda x$
with some $\lambda\not=0$. An atlas of $\mathbb{P}^{n}$ is given by $n+1$
charts $(U_{i},\alpha_{i})$, where $U_{i}$ is the set of equivalence classes
$[x_{1}:\cdots:x_{n+1}]$ with $x_{i}\not=0$ (using homogeneous coordinates)
and $\alpha_{i}:U_{i}\rightarrow\mathbb{R}^{n}$ is defined by

$\alpha_{i}([x_{1}:\cdots:x_{n+1}])=\left(\frac{x_{1}}{x_{i}},\ldots,\widehat{\frac{x_{i}}{x_{i}}},\ldots,\frac{x_{n+1}}{x_{i}}\right);$

here the hat means that the $i$-th entry is omitted. In homogeneous
coordinates $\mathbb{P}^{n,1}$ is described by

$\mathbb{P}^{n,1}=\left\\{[x_{1}:\cdots:x_{n}:1]\left|(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\right.\right\\}.$

Note that $U_{n+1}=\mathbb{P}^{n,1}$ and
$\mathbb{P}h(x)=\left(\alpha_{n+1}\right)^{-1}(x),x\in\mathbb{R}^{n}$. A
trivial atlas for $\mathbb{P}^{n,1}$ is given by
$\left\\{(U_{n+1},\alpha_{n+1})\right\\}$ proving that $\mathbb{P}^{n,1}$ is a
manifold which under $\left(\mathbb{P}h\right)^{-1}$ is diffeomorphic to
$\mathbb{R}^{n}$. ∎

Finally, we analyze when strong chain control sets in $\mathbb{R}^{n}$ are
unbounded.

######  Proposition 2.

Let $E^{\ast}$ be a strong chain control set of the affine system (1). Then
$E^{\ast}$ is unbounded if there are $u\in\mathcal{U}$ and $\tau>0$ with
$\psi(\tau,x,u)=x\in E^{\ast}$ such that the fundamental solution
$X_{u}(t,s),t,s\in\mathbb{R}$, of the homogeneous part (17) satisfies
$1\in\mathrm{spec}(X_{u}(\tau,0))$.

###### Proof.

Extend $u_{\left|[0,\tau]\right.}$ to a $\tau$-periodic control also denoted
by $u$. By the variation-of-parameters formula, this yields the
$\tau$-periodic trajectory $\psi(t,x,u),t\in\mathbb{R}$, satisfying

$\int_{0}^{\tau}X_{u}(\tau,s)\left(a_{0}+\sum_{i=1}^{m}u_{i}(s)a_{i}\right)ds=(I-X_{u}(\tau,0))x.$

The assumption means that the eigenspace $\mathbf{E}(X_{u}(\tau,0);1))$ is
nontrivial. It follows that for every $y$ in the affine subspace
$Y:=x+\mathbf{E}(X_{u}(\tau,0);1))$ there is a $\tau$-periodic solution of (1)
with control $u$ starting in $y$. This implies that $Y\subset\mathbb{R}^{n}$
consists of fixed points of the time $\tau$-map of the differential equation

$\dot{x}(t)=A_{0}x(t)+a_{0}+\sum_{i=1}^{m}u_{i}(t)[A_{i}x(t)+a_{i}].$

Hence $Y$ consists of a continuum of $\tau$-periodic solutions. Fixing any
$\varepsilon\in\mathcal{P}(\mathbb{R}^{n})$ and $T>0$, one finds that the
unbounded set $Y$ is contained in the strong chain control set $E^{\ast}$. ∎

Recall that a set in $\mathbb{R}^{n}$ is unbounded if and only if the closure
of its image in $\mathbb{P}^{n}$ has a nonvoid intersection with the
projective equator $\mathbb{P}^{n,0}$ or, equivalently, if the closure of its
image in $\overline{\mathbb{S}^{n,\pm}}$ has nonvoid intersection with the
equator $\mathbb{S}^{n,0}$.

##  5 Chain control sets on the sphere for bilinear systems

We discuss the relation between chain control sets on projective space
$\mathbb{P}^{n}$ and on the sphere $\mathbb{S}^{n}$ for general bilinear
control systems; cf. Bacciotti and Vivalda [4] for a discussion of complete
controllability on these spaces. Then we apply these results to the system on
the Poincaré sphere induced by an affine control system. In the rest of this
paper, $\varepsilon>0$ denotes constants.

Consider a bilinear control system on $\mathbb{R}^{n+1}$ of the form

$\dot{x}(t)=\left[A_{0}+\sum_{i=1}^{m}u_{i}(t)A_{i}\right]x(t),\,u\in\mathcal{U}.$
(19)

The trajectories are denoted by $\varphi(t,x,u),t\in\mathbb{R}$. Selgrade’s
theorem (cf. Colonius and Kliemann [7, Theorem 7.1.2]) can be applied to the
corresponding linear control flow
$\Phi:\mathbb{R}\times\mathcal{U}\times\mathbb{R}^{n+1}\rightarrow\mathcal{U}\times\mathbb{R}^{n+1}$
and yields the Whitney sum of $\ell$ subbundles

$\mathcal{U}\times\mathbb{R}^{n+1}=\mathcal{V}_{1}\oplus\cdots\oplus\mathcal{V}_{\ell},$

where the projections $\mathbb{P}\mathcal{V}_{i}$ of $\mathcal{V}_{i}$ to the
projective bundle $\mathcal{U}\times\mathbb{P}^{n}$ are the chain recurrent
components of the projectivized flow $\mathbb{P}\Phi$. They yield the chain
control sets ${}_{\mathbb{P}}E_{1},\ldots,\,_{\mathbb{P}}E_{\ell}$ in
$\mathbb{P}^{n}$ satisfying

${}_{\mathbb{P}}E_{i}$
$\displaystyle=\\{\mathbb{P}x\in\mathbb{P}^{n}\left|\exists
u\in\mathcal{U}:(u,\mathbb{P}x)\in\mathbb{P}\mathcal{V}_{i}\right.\\},$ (20)
$\displaystyle\mathcal{V}_{i}$
$\displaystyle=\left\\{(u,x)\in\mathcal{U}\times\mathbb{R}^{n+1}\left|\mathbb{P}\varphi(t,x,u)\in\,_{\mathbb{P}}E_{i}\text{
for all }t\in\mathbb{R}\right.\right\\}.$ (21)

We will analyze the chain control sets in $\mathbb{S}^{n}$. For
$\varepsilon,T>0$ let $\zeta$ be a controlled $(\varepsilon,T)$-chain from
$s\in\mathbb{S}^{n}$ to $s^{\prime}\in\mathbb{S}^{n}$ given by
$T_{0},\ldots,T_{k-1}\allowbreak\geq
T,u^{0},\ldots,u^{k-1},s^{0}=s,s^{1},\ldots,s^{k}=s^{\prime}$ with

$d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),s^{i+1})<\varepsilon\text{ for all
}i.$

Since
$\mathbb{S}\varphi(T_{i},-s^{i},u^{i})=-\mathbb{S}\varphi(T_{i},s^{i},u^{i})$
a controlled $(\varepsilon,T)$-chain from $-s$ to $-s^{\prime}$ is given by
$T_{0},\ldots,T_{k-1}\geq T,-s^{0}=-s,-s^{1},\ldots,-s^{k}=-s^{\prime}$ with

$d(\mathbb{S}\varphi(T_{i},-s^{i},u^{i}),-s^{i+1})<\varepsilon.$

We denote this chain by $-\zeta$. Recall the definition of the chain reachable
set in Definition 1.

######  Lemma 3.

Let $s,s^{\prime}\in\mathbb{S}^{n}$ with $\mathbb{P}s,\mathbb{P}s^{\prime}$ in
a chain control set ${}_{\mathbb{P}}E\subset\mathbb{P}^{n}$. Then for all
$\varepsilon,T>0$ there is a controlled $(\varepsilon,T)$-chain in
$\mathbb{S}^{n}$ from $s$ to $s^{\prime}$ or to $-s^{\prime}$. Thus the chain
reachable set $\mathbf{R}^{c}(s)$ contains $s^{\prime}$ or $-s^{\prime}$.

###### Proof.

Let $s,s^{\prime}\in\mathbb{S}^{n}$ with
$\mathbb{P}s,\mathbb{P}s^{\prime}\in\,_{\mathbb{P}}E$. For $\varepsilon,T>0$
consider a controlled $(\varepsilon,T)$-chain from $\mathbb{P}s$ to
$\mathbb{P}s^{\prime}$ given by $T_{0},\ldots,T_{k-1}\geq
T,p^{0}=\mathbb{P}s,p^{1},\ldots,p^{k}=\mathbb{P}s^{\prime}$ in
${}_{\mathbb{P}}E$ and $u^{0},\ldots,u^{k-1}$ with

$d(\mathbb{P}\varphi(T_{i},p^{i},u^{i}),p^{i+1})<\varepsilon\text{ for all
}i.$

For $s^{i}\in\mathbb{S}^{n}$ with $\mathbb{P}s^{i}=p^{i}$ the definition of
the metric in $\mathbb{P}^{n}$ shows that

$d(\mathbb{P}\varphi(T_{i},p^{i},u^{i}),p^{i+1})=\min\\{d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),s^{i+1}),d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),-s^{i+1})\\}.$

If $\mathbb{S}\varphi(T_{0},s,u^{0})$ is close to the equator
$\mathbb{S}^{n,0}$ it may happen that

$d(\mathbb{P}\varphi(T_{0},p^{0},u^{0}),p^{1})=d(\mathbb{S}\varphi(T_{0},s^{0},u^{0}),-s^{1}).$

Thus one can continue the controlled $(\varepsilon,T)$-chain in
$\mathbb{S}^{n}$ with $-s^{1}$ and

$d(\mathbb{S}\varphi(T_{0},s,u^{0}),-s^{1})<\varepsilon.$

For the next step note that

$d(\mathbb{P}\varphi(T_{1},p^{1},u^{1}),p^{2})=\min\\{d(\mathbb{S}\varphi(T_{1},-s^{1},u^{1}),s^{2}),d(\mathbb{S}\varphi(T_{1},-s^{1},u^{1}),-s^{2})\\}.$

If
$d(\mathbb{P}\varphi(T_{1},p^{1},u^{1}),p^{2})=d(\mathbb{S}\varphi(T_{1},-s^{1},u^{1}),-s^{2})$,
we can continue the chain with $-s^{2}\in\mathbb{S}^{n}$ and

$d(\mathbb{S}\varphi(T_{1},-s^{1},u^{1}),-s^{2})<\varepsilon.$

If $\mathbb{S}\varphi(T_{1},-s^{1},u^{1})$ is close to the equator it may
happen that

$d(\mathbb{P}\varphi(T_{1},p^{1},u^{1}),p^{2})=d(\mathbb{S}\varphi(T_{1},-s^{1},u^{1}),s^{2}).$

Thus we can continue the chain with $s^{2}$ and

$d(\mathbb{S}\varphi(T_{1},-s^{1},u^{1}),s^{2})<\varepsilon.$

Continuing in this way we end up in $s^{k}=s^{\prime}$ or in
$s^{k}=-s^{\prime}$. Thus we have constructed a controlled
$(\varepsilon,T)$-chain in $\mathbb{S}^{n}$ from $s$ to $s^{\prime}$ or to
$-s^{\prime}$. Since $\varepsilon,T>0$ are arbitrary it follows that
$s^{\prime}\in\mathbf{R}^{c}(s)$ or $-s^{\prime}\in\mathbf{R}^{c}(s)$. ∎

######  Lemma 4.

Let ${}_{\mathbb{P}}E$ be a chain control set in $\mathbb{P}^{n}$. Then every
$s\in\mathbb{S}^{n}$ with $\mathbb{P}s\in\,_{\mathbb{P}}E$ is in a chain
control set of the system on $\mathbb{S}^{n}$.

###### Proof.

Let $\varepsilon,T>0$. By Lemma 3 with $s^{\prime}=s$ there are controlled
$(\varepsilon,T)$-chains from $s$ to $s$ or to $-s$. In the second case,
consider the chain $-\zeta$ which starts in $-s$. We end up in $s$.
Concatenation of the chains $\zeta$ from $s$ to $-s$ and $-\zeta$ from $-s$ to
$s$, yields a controlled $(\varepsilon,T)$-chain
$\left(-\zeta\right)\circ\zeta$ from $s$ to $s$. Since $\varepsilon,T>0$ are
arbitrary, this shows that every $s\in\mathbb{S}^{n}$ with
$\mathbb{P}s\in\,_{\mathbb{P}}E$ is in a chain control set of the system on
$\mathbb{S}^{n}$. ∎

The proof of the following lemma is modeled after Bacciotti and Vivalda [4,
Lemma 3], where controllability is analyzed; cf. also Colonius, Santana, Setti
[10, Lemma 3.10] for the case of control sets.

######  Lemma 5.

(i) Let ${}_{\mathbb{S}}E$ be a chain control set in $\mathbb{S}^{n}$. Then
the projection of ${}_{\mathbb{S}}E$ to $\mathbb{P}^{n}$ is contained in a
chain control set ${}_{\mathbb{P}}E$.

(ii) Consider a chain control set ${}_{\mathbb{P}}E$ in $\mathbb{P}^{n}$.
Suppose that there is $s_{0}\in\mathbb{S}^{n}$ such that
$\mathbb{P}s_{0}\in\,_{\mathbb{P}}E$ and $-s_{0}\in\mathbf{R}^{c}(s_{0})$.
Then there exists a chain control set ${}_{\mathbb{S}}E$ in $\mathbb{S}^{n}$
containing
$\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in\,_{\mathbb{P}}E\right.\\}$.

###### Proof.

Assertion (i) is immediate from the definitions. Concerning assertion (ii) let
$s,s^{\prime}\in\mathbb{S}^{n}$ with
$\mathbb{P}s,\mathbb{P}s^{\prime}\in\,_{\mathbb{P}}E$. We have to show that
$s^{\prime}$ is in $\mathbf{R}^{c}(s)$. Lemma 3 shows that
$s^{\prime}\in\mathbf{R}^{c}(s_{0})$ or $s^{\prime}\in\mathbf{R}^{c}(-s_{0})$,
and also that $s_{0}\in\mathbf{R}^{c}(s)$ or $-s_{0}\in\mathbf{R}^{c}(s)$.

If $s^{\prime}\in\mathbf{R}^{c}(s_{0})$ and $s_{0}\in\mathbf{R}^{c}(s)$ it
follows that $s^{\prime}\in\mathbf{R}^{c}(s_{0})\subset\mathbf{R}^{c}(s)$,
and, analogously, the assertion follows if
$s^{\prime}\in\mathbf{R}^{c}(-s_{0})$ and $-s_{0}\in\mathbf{R}^{c}(s)$.
Suppose that $s^{\prime}\in\mathbf{R}^{c}(s_{0})$ and
$-s_{0}\in\mathbf{R}^{c}(s)$. Using the assumption on $s_{0}$ one finds that
$s^{\prime}\in\mathbf{R}^{c}(s_{0})\subset\mathbf{R}^{c}(-s_{0})\subset\mathbf{R}^{c}(s)$.
Analogously, the assertion follows if $s^{\prime}\in\mathbf{R}^{c}(-s_{0})$
and $s_{0}\in\mathbf{R}^{c}(s)$. ∎

The following theorem is the main result of this section. It describes the
chain control sets in $\mathbb{S}^{n}$.

######  Theorem 8.

Consider a bilinear control system on $\mathbb{R}^{n+1}$ of the form (19) and
let ${}_{\mathbb{P}}E$ be a chain control set of the induced control system on
$\mathbb{P}^{n}$.

(i) The set
${}_{\mathbb{S}}E_{0}:=\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in\,_{\mathbb{P}}E\right.\\}$
is the unique chain control set in $\mathbb{S}^{n}$ which projects into
${}_{\mathbb{P}}E$ if and only if there is $s_{0}\in\mathbb{S}^{n}$ with
$\mathbb{P}s_{0}\in\,_{\mathbb{P}}E$ and $-s_{0}\in\mathbf{R}^{c}(s_{0})$.

(ii) There are two chain control sets
${}_{\mathbb{S}}E_{1}=-\,_{\mathbb{S}}E_{2}$ with

${}_{\mathbb{S}}E_{1}\cup\,_{\mathbb{S}}E_{2}=\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in\,_{\mathbb{P}}E\right.\\},$
(22)

if and only if for all $s_{0}\in\mathbb{S}^{n}$ with
$\mathbb{P}s_{0}\in\,_{\mathbb{P}}E$ it holds that
$-s_{0}\not\in\mathbf{R}^{c}(s_{0})$.

(iii) Define for $j\in\\{0,1,2\\}$ and $u\in\mathcal{U}$

$V_{j}(u):=\left\\{x\in\mathbb{R}^{n+1}\left|\mathbb{S}\varphi(t,x,u)\in\,_{\mathbb{S}}E_{j}\text{
for all }t\in\mathbb{R}\right.\right\\}.$

In case (i) it follows that $V_{0}(u)$ is a linear subspace. In case (ii) it
follows for $j=1,2$ that $V_{j}(u)$ is a convex cone.

(iv) There are $\ell_{1}$ chain control sets on $\mathbb{S}^{n}$ denoted by
${}_{\mathbb{S}}E_{1},\ldots,\allowbreak\,_{\mathbb{S}}E_{\ell_{1}}$ with
$1\leq\ell_{1}\leq 2\ell\leq 2(n+1)$.

###### Proof.

(i) Suppose that there is $\mathbb{P}s_{0}\in\,_{\mathbb{P}}E$ with
$-s_{0}\in\mathbf{R}^{c}(s_{0})$. By Lemma 5(ii) there is a chain control set
${}_{\mathbb{S}}E$ on the unit sphere containing
$\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in\,_{\mathbb{P}}E\right.\\}$, hence
the projection of ${}_{\mathbb{S}}E$ to projective space contains
${}_{\mathbb{P}}E$. Using Lemma 5(i) one concludes that
${}_{\mathbb{S}}E=\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in\,_{\mathbb{P}}E\right.\\}$.

(ii) Suppose that for all $s_{0}\in\mathbb{S}^{n}$ with
$\mathbb{P}s_{0}\in\,_{\mathbb{P}}E$ it holds that
$-s_{0}\not\in\mathbf{R}^{c}(s_{0})$. Fix a point $s_{0}\in\mathbb{S}^{n}$
with $\mathbb{P}s_{0}\in\,_{\mathbb{P}}E$ and define

$\displaystyle A^{+}$
$\displaystyle:=\left\\{s\in\mathbb{S}^{n}\left|\,\mathbb{P}s\in\,_{\mathbb{P}}E\text{
and }s\in\mathbf{R}^{c}(s_{0})\cap\mathbf{C}^{c}(s_{0})\right.\right\\},$
$\displaystyle A^{-}$
$\displaystyle:=\left\\{s\in\mathbb{S}^{n}\left|\,\mathbb{P}s\in\,_{\mathbb{P}}E\text{
and }s\in\mathbf{R}^{c}(-s_{0})\cap\mathbf{C}^{c}(-s_{0})\right.\right\\}.$

It follows that $A^{-}=-A^{+}$ and the sets $A^{+}$ and $A^{-}$ are disjoint
since $s\in A^{+}\cap A^{-}$ implies $s\in\mathbf{R}^{c}(s_{0})$ and
$s\in\mathbf{C}^{c}(-s_{0})$ which implies that
$-s_{0}\in\mathbf{R}^{c}(s_{0})$ contradicting the assumption. By Lemma 4
there is a chain control set ${}_{\mathbb{S}}E_{1}$ with
$s_{0}\in\,_{\mathbb{S}}E_{1}$. Lemma 5(i) implies that
$\mathbb{P}(_{\mathbb{S}}E_{1})\subset\,_{\mathbb{P}}E$ and Theorem 2 implies
that ${}_{\mathbb{S}}E_{1}=\mathbf{R}^{c}(s_{0})\cap\mathbf{C}^{c}(s_{0})$,
hence $A^{+}=\,_{\mathbb{S}}E_{1}$.

Since by our assumption $s_{0}\not\in\mathbf{R}^{c}(-s_{0})$ it follows by
Lemma 3 that $-s_{0}\in\mathbf{R}^{c}(-s_{0})$, hence there is a chain control
set ${}_{\mathbb{S}}E_{2}$ with $-s_{0}\in\,_{\mathbb{S}}E_{2}$. By Theorem 2
it follows that
${}_{\mathbb{S}}E_{2}=\mathbf{R}^{c}(-s_{0})\cap\mathbf{C}^{c}(-s_{0})$, hence
$A^{-}=\,_{\mathbb{S}}E_{2}$.

Suppose that $\mathbb{P}s\in\,_{\mathbb{P}}E$. Then Lemma 3 implies that
$s\in\mathbf{R}^{c}(s_{0})$ or $-s\in\mathbf{R}^{c}(s_{0})$ and
$s\in\mathbf{C}^{c}(s_{0})$ or $-s\in\mathbf{C}^{c}(s_{0})$. If
$s\in\mathbf{R}^{c}(s_{0})$ and $-s\in\mathbf{C}^{c}(s_{0})$ it follows that
$s\in\mathbf{C}^{c}(-s_{0})$ which leads to the contradiction
$-s_{0}\in\mathbf{R}^{c}(s_{0})$. Similarly, $-s\in\mathbf{R}^{c}(s_{0})$ and
$s\in\mathbf{C}^{c}(s_{0})$ leads to the contradiction
$-s_{0}\in\mathbf{R}^{c}(s_{0})$. We conclude that
$\mathbb{P}s\in\,_{\mathbb{P}}E$ implies
$s\in\mathbf{R}^{c}(s_{0})\cap\mathbf{C}^{c}(s_{0})$ or
$s\in\mathbf{R}^{c}(-s_{0})\cap\mathbf{C}^{c}(-s_{0})$. This shows that

$\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in\,_{\mathbb{P}}E\right.\\}=A^{+}\cup
A^{-}=\,_{\mathbb{S}}E_{1}\cup\,_{\mathbb{S}}E_{2}.$ (23)

Together with the first part of the proof it follows that the conditions in
(i) and (ii) are also necessary for the existence of a single chain control
set ${}_{\mathbb{S}}E_{0}$ and of two chain control sets
${}_{\mathbb{S}}E_{1}$ and ${}_{\mathbb{S}}E_{2}$, respectively.

(iii) In the situation of case (i), one has

$\left\\{x\in\mathbb{R}^{n+1}\left|\mathbb{S}x\in\,_{\mathbb{S}}E_{0}\right.\right\\}=\left\\{x\in\mathbb{R}^{n+1}\left|\mathbb{P}x\in\,_{\mathbb{P}}E\right.\right\\},$

and the assertion for ${}_{\mathbb{S}}E_{0}$ follows since by Selgrade’s
theorem the fibers of $\mathcal{V}_{i}$ are linear; cf. formula (21). In the
situation of the case (ii), the assertion follows if for $j=1,2$ and
$u\in\mathcal{U}$

$x_{1},x_{2}\in V_{j}(u)\text{ implies }\alpha x_{1}+\beta x_{2}\in
V_{j}(u)\text{ for all }\alpha,\beta>0.$

It suffices to prove that $\mathbb{S}(\alpha x_{1}+\beta
x_{2})\in\,_{\mathbb{S}}E_{j}$ since one may replace this point by
$\mathbb{S}\varphi(t,\alpha x_{1}+\beta x_{2},u),t\in\mathbb{R}$. Note first
that $\mathbb{P}\left(\alpha x_{1}+\beta x_{2}\right)\in\,_{\mathbb{P}}E_{j}$
by Selgrade’s theorem. Furthermore, $\alpha x_{1}+\beta x_{2}\not=0$ since
$\alpha x_{1}+\beta x_{2}=0$ implies
$\mathbb{S}x_{1}=-\mathbb{S}x_{2}\in\,_{\mathbb{S}}E_{1}\cap\,_{\mathbb{S}}E_{2}$
contradicting ${}_{\mathbb{S}}E_{1}\not=\,_{\mathbb{S}}E_{2}$. It follows for
$t\in\mathbb{R}$ that $\mathbb{S}\varphi(t,\alpha x_{1}+\beta x_{2},u)\not=0$.
Since

$\mathbb{S}(\alpha x_{1}+\beta x_{2})=\frac{\alpha x_{1}+\beta
x_{2}}{\left\|\alpha x_{1}+\beta
x_{2}\right\|}=\frac{x_{1}+\frac{\beta}{\alpha}x_{2}}{\left\|x_{1}+\frac{\beta}{\alpha}x_{2}\right\|},$

it suffices to show that for $s_{1},s_{2}\in\mathbb{S}^{n}$ with
$s_{1},s_{2}\in\,_{\mathbb{S}}E_{j}$ and $\alpha\in(0,1)$

$\mathbb{S}(s_{1}+\alpha s_{2})=\frac{s_{1}+\alpha s_{2}}{\left\|s_{1}+\alpha
s_{2}\right\|}\in\,_{\mathbb{S}}E_{j}.$

Note that $\mathbb{P}(s_{1}+\alpha s_{2})\in\,_{\mathbb{P}}E_{j}$. The
Hausdorff distance between the compact sets ${}_{\mathbb{S}}E_{1}$ and
${}_{\mathbb{S}}E_{2}$ is positive, hence (22) implies
$\mathbb{S}(s_{1}+\alpha s_{2})\in\,_{\mathbb{S}}E_{j}$ for small $\alpha>0$.
Let

$\alpha^{0}:=\sup\left\\{\alpha\in(0,1)\left|\mathbb{S}(s_{1}+\alpha^{\prime}s_{2})\in\mathbb{\,}_{\mathbb{S}}E_{j}\text{
for all }\alpha^{\prime}\in(0,\alpha)\right.\right\\}.$

If $\alpha^{0}<1$ there exists $\alpha\in(\alpha^{0},1)$ with
$\mathbb{S}(s_{1}+\alpha
s_{2})\not\in\,_{\mathbb{S}}E_{1}\cup\,_{\mathbb{S}}E_{2}$. This contradicts
(22). Hence it follows that $\mathbb{S}(s_{1}+\alpha
s_{2})\in\,_{\mathbb{S}}E_{j}$ for all $\alpha\in(0,1)$.

(iv) By Selgrade’s theorem the number $\ell$ of chain control sets in
$\mathbb{P}^{n}$ satisfies $1\leq\ell\leq n+1$. Hence assertions (i) and (ii)
imply (iv). ∎

Next, we describe the relation of the chain control sets on the sphere to the
Selgrade bundles $\mathcal{V}_{i}$; cf. (21).

######  Corollary 3.

For a bilinear system of the form (19) the projection to $\mathbb{S}^{n}$ of
the Selgrade bundle $\mathcal{V}_{i}$ given by

$\pi_{\mathbb{S}}\mathcal{V}_{i}:=\left\\{s\in\mathbb{S}^{n}\left|\exists(u,x)\in\mathcal{V}_{i}\text{
with }s=\frac{x}{\left\|x\right\|}\right.\right\\},$

equals ${}_{\mathbb{S}}E_{0}$ or
${}_{\mathbb{S}}E_{1}\cup\,_{\mathbb{S}}E_{2}$, where ${}_{\mathbb{S}}E_{j}$
are the chain control sets from Theorem 8(i) or (ii), respectively.

###### Proof.

The set $\pi_{\mathbb{S}}\mathcal{V}_{i}$ projects onto
${}_{\mathbb{P}}E_{i}\subset\mathbb{P}^{n}$, hence
$\pi_{\mathbb{S}}\mathcal{V}_{c}^{1}=\left\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in\,_{\mathbb{P}}E_{i}\right.\right\\}$
and the assertion follows from Theorem 8. ∎

In the rest of this section, we consider the control system induced by an
affine control system on the Poincaré sphere.

######  Corollary 4.

Consider an affine control system of the form (1) and the induced control
systems on $\mathbb{P}^{n}$ and $\mathbb{S}^{n}$. Let
${}_{\mathbb{P}}E_{c}^{1}$ be the central chain control set on
$\mathbb{P}^{n}$. Then the system on $\mathbb{S}^{n}$ satisfies

$\left\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in
E_{c}^{1}\right.\right\\}=\,_{\mathbb{S}}E_{c,0}\text{ or
}\left\\{s\in\mathbb{S}^{n}\left|\mathbb{P}s\in
E_{c}^{1}\right.\right\\}=\,_{\mathbb{S}}E_{c,1}\cup\,_{\mathbb{S}}E_{c,2},$

where ${}_{\mathbb{S}}E_{c,0}$ is a chain control set with
${}_{\mathbb{S}}E_{c,0}=-\,_{\mathbb{S}}E_{c,0}$ and
${}_{\mathbb{S}}E_{c,2}=-\,_{\mathbb{S}}E_{c,1}$ are two chain control sets,
respectively.

###### Proof.

This is a consequence of Theorem 8(i),(ii). ∎

The chain control sets ${}_{\mathbb{S}}E_{c,j},j\in\\{0,1,2\\}$, are the only
chain control sets on $\mathbb{S}^{n}$, which are not contained in the equator
$\mathbb{S}^{n,0}$. We call them the central chain control sets on
$\mathbb{S}^{n}$.

The following proposition shows that the intersection of a chain control set
${}_{\mathbb{S}}E_{c,j}$ in $\mathbb{S}^{n}$ with the equator
$\mathbb{S}^{n,0}$ contains a chain control set in $\mathbb{S}^{n,0}$. This
result and its proof are similar to Colonius, Santana, and Setti [11, Lemma
6.8], where an analogous result is given on projective spaces. Theorem 8 also
applies to the bilinear control system on $\mathbb{R}^{n}$ given by the
homogeneous part (3). For the induced system on the sphere $\mathbb{S}^{n-1}$
this yields chain control sets ${}_{\mathbb{S}}E^{\hom}$ in
$\mathbb{S}^{n-1}$. We identify $\mathbb{S}^{n-1}\times\\{0\\}$ with
$\left\\{s\in\mathbb{S}^{n}\left|s_{n+1}=0\right.\right\\}$, hence the chain
control sets contained in the equator $\mathbb{S}^{n,0}$ have the form
${}_{\mathbb{S}}E^{\hom}\times\\{0\\}$.

######  Proposition 3.

Consider an affine control system of the form (1) on $\mathbb{R}^{n}$, and let
${}_{\mathbb{S}}E_{c,j},j\in\\{0,1,2\\}$, be the central chain control sets in
the Poincaré sphere $\mathbb{S}^{n}$.

(i) If
${}_{\mathbb{S}}E_{c,j}\cap\left({}_{\mathbb{S}}E^{\hom}\times\\{0\\}\right)\not=\varnothing$
for a chain control set ${}_{\mathbb{S}}E^{\hom}\subset\mathbb{S}^{n-1}$ of
the projectivized homogeneous part, then
${}_{\mathbb{S}}E^{\hom}\times\\{0\\}\subset\,_{\mathbb{S}}E_{c,j}$.

(ii) If ${}_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}\not=\varnothing$ it
contains ${}_{\mathbb{S}}E^{\hom}\times\\{0\\}$ for a chain control set
${}_{\mathbb{S}}E^{\hom}$ in $\mathbb{S}^{n-1}$ of the homogeneous part (3).

(iii) The chain control sets ${}_{\mathbb{S}}E_{c,j}$ and
${}_{\mathbb{S}}E^{\hom}$ are also strong chain control sets.

###### Proof.

(i) We will show that the set
${}_{\mathbb{S}}E_{c,j}\cup\left({}_{\mathbb{S}}E^{\hom}\times\\{0\\}\right)$
is chain controllable. Then the maximality property of the chain control set
${}_{\mathbb{S}}E_{c,j}$ implies that
${}_{\mathbb{S}}E^{\hom}\times\\{0\\}\subset\,_{\mathbb{S}}E_{c,j}$.

Consider $x\in\,_{\mathbb{S}}E_{c,j}$ and
$y\in\,_{\mathbb{S}}E^{\hom}\times\\{0\\}$ and let $\varepsilon,T>0$. Fix
$z\in\,_{\mathbb{S}}E_{c,j}\cap\left({}_{\mathbb{S}}E^{\hom}\times\\{0\\}\right)$.
There are controlled $(\varepsilon,T)$-chains $\zeta_{1}$ and $\zeta_{2}$ from
$x$ to $z$ and from $z$ to $x$, resp. For the system restricted to
$\mathbb{S}^{n,0}$, there exist controlled $(\varepsilon,T)$-chains
$\zeta_{3}$ and $\zeta_{4}$ from $z$ to $y$ and from $y$ to $z$, respectively.
Then the concatenations $\zeta_{3}\circ\zeta_{1}$ and
$\zeta_{2}\circ\zeta_{4}$ are controlled $(\varepsilon,T)$-chains from $x$ to
$y$ and from $y$ to $x$, resp. This concludes the proof of assertion (i) since
$\varepsilon,T>0$ are arbitrary.

(ii) Let $(x,0)\in\,_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}$. Then there
exists a control $u\in\mathcal{U}$ with
$\mathbb{S}\psi^{1}(t,x,0,u)\allowbreak\in\,_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0},t\in\mathbb{R}$,
by Theorem 2 and invariance of $\mathbb{S}^{n,0}$. Since
${}_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}$ is compact, it follows that the
$\omega$-limit set

$\left\\{\left.y=\lim_{k\rightarrow\infty}\mathbb{S}\psi^{1}(t_{k},x,0,u)\right|t_{k}\rightarrow\infty\right\\}\subset\,_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}$

is nonvoid. Hence Colonius and Kliemann [7, Corollary 4.3.12] implies that
there exists a chain control set of the system restricted to
$\mathbb{S}^{n,0}$ containing the $\omega$-limit set. Thus there is a chain
control set ${}_{\mathbb{S}}E^{\hom}$ in $\mathbb{S}^{n-1}$ of the homogeneous
part with
${}_{\mathbb{S}}E_{c,j}\cap\left({}_{\mathbb{S}}E^{\hom}\times\\{0\\}\right)\not=\varnothing$.
Now the assertion follows from (i).

(iii) Since $\mathbb{S}^{n}$ and $\mathbb{S}^{n-1}$ are compact, Proposition
1(iv) implies that ${}_{\mathbb{S}}E$ and ${}_{\mathbb{S}}E^{\hom}$ are also
strong chain control sets. ∎

The following theorem describes the intersection of a chain control set on the
Poincaré sphere $\mathbb{S}^{n}$ with the equator $\mathbb{S}^{n,0}$.

######  Theorem 9.

Consider an affine control system of the form (1) on $\mathbb{R}^{n}$, and
suppose that the central chain control set ${}_{\mathbb{P}}E_{c}$ on the
projective Poincaré sphere $\mathbb{P}^{n}$ has a nonvoid intersection
${}_{\mathbb{P}}E_{c}\cap\left({}_{\mathbb{P}}E^{\hom}\times\\{0\\}\right)\,$
with a chain control set ${}_{\mathbb{P}}E^{\hom}$ in $\mathbb{P}^{n-1}$for
the homogeneous part (3). Denote by
${}_{\mathbb{S}}E_{j}^{\hom}\subset\mathbb{S}^{n-1},j=0$ or $j=1,2$, the chain
control sets mapped onto ${}_{\mathbb{P}}E^{\hom}$.

(i) Consider the case where the central chain control set
${}_{\mathbb{S}}E_{c,0}\ $on $\mathbb{S}^{n}$ exists. Then for the chain
control sets ${}_{\mathbb{S}}E_{j}^{\hom},j=0$ or $j=1,2$, one has
${}_{\mathbb{S}}E_{j}^{\hom}\times\\{0\\}\subset\,_{\mathbb{S}}E_{c,0}\cap\mathbb{S}^{n,0}$.

(ii) Consider the case where the central chain control sets
${}_{\mathbb{S}}E_{c,1}$ and ${}_{\mathbb{S}}E_{c,2}\ $exist. Then the chain
control sets ${}_{\mathbb{S}}E_{1}^{\hom}$ and $\,{}_{\mathbb{S}}E_{2}^{\hom}$
exist, and for $j=1,2$ exactly one of the sets
${}_{\mathbb{S}}E_{1}^{\hom}\times\\{0\\}$ and
${}_{\mathbb{S}}E_{2}^{\hom}\times\\{0\\}$ is contained in
${}_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}$. The case where
${}_{\mathbb{S}}E_{0}^{\hom}$ exists cannot occur.

###### Proof.

(i) The chain control set ${}_{\mathbb{S}}E_{c,0}\ $is the preimage of
${}_{\mathbb{P}}E_{c}\ $and either ${}_{\mathbb{S}}E_{j}^{\hom}$ or the union
${}_{\mathbb{S}}E_{1}^{\hom}\cup\,_{\mathbb{S}}E_{2}^{\hom}$ forms the
preimage of ${}_{\mathbb{P}}E^{\hom}$. Since $\mathbb{S}^{n,0}$ is the
preimage of $\mathbb{P}^{n,0}$ it follows that
${}_{\mathbb{S}}E_{j}^{\hom}\times\\{0\\}\subset\,_{\mathbb{S}}E_{c,0}\cap\mathbb{S}^{n,0}$
for $j=0,1,2$.

(ii) The same argument shows that
${}_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}\not=\varnothing,j=1,2$. Suppose
that ${}_{\mathbb{S}}E_{1}^{\hom}=-\,_{\mathbb{S}}E_{2}^{\hom}$ exist. Then it
holds that

$\left({}_{\mathbb{S}}E_{1}^{\hom}\times\\{0\\}\right)\cup\left(-\,_{\mathbb{S}}E_{2}^{\hom}\times\\{0\\}\right)=\,_{\mathbb{S}}E_{c,1}\cup\,_{\mathbb{S}}E_{c,2}.$

If ${}_{\mathbb{S}}E_{1}^{\hom}\cap\,_{\mathbb{S}}E_{c,1}\not=\varnothing$ it
follows by Proposition 3(i) that
${}_{\mathbb{S}}E_{1}^{\hom}\subset\,_{\mathbb{S}}E_{c,1}$. Then the set
${}_{\mathbb{S}}E_{2}^{\hom}\times\\{0\\}=-\left({}_{\mathbb{S}}E_{1}^{\hom}\times\\{0\\}\right)$
cannot intersect ${}_{\mathbb{S}}E_{c,1}$ since this would lead to an element
$s_{0}$ with $\mathbb{P}s_{0}\in\,_{\mathbb{P}}E_{c,1}$ and
$-s_{0}\in\,_{\mathbb{S}}E_{c,1}\subset\mathbf{R}^{c}(s_{0})$ contradicting
Theorem 8(ii). The same argument applies to ${}_{\mathbb{S}}E_{c,2}$. If
${}_{\mathbb{S}}E_{0}^{\hom}$ exists, then the chain control set
${}_{\mathbb{S}}E_{0}^{\hom}\times\\{0\\}$ would be contained in the
intersection ${}_{\mathbb{S}}E_{c,1}\cap\,_{\mathbb{S}}E_{c,2}$, which is a
contradiction. ∎

##  6 Chain control sets on hemispheres

By Corollary 2 the strong chain control sets on the northern hemisphere
$\mathbb{S}^{n,+}$ of the Poincaré sphere correspond to the strong chain
control sets $E^{\ast}$ of the affine system (1) and are given by
${}_{\mathbb{S}}E^{\ast}=h(E^{\ast})$. By Proposition 1(ii) every strong chain
control set is contained in a chain control set. This implies that the strong
chain control sets ${}_{\mathbb{S}}E^{\ast}$ are contained in chain control
sets in $\mathbb{S}^{n,+}$ and in chain control sets of the system on the
closure $\overline{\mathbb{S}^{n,+}}$. These chain control sets are contained
in one of the central chain control set
${}_{\mathbb{S}}E_{c,j},j\in\\{0,1,2\\}$ on $\mathbb{S}^{n}$. In this section,
we will further analyze the chain control sets in
$\overline{\mathbb{S}^{n,+}}$.

Theorem 3(ii) determines the intersection of the central Selgrade bundle
$\mathcal{V}_{c}^{1}$ with $\mathcal{U}\times\mathbb{S}^{n,0}$. In this
section, we restrict the analysis to the case $\ell^{0}=1$, hence there is a
unique subbundle $\mathcal{V}_{i}^{\infty},i=\ell^{+}+1$, contained in
$\mathcal{V}_{c}^{1}$ and

$\mathcal{V}_{c}^{1}\cap\left(\mathcal{U}\times\mathbb{S}^{n,0}\right)=\mathcal{V}_{\ell^{+}+1}^{\infty}\cap\left(\mathcal{U}\times\mathbb{S}^{n,0}\right)\text{
with }\dim\mathcal{V}_{c}^{1}=1+\dim\mathcal{V}_{\ell^{+}+1}^{\infty}.$

Here
$\mathcal{V}_{\ell^{+}+1}^{\infty}=\mathcal{V}_{\ell^{+}+1}\times\\{0\\}\subset\mathcal{V}_{c}^{1}$,
where $\mathcal{V}_{\ell^{+}+1}$ is a Selgrade bundle of the homogeneous part
(3). The Selgrade bundles $\mathcal{V}_{i}$ are related to the chain control
sets ${}_{\mathbb{P}}E_{i}^{\hom}$ in $\mathbb{P}^{n-1}$ of the homogeneous
part (3) by

$\mathcal{V}_{i}=\left\\{(u,x)\in\mathcal{U}\times\mathbb{R}^{n}\left|\mathbb{P}\varphi(t,x,u)\in\,_{\mathbb{P}}E_{i}^{\hom}\text{
for all }t\in\mathbb{R}\right.\right\\}.$

We will analyze chain controllability properties in the closure
$\overline{\mathbb{S}^{n,+}}=\mathbb{S}^{n,+}\cup\mathbb{S}^{n,0}$ of the
northern hemisphere. On the level of chain control sets this means that for
the central chain control set ${}_{\mathbb{P}}E_{c}$ in $\mathbb{P}^{n}$ there
is a single chain control set ${}_{\mathbb{P}}E^{\hom}$ of the projectivized
homogeneous part with

${}_{\mathbb{P}}E_{c}\cap\mathbb{P}^{n,0}=\,_{\mathbb{P}}E^{\hom}\times\\{0\\}.$

Actually, we even require that ${}_{\mathbb{P}}E^{\hom}$ is the projection of
a single chain control set ${}_{\mathbb{S}}E_{0}^{\hom}$ on
$\mathbb{S}^{n-1}$. With this assumption on the level of chain control sets on
$\mathbb{S}^{n}$ we prove the following lemma.

######  Lemma 6.

Consider an affine control system of the form (1) on $\mathbb{R}^{n}$ and
assume that a central chain control set
${}_{\mathbb{S}}E_{c,j},j\in\\{0,1,2\\}$, in $\mathbb{S}^{n}$ satisfies

${}_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}=\,_{\mathbb{S}}E_{0}^{\hom}\times\\{0\\},$
(24)

where the chain control set ${}_{\mathbb{S}}E_{0}^{\hom}$ is the preimage in
$\mathbb{S}^{n-1}$ of a chain control set ${}_{\mathbb{P}}E^{\hom}$ of the
projectivized homogeneous part (3). Then the intersection
${}_{\mathbb{S}}E_{c,j}\cap\overline{\mathbb{S}^{n,+}}$ is chain controllable
on $\overline{\mathbb{S}^{n,+}}$.

###### Proof.

Let $s,s^{\prime}\in\,_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,+}$. We construct
for $\varepsilon,T>0$ a controlled $(\varepsilon,T)$-chain in
$\overline{\mathbb{S}^{n,+}}$ from $s$ to $s^{\prime}$. Choose a controlled
$(\varepsilon/2,T)$-chain $\zeta$ in $\mathbb{S}^{n}$ from
$s\in\,_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,+}$ to
$s^{\prime}\in\,_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,+}$ given by
$T_{0},\ldots,T_{k-1}\geq
T,u^{0},\ldots,u^{k-1}\in\mathcal{U},s^{0}=s,s^{1},\ldots,s^{k}=s^{\prime}$ in
${}_{\mathbb{S}}E_{c,j}$ with

$d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),s^{i+1})<\varepsilon/2\text{ for all
}i.$

Since $\mathbb{S}^{n}$ is compact, the chain may be chosen in
${}_{\mathbb{S}}E_{c,j}$. If all $s^{i}$ are in $\mathbb{S}^{n,+}$, we are
done. Otherwise there is a minimal $i$, called a crossing index, with

$s^{i}\in S^{n,+}\text{ and }s^{i+1}\in\mathbb{S}^{n,-}\cup\mathbb{S}^{n,0}.$
(25)

Step 1: If $s^{i+1}\in\mathbb{S}^{n,0}$, define $\widetilde{s}:=s^{i+1}$.
Otherwise $s^{i+1}\in\mathbb{S}^{n,-}$. By invariance of $\mathbb{S}^{n,+}$ we
know that $\mathbb{S}\varphi(T_{i},s^{i},u^{i})\in S^{n,+}$ and the estimate

$d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),s^{i+1})<\varepsilon/2$

holds. By Theorem 8(iii) it follows that

$\left\\{\left.\frac{\alpha\mathbb{S}\varphi(T_{i},s^{i},u^{i})+(1-\alpha)s^{i+1}}{\left\|\alpha\mathbb{S}\varphi(T_{i},s^{i},u^{i})+(1-\alpha)s^{i+1}\right\|}\right|\alpha\in[0,1]\right\\}\subset\,_{\mathbb{S}}E_{c,j}.$

The last component of $\mathbb{S}\varphi(T_{i},s^{i},u^{i})$ is positive, and
the last component of $s^{i+1}$ is negative. Hence there is $\alpha\in(0,1)$
such that

$\widetilde{s}:=\frac{\alpha\mathbb{S}\varphi(T_{i},s^{i},u^{i})+(1-\alpha)s^{i+1}}{\left\|\alpha\mathbb{S}\varphi(T_{i},s^{i},u^{i})+(1-\alpha)s^{i+1}\right\|}\in\,_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,0}.$

This implies

$d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),\widetilde{s})<\varepsilon/2\text{ and
}d(\widetilde{s},s^{i+1})<\varepsilon/2.$

By assumption (24) it follows that there is
$\widehat{s}\in\,_{\mathbb{S}}E_{0}^{\hom}$ with
$\widetilde{s}=(\widehat{s},0)$. Since
${}_{\mathbb{S}}E_{0}^{\hom}=-\,_{\mathbb{S}}E_{0}^{\hom}$ there is a
controlled $(\varepsilon/2,T)$-chain of the homogeneous part from
$\widehat{s}$ to $-\widehat{s}$. This defines a controlled
$(\varepsilon/2,T)$-chain $\zeta^{1}$ in $\mathbb{S}^{n}$ from
$(\widehat{s},0)$ to $(-\widehat{s},0)$ and

$d(-\widetilde{s},-s^{i+1})=d(-(\widehat{s},0),-s^{i+1})=d((\widehat{s},0),s^{i+1})<\varepsilon/2.$

Replace the final point $(-\widehat{s},0)$ of $\zeta^{1}$ by $-s^{i+1}$, hence
the jump length is bounded by

$d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),-s^{i+1})\leq
d(\mathbb{S}\varphi(T_{i},s^{i},u^{i}),-\widetilde{s})+d(-\widetilde{s},-s^{i+1})<\varepsilon/2+\varepsilon/2=\varepsilon.$

Then we insert this controlled $(\varepsilon,T)$-chain between
$\mathbb{S}\varphi(T_{i},s^{i},u^{i})$ and $-s^{i+1}$. The concatenation is a
controlled $(\varepsilon,T)$-chain from $s$ to $-s^{i+1}$.

Step 2: Next we consider what happens for $i+2$.

(a) Suppose first that $s^{i+2}\in S^{n,+}$. The construction in Step 1 has
led us to the point $-s^{i+1}\in\mathbb{S}^{n,+}$. We know that

$d(\mathbb{S}\varphi(T_{i+1},s^{i+1},u^{i+1}),s^{i+2})<\varepsilon.$

By invariance of $\mathbb{S}^{n,-}$ we find, similarly as in Step 1 a point
$(\widehat{s},0)$ with $\widehat{s}\in\,_{\mathbb{S}}E^{\hom}$ with

$d(\mathbb{S}\varphi(T_{i+1},s^{i+1},u^{i+1}),(\widehat{s},0))<\varepsilon/2\text{
and }d((\widehat{s},0),s^{i+2})<\varepsilon/2.$

This implies

$d(\mathbb{S}\varphi(T_{i+1},-s^{i+1},u^{i+1}),(-\widehat{s},0))=d(\mathbb{S}\varphi(T_{i+1},s^{i+1},u^{i+1}),(\widehat{s},0))<\varepsilon/2.$

We insert a controlled $(\varepsilon/2,T)$-chain $\zeta^{2}$ from
$(-\widehat{s},0)$ to $(\widehat{s},0)$ and replace the final point by
$s^{i+2}$. The concatenation is a controlled $(\varepsilon,T)$-chain from $s$
to $s^{i+2}\in\mathbb{S}^{n,+}$.

(b) Suppose that $s^{i+2}\in S^{n,-}\cup\mathbb{S}^{n,0}$. Since

$d(\mathbb{S}\varphi(T_{i+1},-s^{i+1},u^{i+1}),-s^{i+2})=d(\mathbb{S}\varphi(T_{i+1},s^{i+1},u^{i+1}),s^{i+2})<\varepsilon$
(26)

we have a controlled $(\varepsilon,T)$-chain from $s$ to
$-s^{i+2}\in\mathbb{S}^{n,+}\cup\mathbb{S}^{n,0}$. If
$s^{i+3}\in\mathbb{S}^{n,-}\cup\mathbb{S}^{n,0}$ we obtain a controlled
$(\varepsilon,T)$-chain from $s$ to
$-s^{i+3}\in\mathbb{S}^{n,+}\cup\mathbb{S}^{n,0}$. If
$s^{i+3}\in\mathbb{S}^{n,+}$ we have

$d(\mathbb{S}\varphi(T_{i+2},-s^{i+2},u^{i+2}),-s^{i+3})<\varepsilon.$

The construction in Step 1 yields a point $\widetilde{s}=(\widehat{s},0)$ with
$\widehat{s}\in E_{0}^{\hom}$ with

$d(\mathbb{S}\varphi(T_{i+2},-s^{i+2},u^{i+2}),\widetilde{s})<\varepsilon/2\text{
and }d(\widetilde{s},-s^{i+3})<\varepsilon/2.$

Then one obtains a controlled $(\varepsilon,T)$\- chain from $-s^{i+2}$ to
$s^{i+3}\in\mathbb{S}^{n,+}\cup\mathbb{S}^{n,0}$.

This “spatchcocking procedure” can be repeated for every crossing index. Thus
we get a controlled $(\varepsilon,T)$-chain in $\overline{\mathbb{S}^{n,+}}$
from $s$ to $s^{\prime}$. This shows that
${}_{\mathbb{S}}E_{c,j}\cap\mathbb{S}^{n,+}$ is chain controllable in
$\overline{\mathbb{S}^{n,+}}$. Since chain control sets are closed this
implies that also ${}_{\mathbb{S}}E_{c,j}\cap\overline{\mathbb{S}^{n,+}}$ is
chain controllable in $\overline{\mathbb{S}^{n,+}}$. ∎

The next theorem describes in a case with $\ell^{0}=1$ the chain control sets
in the closure of the northern hemisphere.

######  Theorem 10.

Consider an affine control system of the form (1) on $\mathbb{R}^{n}$. Assume
for the central chain control set ${}_{\mathbb{P}}E_{c}$ in $\mathbb{P}^{n}$
that the intersection with $\mathbb{P}^{n,0}$ is a single chain control set
${}_{\mathbb{P}}E^{\hom}\times\\{0\\}$ of the homogeneous part (3) and that
the preimage in $\mathbb{S}^{n-1}$ of ${}_{\mathbb{P}}E^{\hom}$ is a single
chain control set ${}_{\mathbb{S}}E_{0}^{\hom}$.

(i) Suppose that the preimage in $\mathbb{S}^{n}$ of ${}_{\mathbb{P}}E_{c}$ is
a single chain control set ${}_{\mathbb{S}}E_{c,0}$. Then the intersection
${}_{\mathbb{S}}E_{c,0}\cap\overline{\mathbb{S}^{n,+}}$ is the unique chain
control set in $\overline{\mathbb{S}^{n,+}}$, which is not contained in the
equator $\mathbb{S}^{n,0}\,.$

(ii) Suppose that the preimage in $\mathbb{S}^{n}$ of ${}_{\mathbb{P}}E_{c}$
is the union of two chain control sets ${}_{\mathbb{S}}E_{c,1}$ and
${}_{\mathbb{S}}E_{c,2}$. Then every chain control set in
$\overline{\mathbb{S}^{n,+}}$, which is not contained in $\mathbb{S}^{n,0}$,
coincides with ${}_{\mathbb{S}}E_{c,1}\cap\overline{\mathbb{S}^{n,+}}$ or
${}_{\mathbb{S}}E_{c,2}\cap\overline{\mathbb{S}^{n,+}}$.

###### Proof.

For assertions (i) and (ii) the assumptions of Lemma 6 hold. Hence the
intersections ${}_{\mathbb{S}}E_{c,j}\cap\overline{\mathbb{S}^{n,+}}$ are
chain controllable in $\overline{\mathbb{S}^{n,+}}$. Since any chain control
set in $\overline{\mathbb{S}^{n,+}}$ having nonvoid intersection with
$\mathbb{S}^{n,+}$ is contained in some ${}_{\mathbb{S}}E_{c,j}$ this are the
chain control sets in $\overline{\mathbb{S}^{n,+}}$. ∎

If the assumptions of Theorem 10 are satisfied it follows that the images
${}_{\mathbb{S}}E^{\ast}$ of the strong chain control sets $E^{\ast}$ of the
affine system are, in case (i) contained in
${}_{\mathbb{S}}E_{c,0}\cap\mathbb{S}^{n,+}$ and, in case (ii) in
${}_{\mathbb{S}}E_{c,1}\cap\mathbb{S}^{n,+}$ or
${}_{\mathbb{S}}E_{c,2}\cap\mathbb{S}^{n,+}$. If the strong chain control sets
$E^{\ast}$ are unbounded (cf. Proposition 2), the closures of the strong chain
control sets ${}_{\mathbb{S}}E^{\ast}$ in
${}_{\mathbb{S}}E_{c,0}\cap\mathbb{S}^{n,+}$ also intersect
${}_{\mathbb{S}}E_{0}^{\hom}\times\\{0\\}$.

######  Remark 3.

The assertions of Theorem 10 also hold when the northern hemisphere
$\mathbb{S}^{n,+}$ is replaced by the southern hemisphere $\mathbb{S}^{n,-}$.

Next, we consider a situation where again the intersection of the central
chain control set ${}_{\mathbb{P}}E_{c}$ with $\mathbb{P}^{n,0}$ is a single
chain control set ${}_{\mathbb{P}}E^{\hom}$ of the homogeneous part. But now
we suppose that the preimage in $\mathbb{S}^{n-1}$ of
${}_{\mathbb{P}}E^{\hom}$ consists of two chain control set
${}_{\mathbb{S}}E_{1}^{\hom}$ and ${}_{\mathbb{S}}E_{2}^{\hom}$. On the
northern and southern hemispheres $\mathbb{S}^{n,+}$ and $\mathbb{S}^{n,-}$,
respectively, two chain control sets exist. They yield two chain control sets
on $\mathbb{S}^{n}$ and each of them intersects both hemispheres.

######  Theorem 11.

Consider an affine control system of the form (1) on $\mathbb{R}^{n}$. Assume
for the central chain control set ${}_{\mathbb{P}}E_{c}$ in $\mathbb{P}^{n}$
that the intersection with $\mathbb{P}^{n,0}$ is a single chain control set
${}_{\mathbb{P}}E^{\hom}\times\\{0\\}$ of the homogeneous part and that the
preimage in $\mathbb{S}^{n-1}$ of ${}_{\mathbb{P}}E^{\hom}$ is the union of
two chain control sets ${}_{\mathbb{S}}E_{1}^{\hom}$ and
${}_{\mathbb{S}}E_{2}^{\hom}$. Furthermore, let ${}_{\mathbb{S}}E_{1}^{+}$ and
$\,{}_{\mathbb{S}}E_{2}^{+}$ be two chain control sets in the northern
hemisphere $\mathbb{S}^{n,+}$ such that

$\overline{{}_{\mathbb{S}}E_{1}^{+}}\cap\left({}_{\mathbb{S}}E_{1}^{\hom}\times\\{0\\}\right)\not=\varnothing\text{
and
}\overline{{}_{\mathbb{S}}E_{2}^{+}}\cap\left({}_{\mathbb{S}}E_{2}^{\hom}\times\\{0\\}\right)\not=\varnothing.$

Then, with the chain control sets
${}_{\mathbb{S}}E_{1}^{-}:=-\,_{\mathbb{S}}E_{1}^{+}$ and
${}_{\mathbb{S}}E_{2}^{-}:=-\,_{\mathbb{S}}E_{2}^{+}$ in the southern
hemisphere $\mathbb{S}^{n,-}$, both sets
${}_{\mathbb{S}}E_{1}^{+}\cup\,_{\mathbb{S}}E_{2}^{-}$ and
${}_{\mathbb{S}}E_{2}^{+}\cup\,_{\mathbb{S}}E_{1}^{-}$ are contained in chain
control sets in $\mathbb{S}^{n}$.

###### Proof.

Since ${}_{\mathbb{S}}E_{2}^{-}=-\,_{\mathbb{S}}E_{2}^{+}$ and
${}_{\mathbb{S}}E_{1}^{\hom}=-\,_{\mathbb{S}}E_{2}^{\hom}$ it follows that

$\overline{{}_{\mathbb{S}}E_{2}^{-}}\cap\left({}_{\mathbb{S}}E_{1}^{\hom}\times\\{0\\}\right)=-\overline{\,{}_{\mathbb{S}}E_{2}^{+}}\cap\left(-\,_{\mathbb{S}}E_{2}^{\hom}\times\\{0\\}\right)=-\left(\overline{{}_{\mathbb{S}}E_{2}^{+}}\cap\left({}_{\mathbb{S}}E_{2}^{\hom}\times\\{0\\}\right)\right)\not=\varnothing.$

This implies that the union

$\,{}_{\mathbb{S}}E_{1}^{+}\cup\,_{\mathbb{S}}E_{2}^{-}\cup\left({}_{\mathbb{S}}E_{1}^{\hom}\times\\{0\\}\right)$

is a chain controllable set, hence contained in a chain control set in
$\mathbb{S}^{n}$. The assertion for
${}_{\mathbb{S}}E_{2}^{+}\cup\,_{\mathbb{S}}E_{1}^{-}$ follows analogously. ∎

The following example (cf. Colonius, Santana, and Setti [10, Example 5.17],
[11, Example 6.10]) illustrates Theorem 11. It shows the delicate relations
between chain control sets and strong chain control sets in the various spaces
$\mathbb{S}^{n}$ and $\mathbb{P}^{n}$ as well as in the hemispheres
$\mathbb{S}^{n,+}$ and $\mathbb{S}^{n,-}$. The affine systems on
$\mathbb{R}^{2}$ has two strong chain control sets $E_{1}^{\ast}$ and
$E_{2}^{\ast}$, hence also the induced systems on $\mathbb{S}^{n,+}$ and
$\mathbb{P}^{n,1}$ have two strong chain control sets, while the system on
$\mathbb{P}^{2}$ has a single (strong) chain control set. See Figure 2.

######  Example 3.

Consider the affine control system

$\left(\begin{array}[c]{c}\dot{x}\\\
\dot{y}\end{array}\right)=\left(\begin{array}[c]{cc}0&1\\\
-1-u(t)&-3\end{array}\right)\left(\begin{array}[c]{c}x\\\
y\end{array}\right)+u(t)\left(\begin{array}[c]{c}0\\\
1\end{array}\right)+\left(\begin{array}[c]{c}0\\\ d\end{array}\right),\quad
u(t)\in\Omega:=[-\rho,\rho],$

where we suppose $\rho\in\left(1,\frac{5}{4}\right)$ and $d<1$. The equilibria
are

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=\left\\{\left.\left(\begin{array}[c]{c}x\\\
0\end{array}\right)\right|x\in\left[\frac{d-\rho}{1-\rho},\infty\right)\right\\}=\left\\{\left.\left(\begin{array}[c]{c}\frac{d+u}{1+u}\\\
0\end{array}\right)\right|u\in[-\rho,-1)\right\\},$
$\displaystyle\mathcal{C}_{2}$
$\displaystyle=\left\\{\left.\left(\begin{array}[c]{c}x\\\
0\end{array}\right)\right|x\in\left(-\infty,\frac{d+\rho}{1+\rho}\right]\right\\}=\left\\{\left.\left(\begin{array}[c]{c}\frac{d+u}{1+u}\\\
0\end{array}\right)\right|u\in(-1,\rho]\right\\},$

respectively. There is no equilibrium for $u=-1$. Note that
$\frac{d-\rho}{1-\rho}>\frac{d+\rho}{1+\rho}>0$, hence
$\mathcal{C}_{1}\cap\mathcal{C}_{2}=\varnothing$. The eigenvalues of the
linear part are
$\lambda_{1,2}(u)=-\frac{3}{2}\pm\sqrt{\frac{5}{4}-u},u\in[-\rho,\rho]$. For
$u=-1$ the eigenvalues are $\lambda_{1}(-1)=0,\lambda_{2}(-1)=-3$ and the
eigenspace of $\lambda_{1}(-1)$ is $\mathbb{R}\times\\{0\\}$. For
$u^{k}\nearrow u^{0}=-1$, the equilibria in $\mathcal{C}_{1}$ satisfy
$(x_{u^{k}},0)\rightarrow(\infty,0)$ and for $u^{k}\searrow u^{0}=-1$, the
equilibria in $\mathcal{C}_{2}$ satisfy $(x_{u^{k}},0)\rightarrow(-\infty,0)$
for $k\rightarrow\infty$. The equilibria in $\mathcal{C}_{1}$ are hyperbolic
since for $u\in[-\rho,-1)\subset(-\frac{5}{4},-1)$ one obtains
$\lambda_{1}(u)>0>\lambda_{2}(u)$. The equilibria in $\mathcal{C}_{2}$ are
stable nodes since for $u\in(-1,\rho]$ one obtains
$0>\lambda_{1}(u)>\lambda_{2}(u)$. For constant $u$ the phase portrait with
$y:=x-\frac{u+d}{1+u}$ is the shifted phase portrait of

$\ddot{y}+3\dot{y}+(1+u)y=0.$

There are control sets $D_{1}$ and $D_{2}$ containing the equilibria in
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, respectively, in the interior.

While the asymptotic stability of the equilibria in $\mathcal{C}_{2}$ implies
that one can steer the system from $D_{1}$ to $D_{2}$, the converse does not
hold which follows by inspection of the phase portraits for the controls in
$[-\rho,-1]$ and $\left[-1,\rho\right]$, hence $D_{1}\not=D_{2}$. One also
sees that the unbounded chain control sets $E_{1}$ and $E_{2}$ with
$D_{1}\subset E_{1}$ and $D_{2}\subset E_{2}$ are disjoint. Note that
according to Gayer [16, Theorem 19 and Remark 20] the system above satisfies
the so-called inner-pair condition, which in the compact case guarantees that,
generically, the closures of control sets are chain control sets. The continua
of equilibria $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ in $\mathrm{int}D_{1}$
and $\mathrm{int}D_{2}$ are contained in strong chain control sets with
$E_{1}^{\ast}\cap D_{1}\not=\varnothing$ and $E_{2}^{\ast}\cap
D_{2}\not=\varnothing$, respectively. It follows that

$D_{1}\subset E_{1}^{\ast}\subset E_{1}\text{ and }D_{2}\subset
E_{2}^{\ast}\subset E_{2}.$

Hence the images $h(E_{1}^{\ast})$ and $h(E_{2}^{\ast})$ are in different
strong chain control sets in $\mathbb{S}^{2,+}$. With
$h_{-}:\mathbb{R}^{2}\rightarrow\mathbb{S}^{2,-}$ defined in Remark 2 one
similarly obtains that $h_{-}(E_{1}^{\ast})$ and $h_{-}(E_{2}^{\ast})$ are in
different strong chain control sets in $\mathbb{S}^{2,-}$.

On the other hand, one has $\lim_{x\rightarrow\pm\infty}h(x,0)=(\pm 1,0,0)$,
and

$\mathbb{S}^{2,0}\ni(1,0,0)\in\overline{h(E_{1}^{\ast})}\cap\overline{h_{-}(E_{2}^{\ast})}\text{
and
}\mathbb{S}^{2,0}\ni(-1,0,0)\in\overline{h(E_{2}^{\ast})}\cap\overline{h_{-}(E_{1}^{\ast})}.$

Thus the subsets of $\mathbb{S}^{n}$ given by

$\overline{h(E_{1}^{\ast})}\cup\overline{h_{-}(E_{2}^{\ast})}\text{ and
}\overline{h(E_{2}^{\ast})}\cup\overline{h_{-}(E_{1}^{\ast})}$

are strongly chain controllable. Each of these two subsets intersects both
hemispheres $\mathbb{S}^{2,+}$ and $\mathbb{S}^{2,-}$. In projective space
$\mathbb{P}^{2,0}$, the points $\mathbb{P}(1,0,0)$ and $\mathbb{P}(-1,0,0)$
coincide. It follows for the images of the strong chain control sets
$E_{1}^{\ast}$ and $E_{2}^{\ast}$ that their closures in
$\mathbb{P}^{2}=\mathbb{P}^{2,1}\cup\mathbb{P}^{2,0}$ intersect, hence they
are contained in a single strong chain control set of the compact space
$\mathbb{P}^{2}$, i.e., in the central chain control set
${}_{\mathbb{P}}E_{c}$.

Figure 1: Control set $D_{0}=\\{(0,0)\\}\times(-1,1)$, strong chain control
set $E^{\ast}=\mathbb{R}\times\\{0\\}\times[-1,1]$ and chain control set
$E=\mathbb{R}^{2}\times[-1,1]$ in Example 2. Figure 2: Phase portraits in
Example 3 for constant $u$-values yielding equilibria in $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$.

## References

  * [1] V. Ayala, A. Da Silva, and L.A.B. San Martin, Control systems on flag manifolds and their chain control sets, Disc. Cont. Dyn. Syst. 37(5) (2017), pp. 2301-2313. doi:10.3934/dcds.2017101.
  * [2] S.A. Ahmadi, Strong chain transitivity via uniformity, Qualitative Theory of Dynamical Systems 23-123 (2024), https://doi.org/10.1007/s12346-024-00983-4.
  * [3] J.M Alongi and G.S. Nelson, Recurrence and Topology, Graduate Studies in Math. Vol. 85, Amer. Math. Soc. 2007.
  * [4] A. Bacciotti and J.-C. Vivalda, On radial and directional controllability of bilinear systems, Systems Control Lett., 62(7) (2013), pp. 575-580.
  * [5] D. Cannarsa and M. Sigalotti, Approximately controllable finite-dimensional bilinear systems are controllable, Systems Control Lett., 157 (2021), article 105028.
  * [6] S.K. Choi, C-K. Chu, and J.S. Park, Chain recurrent sets for flows on non-compact spaces, J. Dyn. Diff. Equations, 14(3) (2002), pp. 597-611.
  * [7] F. Colonius and W. Kliemann,  The Dynamics of Control, Birkhäuser 2000.
  * [8] F. Colonius and W. Kliemann,  Dynamical Systems and Linear Algebra, Graduate Studies in Math. Vol. 156, Amer. Math. Soc. 2014.
  * [9] F. Colonius and A.J. Santana, Chain recurrence and Selgrade’s theorem for affine flows, J. Dyn. Diff. Equations (2024), doi.org/10.1007/s10884-024-10351-0.
  * [10] F. Colonius, A.J. Santana and J. Setti, Control sets for bilinear and affine systems, Mathematics of Control, Signals and Systems, 34 (2022), pp. 1-35.
  * [11] F. Colonius, A.J. Santana and J. Setti, Control sets for affine systems, spectral properties, and projective spaces, SIAM J. Control Optim., 61(2) (2023), pp. 647-673.
  * [12] F. Colonius, A.J. Santana, and E.C. Viscovini,  Chain control sets for linear control systems, to appear in SIAM J. Control Optim. (2024).
  * [13] O.G. Do Rocio, A.J. Santana, and M.A. Verdi, Semigroups of affine groups, controllability of affine systems and affine bilinear systems in $Sl(2,\mathbb{R})\rtimes\mathbb{R}^{2}$_,_ SIAM J. Control Optim. 48 (2009), pp. 1080-1088.
  * [14] R.W. Easton, Chain transitivity and the domain of influence of an invariant set, in: The Structure of Attractors in Dynamical Systems, 1978, pp.. 95-102.
  * [15] D.L. Elliott, Bilinear Control Systems, Matrices in Action, Kluwer Academic Publishers, 2008.
  * [16] T. Gayer, Control sets and their boundaries under parameter variation, J. Diff. Equations 201(2004), pp. 177-200.
  * [17] M. Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc. 115(4) (1992), pp. 1139-1148.
  * [18] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dyn. Diff. Equations 7(3) (1995), pp. 437-456.
  * [19] V. Jurdjevic, Controllability properties of affine systems, Proceedings of 23rd Conference on Decision and Control, Las Vegas, NV, December 1984, pp. 787-789.
  * [20] V. Jurdjevic, Geometric Control Theory, Cambridge University Press, 1997.
  * [21] V. Jurdjevic and G. Sallet, Controllability properties of affine systems, SIAM J. Control Optim. 22(3) (1984), pp. 501-508.
  * [22] C. Kawan, On the structure of uniformly hyperbolic chain control sets, Systems Control Lett. 90 (2016), pp. 71-75.
  * [23] C. Kawan,  Invariance Entropy for Deterministic Control Systems, Lect. Notes Math. Vol 2089, Springer 2013.
  * [24] R.R. Mohler, Bilinear Control Processes, Academic Press, New York and London, 1973.
  * [25] M. Patrão and L.A.B. San Martin, Semiflows on topological spaces: chain transitivity and shadowing semigroups. J. Dyn. Diff. Equations 19(1) (2007), p. 155-180.
  * [26] L. Perko, Differential Equations and Dynamical Systems, Springer, 3rd ed., 2001.
  * [27] H. Poincare, Memoire sur les courbes definies par une equation differentielle, J. Mathematiques, 7 (1881), pp. 375-422; Ouevre Tome I, Gauthier-Villar, Paris 1928, pp. 3-84.
  * [28] R.E. Rink and R.R. Mohler, Completely controllable bilinear systems, SIAM J. Control Optim. 6(3) (1968), pp. 477- 486.
  * [29] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Taylor & Francis Inc., 2nd ed., 1998.
  * [30] D. Salamon and E. Zehnder, Flows on vector bundles and hyperbolic sets, Trans. Amer. Math. Soc., 306 (1988), pp. 623-649.
  * [31] J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203 (1975), pp. 259-390.
  * [32] E. Sontag, Mathematical Control Theory, Springer-Verlag 1998 _._