# Control Sets for Affine Systems, Spectral Properties and Projective Spaces

Fritz Colonius
Institut für Mathematik  Universität Augsburg  Augsburg  Germany  Alexandre J.
Santana and Juliana Setti
Departamento de Matemática  Universidade Estadual de Maringá
Maringá  Brazil

Abstract: For affine control systems with bounded control range the control
sets, i.e., the maximal subsets of complete approximate controllability, are
studied using spectral properties. For hyperbolic systems there is a unique
control set with nonvoid interior and it is bounded. For nonhyperbolic
systems, these control sets are unbounded. In an appropriate compactification
of the state space there is a unique chain control set, and the relations to
the homogenous part of the control system are worked out.

Key words. affine control systems, control sets, boundary at infinity

AMS subject classification. 93B05, 34H05

##  1 Introduction

We study controllability properties for affine control systems of the form

$\dot{x}(t)=Ax(t)+\sum_{i=1}^{m}u_{i}(t)(B_{i}x(t)+c_{i})+d,\quad
u(t)\in\Omega,$ (1)

where $A,B_{1},\ldots,B_{m}\in\mathbb{R}^{n\times n}$ and
$c_{1},\ldots,c_{m},d\in\mathbb{R}^{n}$. The controls $u=(u_{1},\ldots,u_{m})$
have values in a bounded set $\Omega\subset\mathbb{R}^{m}$ with $0\in\Omega$.
The set of admissible controls is $\mathcal{U}=\\{u\in
L^{\infty}(\mathbb{R},\mathbb{R}^{m})\left|u(t)\in\Omega\text{ for almost all
}t\right.\\}$ or the set $\mathcal{U}_{pc}$ of all piecewise constant
functions defined on $\mathbb{R}$ with values in $\Omega$. We also write (1)
as

$\dot{x}(t)=A(u(t))x(t)+Cu(t)+d,\quad u(t)\in\Omega,$

with $A(u):=A+\sum_{i=1}^{m}u_{i}B_{i}$ for $u\in\Omega$ and
$C:=(c_{1},\ldots,c_{m})$. With the vector fields $f_{0}(x)=Ax+d$ and
$f_{i}(x)=B_{i}x+c_{i}$ on $\mathbb{R}^{n}$ we assume throughout that the
following accessibility (or Lie algebra) rank condition holds,

$\dim\mathcal{LA}(f_{0},f_{1},\ldots,f_{m})(x)=n\text{ for all
}x\in\mathbb{R}^{n},$ (2)

where $\mathcal{LA}(f_{0},f_{1},\ldots,f_{m})$ is the set of vector fields in
the Lie algebra generated by $f_{0},f_{1},\ldots,f_{m}$.

Controllability properties of bilinear and affine systems have been studied
since more than fifty years. Early contributions are due to Rink and Mohler
[21] who took the set of equilibria as a starting point for establishing
results on complete controllability. On the other hand, Lie-algebraic methods
have yielded important insights. A classical result due to Jurdjevic and
Sallet [15, Theorem 2] (cf. also Do Rocio, Santana, and Verdi [11]) shows that
affine system (1) is controllable on $\mathbb{R}^{n}$ if it has no fixed
points and its homogeneous part, the bilinear system

$\dot{x}(t)=A(u(t))x(t),\quad u(t)\in\Omega,$ (3)

is controllable on $\mathbb{R}^{n}\setminus\\{0\\}$ (only the first condition
is necessary). Initially, bilinear control systems were considered as “nearly
linear” (cf. Bruni, Di Pillo, and Koch [4] containing also many early
references). However it has turned out that characterizing controllability of
such systems (even with unrestricted controls) is a very difficult problem. As
Jurdjevic [14, p. 182] emphasizes, the controllability properties of affine
systems are substantially richer and may require “entirely different
geometrical considerations” (based on Lie-algebraic methods). There is a
substantial literature on controllability properties of bilinear and affine
control systems. Here we only refer to the monographs Mohler [18], Elliott
[12], and Jurdjevic [14]. Further references are also included in [9] where we
analyze controllability properties near equilibria.

The present paper concentrates on control sets, i.e., maximal subsets of
approximate controllability; cf. Definition 1 (another relaxation of complete
controllability is the notion of “near controllability” studied by Nie [19]
for discrete-time bilinear control systems). The key to our analysis are
properties of the interior of the system semigroup and spectral properties of
the homogeneous part (3). In the hyperbolic case (cf. Definition 19) Theorem
22 shows that an affine control system has a unique control set $D$ with
nonvoid interior. In the uniformly hyperbolic case (cf. Definition 23) Theorem
26 shows that $D$ is bounded. Hence these systems enjoy similar
controllability properties as linear control systems of the form
$\dot{x}=Ax+Bu$ with hyperbolic matrix $A$; cf. Colonius and Kliemann [7,
Example 3.2.16]. We remark that a generalization in another direction is given
for control sets of linear control systems on Lie groups by Ayala and Da Silva
[2].

Nonhyperbolic affine systems may possess several control sets with nonvoid
interior. By Theorem 33 each of them is unbounded. The proof takes up ideas
from Rink and Mohler [21], replacing the set of equilibria by the set of
periodic solutions. Then we compactify the state space using an embedding into
projective space $\mathbb{P}^{n}$. This is in line with the study of the
behavior at infinity for ordinary differential equations based on the Poincaré
sphere; cf. Perko [20, Section 3.10]. Already the special case of linear
control systems discussed in the beginning of Section 6 shows that here chain
transitivity (a classical notion in the theory of dynamical systems; cf.
Robinson [22]) plays an important role. Theorem 35 shows that for a control
set with nonvoid interior the “boundary at infinity” (cf. Definition 30)
intersects a chain control set of the projectivized homogeneous part (here
small jumps in the trajectories are allowed; cf. Definition 5). The main
result on the nonhyperbolic case is Theorem 41 showing that there is a single
chain control set in $\mathbb{P}^{n}$ containing the images of all control
sets $D$ with nonvoid interior in $\mathbb{R}^{n}$. The boundary at infinity
of this chain control set contains all chain control sets of the projectivized
homogeneous part having nonvoid intersection with the boundary at infinity of
one of the control sets $D$. These results cast new light on the relations
between affine systems and their homogeneous parts and are intuitively
appealing, since one may expect that for unbounded $x$-values the
inhomogeneous part $Cu(t)+d$ becomes less relevant.

The contents of the present paper are as follows. In Section 2 we first recall
some notation and general properties of nonlinear control systems and cite
results on spectral properties and controllability for homogeneous bilinear
control systems. A result on periodic solutions of periodic inhomogeneous
differential equations is stated. Its proof is given in the appendix. Section
3 analyzes the system semigroup of affine control systems and their
homogeneous parts. Section 4 shows that for hyperbolic systems a unique
control set exists and that it is bounded. Section 5 prepares the discussion
of nonhyperbolic systems by embedding affine control systems into homogeneous
bilinear systems and associated systems in projective spaces. Section 6
describes the control sets and their boundaries at infinity for nonhyperbolic
systems.

Notation. The set of eigenvalues of a matrix $A\in\mathbb{R}^{n\times n}$ is
$\mathrm{spec}(A)$ and the real eigenspace for an eigenvalue
$\mu\in\mathbb{C}$ is $\mathbf{E}(A;\mu)$. In a metric space $X$ with distance
$d$ the distance of $x\in X$ to a nonvoid subset $A\subset X$ is
$d(x,A)=\inf\\{d(x,a)\left|a\in A\right.\\}$.

##  2 Preliminaries

In Subsection 2.1 notation and some basic properties of control systems are
recalled. Subsection 2.2 cites results on control sets for homogeneous
bilinear control systems and Subsection 2.3 characterizes periodic solutions
of inhomogeneous periodic linear differential equations.

###  2.1 Basic properties of nonlinear control systems

In this subsection we introduce some terminology and notation for control-
affine systems including control sets and chain control sets.

We will consider control-affine systems on a smooth (real analytic) manifold
$M$ of the form

$\dot{x}(t)=f_{0}(x(t))+\sum\nolimits_{i=1}^{m}u_{i}(t)f_{i}(x(t)),\quad
u\in\mathcal{U}\text{ or }u\in\mathcal{U}_{pc},$ (4)

where $f_{0},f_{1},\ldots,f_{m}$ are smooth vector fields on $M$ and the
control range $\Omega\subset\mathbb{R}^{m}$ is bounded with $0\in\Omega$. We
assume that for every initial state $x\in M$ and every control function
$u\in\mathcal{U}$ there exists a unique solution
$\varphi(t,x,u),t\in\mathbb{R}$, with $\varphi(0,x,u)=x$ of (4) depending
continuously on $x$. For the general theory of nonlinear control systems we
refer to Sontag [23] and Jurdjevic [14].

The set of points reachable from $x\in M$ and controllable to $x\in M$ up to
time $T>0$ are defined by

$\displaystyle{\mathcal{O}}_{\leq T}^{+}(x)$ $\displaystyle:=\\{y\in
M\left|\text{there are}\;0\leq t\leq
T\;\text{and}\;u\in\mathcal{U}\;\text{with}\;y=\varphi(t,x,u)\right.\\},$
$\displaystyle{\mathcal{O}}_{\leq T}^{-}(x)$ $\displaystyle:=\\{y\in
M\left|\text{there are}\;0\leq t\leq
T\;\text{and}\;u\in\mathcal{U}\;\text{with}\;x=\varphi(t,y,u)\right.\\},$

resp. Furthermore, the reachable set (or “positive orbit”) from $x$ and the
set controllable to $x$ (or “negative orbit” of $x$) are

$\mathcal{O}^{+}(x)=\bigcup\nolimits_{T>0}O_{\leq
T}^{+}(x),\quad\mathcal{O}^{-}(x)=\bigcup\nolimits_{T>0}O_{\leq T}^{-}(x),$

resp. The system is called locally accessible in $x$ if $\mathcal{O}_{\leq
T}^{+}(x)$ and $\mathcal{O}_{\leq T}^{-}(x)$ have nonvoid interior for all
$T>0$, and the system is called locally accessible if this holds in every
point $x\in M$. This is equivalent to the following accessibility rank
condition

$\dim\mathcal{LA}\left\\{f_{0},f_{1},\ldots,f_{m}\right\\}(x)=\dim M\text{ for
all }x\in M;$ (5)

here $\mathcal{LA}\left\\{f_{0},f_{1},\ldots,f_{m}\right\\}(x)$ is the
subspace of the tangent space $T_{x}M$ corresponding to the vector fields,
evaluated in $x$, in the Lie algebra generated by $f_{0},f_{1},\ldots,f_{m}$.
The sets $\mathcal{O}^{-}(x)$ are the reachable sets of the time reversed
system given by
$\dot{x}(t)=-f_{0}(x(t))-\sum\nolimits_{i=1}^{m}u_{i}(t)f_{i}(x(t))$. The
trajectories for controls in $\mathcal{U}$ can be uniformly approximated on
bounded intervals by trajectories for controls in $\mathcal{U}_{pc}$.

The following definition introduces sets of complete approximate
controllability.

######  Definition 1.

A nonvoid set $D\subset M$ is called a control set of system (4) if it has the
following properties: (i) for all $x\in D$ there is a control function
$u\in\mathcal{U}$ such that $\varphi(t,x,u)\in D$ for all $t\geq 0$, (ii) for
all $x\in D$ one has $D\subset\overline{\mathcal{O}^{+}(x)}$, and (iii) $D$ is
maximal with these properties, that is, if $D^{\prime}\supset D$ satisfies
conditions (i) and (ii), then $D^{\prime}=D$. A control set $D\subset M$ is
called an invariant control set if
$\overline{D}=\overline{\mathcal{O}^{+}(x)}$ for all $x\in D$. All other
control sets are called variant.

We recall some properties of control sets; cf. Colonius and Kliemann [7, Chp.
3].

######  Remark 2.

If the intersection of two control sets is nonvoid, the maximality property
(iii) implies that they coincide. If the system is locally accessible in all
$x\in\overline{D}$, then by [7, Lemma 3.2.13(i)]
$\overline{D}=\overline{\mathrm{int}(D)}$ and
$D=\mathcal{O}^{-}(x)\cap\overline{\mathcal{O}^{+}(x)}$ for all
$x\in\mathrm{int}(D)$ and $\mathrm{int}(D)\subset\mathcal{O}^{+}(x)$ for all
$x\in D$ (here it suffices to consider controls in $\mathcal{U}_{pc}$). If
local accessibility holds on $M$, the control sets with nonvoid interior for
controls in $\mathcal{U}$ coincide with those defined analogously for controls
in $\mathcal{U}_{pc}$. This is proved using the approximation by trajectories
for controls in $\mathcal{U}_{pc}$.

######  Lemma 3.

Suppose that the system is locally accessible and there is
$x\in\mathrm{int}(\mathcal{O}^{-}(x))\cap\mathrm{int}(\mathcal{O}^{+}(x))$.
Then $D:=\mathcal{O}^{-}(x)\cap\overline{\mathcal{O}^{+}(x)}$ is a control set
and $x\in\mathrm{int}(D)$.

###### Proof.

††margin:  10.5.

The set $\mathrm{int}(\mathcal{O}^{-}(x))\cap\mathrm{int}(\mathcal{O}^{+}(x))$
satisfies properties (i) and (ii) of control sets, hence it is contained in a
control set $D^{\prime}$ and $x\in\mathrm{int}(D^{\prime})$. Then
$D^{\prime}=\mathcal{O}^{-}(x)\cap\overline{\mathcal{O}^{+}(x)}$ by Remark 2.
∎

Next we introduce a notion of controllability allowing for (small) jumps
between pieces of trajectories. Here we fix a metric $d$ on $M$.

######  Definition 4.

Fix $x,y\in M$ and let $\varepsilon,T>0.$ A controlled $(\varepsilon,T)$-chain
$\zeta$ from $x$ to $y$ is given by $n\in\mathbb{N},\
x_{0}=x,\ldots,x_{n-1},x_{n}=y\in M,\ u_{0},\ldots,u_{n-1}\in\mathcal{U}$, and
$t_{0},\ldots,t_{n-1}\geq T$ with

$d(\varphi(t_{j},x_{j},u_{j}),x_{j+1})\leq\varepsilon\text{
}\,\text{for\thinspace all}\,\,\,j=0,\ldots,n-1.$

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

In analogy to control sets, chain control sets are defined as maximal regions
of chain controllability; cf. [7, Chapter 4].

######  Definition 5.

A nonvoid set $E\subset M$ is called a chain control set of system (4) if (i)
for all $x\in E$ there is $u\in\mathcal{U}$ such that $\varphi$$(t,x,u)\in E$
for all $t\in\mathbb{R}$, (ii) for all $x,y\in E$ and $\varepsilon,\,T>0$
there is a controlled $(\varepsilon,T)$-chain from $x$ to $y$, and (iii) $E$
is maximal (with respect to set inclusion) with these properties.

Obviously, every equilibrium and every periodic trajectory is contained in a
control set and a chain control set. Since the concatenation of two controlled
$(\varepsilon,T)$-chains again yields a controlled $(\varepsilon,T)$-chain,
two chain control sets coincide if their intersection is nonvoid.

For a continuous dynamical system $\psi:\mathbb{R}\times X\rightarrow X$ on a
metric space $X$ a subset $Y\subset X$ is called chain transitive, if for all
$x,y\in Y$ and all $\varepsilon,T>0$ there is an $(\varepsilon,T)$-chain from
$x$ to $y$ given by $n\in\mathbb{N},\,x_{0}=x,x_{1},\ldots,x_{n}=y\in X$, and
$t_{0},\ldots,t_{n-1}\geq T$ with
$d(\psi(t_{j},x_{j}),x_{j+1})\leq\varepsilon$ for all$\,\,\,j=0,\ldots,n-1$.

For compact and convex control range $\Omega$, a control system of the form
(4) defines a continuous dynamical system, the control flow, given by
$\mathbf{\Psi}:\mathbb{R}\times\mathcal{U}\times M\rightarrow\mathcal{U}\times
M,\,(t,u,x)\mapsto(u(t+\cdot),\varphi(t,x,u))$, where
$u(t+\cdot)(s):=u(t+s),\,s\in\mathbb{R}$, and $\mathcal{U}\subset
L^{\infty}(\mathbb{R},\mathbb{R}^{m})$ considered in a metric for the weak∗
topology is compact; cf. Kawan [16, Proposition 1.17]. The following
assertions are shown in [16, Proposition 1.24]: chain control sets are closed
and for locally accessible systems every control set with nonvoid interior is
contained in a chain control set. The chain control sets $E$ uniquely
correspond to the maximal invariant chain transitive sets $\mathcal{E}$ of the
control flow $\mathbf{\Psi}$ via

$\mathcal{E}:=\\{(u,x)\in\mathcal{U}\times E\left|\varphi(t,x,u)\in E\text{
for all }t\in\mathbb{R}\right.\\}.$ (6)

###  2.2 Control sets for homogeneous bilinear systems

In this subsection we cite several results on control sets and chain control
sets for homogeneous bilinear control systems of the form

$\dot{x}(t)=A(u(t))x(t),\quad u(t)\in\Omega,\text{ with
}A(u):=A+\sum_{i=1}^{m}u_{i}B_{i},u\in\Omega,$ (7)

where $A,B_{1},\ldots,B_{m}\in\mathbb{R}^{n\times n}$ and the controls
$u=(u_{1},\ldots,u_{m})$ have values in a compact convex neighborhood $\Omega$
of the origin in $\mathbb{R}^{m}$. The solutions are denoted by
$\varphi_{\hom}(t,x,u),t\in\mathbb{R}$.

By homogeneity a system of the form (7) induces control systems on the unit
sphere $\mathbb{S}^{n-1}$ and on projective space $\mathbb{P}^{n-1}$. The
projections of $\mathbb{R}^{n}\setminus\\{0\\}$ to the unit sphere
$\mathbb{S}^{n-1}$ and projective space $\mathbb{P}^{n-1}$ are denoted by
$\pi_{\mathbb{S}}$ and $\pi_{\mathbb{P}}$, resp.

######  Definition 6.

Let
$\lambda(u,x)=\lim\sup_{t\rightarrow\infty}\frac{1}{t}\log\left\|\varphi_{\hom}(t,x,u)\right\|$
be the Lyapunov exponent for
$(u,x)\in\mathcal{U}\times\left(\mathbb{R}^{n}\setminus\\{0\\}\right)$.

(i) The Floquet spectrum of a control set ${}_{\mathbb{S}}D$ on the unit
sphere $\mathbb{S}^{n-1}$ is

$\Sigma_{Fl}(_{\mathbb{S}}D)=\left\\{\lambda(u,x)\left|\pi_{\mathbb{S}}x\in\mathrm{int}(_{\mathbb{S}}D),\,u\in\mathcal{U}_{pc}\text{
}\tau\text{-periodic with
}\pi_{\mathbb{S}}\varphi_{\hom}(\tau,x,u)=\pi_{\mathbb{S}}x\right.\right\\}.$

(ii) The Floquet spectrum of a control set ${}_{\mathbb{P}}D$ on projective
space $\mathbb{P}^{n-1}$ is

$\Sigma_{Fl}(_{\mathbb{P}}D)=\left\\{\lambda(u,x)\left|\pi_{\mathbb{P}}x\in\mathrm{int}(_{\mathbb{P}}D),\,u\in\mathcal{U}_{pc}\text{
}\tau\text{-periodic with
}\pi_{\mathbb{P}}\varphi_{\hom}(\tau,x,u)=\pi_{\mathbb{P}}x\right.\right\\}.$

(iii) The Lyapunov spectrum of a control set ${}_{\mathbb{P}}D$ on projective
space $\mathbb{P}^{n-1}$ is

$\Sigma_{Ly}(_{\mathbb{P}}D)=\left\\{\lambda(u,x)\left|u\in\mathcal{U}\text{
and }\pi_{\mathbb{P}}\varphi(t,x,u)\in\overline{{}_{\mathbb{P}}D}\text{ for
all }t\geq 0\right.\right\\}.$

######  Remark 7.

The Floquet spectrum can be characterized using the system semigroups
${}_{\mathbb{R}}\mathcal{S}^{\hom}$ and ${}_{\mathbb{P}}\mathcal{S}^{\hom}$ of
the systems on $\mathbb{R}^{n}\setminus\\{0\\}$ and on $\mathbb{P}^{n-1}$,
resp. (cf. Section 3 for the definition of system semigroups). Suppose that
the accessibility rank condition in $\mathbb{P}^{n-1}$ holds. Corollary 7.3.18
in Colonius and Kliemann [7] implies that the Floquet spectrum of a control
set ${}_{\mathbb{P}}D$ consists of the numbers
$\frac{1}{\tau}\log\left|\rho\right|$ where $\rho$ is a real eigenvalue of an
element $\Phi_{u}(\tau,0)\in\,_{\mathbb{R}}\mathcal{S}_{\tau}^{\hom}$ with
eigenspace
$\pi_{\mathbb{P}}\left(\mathbf{E}(\Phi_{u}(\tau,0);\rho)\right)\subset\mathrm{int}(_{\mathbb{P}}D)$
and such that $\Phi_{u}(\tau,0)$ induces an element of the system semigroup in
$\mathrm{int}(_{\mathbb{P}}\mathcal{S}_{\leq\tau+1}^{\hom})$.

The following theorem analyzes the control sets of the homogeneous system (7).

######  Theorem 8.

Consider the systems on the unit sphere $\mathbb{S}^{n-1}$ and on projective
space $\mathbb{P}^{n-1}$ obtained by projection of the homogeneous bilinear
control system (7). Assume that the accessibility rank condition on
$\mathbb{P}^{n-1}$ is satisfied.

(i) There are $1\leq k_{0}\leq n$ control sets ${}_{\mathbb{P}}D_{j}$ with
nonvoid interior in $\mathbb{P}^{n-1}$ and exactly one of these control sets
is an invariant control set.

(ii) There are $1\leq k_{1}\leq 2k_{0}$ control sets ${}_{\mathbb{S}}D_{i}$ on
$\mathbb{S}^{n-1}$. In each case, one or two of the control sets
${}_{\mathbb{S}}D_{i}$ on $\mathbb{S}^{n-1}$ project to a single control set
${}_{\mathbb{P}}D_{j}$ on $\mathbb{P}^{n-1}$ and then
$\Sigma_{Fl}(_{\mathbb{S}}D_{i})=\Sigma_{Fl}(_{\mathbb{P}}D_{j})$ and
$\Sigma_{Ly}(_{\mathbb{S}}D_{i})=\Sigma_{Ly}(_{\mathbb{P}}D_{j})$.

(iii) If $0$ is in the interior of the Floquet spectrum
$\Sigma_{Fl}(_{\mathbb{S}}D_{i})$ the cone

${}_{\mathbb{R}}D_{i}:=\\{\alpha x\in\mathbb{R}^{n}\left|\alpha>0\text{ and
}x\in\,_{\mathbb{S}}D_{i}\right.\\}$

generated by ${}_{\mathbb{S}}D_{i}$ is a control set with nonvoid interior in
$\mathbb{R}^{n}\setminus\\{0\\}$. Conversely, if ${}_{\mathbb{R}}D$ is control
set with nonvoid interior in $\mathbb{R}^{n}\setminus\\{0\\}$, then
$0\in\overline{\Sigma_{Ly}(_{\mathbb{P}}D)}$, where
${}_{\mathbb{P}}D\supset\pi_{\mathbb{P}}(_{\mathbb{R}}D)$.

###### Proof.

For (i) see [7, Theorem 7.1.1]. Assertion (ii) follows from Colonius, Santana,
and Setti [9, Theorem 3.15] and the observation that the equality of the
Lyapunov spectra is obvious. (iii) follows from [9, Proposition 3.18] and
(ii). ∎

We turn to the chain control sets in projective space. By [7, Theorem 7.1.2]
every chain control set ${}_{\mathbb{P}}E_{j}$ contains a control set
${}_{\mathbb{P}}D_{i}$ with nonvoid interior, hence the number $l$ of chain
control sets satisfies $1\leq l\leq k_{0}$. Furthermore, [7, Theorem 7.3.16]
shows that for every chain control set ${}_{\mathbb{P}}E_{j}$ in
$\mathbb{P}^{n-1}$ and every $u\in\mathcal{U}$

$\\{x\in\mathbb{R}^{n}\left|x\not=0\text{ implies
}\pi_{\mathbb{P}}\varphi_{\hom}(t,x,u)\in\,_{\mathbb{P}}E_{j}\text{ for all
}t\in\mathbb{R}\right.\\}$

is a linear subspace and its dimension is independent of $u\in\mathcal{U}$. By
(6) the chain control sets ${}_{\mathbb{P}}E_{j}$ uniquely correspond to the
maximal chain transitive subsets ${}_{\mathbb{P}}\mathcal{E}_{j}$ of the
control flow on $\mathcal{U}\times\mathbb{P}^{n-1}$ via

${}_{\mathbb{P}}\mathcal{E}_{j}:=\\{(u,\pi_{\mathbb{P}}x)\in\mathcal{U}\times\mathbb{P}^{n-1}\left|\pi_{\mathbb{P}}\varphi_{\hom}(t,x,u)\in\,_{\mathbb{P}}E_{j}\text{
for all }t\in\mathbb{R}\right.\\}.$ (8)

###  2.3 Periodic solutions

We state some facts on periodic solutions of inhomogeneous periodic
differential equations of the form

$\dot{x}(t)=P(t)x(t)+z(t),$ (9)

where $P(\cdot)\in L^{\infty}(\mathbb{R},\mathbb{R}^{n\times n})$ and
$z(\cdot)\in L^{\infty}(\mathbb{R},\mathbb{R}^{n})$ are $\tau$-periodic, i.e.,
$P(t+\tau)=P(t)$ and $z(t+\tau)=z(t)$ for almost all $t\in\mathbb{R}$. The
principal fundamental solution $\Phi(t,s)\in\mathbb{R}^{n\times
n},t,s\in\mathbb{R}$, is given by

$\frac{d}{dt}\Phi(t,s)=P(t)\Phi(t,s)\text{ with }\Phi(s,s)=I.$

The homogeneous equation with $z(t)\equiv 0$ has nontrivial (non-unique)
$\tau$-periodic solutions if and only if $1$ is an eigenvalue of
$\Phi(\tau,0)$. Here and in the following, uniqueness of a periodic solution
means that it is unique up to time shifts.

The Floquet multipliers $\rho_{j}\in\mathbb{C},j=1,\ldots,n$, are defined as
the eigenvalues of $\Phi(\tau,0)$ and the Floquet exponents are
$\lambda_{j}:=\frac{1}{\tau}\log\left|\rho_{j}\right|$. ††margin:  10.5. They
coincide with the Lyapunov exponents; cf. Chicone [5, Proposition 2.61],
Colonius and Kliemann [8, Theorem 7.2.9]. In particular, $0$ is a Floquet
exponent if $1$ is a Floquet multiplier. We also refer to Chicone [5, Section
2.4] and Teschl [24, Section 3.6] for background on Floquet theory (note that
the Floquet exponents as defined above are the real parts of the Floquet
exponents defined in [5] and [24]).

We need the following results on periodic solutions.

######  Proposition 9.

Consider the $\tau$-periodic differential equation (9).

(i) There is a unique $\tau$-periodic solution if and only if
$1\not\in\mathrm{spec}(\Phi(\tau,0))$. Its initial value (at time $0$) is
$x^{0}=(I-\Phi(\tau,0))^{-1}\int_{0}^{\tau}\Phi(\tau,s)z(s)ds$.

(ii) If for $z(t)\not\equiv 0$ there does not exist a $\tau$-periodic
solution, then the principal fundamental solution satisfies
$1\in\mathrm{spec}(\Phi(\tau,0))$ and
$\int_{0}^{\tau}\Phi(\tau,s)z(s)ds\not\in\operatorname{Im}(I-\Phi(\tau,0))$.

(iii) For $k=0,1,\ldots$, let $P^{k}(\cdot)$ and $z^{k}(\cdot)$ be
$\tau_{k}$-periodic and suppose that for a $c>0$ the norms in
$L^{\infty}(\left[0,\tau_{0}+1\right];\mathbb{R}^{n})$ satisfy
$\left\|P^{k}(\cdot)\right\|_{\infty},\allowbreak\left\|z^{k}(\cdot)\right\|_{\infty}\leq
c$ for all $k$. Assume that
$\tau_{k}\rightarrow\tau_{0},~{}P^{k}(\cdot)\rightarrow P^{0}(\cdot)$ in
$L^{1}([0,\tau_{0}+1],\mathbb{R}^{n\times n})$, and $z^{k}(\cdot)\rightarrow
z^{0}(\cdot)$ in $L^{1}(\left[0,\tau_{0}+1\right];\mathbb{R}^{n})$ for
$k\rightarrow\infty$. Then for $k\rightarrow\infty$ the corresponding
principal fundamental matrices $\Phi^{k}(t,s)$ converge to $\Phi^{0}(t,s)$
uniformly in $t,s\in[0,\tau_{0}+1]$.

(iv) In the situation of (iii) assume additionally that for $k=0,1,\ldots$ the
corresponding principal fundamental solutions $\Phi^{k}(t,s)$ satisfy
$1\not\in\mathrm{spec}(\Phi^{k}(\tau_{k},0))$. Then the initial values $x^{k}$
of the corresponding unique $\tau_{k}$-periodic solutions converge for
$k\rightarrow\infty$ to the initial value $x^{0}$ of the unique
$\tau_{0}$-periodic solution for $P^{0}(\cdot)$ and $z^{0}(\cdot)$.

###### Proof.

see Appendix. ∎

##  3 System semigroups of affine systems in $\mathbb{R}^{n}$

In this section we analyze system semigroups for affine systems of the form
(1); cf. also Jurdjevic and Sallet [15].

We start with the following general remarks on the relevant Lie group which is
the semidirect product $G=\mathbb{R}^{n}\ltimes GL(\mathbb{R}^{n})$ with
product given by

$(v,g)\cdot(w,h)=(v+gw,gh).$

Its Lie algebra is given by the semidirect product
$\mathfrak{g}=\mathbb{R}^{n}\ltimes\mathfrak{gl}(\mathbb{R}^{n})$, where the
Lie bracket is

$[(a,A),(b,B)]=(Ab-Ba,AB-BA).$

If we consider $X=(a,A),\,Y=(b,B)$ as vector fields on $\mathbb{R}^{n}$
through the relation $X(x)=Ax+a,\,Y(x)=Bx+b$, the Lie bracket is

$[X,Y](x)=-(AB-BA)x-(Ab-Ba).$

If we denote by $e=(0,I)\in G$ the identity element, then the tangent space is
$T_{e}G=\mathfrak{g}$ and an element $X\in\mathfrak{g}$ can be identified with
a right-invariant vector field, a smooth vector field on $G$, given by
$X^{R}(g):=(dR_{g})_{e}X$, where $R_{g}$ stands for the right-translation and
$(dR_{g})_{e}$ is its differential at the identity element. By standard
results, the vector fields $X^{R}$ are complete and their flows satisfy

$\phi_{t}^{X^{R}}(g)=R_{g}(\phi_{t}^{X^{R}}(e))\text{ for all }g\in G.$ (10)

Note also that

$(\exp tX)(x)=e^{tA}x+\int_{0}^{t}e^{(t-s)A}a\,ds\text{ for
}x\in\mathbb{R}^{n}\text{ and }X=(a,A),$

gives us exactly the expression for the exponential map for
$G=\mathbb{R}^{n}\ltimes GL(\mathbb{R}^{n})$ and
$\mathfrak{g}=\mathbb{R}^{n}\ltimes\mathfrak{gl}(\mathbb{R}^{n})$, meaning
that the Lie group exponential is given by

$\exp tX=\left(\int_{0}^{t}e^{(t-s)A}a\,ds,e^{tA}\right)\text{ for }X=(a,A).$

If $\mathcal{F}\subset\mathfrak{g}$ is a nonempty subset, consider the
subgroups of $G$

$\displaystyle\mathcal{G}(\mathcal{F})$ $\displaystyle:=\left\\{\exp
t_{1}X_{1}\cdot\exp t_{2}X_{2}\cdots\exp
t_{k}X_{k}\left|t_{i}\in\mathbb{R}\text{ and
}X_{i}\in\mathcal{F}\right.\right\\},$
$\displaystyle\mathcal{G}^{R}(\mathcal{F})$
$\displaystyle:=\\{(\phi_{t_{1}}^{X_{1}^{R}}\circ\phi_{t_{2}}^{X_{2}^{R}}\circ\cdots\circ\phi_{t_{k}}^{X_{k}^{R}})(e)\left|t_{i}\in\mathbb{R}\text{
and }X_{i}\in\mathcal{F}\right.\\},$

and the semigroups $\mathcal{S}(\mathcal{F}),\,\mathcal{S}^{R}(\mathcal{F})$
where only $t_{i}>0$ are allowed. Thus $\mathcal{S}^{R}(\mathcal{F})$ is the
set of points on $G$ that can be attained from $e\in G$ by concatenations of
the flows of $\mathcal{F}^{R}=\\{X^{R}\left|X\in\mathcal{F}\right.\\}$.

######  Proposition 10.

If $\mathcal{F}\subset\mathfrak{g}$ is a nonempty subset, the groups
$\mathcal{G}(\mathcal{F})$ and $\mathcal{G}^{R}(\mathcal{F})$ as well as the
semigroups $\mathcal{S}(\mathcal{F})$ and $\mathcal{S}^{R}(\mathcal{F})$,
resp., coincide.

###### Proof.

Since $X^{R}(e)=X$, the exponential map $\exp:\mathfrak{g}\rightarrow G$ is
defined by $\exp X=\phi_{1}^{X^{R}}(g)$. Consequently we get for all
$t_{i}\in\mathbb{R}$ and $X_{i}\in\mathcal{F}\subset\mathfrak{g}$, using (10),

$\displaystyle\exp t_{1}X_{1}\cdot\exp t_{2}X_{2}\cdots\exp t_{k}X_{k}$
$\displaystyle=R_{e^{t_{1}X_{1}}e^{t_{2}X_{2}}\cdots
e^{t_{k-1}X_{k-1}}}(e^{t_{k}X_{k}})=R_{e^{t_{1}X_{1}}e^{t_{2}X_{2}}\cdots
e^{t_{k-1}X_{k-1}}}(\phi_{t_{k}}^{X_{k}^{R}}(e))$
$\displaystyle=\phi_{t_{k}}^{X_{k}^{R}}(e^{t_{1}X_{1}}e^{t_{2}X_{2}}\cdots
e^{t_{k-1}X_{k-1}})=\phi_{t_{k}}^{X_{k}^{R}}\left(R_{e^{t_{1}X_{1}}\cdots
e^{t_{k-2}X_{k-2}}}(e^{t_{k-1}X_{k-1}})\right)$
$\displaystyle=\phi_{t_{k}}^{X_{k}^{R}}\left(R_{e^{t_{1}X_{1}}e^{t_{2}X_{2}}\cdots
e^{t_{k-2}X_{k-2}}}(\phi_{t_{k-1}}^{X_{k-1}^{R}}(e))\right)=\phi_{t_{k}}^{X_{k}^{R}}\circ\phi_{t_{k-1}}^{X_{k-1}^{R}}(e^{t_{1}X_{1}}\cdots
e^{t_{k-2}X_{k-2}})$
$\displaystyle=\,\cdots\,=\left(\phi_{t_{k}}^{X_{k}^{R}}\circ\phi_{t_{k-1}}^{X_{k-1}^{R}}\circ\cdots\circ\phi_{t_{1}}^{X_{1}^{R}}\right)(e).$

This implies the assertion. ∎

The family of affine vector fields on $\mathbb{R}^{n}$ associated with (1) is
given by

$\mathcal{F}=\left\\{X^{u}(x)=A(u)x+Cu+d\left|u\in\Omega\right.\right\\}.$
(11)

Then $\mathcal{LA}(\mathcal{F})=\mathcal{LA}(f_{0},f_{1},\ldots,f_{m})$, cf.
(2). The system group $\mathcal{G}=\mathcal{G}(\mathcal{F})$ is a subgroup of
the semidirect product $\mathbb{R}^{n}\ltimes GL(\mathbb{R}^{n})$.

Since we assume the accessibility rank condition (2), Jurdjevic [14, Theorem 3
on p. 44] implies that the set $\mathcal{F}$ of vector fields is transitive on
$\mathbb{R}^{n}$, i.e., for all $x,y\in\mathbb{R}^{n}$ there is
$g\in\mathcal{G}$ with $y=gx$. Denote by
$\mathcal{S}_{\tau}=\mathcal{S}_{\tau}(\mathcal{F})$ the set of those elements
of $\mathcal{S}(\mathcal{F})$ with $t_{1}+\cdots+t_{k}=\tau$, and let
$\mathcal{S}_{\leq T}\mathcal{=}\bigcup_{\tau\in[0,T]}\mathcal{S}_{\tau}$,
analogously for $\mathcal{S}_{\tau}(\mathcal{F}^{R})$. The trajectories for
$u\in\mathcal{U}_{pc}$ of control system (1) are given by the action of the
semigroup $\mathcal{S}$ on $\mathbb{R}^{n}$: For $g\in\mathcal{S}_{\tau}$ and
$\tau=t_{k}+\cdots+t_{1},t_{i}>0$,

$gx=g(u)x=\exp(t_{k}X^{u^{k}})\cdots\exp(t_{1}X^{u^{1}})x=\varphi(\tau,x,u),$

where $u^{j}\in\Omega,X^{u^{j}}\in\mathcal{F}$, and
$\varphi(t,x,u),t\in[0,\tau]$, is the solution of (1) with piecewise constant
control $u$ defined, with $t_{0}=0$, by

$u(t):=u^{j+1}\text{ for
}t\in\left[\sum\nolimits_{i=0}^{j}t_{i},\sum\nolimits_{i=0}^{j+1}t_{i}\right),\quad
j=0,\ldots,k-1.$ (12)

Note that $u$ is not uniquely determined by $g$. We will always consider the
interior of $\mathcal{S}$ in the system group $\mathcal{G}$.

######  Theorem 11.

(i) The system semigroup $\mathcal{S}=\mathcal{S}(\mathcal{F})$ of (1)
satisfies
$\mathcal{S}_{\leq\tau}\subset\overline{\mathrm{int}(\mathcal{S}_{\leq\tau})}$
in $\mathcal{G}$ for every $\tau>0$.

(ii) If $g\in\mathrm{int}(\mathcal{S}_{\leq\tau})$ for a $\tau>0$ then
$gx\in\mathrm{int}(\mathcal{O}_{\leq\tau}^{+}(x))$ for every
$x\in\mathbb{R}^{n}$.

###### Proof.

(i) The right invariant vector fields in $\mathcal{F}^{R}$ on $\mathcal{G}$
are real analytic. Since this implies that they are Lie-determined we can
apply Jurdjevic [14, Corollary on p. 67] which shows that for every open set
$U$ in an orbit of $\mathcal{F}_{r}$, any $y\in U$, and any $\tau>0$, the
reachable set $\mathcal{S}_{\leq\tau}(\mathcal{F}^{R})(y)\cap U$ contains an
open set in the orbit topology. In particular, this applies to the reachable
set up to time $\tau$ of the identity which by Proposition 10 coincides with
$\mathcal{S}_{\leq\tau}$. Furthermore, [14, Corollary 1 on p. 68] implies
$\mathcal{S}_{\leq\tau}\subset\overline{\mathrm{int}(\mathcal{S}_{\leq\tau})}$.

(ii) The maps $\mathcal{G}\rightarrow\mathbb{R}^{n}:g\mapsto gx$ are open,
hence $g\in\mathrm{int}\left(\mathcal{S}_{\leq\tau}\right)$ implies
$gx\in\mathrm{int}(\mathcal{O}_{\leq\tau}^{+}(x))$. ∎

Next we relate the system semigroup and fixed points in control sets.

######  Proposition 12.

(i) Let $D\subset\mathbb{R}^{n}$ be a control set with nonvoid interior for
(1). Then for every $x\in\mathrm{int}(D)$ there are $\tau>0$ and
$g\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S}_{\leq\tau+1})$ such that
$gx=x$.

(ii) Conversely, let $g\in\mathrm{int}(\mathcal{S})$ with $gx=x$ for some
point $x\in\mathbb{R}^{n}$. Then $x\in\mathrm{int}(D)$ for some control set
$D\subset\mathbb{R}^{n}$.

###### Proof.

(i) Let $x\in\mathrm{int}(D)$. By continuity of the action the set
$H=\\{h\in\mathcal{G}\left|hx\in\mathrm{int}(D)\right.\\}$ is open in
$\mathcal{G}$, hence for all $t>0$ small enough there exists an element
$h\in\mathcal{S}_{t}\cap H$. By Proposition 10(i) it follows that for
$\sigma_{k}\rightarrow 0^{+}$ there are elements
$g_{k}\in\mathrm{int}(\mathcal{S}_{\leq\sigma_{k}})$ converging to the
identity in $\mathcal{G}$, and hence
$h_{k}:=g_{k}h\in\mathrm{int}\left(\mathcal{S}_{\leq
t+\sigma_{k}}\right)\rightarrow h$. Since $H$ is open there is
$k\in\mathbb{N}$ large enough such that $h_{k}\in
H\cap\mathcal{S}_{t+\sigma}\cap\mathrm{int}(\mathcal{S}_{\leq t+\sigma_{k}})$
for some $\sigma\in[0,\sigma_{k}]$, hence $h_{k}x\in\mathrm{int}(D)$. By
Remark 2, exact controllability in $\mathrm{int}(D)$ holds. Thus we find
$h_{0}\in\mathcal{S}_{s},s>0$, such that $h_{0}h_{k}x=x$. It follows that
$g:=h_{0}h_{k}\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S}_{\leq\tau+1})$
with $\tau:=t+\sigma+s$ and $gx=x$.

(ii) Every $g\in\mathrm{int}(\mathcal{S})$ satisfies
$gx\in\mathrm{int}(\mathcal{O}^{+}(x))$ and
$x\in\mathrm{int}(\mathcal{O}^{-}(gx))$ for all $x$. Now $gx=x$ implies that
$x\in\mathrm{int}(\mathcal{O}^{+}(x))\cap\mathrm{int}(\mathcal{O}^{-}(x))$,
and hence Lemma 3 shows that
$D=\mathcal{O}^{-}(x)\cap\overline{\mathcal{O}^{+}(x)}$ is a control set with
$x\in\mathrm{int}(D)$. ∎

Note that $g\in\mathrm{int}(\mathcal{S})$ implies that
$g\in\mathrm{int}(\mathcal{S}_{\leq\tau})$ for some $\tau>0$; cf. Colonius and
Kliemann [7, Lemma 4.5.2]. The control $u$ in (12) also determines the element
$\Phi_{u}(\tau,0)$ of the system semigroup of the homogeneous bilinear control
system (7). We denote the corresponding semigroup by
${}_{\mathbb{R}}\mathcal{S}^{\hom}\subset GL(n,\mathbb{R})$. Theorem 11 and
Proposition 12 are also valid for system (7) and
${}_{\mathbb{R}}\mathcal{S}^{\hom}$ provided that the corresponding
accessibility rank condition in $\mathbb{R}^{n}\setminus\\{0\\}$ holds.

Next we describe the relation between the action of the system semigroup and
periodic control functions. We may extend the control defined by (12) to a
$\tau$-periodic control function in $\mathcal{U}_{pc}$. The next lemma follows
immediately from Proposition 9(i).

######  Lemma 13.

Let $g(u)\in\mathcal{S}_{\tau}$. Then $g(u)x=x$ for some $x\in\mathbb{R}^{n}$
if and only if the corresponding $\tau$-periodic differential equation in (1)
has a $\tau$-periodic solution with initial value $x(0)=x$. This solution is
unique if and only if $1\not\in\mathrm{spec}(\Phi_{u}(\tau,0))$.

Note the following results on continuous dependence.

######  Lemma 14.

Let $u,v\in\mathcal{U}_{pc}$ be $\sigma$-periodic and $\tau$-periodic, resp.,
for some $\sigma,\tau>0$. Define for $\alpha\in[\sigma,\sigma+\tau]$

$u^{\alpha}(t):=u(t)\text{ for
}t\in[0,\sigma],\,u^{\alpha}(t):=v(t-\sigma)\text{ for }t\in[\sigma,\alpha],$

and extend $u^{\alpha}$ to an $\alpha$-periodic control
$u^{\alpha}\in\mathcal{U}_{pc}$. Then the controls
$u_{\left|[0,\sigma+\tau]\right.}^{\alpha}$ depend continuously on $\alpha$ as
elements of $L^{1}([0,\sigma+\tau],\mathbb{R}^{m})$. Furthermore, the
principal fundamental solutions
$\Phi_{u^{\alpha}}(\alpha,0)\in\mathbb{R}^{n\times n}$ and the elements
$g(u^{\alpha})\in\mathbb{R}^{n}\ltimes GL(\mathbb{R}^{n})$ depend, in a
continuous and piecewise analytic way, on $\alpha$.

###### Proof.

We may write the control $v$ as $v(t)=v^{j}$ for
$t\in[t_{0}+\cdots+t_{j-1},t_{0}+\cdots+t_{j}),$ where
$v^{j}\in\Omega,t_{0}:=0$ and $t_{j}>0$ for $j=1,\ldots,\ell$ with
$\tau=t_{1}+\cdots+t_{\ell}$. The assertions follow from the explicit
expressions for
$\alpha\in[\sigma+\sum_{i=0}^{j-1}t_{i},\sigma+\sum_{i=0}^{j}t_{i})$ and
$j=1,\ldots,\ell$,

$\displaystyle\Phi_{u^{\alpha}}(\alpha,0)$
$\displaystyle=e^{(\alpha-\sigma-\sum_{i=0}^{j-1}t_{i})A(v^{j})}e^{t_{i-1}A(v^{j-1})}\cdots
e^{t_{1}A(v^{1})}\Phi_{u}(\sigma,0),$ $\displaystyle g(u^{\alpha})$
$\displaystyle=\exp\left((\alpha-\sigma-\sum\nolimits_{i=0}^{j-1}t_{i})X^{v^{j}}\right)\exp\left(t_{j-1}X^{v^{j-1}}\right)\cdots\exp\left(t_{1}X^{v^{1}}\right)g(u),$

where $X^{v^{j}}$ is the affine vector field
$X^{v^{j}}(x)=A(v^{j})x+Cv^{j}+d$. ∎

This result can be used in order to analyze the system semigroup $\mathcal{S}$
of the affine system and the system semigroup
${}_{\mathbb{R}}\mathcal{S}^{\hom}$ of the homogeneous part.

######  Lemma 15.

Let $g(u)\in\mathcal{S}_{\sigma}\cap\mathrm{int}(\mathcal{S})$ and
$g(v)\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S})$ for some
$\sigma,\tau>0$. Then there exist for $\alpha\in[0,1]$
$\tau_{\alpha}$-periodic controls $u^{\alpha}$ with $u^{0}=u,\tau_{0}=\sigma$
and $u^{1}=v,\tau_{1}=\tau$ such that the maps
$p:[0,1]\rightarrow\mathrm{int}(\mathcal{S})$ and
$p^{\hom}:[0,1]\rightarrow\,_{\mathbb{R}}\mathcal{S}^{\hom}$,

$p(\alpha):=g(u^{\alpha})\in\mathrm{int}(\mathcal{S})\text{ and
}p^{\hom}(\alpha):=\Phi_{u^{\alpha}}(\tau_{\alpha},0)\in\,_{\mathbb{R}}\mathcal{S}^{\hom},\,\alpha\in[0,1],$

are continuous paths. Furthermore the continuity and smoothness properties
from Lemma 14 hold for
$u^{a},\Phi_{u^{\alpha}}(\tau_{\alpha},0)\in\,_{\mathbb{R}}\mathcal{S}^{\hom}$
and $g(u^{\alpha})\in\mathcal{S}$ and also for $\tau_{\alpha}$.

###### Proof.

By Lemma 14 one finds a path in $\mathcal{S}$ from $g(u)$ to $g(v)g(u)$ and an
analogous construction yields a path in $\mathcal{S}$ from $g(v)$ to
$g(v)g(u)$. Since for any $g^{\prime}\in\mathcal{S}$ and
$g^{\prime\prime}\in\mathrm{int}(\mathcal{S})$ it follows that
$g^{\prime}g^{\prime\prime},g^{\prime\prime}g^{\prime}\in\mathrm{int}(\mathcal{S})$,
the elements on the paths are in $\mathrm{int}(\mathcal{S})$. Combining these
paths one obtains a path from $g(u)$ to $g(v)$. This can be reparametrized to
obtain a path with $\alpha\in[0,1]$ and periods $\tau_{\alpha}$ in
$[0,\sigma+\tau]$. Analogously one obtains the path $p^{\hom}$. The smoothness
properties remain valid. ∎

Next we use these lemmas to prove spectral properties of elements in the
interior of the system semigroup and corresponding periodic solutions.

######  Proposition 16.

Let $g(u)\in\mathcal{S}_{\sigma}\cap\mathrm{int}(\mathcal{S})$ for some
$\sigma>0$ with $1\not\in\mathrm{spec}(\Phi_{u}(\sigma,0))$ and
$g(v)\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S})$ for some $\tau>0$.
Consider the paths $p$ in $\mathrm{int}(\mathcal{S})$ and $p^{\hom}$ in
${}_{\mathbb{R}}\mathcal{S}^{\hom}$ constructed in Lemma 15.

(i) For every $\varepsilon>0$ there are $\tau_{\alpha}$-periodic controls
$w^{\alpha}\in\mathcal{U}_{pc}$ and continuous paths
$p_{1}:[0,1]\rightarrow\mathrm{int}(\mathcal{S})$ with
$p_{1}(\alpha)=g(w^{\alpha})$ for $\alpha\in[0,1]$ and
$p_{1}^{\hom}:[0,1]\rightarrow\,_{\mathbb{R}}\mathcal{S}^{\hom}$ with
$p_{1}^{\hom}(\alpha)=\Phi_{w^{\alpha}}(\tau_{\alpha},0)$ with

$\displaystyle\left\|p_{1}(1)-g(v)\right\|$
$\displaystyle=\left\|g(w^{1})-g(v)\right\|<\varepsilon,$
$\displaystyle\left\|p_{1}^{\hom}(1)-\Phi_{v}(\tau,0)\right\|$
$\displaystyle=\left\|\Phi_{w^{1}}(\tau_{1},0)-\Phi_{v}(\tau,0)\right\|<\varepsilon,\,$

such that $1\not\in\mathrm{spec}(\Phi_{w^{\alpha}}(\tau_{\alpha},0)$ for all
but at most finitely many $\alpha\in[0,1]$, and the continuity and smoothness
properties from Lemma 14 hold for
$u^{\alpha},\tau_{\alpha},\Phi_{w^{\alpha}}(\alpha,0)\in\,_{\mathbb{R}}\mathcal{S}^{\hom}$,
and $g(w^{\alpha})\in\mathcal{S}$.

(ii) If $1\not\in\mathrm{spec}(\Phi_{u^{\alpha_{0}}}(\tau_{\alpha_{0}},0))$
(or $1\not\in\mathrm{spec}(\Phi_{w^{\alpha_{0}}}(\tau_{\alpha_{0}},0))$) for
some $\alpha_{0}\in[0,1]$, then for all $\alpha$ in a neighborhood of
$\alpha_{0}$ there are unique $\tau^{\alpha}$-periodic solutions with initial
values $x^{\alpha}$ depending continuously on $\alpha$.

###### Proof.

(i) First we prove these properties for the $\alpha$-periodic controls
$u^{\alpha}$ constructed in Lemma 14. The proof will proceed inductively for
$j=0,\ldots,\ell-1$ and $t\in[t_{j},t_{j+1}]$. Since
$1\not\in\mathrm{spec}(\Phi_{u}(\sigma,0)$ it follows that the analytic
function $\alpha\mapsto\det(I-\Phi_{u^{\alpha}}(\tau^{\alpha},0))$ is not
identically $0$. Hence there are at most finitely many
$\alpha_{i}\in[0,t_{1}]$ with
$1\in\mathrm{spec}(\Phi_{u^{\alpha_{i}}}(\tau^{\alpha_{i}},0)$. If
$\det(I-\Phi_{u^{t_{1}}}(\tau_{t_{1}},0))\not=0$, choose $s_{1}=t_{1}$,
otherwise choose $s_{1}<t_{1}$ arbitrarily close to $t_{1}$ with
$\det(I-\Phi_{u^{s_{1}}}(\tau_{s_{1}},0))\not=0$, and define for
$\alpha\in[\sigma,\sigma+s_{1}]$

$w^{\alpha}(t)=u^{\alpha}(t),\,t\in[0,\alpha].$

Then the $\alpha$-periodic extension of $w^{\alpha}$ satisfies
$p_{1}(\alpha):=g(w^{\alpha})\in\mathrm{int}(\mathcal{S})$ and
$p_{1}^{\hom}(\alpha):=\Phi_{w^{\alpha}}(\alpha,0)\in\,_{\mathbb{R}}\mathcal{S}^{\hom}$
with $1\not\in\mathrm{spec}(\Phi_{w^{\alpha}}(\alpha,0))$ for all but at most
finitely many $\alpha\in[\sigma,\sigma+s_{1}]$. Consider

$I-e^{(\alpha-\sigma-s_{1})A(v^{2})}e^{s_{1}A(v^{1})}\Phi_{u}(\sigma,0)\text{
for }\alpha\in[\sigma+s_{1},\sigma+t_{2}].$

For $\alpha=\sigma+s_{1}$ the determinant of this matrix is unequal zero,
hence it has at most finitely many zeros in $[s_{1},t_{2}]$.

Proceeding in this way up to $j=\ell-1$ one constructs $\alpha$-periodic
controls $w^{\alpha}$ such that
$p_{1}(\alpha):=g(w^{\alpha})\in\mathrm{int}(\mathcal{S})$ and
$p_{1}^{\hom}(\alpha):=\Phi_{w^{\alpha}}(\alpha,0)\in\,_{\mathbb{R}}\mathcal{S}^{\hom}$
with $1\not\in\mathrm{spec}(\Phi_{w^{\alpha}}(\alpha,0))$ for all but at most
finitely many $\alpha\in[\sigma,\sigma+\tau]$. Since $t_{i}-s_{i}>0$ is
arbitrarily small it also follows that

$\left\|p_{1}^{\hom}(\sigma+\tau)-p^{\hom}(\sigma+\tau)\right\|<\varepsilon,\,\left\|p_{1}(\sigma+\tau)-p(\sigma+\tau)\right\|<\varepsilon.$

The same constructions as in the proof of Lemma 15 can also be applied here
and yield assertion (i).

(ii) Continuous dependence on $\alpha$ of $\Phi_{w^{\alpha}}(\tau_{\alpha},0)$
implies that $1\not\in\mathrm{spec}(\Phi_{w^{\alpha}}(\tau_{\alpha},0)$ for
all $\alpha$ in a neighborhood of $\alpha_{0}$, hence by Proposition 9(i)
there are unique $\tau_{\alpha}$-periodic solutions. Since the controls
$w^{\alpha}$ depend continuously on $\alpha$ as elements of
$L^{1}([0,1],\mathbb{R}^{m}\dot{)}$ also

$A(w^{\alpha}(\cdot))=A+\sum_{i=1}^{m}w_{i}^{\alpha}(\cdot)B_{i}\in
L^{1}(\left[0,\sigma+\tau\right];\mathbb{R}^{n\times
n}),\,Cw^{\alpha}(\cdot)+d\in
L^{1}(\left[0,\sigma+\tau\right];\mathbb{R}^{n})$

depend continuously on $\alpha$. Thus Proposition 9(iv) shows that their
initial values $x^{\alpha}$ depend continuously on $\alpha$. ∎

The next two lemmas discuss the periodic solutions when $1$ is in the
spectrum.

######  Lemma 17.

Consider, for $k=0,1,\ldots$, the differential equations

$\dot{x}(t)=A(u^{k}(t))x(t)+Cu^{k}(t)+d,$

where $u^{k}$ is $\tau_{k}$-periodic with $\tau_{k}\rightarrow\tau_{0}>0$ and
$u^{k}\rightarrow u^{0}$ in $L^{1}(\left[0,\tau_{0}+1\right];\mathbb{R}^{m})$
with $\left\|u_{k}\right\|_{\infty}\leq c,k\in\mathbb{N}$, for some $c>0$, and

(i) the principal fundamental solutions $\Phi_{u^{k}}(t,s)$ of
$\dot{x}(t)=A(u^{k}(t))x(t)$ satisfy
$1\in\mathrm{spec}(\Phi_{u^{0}}(\tau_{0},0))$ and
$1\not\in\mathrm{spec}(\Phi_{u^{k}}(\tau_{k},0))$ for $k=1,2,\ldots$,

(ii)
$\int_{0}^{\tau_{0}}\Phi_{u^{0}}(\tau_{0},s)\left(Cu^{0}(s)+d\right)ds\not\in\operatorname{Im}(I-\Phi_{u^{0}}(\tau_{0},0))$.

Then there are unique $\tau_{k}$-periodic solutions for $u^{k}$ and their
initial values $x^{k}$ satisfy

$\left\|x^{k}\right\|\rightarrow\infty\text{ and
}\frac{x^{k}}{\left\|x^{k}\right\|}\rightarrow\ker(I-\Phi_{u^{0}}(\tau_{0},0))=\mathbf{E}(\Phi_{u^{0}}(\tau_{0},0);1)\text{
for }k\rightarrow\infty.$ (13)

###### Proof.

By Proposition 9(i) it follows that for every $k=1,2,\ldots$ there is a unique
$\tau_{k}$-periodic solution with initial value $x^{k}$ satisfying

$(I-\Phi_{u^{k}}(\tau_{k},0))x^{k}=\int_{0}^{\tau_{k}}\Phi_{u^{k}}(\tau_{k},s)\left(Cu^{k}(s)+d\right)ds.$
(14)

If $x^{k}$ remains bounded we may suppose that there is
$x^{0}\in\mathbb{R}^{n}$ with $x^{k}\rightarrow x^{0}$. Since
$u^{k}\rightarrow u^{0}$ in $L^{1}(\left[0,\tau_{0}+1\right];\mathbb{R}^{m})$
it follows that also

$A(u^{k}(\cdot))\rightarrow A(u^{0}(\cdot))\text{ and
}Cu^{k}(\cdot)+d\rightarrow Cu^{0}(\cdot)+d$

in $L^{1}(\left[0,\tau_{0}+1\right];\mathbb{R}^{n\times n})$ and in
$L^{1}(\left[0,\tau_{0}+1\right];\mathbb{R}^{n})$, resp. Thus, by Proposition
9(iii), the right hand sides of (14) converge to
$\int_{0}^{\tau_{0}}\Phi_{u^{0}}(\tau_{0},s)\left(Cu^{0}(s)+d\right)ds$, and
one obtains a contradiction to assumption (ii). This shows that
$\left\|x^{k}\right\|\rightarrow\infty$. Similarly also the second assertion
in (13) follows when we divide (14) by $\left\|x^{k}\right\|$. ∎

The next lemma describes the case where assumption (ii) above is violated.

######  Lemma 18.

Consider for a $\tau_{0}$-periodic control $u^{0}$

$\dot{x}(t)=A(u^{0}(t))x(t)+Cu^{0}(t)+d,$ (15)

and suppose that the principal fundamental solution of
$\dot{x}(t)=A(u^{0}(t))x(t)$ satisfies
$1\in\mathrm{spec}(\Phi_{u^{0}}(\tau_{0},0))$ and
$\int_{0}^{\tau_{0}}\Phi_{u^{0}}(\tau_{0},s)\left(Cu^{0}(s)+d\right)ds=(I-\Phi_{u^{0}}(\tau_{0},0))y^{0}$
for some $y^{0}\in\mathbb{R}^{n}$. Then the nontrivial affine subspace
$Y:=y^{0}+\mathbf{E}(\Phi_{u^{0}}(\tau_{0},0);1))$ has the property that there
is a $\tau_{0}$-periodic solution of (15) starting in $y$ if and only if $y\in
Y$, and there are $x^{k}\in Y,k\in\mathbb{N}$, satisfying the conditions in
(13).

###### Proof.

The first assertion is clear by the definitions. The second assertion follows
by choosing $x^{k}:=y^{0}+kz,k\in\mathbb{N}$, where
$0\not=z\in\mathbf{E}(\Phi_{u^{0}}(\tau_{0},0);1)$. ∎

##  4 Control sets for hyperbolic systems

In this section we present definitions of hyperbolicity for affine control
systems and show that hyperbolic systems have a unique control set with
nonvoid interior and that it is bounded.

Since for any $\tau$-periodic control, the homogeneous part (3) of affine
system (1) is a homogeneous periodic differential equation, we can define
corresponding Floquet multipliers which are the eigenvalues of the principal
fundamental solution $\Phi_{u}(\tau,0)$; cf. Subsection 2.3.

######  Definition 19.

An affine system of the form (1) is hyperbolic if

$1\not\in\mathrm{spec}(\Phi_{u}(\tau,0))\text{ for all }\tau\text{-periodic
}u\in\mathcal{U}_{pc}\text{ with }\tau>0\text{,
}g(u)\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S}).$ (16)

Otherwise it is called nonhyperbolic.

######  Remark 20.

If there is a $\tau$-periodic control $u$ with
$g(u)\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S})$ and
$\rho\in\mathrm{spec}(\Phi_{u}(\tau,0))$ with $\rho^{k}=1$ for some
$k\in\mathbb{N}$, then the system is nonhyperbolic. In fact, we may consider
$u$ as a $k\tau$-periodic control and find that
$1\in\mathrm{spec}(\Phi_{u}(k\tau,0))$ with
$\hat{g}(u)\in\mathcal{S}_{k\tau}\cap\mathrm{int}(\mathcal{S})$.

######  Remark 21.

If our sufficient condition for the existence of a control set
${}_{\mathbb{R}}D^{\hom}$ of the homogeneous part of (1) holds (cf. Theorem
8(iii)) there is a Floquet exponent $0=\frac{1}{\tau}\log\left|\rho\right|$
for a Floquet multiplier $\rho\in\mathrm{spec}(\Phi_{u}(\tau,0))$, hence
$\left|\rho\right|=1$. According to the preceding remark, the system can only
be hyperbolic, if $\rho$ is not a root of unity.

Next we show that, for hyperbolic affine systems, there is a unique control
set with nonvoid interior.

######  Theorem 22.

Suppose that the affine system (1) is hyperbolic. Then there is a unique
control set $D$ with nonvoid interior, and for every
$g\in\mathrm{int}(\mathcal{S})$ there is a unique $x\in\mathbb{R}^{n}$ with
$x=gx$ and

$\mathrm{int}(D)=\left\\{x\in\mathbb{R}^{n}\left|\text{there is
}g\in\mathrm{int}(\mathcal{S})\text{ with }x=gx\right.\right\\}.$

###### Proof.

Let $g=g(u)\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S})$. By
hyperbolicity, $1$ is not an eigenvalue of the principal fundamental solution
$\Phi_{u}(\tau,0)$. Proposition 9(i) implies that there is a unique
$\tau$-periodic solution starting in some $x\in\mathbb{R}^{n}$, hence $gx=x$
by Lemma 13. By Proposition 12(ii) it follows that $x\in\mathrm{int}(D)$ for
some control set $D$. In order to show that $D$ does not depend on $g$,
consider $g,h\in\mathrm{int}(\mathcal{S})$. By Lemma 15, one finds a
continuous path $p$ in $\mathrm{int}(\mathcal{S})$ from $g$ to $h$. For all
$\alpha\in[0,1]$ hyperbolicity implies that
$1\not\in\mathrm{spec}(\Phi_{u^{\alpha}}(\tau_{\alpha},0))$ and hence there
are unique fixed point $x^{\alpha}$ of $g(u^{\alpha})$ and a control set
$D^{\alpha}$ with $x^{\alpha}\in\mathrm{int}(D^{\alpha})$. As in Proposition
16(ii) it follows that also $x^{\alpha}$ depends continuously on $\alpha$.
Hence for small $\alpha>0$ all points $x^{\alpha}$ are contained in a single
control set $D$ showing

$\alpha^{\ast}:=\sup\left\\{\alpha\left|x^{\alpha^{\prime}}\in D\text{ for all
}\alpha^{\prime}\in[0,\alpha]\right.\right\\}>0.$

Since $x^{\alpha^{\ast}}\in\mathrm{int}(D^{\alpha^{\ast}})$ it follows from
Remark 2 that $D^{\alpha^{\ast}}=D$ which shows that
$x^{\alpha^{\ast}}\in\mathrm{int}(D)$. If $\alpha^{\ast}<1$ this implies that
$D^{\alpha}=D$ for all $\alpha\in[\alpha^{\ast},\alpha^{\ast}+\varepsilon]$
for some $\varepsilon>0$ contradicting the definition of $\alpha^{\ast}$. It
follows that $\alpha^{\ast}=1$ and hence there is a single control set $D$
containing all $x=gx$ for $g\in\mathrm{int}(\mathcal{S})$. The corresponding
periodic controls generate periodic solution which also are contained in $D$.

It remains to show that $D$ is the unique control set with nonvoid interior.
By Proposition 12(i), for a point $x$ in the interior of any control set,
there are $\tau>0$ and
$g\in\mathcal{S}_{\tau}\cap\mathrm{int}\left(\mathcal{S}_{\leq\tau+1}\right)$
with $gx=x$. Hence it follows that $x\in D$. ∎

The question arises if the control set $D$ is bounded. We will give a positive
answer provided that the following uniform hyperbolicity condition holds
assuming that the control range $\Omega$ is a compact and convex neighborhood
of the origin in $\mathbb{R}^{m}$ and hence, for system (7), the control flow
$\mathbf{\Psi}$ on $\mathcal{U}\times\mathbb{R}^{n}$ is well defined (cf.
Subsection 2.1).

######  Definition 23.

The homogeneous bilinear system (7) is uniformly hyperbolic if the vector
bundle $\mathcal{U}\times\mathbb{R}^{n}$ can be decomposed into the Whitney
sum of two invariant subbundles $\mathcal{V}^{1}$ and $\mathcal{V}^{2}$ such
that the restrictions $\mathbf{\Psi}^{1}$ and $\mathbf{\Psi}^{2}$ of the
control flow $\mathbf{\Psi}$ to $\mathcal{V}^{1}$ and $\mathcal{V}^{2}$,
resp., satisfy for constants $\alpha>0$ and $K\geq 1$ and for all
$(u,x_{i})\in\mathcal{V}^{i}$

$\displaystyle\left\|\varphi(t,x_{1},u)\right\|$
$\displaystyle=\left\|\mathbf{\Psi}_{t}^{1}(u,x_{1})\right\|\leq Ke^{-\alpha
t}\left\|x_{1}\right\|\text{ for }t\geq 0\text{,}$
$\displaystyle\left\|\varphi(t,x_{2},u)\right\|$
$\displaystyle=\left\|\mathbf{\Psi}_{t}^{2}(u,x_{2})\right\|\leq Ke^{\alpha
t}\left\|x_{2}\right\|\text{ for }t\leq 0.$

######  Remark 24.

The uniform hyperbolicity condition is also used in Kawan [17], Da Silva and
Kawan [10]. It is equivalent to the condition that $0$ is not in the Sacker-
Sell spectrum of the linear flow $\mathbf{\Psi}$; cf. Colonius and Kliemann
[7, Section 5.5].

Then, for $i=1,2\,$, one obtains that
$\mathcal{V}^{i}(u):=\\{x\in\mathbb{R}^{n}\left|(u,x)\in\mathcal{V}^{i}\right.\\}$
is a subspace of $\mathbb{R}^{n}$ and its dimension is independent of
$u\in\mathcal{U}$. For all $u\in\mathcal{U}$

$\mathbb{R}^{n}=\mathcal{V}^{1}(u)\oplus\mathcal{V}^{2}(u)\text{ and
}\varphi(t,x_{i},u)\in\mathcal{V}^{i}(u(t+\cdot))\text{ for all
}t\in\mathbb{R},$

hence, for $x=x_{1}\oplus x_{2}$ with $x_{i}\in\mathcal{V}^{i}(u)$ and
$\Phi_{u}^{i}(t,s):=\Phi_{u}(t,s)_{\left|\mathcal{V}_{i}(u(s+\cdot))\right.}$
for $t,s\in\mathbb{R}$,

$\varphi(t,x,u)=\varphi(t,x_{1},u)\oplus\varphi(t,x_{2},u)\text{ and
}\Phi_{u}(t,s)=\Phi_{u}^{1}(t,s)+\Phi_{u}^{2}(t,s).$

The uniform hyperbolicity condition above implies that system (1) is
hyperbolic in the sense of Definition 19 since

$\mathrm{spec}(\Phi_{u}(\tau,0))=\mathrm{spec}(\Phi_{u}^{1}(\tau,0))\cup\mathrm{spec}(\Phi_{u}^{2}(\tau,0)),$

and $\rho\in\mathrm{spec}(\Phi_{u}^{1}(\tau,0))$ implies
$\left|\rho\right|\leq e^{-\alpha\tau}$,
$\rho\in\mathrm{spec}(\Phi_{u}^{2}(\tau,0))$ implies $\left|\rho\right|\geq
e^{\alpha\tau}$.

######  Lemma 25.

Suppose that the uniform hyperbolicity assumption holds. Then there is $c>0$
such that for $(u,x_{1})\in\mathcal{V}^{1}$ and $(u,x_{2})\in\mathcal{V}^{2}$

$\left\|\varphi(t,x_{1},u)\right\|\leq
K\left\|x_{1}\right\|+\frac{Kc}{\alpha}\text{ for }t\geq
0,\quad\left\|\varphi(t,x_{2},u)\right\|\leq
K\left\|x_{2}\right\|+\frac{Kc}{\alpha}\text{ for }t\leq 0.$

###### Proof.

Denote the projections of $\mathbb{R}^{n}$ to $\mathcal{V}^{1}(u)$ along
$\mathcal{V}^{2}(u)$ by $P_{u}$ and choose $c>0$ such that
$\left\|P_{u}\right\|\left\|Cv+d\right\|\leq c$ for all
$u\in\mathcal{U},v\in\Omega$. By invariance of $\mathcal{V}^{1}$,
$P_{u(t+\cdot)}\Phi_{u}(t,s)=\Phi_{u}(t,s)P_{u(s+\cdot)}$, and hence

$\displaystyle\varphi(t,x_{1},u)$
$\displaystyle=P_{u(t+\cdot)}\varphi(t,x_{1},u)=P_{u(t+\cdot)}\Phi_{u}(t,0)x_{1}+\int_{0}^{t}P_{u(t+\cdot)}\Phi_{u}(t,s)[Cu(s)+d]ds$
$\displaystyle=\Phi_{u}^{1}(t,0)x_{1}+\int_{0}^{\tau}\Phi_{u}^{1}(t,s)P_{u(s+\cdot)}[Cu(s)+d]ds.$

Then it follows for all $u\in\mathcal{U}$ and $t\geq 0$ that

$\displaystyle\left\|\varphi(t,x_{1},u)\right\|$
$\displaystyle\leq\left\|\Phi_{u}^{1}(t,0)x_{1}\right\|+\int_{0}^{t}\left\|\Phi_{u}^{1}(t,s)P_{u(s+\cdot)}[Cu(s)+d]\right\|ds$
$\displaystyle\leq Ke^{-\alpha
t}\left\|x_{1}\right\|+Kc\int_{0}^{t}e^{-\alpha(t-s)}ds\leq
K\left\|x_{1}\right\|+\frac{Kc}{\alpha}.$

The second assertion is shown analogously. ∎

######  Theorem 26.

Let $\Omega$ be a compact and convex neighborhood of the origin in
$\mathbb{R}^{m}$. Suppose that the homogeneous part (3) of affine system (1)
satisfies the uniform hyperbolicity condition in Definition 23. Then the
unique control set $D$ with nonvoid interior is bounded.

###### Proof.

We show that $\mathrm{int}(D)$ is bounded. This will yield the assertion,
since by Remark 2 the accessibility rank condition implies that
$D\subset\overline{\mathrm{int}(D)}$. Fix $x\in\mathrm{int}(D)$ and consider
an arbitrary point
$y\in\mathrm{int}(D)=\mathcal{O}^{+}(x)\cap\mathcal{O}^{-}(x)$. Thus there are
controls $u^{1},u^{2}\in\mathcal{U}$ and times $t_{1},t_{2}>0$ with
$y=\varphi(t_{1},x,u^{1})=\varphi(-t_{2},x,u^{2})$. Define

$u(t)=\left\\{\begin{array}[c]{lll}u^{1}(t)&\text{for}&t\in[0,t_{1}]\\\
u^{2}(t)&\text{for}&t\in(-t_{2},0)\end{array}\right.$

and extend $u$ to a $t_{1}+t_{2}$-periodic function on $\mathbb{R}$. Then
$u(t_{1}+\cdot)=u(-t_{2}+\cdot)$ in $\mathcal{U}$ implying

$\mathcal{V}^{i}(u(t_{1}+\cdot))=\mathcal{V}^{i}(u(-t_{2}+\cdot))\text{ for
}i=1,2.$ (17)

We decompose

$x=x_{1}\oplus x_{2}\text{ with }x_{i}\in\mathcal{V}^{i}(u)\text{ and
}y=y_{1}\oplus y_{2}\text{ with }y_{i}\in\mathcal{V}^{i}(u(t_{1}+\cdot))\text{
for }i=1,2.$

The invariance of the complementary subbundles $\mathcal{V}^{i}$ together with
(17) shows that

$y_{1}=\varphi(t_{1},x_{1},u)=\varphi(-t_{2},x_{1},u)\text{ and
}y_{2}=\varphi(t_{1},x_{2},u)=\varphi(-t_{2},x_{2},u),$

and $y=\varphi(t_{1},x_{1},u)\oplus\varphi(-t_{2},x_{2},u)$. By Lemma 25
$\left\|\varphi(t_{1},x_{1},u)\right\|$ and
$\left\|\varphi(-t_{2},x_{2},u)\right\|$ and hence all $y\in\mathrm{int}(D)$
satisfy bounds which are independent of $t_{1}$ and $t_{2}$. ∎

Next we present a simple example of a uniformly hyperbolic affine system.

######  Example 27.

Consider the following system with control range $\Omega=[-1,1]$

$\left(\begin{array}[c]{c}\dot{x}\\\
\dot{y}\end{array}\right)=\left(\begin{array}[c]{cc}2&0\\\
0&-2\end{array}\right)\left(\begin{array}[c]{c}x\\\
y\end{array}\right)+u(t)\left(\begin{array}[c]{cc}1&0\\\
0&1\end{array}\right)\left(\begin{array}[c]{c}x\\\
y\end{array}\right)+\left(\begin{array}[c]{c}3\\\
3\end{array}\right)u(t)+\left(\begin{array}[c]{c}3\\\ 0\end{array}\right).$

For the homogeneous part $A(u)=A+uB$ the exponential growth rates of the $x$\-
and the $y$-component are for $u\in[-1,1]$ given by $\lambda_{1}(u)=2+u\geq 1$
and $\lambda_{2}(u)=-2+u\leq-1$, resp. Thus the system is uniformly hyperbolic
with $\mathcal{V}^{1}=\mathcal{U}\times(\\{0\\}\times\mathbb{R})$ and
$\mathcal{V}^{2}=\mathcal{U}\times(\mathbb{R}\times\\{0\\})$. We claim that
the unique control set with nonvoid interior is
$D=\left(-2,0\right)\times[-1,3]$. For the proof, first observe that the
equilibria are given by

$0=(2+u)x_{u}+3u+3,~{}0=(-2+u)y_{u}+3u\text{, hence
}x_{u}=-\frac{3u+3}{2+u}\text{ and }y_{u}=\frac{3u}{2-u}.$

The maps $u\mapsto x_{u}$ and $u\mapsto y_{u}$ are monotonically decreasing
and increasing, resp., since $\frac{d}{du}x_{u}<0$ and $\frac{d}{du}y_{u}>0$.
This implies that the set of equilibria is contained in

$[x_{1},x_{-1}]\times[y_{-1},y_{1}]=\left[-2,0\right]\times[-1,3].$

Inspection of the phase portraits for constant $u$ shows that any control set
is contained in this set, and the phase portraits for $u=-1$ and $u=1$ show
that one can approximately reach (with a combination of these controls) from
any point $(x,y)^{\top}$ in $\left(-2,0\right)\times[-1,3]$ any other point in
this set while this is not possible from points $(-2,y)^{\top}$ and
$(0,y)^{\top},y\in[-1,3]$. This proves the claim.

##  5 Affine control systems and projective spaces

In this section we construct for affine control systems and their homogeneous
parts induced systems on projective spaces. In order to distinguish explicitly
between control sets and chain control sets referring to the affine system and
its homogenous part, we will mark the latter by the suffix “$\hom$” in the
rest of this paper.

System (1) and (3) can be embedded into a homogeneous bilinear control system
in $\mathbb{R}^{n+1}$ of the form (cf. Elliott [12, Subsection 3.8.1])

$\left(\begin{array}[c]{c}\dot{x}(t)\\\
\dot{z}(t)\end{array}\right)=\left(\begin{array}[c]{cc}A&d\\\
0&0\end{array}\right)\left(\begin{array}[c]{c}x(t)\\\
z(t)\end{array}\right)+\sum_{i=1}^{m}u_{i}(t)\left(\begin{array}[c]{cc}B_{i}&c_{i}\\\
0&0\end{array}\right)\left(\begin{array}[c]{c}x(t)\\\ z(t)\end{array}\right).$
(18)

Denote the solutions of (18) with initial condition
$(x(0),z(0))=(x^{0},z^{0})\in\mathbb{R}^{n}\times\mathbb{R}$ by
$\psi(t,\left(x^{0},z^{0}\right),u),t\in\mathbb{R}$. For initial values of the
form $(x^{0},1)\in\mathbb{R}^{n+1}$ one finds

$\psi(t,\left(x^{0},1\right),u)=\left(\varphi(t,x^{0},u),1\right)\text{ in
}\mathbb{R}^{n+1},$ (19)

and for initial values of the form $(x^{0},0)\in\mathbb{R}^{n+1}$ one finds

$\psi(t,\left(x^{0},0\right),u)=\left(\varphi_{\hom}(t,x^{0},u),0\right)\text{
in }\mathbb{R}^{n+1}.$ (20)

Thus the trajectories (19) and (20) are copies of the trajectories of (1) and
of its homogeneous part (3), resp., obtained by adding a trivial
$\left(n+1\right)$st. component. An immediate consequence is the following
proposition.

######  Proposition 28.

(i) A subset $D\subset\mathbb{R}^{n}$ is a control set of (1) if and only if
the set $D^{1}:=\\{(x,1)\left|x\in D\right.\\}$ is a control set of (18) in
$\mathbb{R}^{n+1}\setminus\\{0\\}$.

(ii) A subset ${}_{\mathbb{R}}D^{\hom}\subset\mathbb{R}^{n}\setminus\\{0\\}$
is a control set of (3) if and only if the set
$D^{0}:=\\{(x,0)\left|x\in\,_{\mathbb{R}}D^{\hom}\right.\\}$ is a control set
of (18) in $\mathbb{R}^{n+1}\setminus\\{0\\}$.

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

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

here the hat means that the $i$-th entry is missing. Denote by
$\pi_{\mathbb{P}}$ both projections
$\mathbb{R}^{n}\rightarrow\mathbb{P}^{n-1}$ and
$\mathbb{R}^{n+1}\rightarrow\mathbb{P}^{n}$. A metric on $\mathbb{P}^{n}$ is
given by defining for elements
$p_{1}=\pi_{\mathbb{P}}x,p_{2}=\pi_{\mathbb{P}}y$

$d(p_{1},p_{2})=\min\left\\{\left\|\frac{x}{\left\|x\right\|}-\frac{y}{\left\|y\right\|}\right\|,\left\|\frac{x}{\left\|x\right\|}+\frac{y}{\left\|y\right\|}\right\|\right\\}.$
(21)

Projecting the homogeneous bilinear control system (18) in $\mathbb{R}^{n+1}$
to $\mathbb{P}^{n}$ one obtains the following system given in homogeneous
coordinates by

$\left[\begin{array}[c]{c}\dot{x}(t)\\\
\dot{z}(t)\end{array}\right]=\left(\left(\begin{array}[c]{cc}A&d\\\
0&0\end{array}\right)+\sum_{i=1}^{m}u_{i}(t)\left(\begin{array}[c]{cc}B_{i}&c_{i}\\\
0&0\end{array}\right)\right)\left[\begin{array}[c]{c}x(t)\\\
z(t)\end{array}\right].$ (22)

Projective space $\mathbb{P}^{n}$ can be written as the disjoint union
$\mathbb{P}^{n}=\mathbb{P}^{n,1}\dot{\cup}\mathbb{P}^{n,0}$, where, in
homogeneous coordinates, the levels $\mathbb{P}^{n,i}$ are given by

$\mathbb{P}^{n,i}:=\left\\{[x_{1}:\cdots:x_{n}:i]\left|(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\right.\right\\}\text{
for }i=0,1.$

Observe that, by homogeneity,
$\mathbb{P}^{n,0}=\left\\{[x_{1}:\cdots:x_{n}:0]\left|\text{
}\left\|(x_{1},\ldots,x_{n})\right\|=1\right.\right\\}$. Any trajectory of
system (22) is obtained as the projection of a trajectory of (18) with initial
condition satisfying $z^{0}=0$ or $1$, since any initial value
$[x_{1}^{0}:\cdots:x_{n}^{0}:z^{0}]$ with $z^{0}\not=0$ coincides with
$[\frac{x_{1}^{0}}{z^{0}}:\cdots:\frac{x_{n}^{0}}{z^{0}}:1]$.

Loosely speaking, $\mathbb{P}^{n,0}$ is projective space $\mathbb{P}^{n-1}$
(embedded into $\mathbb{P}^{n}$) and $\mathbb{P}^{n,1}$ is $\mathbb{P}^{n}$
without $\mathbb{P}^{n-1}$. In fact, as noted above, an atlas of
$\mathbb{P}^{n}$ is given by $n+1$ charts $(U_{i},\psi_{i})$. A trivial atlas
for $\mathbb{P}^{n,1}$ is given by $\left\\{(U_{n+1},\psi_{n+1})\right\\}$
proving that $\mathbb{P}^{n,1}$ is a manifold which is diffeomorphic to
$\mathbb{R}^{n}$. The space $\mathbb{P}^{n,0}$ is closed in $\mathbb{P}^{n}$,
and the spaces $\mathbb{P}^{n-1}$ and $\mathbb{P}^{n,0}$ are diffeomorphic
under the map

$e:\mathbb{P}^{n-1}\rightarrow\mathbb{P}^{n,0}:[x_{1}:\cdots:x_{n}]\mapsto[x_{1}:\cdots:x_{n}:0].$
(23)

For any trajectory
$\psi(t,\left(x^{0},1\right),u)=\left(\varphi_{1}(t,x^{0},u),\ldots,\varphi_{n}(t,x^{0},u),1\right)^{\top}$
of system (18) in $\mathbb{R}^{n+1}\setminus\\{0\\}$, the projection to
$\mathbb{P}^{n,1}\subset\mathbb{P}^{n}$ is
$[\varphi_{1}(t,x^{0},u):\cdots:\varphi_{n}(t,x^{0},u):1]$.

The proof of the following proposition is straightforward and we omit it.

######  Proposition 29.

Consider in $\mathbb{R}^{n}$ the affine control system (1), its homogeneous
part (3), and in $\mathbb{R}^{n+1}$ the homogeneous bilinear control system
(18) as well as the system in $\mathbb{P}^{n-1}$ induced by (3) and the system
(22) in $\mathbb{P}^{n}$ induced by (18).

(i) Every control set $D\subset\mathbb{R}^{n}$ of the affine system (1) yields
a control set $\pi_{\mathbb{P}}D^{1}$ of the system (22) in $\mathbb{P}^{n}$
via the map

$(x_{1},\ldots,x_{n})\longmapsto[x_{1}:\cdots:x_{n}:1]:\mathbb{R}^{n}\rightarrow\mathbb{P}^{n,1}\subset\mathbb{P}^{n}.$

Furthermore, $D$ is an invariant control set if and only if
$\pi_{\mathbb{P}}D^{1}$ is an invariant control set. The control set $D$ is
unbounded if and only
$\partial\left(\pi_{\mathbb{P}}D^{1}\right)\cap\mathbb{P}^{n,0}\not=\varnothing$.
More precisely, if $x^{k}\in D$ with $\left\|x^{k}\right\|\rightarrow\infty$,
then every cluster point $y$ of $\frac{x^{k}}{\left\|x^{k}\right\|}$
satisfies, in homogeneous coordinates,

$[x_{1}^{k_{i}}:\cdots:x_{n}^{k_{i}}:1]\rightarrow[y_{1}:\cdots:y_{n}:0]\in\mathbb{P}^{n,0}\text{
for a subsequence }k_{i}\rightarrow\infty\text{.}$

(ii) Every control set ${}_{\mathbb{P}}D^{\hom}$ and every chain control set
${}_{\mathbb{P}}E^{\hom}$ of the system in $\mathbb{P}^{n-1}$ induced by (3)
corresponds to a unique control set $e(_{\mathbb{P}}D^{\hom})$ and chain
control set $e(_{\mathbb{P}}E^{\hom})$, resp., of the system (22) restricted
to $\mathbb{P}^{n,0}$ and conversely via the map $e$. The invariant control
sets in $\mathbb{P}^{n-1}$ correspond to the invariant control sets in
$\mathbb{P}^{n,0}$.

We remark that the assertions in Proposition 29(ii) also hold, if the
accessibility rank condition in $\mathbb{P}^{n-1}$ is violated (this is the
case in Example 37).

The intersection
$\partial\left(\pi_{\mathbb{P}}D^{1}\right)\cap\mathbb{P}^{n,0}$ will be of
relevance below. Hence we give it a suggestive name.

######  Definition 30.

For a control set $D\subset\mathbb{R}^{n}$ with associated control set
$\pi_{\mathbb{P}}D^{1}$ in $\mathbb{P}^{n,1}$ the set
$\partial_{\infty}(D):=\partial\left(\pi_{\mathbb{P}}D^{1}\right)\cap\mathbb{P}^{n,0}$
is the boundary at infinity of $D$.

Proposition 29(i) shows, in particular, that the boundary at infinity
$\partial_{\infty}(D)$ is nonvoid if and only if $D$ is unbounded.

######  Remark 31.

The construction of the boundary at infinity of a control set bears some
similarity to the ideal boundary used by Firer and do Rocio [13] in the
analysis of invariant control sets for sub-semigroups of a semisimple Lie
group.

Next we clarify the relations between the accessibility rank conditions on the
relevant spaces.

######  Theorem 32.

(i) If the accessibility rank condition holds for affine system (1) on
$\mathbb{R}^{n}$, then it also holds for the system on the submanifold
$\mathbb{P}^{n,1}\subset\mathbb{P}^{n}$ induced by the bilinear system (18) on
$\mathbb{R}^{n+1}$.

(ii) If the accessibility rank condition holds for the system on
$\mathbb{P}^{n-1}$ induced by the homogeneous part (3) of system (1), then it
holds for the system on the invariant submanifold
$\mathbb{P}^{n,0}\subset\mathbb{P}^{n}$ induced by the bilinear system (18) on
$\mathbb{R}^{n+1}$.

###### Proof.

The proof is based on the local coordinate description of vector fields in
projective space obtained by projection of linear vector fields; cf. Bacciotti
and Vivalda [3, Section 4]. We omit the details. ∎

##  6 Control sets for nonhyperbolic systems

This section shows that all control sets with nonvoid interior are unbounded
if the hyperbolicity condition specified in Definition 19 is violated. Using
the compactification of the state space constructed in the previous section,
we show that there is a single chain control set in $\mathbb{P}^{n}$
containing the images of all control sets $D$ with nonvoid interior in
$\mathbb{R}^{n}$ and the boundary at infinity of this chain control set
contains all chain control sets of the homogeneous part having nonvoid
intersection with the boundary at infinity of one of the control sets $D$.

We begin with the following motivation. Consider a linear control system

$\dot{x}(t)=Ax(t)+Bu(t),\quad u(t)\in\Omega,$ (24)

where the control range $\Omega\subset\mathbb{R}^{m}$ is a compact convex
neighborhood of the origin. This is a special case of system (1) for
$B_{1}=\cdots=B_{m}=0$ and $d=0$. We assume that the system without control
restriction is controllable. By Colonius and Kliemann [7, Example 3.2.16]
there is a unique control set $D$ with nonvoid interior, and
$0\in\mathrm{int}(D)$. Let $\mathbf{GE}(A;\mu)$ denote the real generalized
eigenspace for an eigenvalue $\mu$ of $A$. Then

$\mathbf{E}^{0}\subset D\subset\overline{K}+\mathbf{E}^{0}+F,$ (25)

where $\mathbf{E}^{0}:=\bigoplus_{\operatorname{Re}\mu=0}\mathbf{GE}(A;\mu)$
is the central spectral subspace and the sets
$K\subset\mathbf{E}^{+}:=\bigoplus_{\operatorname{Re}\mu<0}\mathbf{GE}(A;\mu)$
and
$F\subset\mathbf{E}^{-}:=\bigoplus_{\operatorname{Re}\mu>0}\mathbf{GE}(A;\mu)$
are bounded. This follows from Sontag [23, Corollary 3.6.7] showing that
$\mathcal{O}^{+}(0)=K+\mathbf{E}_{0}$. Then time reversal yields
$\mathcal{O}^{-}(0)=\mathbf{E}_{0}+F$ and hence, by Remark 2,

$D=\overline{\mathcal{O}^{+}(0)}\cap\mathcal{O}^{-}(0)=\left(\overline{K}+\mathbf{E}^{0}\right)\cap\left(\mathbf{E}^{0}+F\right).$

Due to the decomposition
$\mathbb{R}^{n}=\mathbf{E}^{+}\oplus\mathbf{E}^{0}\oplus\mathbf{E}^{-}$ this
implies (25). In particular, $D$ is bounded if and only if
$\mathbf{E}^{0}=\\{0\\}$, i.e., if $A$ is a hyperbolic matrix. If $A$ is
nonhyperbolic we embed system (24) into a homogeneous bilinear control system
in $\mathbb{R}^{n+1}$ as explained in Section 5 and find that the boundary at
infinity satisfies

$\partial_{\infty}(D)=\partial\left(\pi_{\mathbb{P}}D^{1}\right)\cap\mathbb{P}^{n,0}=\\{[x_{1}:\cdots:x_{n}:0]\left|[x_{1}:\cdots:x_{n}]\in\pi_{\mathbb{P}}\mathbf{E}^{0}\right.\\}.$
(26)

This follows from (25) noting that for $0\not=x\in\mathbf{E}^{0}$ and every
$j\in\mathbb{N}$ one obtains an element of $D$ given by $k_{j}+jx+f_{j}$ with
$k_{j}\in\overline{K}$ and $f_{j}\in F$. Considering the homogeneous
coordinates and dividing by $j$ one finds for $j\rightarrow\infty$ that (26)
holds. The set $\pi_{\mathbb{P}}\mathbf{E}^{0}$ is a maximal invariant chain
transitive set for the flow induced by the homogeneous part $\dot{x}=Ax$ on
$\mathbb{P}^{n-1}$ (cf. Colonius and Kliemann [8, Theorem 4.1.3]). Thus the
boundary at infinity $\partial_{\infty}(D)$ is a maximal invariant chain
transitive set for the induced flow on $\mathbb{P}^{n,0}$.

For general affine control systems of the form (1) it stands to reason to
replace the maximal chain transitive set $\pi_{\mathbb{P}}\mathbf{E}^{0}$ by
maximal chain transitive sets of the control flow in
$\mathcal{U}\times\mathbb{P}^{n-1}$ associated with the homogeneous part or,
equivalently, by chain control sets in $\mathbb{P}^{n-1}$ (cf. (8)) and to
replace the spectral property of $\mathbf{E}^{0}$ by appropriate generalized
spectral properties. However, the situation for general affine control systems
will turn out to be more intricate than for linear control systems.

Now we start our discussion of the nonhyperbolic case for (1). Here several
control sets with nonvoid interior may coexist as illustrated by Example 42
and [9, Example 5.16 and Example 5.17]. The following theorem shows that in
the nonhyperbolic case all control sets with nonvoid interior are unbounded.

######  Theorem 33.

Assume that the affine control system (1) on $\mathbb{R}^{n}$ is
nonhyperbolic.

(i) If there is a $\sigma$-periodic control $u\in\mathcal{U}_{pc}$ with
$g(u)\in\mathcal{S}_{\sigma}\cap\mathrm{int}(\mathcal{S})$ and
$1\not\in\mathrm{spec}(\Phi_{u}(\sigma,0))$, then there exists a control set
$D$ with nonvoid interior.

(ii) Every control set $D$ with nonvoid interior is unbounded. More precisely,
there are $x^{k}\in\mathrm{int}(D),k\in\mathbb{N}$ and
$g(v)\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S}),\tau>0$, with
$1\in\mathrm{spec}(\Phi_{v}(\tau,0))$ and

$\left\|x^{k}\right\|\rightarrow\infty\text{ and
}d\left(\frac{x^{k}}{\left\|x^{k}\right\|},\mathbf{E}(\Phi_{v}(\tau,0);1)\right)\rightarrow
0\text{ for }k\rightarrow\infty.$ (27)

###### Proof.

(i) By Lemma 13 there is a unique $\sigma$-periodic trajectory of (1) for $u$.
Proposition 12(ii) implies that it is contained in the interior of a control
set $D$.

(ii) Let $x$ be in the interior of a control set $D$. By Proposition 12(i)
there are $\sigma>0$ and
$g(u)\in\mathcal{S}_{\sigma}\cap\mathrm{int}(\mathcal{S})$ such that
$g(u)x=x$. Then the $\sigma$-periodic control $u$ yields the $\sigma$-periodic
trajectory $\varphi(\cdot,x,u)\subset\mathrm{int}(D)$ and hence

$\int_{0}^{\sigma}\Phi_{u}(\sigma,s)\left(Cu(s)+d\right)ds=(I-\Phi_{u}(\sigma,0))x.$

Case 1: If $1\in\mathrm{spec}(\Phi_{u}(\sigma,0))$ the affine subspace
$Y=x+\mathbf{E}(\Phi_{u}(\sigma,0);1)$ is contained in $\mathrm{int}(D)$. For
the proof, an application of Lemma 18 shows that there is a $\sigma$-periodic
solution of (15) starting in $y$ if and only if $y\in
Y=x+\mathbf{E}(\Phi_{u}(\sigma,0);1)$. Thus $g(u)y=y\,$ for all $y\in Y$.
Proposition 12(ii) implies that every $y$ is in the interior of some control
set, hence $Y\subset\mathrm{int}(D)$. Furthermore, Lemma 18 also yields points
$x^{k}\in Y$ such that assertion (27) holds with $v:=u$ and $\tau:=\sigma$.

Case 2: Suppose that $1\not\in\mathrm{spec}(\Phi_{u}(\sigma,0))$. Since the
system is nonhyperbolic there is a $\tau^{\ast}$-periodic control $v^{\ast}$
with $1\in\mathrm{spec}(\Phi_{v^{\ast}}(\tau^{\ast},0))$ and
$g(v^{\ast})\in\mathcal{S}_{\tau^{\ast}}\cap\mathrm{int}(\mathcal{S})$.

Consider the continuous paths $p$ and $p^{\hom}$ from $g(u)$ to $g(v^{\ast})$
and $\Phi_{u}(\sigma,0)$ to $\Phi_{v^{\ast}}(\tau^{\ast},0)$, resp., given by
Lemma 15. Let

$\alpha_{0}:=\sup\\{\alpha\in[0,1]\left|\forall\alpha^{\prime}\in[0,\alpha):1\not\in\mathrm{spec}(\Phi_{u^{\alpha^{\prime}}}(\tau_{\alpha^{\prime}},0))\right.\\}.$

Hence, for $\alpha\in[0,\alpha_{0})$, Proposition 9(i) shows that there are
unique $\tau_{\alpha}$-periodic trajectories for $u^{\alpha}$ which by
Proposition 12(ii) are in the interior of a control set. By Proposition 16(ii)
$x^{\alpha}$ depends continuously on $\alpha\in[0,\alpha_{0})$ and then the
arguments in the proof of Theorem 22 shows that the initial values satisfy
$x^{\alpha}\in\mathrm{int}(D)$ for all $\alpha\in[0,\alpha_{0})$.

Now consider a sequence $\alpha_{k}\rightarrow\alpha_{0}$ with
$\alpha_{k}<\alpha_{0}$. Suppose first that

$\int_{0}^{\tau_{\alpha_{0}}}\Phi_{u^{\alpha_{0}}}(\tau_{\alpha_{0}},s)\left(Cu^{\alpha_{0}}(s)+d\right)ds\in\operatorname{Im}(I-\Phi_{u^{\alpha_{0}}}(\tau_{\alpha_{0}},0)).$
(28)

Let for $k=0,1,2,\ldots$

$b_{k}:=\int_{0}^{\tau_{\alpha_{k}}}\Phi_{u^{\alpha_{k}}}(\tau_{\alpha_{k}},s)\left(Cu^{\alpha_{k}}(s)+d\right)ds,\quad
A_{k}:=I-\Phi_{u^{\alpha_{k}}}(\tau_{\alpha_{k}},0).$

Then $A_{k}x^{\alpha_{k}}=b_{k}$ and $A_{k}\rightarrow A_{0},b_{k}\rightarrow
b_{0}$ for $k\rightarrow\infty$, and
$\mathrm{\ker}A_{0}=\mathbf{E}(\Phi_{u^{\alpha_{0}}}(\tau_{\alpha_{0}},0);1)$.
If $x^{\alpha_{k}}$ remains bounded, we may assume that
$x^{\alpha_{k}}\rightarrow x^{0}$ for some $x^{0}\in\mathbb{R}^{n}$ and hence
$A_{0}x^{0}=b_{0}$. Since $1\in\mathrm{spec}(\Phi_{u^{\alpha_{0}}},0))$ Lemma
18 implies assertion (27) for $v:=u^{\alpha_{0}}$, as in Case 1. If
$x^{\alpha_{k}}$ becomes unbounded then

$\left\|A_{0}\frac{x^{\alpha_{k}}}{\left\|x^{\alpha_{k}}\right\|}\right\|\leq\left\|A_{0}-A_{k}\right\|+\left\|A_{k}\frac{x^{\alpha_{k}}}{\left\|x^{\alpha_{k}}\right\|}\right\|\leq\left\|A_{0}-A_{k}\right\|+\frac{b_{k}}{\left\|x^{\alpha_{k}}\right\|}\rightarrow
0\text{ for }k\rightarrow\infty,$

and again (27) follows.

If (28) does not hold, Lemma 17 implies that, for $k=1,2,\ldots$, the initial
values $x^{k}$ of the $\tau_{\alpha_{k}}$-periodic solutions satisfy (27) with
$v:=u^{\alpha_{0}},\tau=\tau_{\alpha_{0}}$. ∎

Next we discuss the relation of the boundary at infinity to control sets of
the homogeneous part of the affine control system, motivated by the case of
linear control systems exposed in the beginning of this section. First we
obtain the following result for invariant control sets.

######  Theorem 34.

Assume that the affine system (1) is nonhyperbolic and suppose that $D$ is an
invariant control set.

(i) Then the interior of $D$ is nonvoid, the set $D$ is unbounded in
$\mathbb{R}^{n}$, and the boundary at infinity $\partial_{\infty}(D)$ contains
an invariant control set $e(_{\mathbb{P}}D^{\hom})$ of the system restricted
to $\mathbb{P}^{n,0}$.

(ii) If the control range $\Omega$ is a compact convex neighborhood of the
origin and the system on $\mathbb{P}^{n-1}$ satisfies the accessibility rank
condition, then the boundary at infinity $\partial_{\infty}(D)$ contains the
unique invariant control set $e(_{\mathbb{P}}D^{\hom})$ where
${}_{\mathbb{P}}D^{\hom}$ the unique invariant control set on
$\mathbb{P}^{n-1}$.

###### Proof.

(i) By local accessibility, the interior of the invariant control set $D$ is
nonvoid and hence Theorem 33 shows that $D$ is unbounded. It follows that
there is a point $\pi_{\mathbb{P}}(x,0)\in$
$\overline{\pi_{\mathbb{P}}D^{1}}\cap\mathbb{P}^{n,0}$ and by Proposition
29(i) $\pi_{\mathbb{P}}D^{1}$ is an invariant control set contained in
$\mathbb{P}^{n,1}$. Since $\mathbb{P}^{n,0}$ is invariant and closed in
$\mathbb{P}^{n}$ it follows that
$\overline{\mathcal{O}^{+}(\pi_{\mathbb{P}}(x,0))}\subset\mathbb{P}^{n,0}$,
hence every point in this set has the form $\pi_{\mathbb{P}}(y,0)$. For fixed
$\varepsilon>0$ there are $T>0$ and $u\in\mathcal{U}$ with
$d(\pi_{\mathbb{P}}\psi(T,(x,0),u),\pi_{\mathbb{P}}(y,0))<\varepsilon$ (recall
(20) and (19)). Furthermore, there are
$\pi_{\mathbb{P}}(x^{k},1)\in\,\pi_{\mathbb{P}}D^{1}$ with
$\pi_{\mathbb{P}}(x^{k},1)\rightarrow\pi_{\mathbb{P}}(x,0)$. Since
$\pi_{\mathbb{P}}D^{1}$ is an invariant control set it follows that
$\pi_{\mathbb{P}}\psi(T,(x^{k},1),u)\in\overline{\pi_{\mathbb{P}}D^{1}}$, and
continuous dependence on the initial values implies that
$d(\pi_{\mathbb{P}}\psi(T,(x^{k},1),u),\pi_{\mathbb{P}}(y,0))<\varepsilon$ for
$k$ large enough. Since $\varepsilon>0$ is arbitrary, we have shown that
$\pi_{\mathbb{P}}(y,0)\in\overline{\pi_{\mathbb{P}}D^{1}}$ and hence
$\overline{\mathcal{O}^{+}(\pi_{\mathbb{P}}(x,0))}\subset\partial\left(\pi_{\mathbb{P}}D^{1}\right)\cap\mathbb{P}^{n,0}=\partial_{\infty}(D)$.
By Colonius and Kliemann [7, Theorem 3.2.8], for every point
$\pi_{\mathbb{P}}(x,0)$ in the compact space $\mathbb{P}^{n,0}$ there is an
invariant control set contained in the closure of the reachable set
$\overline{\mathcal{O}^{+}(\pi_{\mathbb{P}}(x,0))}$. By Proposition 29(ii)
this invariant control set has the form $e(_{\mathbb{P}}D^{\hom})$ implying
(i).

(ii) This follows from Proposition 29(ii) since the accessibility rank
condition implies by Theorem 8(i) that the invariant control set of the system
on $\mathbb{P}^{n-1}$ is unique. ∎

We proceed to prove the following result on the relation between the boundary
at infinity of a control set in $\mathbb{R}^{n}$ and the chain control sets of
the homogeneous part in $\mathbb{P}^{n-1}$.

######  Proposition 35.

Assume that the affine control system (1) on $\mathbb{R}^{n}$ is
nonhyperbolic. Then for every control set $D\subset\mathbb{R}^{n}$ with
nonvoid interior of (1) there is a chain control set
${}_{\mathbb{P}}E^{\hom}\subset\mathbb{P}^{n-1}$ such that
$\partial_{\infty}(D)\cap e(_{\mathbb{P}}E^{\hom})\not=\varnothing$.

###### Proof.

Theorem 33 shows that $D$ is unbounded and that there are
$g(v)\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S})$ with
$1\in\mathrm{spec}(\Phi_{v}(\tau,0))$ and $x^{k}\in\mathrm{int}(D)$ satisfying
$\left\|x^{k}\right\|\rightarrow\infty$ and
$d(\frac{x^{k}}{\left\|x^{k}\right\|},\mathbf{E}(\Phi_{v}(\tau,0);1))\rightarrow
0$ for $k\rightarrow\infty$. Since $x=\Phi_{v}(\tau,0)x$ for all
$x\in\mathbf{E}(\Phi_{v}(\tau,0);1)$ the path connected set
$\pi_{\mathbb{P}}\mathbf{E}(\Phi_{v}(\tau,0);1)$ consists of points on
$\tau$-periodic solutions for the $\tau$-periodic control $v$ and hence is
contained in a chain control set ${}_{\mathbb{P}}E^{\hom}$. Hence one obtains
$\partial_{\infty}(D)\cap e(_{\mathbb{P}}E^{\hom})\not=\varnothing$ in
$\mathbb{P}^{n,0}$. ∎

The following theorem presents a partial converse of Theorem 34(i).

######  Theorem 36.

Assume that the homogeneous part (3) of the affine system (1) satisfies the
accessibility rank condition on $\mathbb{R}^{n}\setminus\\{0\\}$, that there
is $g(u)\in S_{\sigma}\cap\mathrm{int}(\mathcal{S})$ with
$1\not\in\mathrm{spec}(\Phi_{u}(\sigma,0))$ for some $\sigma>0$, and that
there are at most finitely many control sets with nonvoid interior of system
(1). Then for every control set ${}_{\mathbb{R}}D_{i}^{\hom}$ with nonvoid
interior there exists such a control set $D$ of (1) with boundary at infinity
satisfying

$\partial_{\infty}\left(D\right)\cap
e(_{\mathbb{P}}D_{i}^{\hom})\not=\varnothing\text{ for
}_{\mathbb{P}}D_{i}^{\hom}\supset\pi_{\mathbb{P}}(_{\mathbb{R}}D_{i}^{\hom}).$
(29)

###### Proof.

Fix a point $x\in\mathrm{int}(_{\mathbb{R}}D_{i}^{\hom})$. Since by Theorem 11
$\mathrm{int}(\mathcal{S}_{\leq\tau})\not=\varnothing$ for all $\tau>0$ there
are $\tau_{0}>0$ small enough and $u^{0}\in\mathcal{U}_{pc}$ with
$g(u^{0})\in\mathcal{S}_{\tau_{0}}\cap\mathrm{int}(\mathcal{S})$ and
$x^{0}:=\Phi_{u^{0}}(\tau_{0},0)x\in\mathrm{int}(_{\mathbb{R}}D_{i}^{\hom})$.
Since also
$\mathrm{int}(_{\mathbb{R}}\mathcal{S}_{\leq\tau}^{\hom})\not=\varnothing$ for
all $\tau>0$ there are $\tau_{1}>0$ small enough and
$u^{1}\in\mathcal{U}_{pc}$ such that the corresponding element
$\Phi_{u^{1}}(\tau_{1},0)\in\,_{\mathbb{R}}\mathcal{S}_{\tau_{1}}^{\hom}\cap\mathrm{int}(_{\mathbb{R}}\mathcal{S}^{\hom})$
satisfies

$x^{1}:=\Phi_{u^{1}}(\tau_{1},0)x^{0}=\Phi_{u^{1}}(\tau_{1},0)\Phi_{u^{0}}(\tau_{0},0)x\in\mathrm{int}(_{\mathbb{R}}D_{i}^{\hom}).$

By Remark 2 controllability in the interior of ${}_{\mathbb{R}}D_{i}^{\hom}$
holds, hence there are $\tau_{2}>0$ and $u^{2}\in\mathcal{U}_{pc}$ satisfying
$\Phi_{u^{2}}(\tau_{2},0)x^{1}=x$. Define $\tau:=\tau_{0}+\tau_{1}+\tau_{2}$
and a control $u\in\mathcal{U}_{pc}$ by $\tau$-periodic extension of

$u(t):=\left\\{\begin{array}[c]{lll}u^{0}(t)&\text{for}&t\in[0,\tau_{0})\\\
u^{1}(t-\tau_{0})&\text{for}&t\in[\tau_{0},\tau_{0}+\tau_{1})\\\
u^{2}(t-\tau_{0}-\tau_{1})&\text{for}&t\in[\tau_{0}+\tau_{1},\tau_{0}+\tau_{1}+\tau_{2})\end{array}\right..$

Then $\Phi_{u}(\tau,0)x=x$, hence $1\in\mathrm{spec}(\Phi_{u}(\tau,0))$, and

$\displaystyle g(u)$
$\displaystyle=g(u^{2})g(u^{1})g(u^{0})\in\mathcal{S}_{\tau}\cap\mathrm{int}(\mathcal{S}),$
$\displaystyle\Phi_{u}(\tau,0)$
$\displaystyle=\Phi_{u^{2}}(\tau_{2},0)\Phi_{u^{1}}(\tau_{1},0)\Phi_{u^{0}}(\tau_{0},0)\in\mathrm{int}(_{\mathbb{R}}\mathcal{S}^{\hom}).$

By Proposition 12(ii) (for ${}_{\mathbb{R}}\mathcal{S}^{\mathrm{hom}}$) this
implies that the eigenspace $\mathbf{E}(\Phi_{u}(\tau,0);1)$ of
$\Phi_{u}(\tau,0)$ for the eigenvalue $1$ is contained in the interior of some
control set in $\mathbb{R}^{n}\setminus\\{0\\}$. Since
$x\in\mathbf{E}(\Phi_{u}(\tau,0);1)\cap\,_{\mathbb{R}}D_{i}^{\hom}$ it follows
that

$\mathbf{E}(\Phi_{u}(\tau,0);1)\subset\mathrm{int}(_{\mathbb{R}}D_{i}^{\hom})\text{
and hence
}\pi_{\mathbb{P}}\mathbf{E}(\Phi_{u}(\tau,0);1)\subset\mathrm{int}(_{\mathbb{P}}D_{i}^{\hom}).$
(30)

By Proposition 16(i) there are
$g(u^{k})\in\mathcal{S}_{\tau_{k}}\cap\mathrm{int}(\mathcal{S})$ with
$1\not\in\mathrm{spec}(\Phi_{u^{k}}(\tau_{k},0))$ and
$\Phi_{u^{k}}(\tau_{k},0)\rightarrow\Phi_{u}(\tau,0)$ for
$k\rightarrow\infty$. Proposition 9(i) implies that there are unique
$\tau_{k}$-periodic solutions denoted by $\varphi(\cdot,x^{k},u^{k})$ of the
affine equation for the $\tau_{k}$-periodic extension of $u^{k}$. By
Proposition 12(ii) each of them is in the interior of a control set for the
affine system (1). Since, by assumption, there are only finitely many of them,
infinitely many $x^{k}$ are contained in the interior of a single control set
$D$. We may assume that all $x^{k}$ are in $\mathrm{int}(D)$.

Suppose that assumption (iii) in Lemma 17 is satisfied. Then it follows that

$\left\|x^{k}\right\|\rightarrow\infty\text{ and
}\frac{x^{k}}{\left\|x^{k}\right\|}\rightarrow\mathbf{E}(\Phi_{u}(\tau,0);1)\text{
for }k\rightarrow\infty.$ (31)

Hence, for $k\rightarrow\infty$, the points
$\pi_{\mathbb{P}}(x^{k},1)\in\,\pi_{\mathbb{P}}D^{1}$ converge to
$e\left(\pi_{\mathbb{P}}\mathbf{E}(\Phi_{u}(\tau,0);1)\right)$ showing that

$\overline{\pi_{\mathbb{P}}D^{1}}\cap
e\left(\pi_{\mathbb{P}}\mathbf{E}(\Phi_{u}(\tau,0);1)\right)\not=\varnothing.$

Together with (30) this implies that the boundary at infinity of $D$ satisfies
(29).

If assumption (iii) in Lemma 17 does not hold Lemma 18 shows that there are
$x^{k}\in\mathbf{E}(\Phi_{u}(\tau,0);1)$ with (31) for $k\rightarrow\infty$.
Then the assertion follows also in this case. ∎

The following examples (cf. Mohler [18, Example 2 on page 32] and Colonius,
Santana, Setti [9, Example 5.16]) show that, in general, the boundary at
infinity of a control set $D$ may intersect more than one control set of the
projectivized homogeneous part.

######  Example 37.

Consider the affine control system

$\left(\begin{array}[c]{c}\dot{x}\\\
\dot{y}\end{array}\right)=\left(\begin{array}[c]{cc}2u&1\\\
1&2u\end{array}\right)\left(\begin{array}[c]{c}x\\\
y\end{array}\right)+\left(\begin{array}[c]{c}0\\\ 1\end{array}\right)u,\quad
u(t)\in\Omega=[-1,1].$

The eigenvalues of $A(u)=A+uB$ are given by
$\lambda_{1}(u)=2u+1>\lambda_{2}(u)=2u-1$ and
$\lambda_{1}(-1/2)=\lambda_{2}(1/2)=0$. For every $u\in\mathbb{R}$, the
eigenspaces for $\lambda_{1}(u)$ and $\lambda_{2}(u)$ are
$\mathbf{E}(A+uB;\lambda_{1}(u))=\\{(z,z)^{\top}\left|z\in\mathbb{R}\right.\\}$
and
$\mathbf{E}(A+uB;\lambda_{2}(u))=\\{(z,-z)^{\top}\left|z\in\mathbb{R}\right.\\}$,
resp. In the northern part of the unit circle (hence in $\mathbb{P}^{1}$) this
yields the two one-point control sets given by the equilibria for any
$u\in[-1,1]$,

${}_{\mathbb{P}}D_{1}^{\hom}=\left\\{\left(1/\sqrt{2},1/\sqrt{2}\right)^{\top}\right\\}\text{
and
}_{\mathbb{P}}D_{2}^{\hom}=\left\\{\left(-1/\sqrt{2},1/\sqrt{2}\right)^{\top}\right\\}.$

Here the trajectories not starting in one of these equilibria approach
${}_{\mathbb{P}}D_{1}^{\hom}$ and ${}_{\mathbb{P}}D_{2}^{\hom}$ for
$t\rightarrow+\infty$ and $t\rightarrow-\infty$, resp. As shown in [9, Example
4.4/5.16] there is a connected branch of equilibria of the affine system

$\mathcal{B}_{1}=\left\\{\left(x_{u},y_{u})^{\top}\right)\left|u\in\left(-1/2,1/2\right)\right.\right\\}\text{
with
}\left(x_{0},y_{0}\right)^{\top}=\left(0,0\right)^{\top}\in\mathcal{B}_{1}.$

They become unbounded for $\left|u\right|\rightarrow\frac{1}{2}$, and there is
a single control set $D$ containing the equilibria in $\mathcal{B}_{1}$ in the
interior. The equilibria in $\mathcal{B}_{1}$ satisfy
$\frac{(x_{u},y_{u})}{\left\|(x_{u},y_{u})\right\|}\rightarrow\left(\mp\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$
for $u\rightarrow\pm\frac{1}{2}$. Consequently, one obtains for the control
sets of the homogeneous part

$e(_{\mathbb{P}}D_{1}^{\hom})\cup\,e(_{\mathbb{P}}D_{2}^{\hom})\subset\partial(\pi_{\mathbb{P}}D^{1})\cap\mathbb{P}^{2,0}=\partial_{\infty}(D).$

The homogeneous part of Example 37 violates the accessibility rank condition
in $\mathbb{P}^{1}$ and the control sets ${}_{\mathbb{P}}D_{1}^{\hom}$ and
${}_{\mathbb{P}}D_{2}^{\hom}$ in $\mathbb{P}^{1}$ have void interiors. We
modify this example in order to get control sets in $\mathbb{P}^{1}$ with
nonvoid interior. Note that here an arbitrarily small perturbation suffices to
change the system behavior drastically.

######  Example 38.

Consider for small $\varepsilon>0$ and $\Omega=[-1,1]$

$\left(\begin{array}[c]{c}\dot{x}\\\
\dot{y}\end{array}\right)=\left(\begin{array}[c]{cc}2u&1\\\
1&(2+\varepsilon)u\end{array}\right)\left(\begin{array}[c]{c}x\\\
y\end{array}\right)+\left(\begin{array}[c]{c}0\\\
1\end{array}\right)u=\left(A+uB(\varepsilon)\right)\left(\begin{array}[c]{c}x\\\
y\end{array}\right)+Cu,$

We will show that there is a control set $D$ in $\mathbb{R}^{2}$ such that the
boundary at infinity $\partial_{\infty}(D)$ intersects two control sets with
nonvoid interior for the homogeneous part.

Step 1: The eigenvalues of $A+uB(\varepsilon)$ are given by

$\lambda_{1,2}(u,\varepsilon)=\frac{u}{2}(4+\varepsilon)\pm\frac{1}{2}\sqrt{4+u^{2}\left[(4+\varepsilon)^{2}-4(4+2\varepsilon)\right]}.$

Note that $\lambda_{1}(u,\varepsilon)>\lambda_{2}(u,\varepsilon)$ for all
$u\in[-1,1]$. For $\varepsilon=0$, it is clear that the functions
$u\mapsto\lambda_{1,2}(u,0)=2u\pm 1$ are strictly increasing, hence this also
holds for small $\varepsilon>0$. Thus there are unique values
$u^{1}(\varepsilon),u^{2}(\varepsilon)\in(-1,1)$ with
$\lambda_{1}(u^{1}(\varepsilon),\varepsilon)=0$ and
$\lambda_{2}(u^{2}(\varepsilon),\varepsilon)=0$, and
$u^{1}(\varepsilon)\rightarrow-\frac{1}{2}$ and
$u^{2}(\varepsilon)\rightarrow\frac{1}{2}$ for $\varepsilon\rightarrow 0$. The
eigenvectors $(x,y)^{\top}$ satisfy
$y=\left(\lambda_{1,2}(u,\varepsilon)-2u\right)x$. For $\varepsilon\rightarrow
0$ and all $u\in[-1,1]$ the eigenspace
$\mathbf{E}(A+uB(\varepsilon);\lambda_{i}(u,\varepsilon))$ converges to the
eigenspace $\mathbf{E}(A+uB(0);\lambda_{i}(u,0))$. In the northern part of the
unit circle (hence in $\mathbb{P}^{1}$) this yields two equilibria
$e_{1}(u,\varepsilon)$ and $e_{2}(u,\varepsilon)$, and the other trajectories
in $\mathbb{P}^{1}$ converge for $t\rightarrow\infty$ to
$e_{1}(u,\varepsilon)$ and for $t\rightarrow-\infty$ to
$e_{2}(u,\varepsilon)$. Hence there are control sets
${}_{\mathbb{P}}D_{1}^{\hom}$ and ${}_{\mathbb{P}}D_{2}^{\hom}$ (depending on
$\varepsilon$) with nonvoid interior consisting of the equilibria
$e_{1}(u,\varepsilon)$ and $e_{2}(u,\varepsilon),u\in[-1,1]$, resp. The
control set ${}_{\mathbb{P}}D_{1}^{\hom}$ is invariant. One easily verifies
the accessibility rank condition in $\mathbb{R}^{2}\setminus 0\\}$. Since
$0\in\mathrm{int}(\Sigma_{Fl}(_{\mathbb{P}}D_{i}^{\hom}))$ it follows that
${}_{\mathbb{P}}D_{i}^{\hom}$ is the projection to $\mathbb{P}^{1}$ of a
control set ${}_{\mathbb{R}}D_{i}^{\hom}$ in $\mathbb{R}^{2}\setminus\\{0\\}$,
$i=1,2$.

Step 2: The equilibria $(x_{u}(\varepsilon),y_{u}(\varepsilon))^{\top}$
approach $\mathbf{E}(A+u^{i}(\varepsilon)B(\varepsilon);0)$ for $u\rightarrow
u^{i}(\varepsilon),i=1,2$. In both cases, the equilibria become unbounded. In
particular, there is a connected unbounded branch of equilibria

$\mathcal{B}_{1}(\varepsilon)=\left\\{(x_{u}(\varepsilon),y_{u}(\varepsilon))^{\top}\left|u\in\left(u^{1}(\varepsilon),u^{2}(\varepsilon)\right)\right.\right\\}$

and a single control set $D$ (again depending on $\varepsilon$) containing the
equilibria in $\mathcal{B}_{1}(\varepsilon)$.

Step 3: Embedding the control system into a homogeneous bilinear system in
$\mathbb{R}^{3}$ and projecting it to $\mathbb{P}^{2}$ one obtains from the
control set $D$ a control set in $\mathbb{P}^{2,1}$ given by
$\pi_{\mathbb{P}}D^{1}=\left\\{[x:y:1]\left|(x,y)^{\top}\in
D\right.\right\\}$. As the equilibria
$(x_{u}(\varepsilon),y_{u}(\varepsilon))^{\top}\in\mathcal{B}_{1}$ become
unbounded for $u\rightarrow u^{i}(\varepsilon)$ they approach the eigenspace
$\mathbf{E}(A+u^{i}(\varepsilon)B(\varepsilon);0)$, hence

$e(_{\mathbb{P}}D_{1}^{\hom})\cap\partial_{\infty}(D)\not=\varnothing\text{
and }e(_{\mathbb{P}}D_{2}^{\hom})\cap\partial_{\infty}(D)\not=\varnothing.$

In the following we require that $\Omega$ is a convex and compact neighborhood
of $0\in\mathbb{R}^{m}$ and consider chain control sets of the affine system
in $\mathbb{P}^{n}$.

######  Definition 39.

The boundary at infinity of a chain control set ${}_{\mathbb{P}}E$ for the
affine system (22) in $\mathbb{P}^{n}$ is
$\partial_{\infty}(_{\mathbb{P}}E):=\partial(_{\mathbb{P}}E)\cap\mathbb{P}^{n,0}$.

This definition is similar to the boundary at infinity for control sets but it
refers to chain control sets in $\mathbb{P}^{n}$ not requiring that they are
obtained from chain control sets in $\mathbb{R}^{n}$.

######  Lemma 40.

Let ${}_{\mathbb{P}}E$ be a chain control set in $\mathbb{P}^{n}$.

(i) If $\partial_{\infty}(_{\mathbb{P}}E)\cap
e(_{\mathbb{P}}E_{j}^{\hom})\not=\varnothing$ for a chain control set
${}_{\mathbb{P}}E_{j}^{\hom}$ in $\mathbb{P}^{n-1}$ of the homogeneous part,
then $e(_{\mathbb{P}}E_{j}^{\hom})\subset\partial_{\infty}(_{\mathbb{P}}E)$.

(ii) If $\partial_{\infty}(_{\mathbb{P}}E)$ is nonvoid, it contains a chain
control set $e(_{\mathbb{P}}E_{j}^{\hom})$ for a chain control set
${}_{\mathbb{P}}E_{j}^{\hom}$ of the homogeneous part.

###### Proof.

(i) Recall from Proposition 29(ii) that $e(_{\mathbb{P}}E_{j}^{\hom})$ is a
chain control set of the system restricted to $\mathbb{P}^{n,0}$. We will show
that the set ${}_{\mathbb{P}}E^{\prime}:=\,_{\mathbb{P}}E\cup
e(_{\mathbb{P}}E_{j}^{\hom})$ satisfies the properties (i) and (ii) of a chain
control set in $\mathbb{P}^{n}$. Then the maximality property (iii) of the
chain control set ${}_{\mathbb{P}}E$ implies that
${}_{\mathbb{P}}E^{\prime}=\,_{\mathbb{P}}E$ showing that
$e(_{\mathbb{P}}E_{j}^{\hom})\subset\partial_{\infty}(_{\mathbb{P}}E)$.

It is clear that ${}_{\mathbb{P}}E^{\prime}$ satisfies (i), since this holds
for ${}_{\mathbb{P}}E$ and $e(_{\mathbb{P}}E_{j}^{\hom})$. For property (ii),
consider $x\in\,_{\mathbb{P}}E$ and $y\in e(_{\mathbb{P}}E_{j}^{\hom})$ and
$\varepsilon,T>0$. Fix $z\in\partial_{\infty}(_{\mathbb{P}}E)\cap
e(_{\mathbb{P}}E_{j}^{\hom})=\,_{\mathbb{P}}E\cap
e(_{\mathbb{P}}E_{j}^{\hom})$. 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 $\mathbb{P}^{n,0}$, there exist controlled
$(\varepsilon,T)$-chains $\zeta_{3}$ and $\zeta_{4}$ from $z$ to $y$ and from
$y$ to $z$, resp. Then the concatenations $\zeta_{3}\circ\zeta_{1}$ and
$\zeta_{3}\circ\zeta_{4}$ 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 $x\in\partial_{\infty}(_{\mathbb{P}}E)$. Then there exists a control
$u\in\mathcal{U}$ with
$\pi_{\mathbb{P}}\varphi(t,x,u)\in\partial_{\infty}(_{\mathbb{P}}E)=\,_{\mathbb{P}}E\cap\mathbb{P}^{n,0}$
for all $t\geq 0$ by property (i) of chain control sets and invariance of
$\mathbb{P}^{n,0}$. Since
${}_{\mathbb{P}}E\cap\mathbb{P}^{n,0}=\partial(_{\mathbb{P}}E)\cap\mathbb{P}^{n,0}$
is compact, it follows that the $\omega$-limit set

$\omega_{\mathbb{P}}(u,x):=\left\\{\left.y=\lim\nolimits_{k\rightarrow\infty}\pi_{\mathbb{P}}\varphi(t_{k},x,u)\right|t_{k}\rightarrow\infty\right\\}\subset\,_{\mathbb{P}}E\cap\mathbb{P}^{n,0}.$

is nonvoid. Hence Colonius and Kliemann [7, Corollary 4.3.12] implies that
there exists a chain control set of the system restricted to
$\mathbb{P}^{n,0}$ containing $\omega_{\mathbb{P}}(u,x)$. Thus there is a
chain control set ${}_{\mathbb{P}}E_{j}^{\hom}$ in $\mathbb{P}^{n-1}$ of the
homogeneous part with $\partial_{\infty}(_{\mathbb{P}}E)\cap
e(_{\mathbb{P}}E_{j}^{\hom})\not=\varnothing$. Now the assertion follows from
(i). ∎

The next theorem is the main result on the control sets $D$ with nonvoid
interior in $\mathbb{R}^{n}$ in the nonhyperbolic case.

######  Theorem 41.

Assume that the affine control system (1) is nonhyperbolic. Furthermore, let
the control range $\Omega$ be a compact convex neighborhood of the origin and
assume that there is $g(u)\in S_{\sigma}\cap\mathrm{int}(\mathcal{S})$ with
$1\not\in\mathrm{spec}(\Phi_{u}(\sigma,0))$ for some $\sigma>0$.

Then there exists a single chain control set ${}_{\mathbb{P}}E$ in
$\mathbb{P}^{n}$ containing the control sets $\pi_{\mathbb{P}}D^{1}$ for all
control sets $D$ with nonvoid interior in $\mathbb{R}^{n}$. Furthermore, the
boundary at infinity $\partial_{\infty}(_{\mathbb{P}}E)$ contains all
$\partial_{\infty}(D)$ and the chain control sets
$e(_{\mathbb{P}}E_{j}^{\hom})$ where ${}_{\mathbb{P}}E_{j}^{\hom}$ are the
chain control sets in $\mathbb{P}^{n-1}$ for the homogeneous part (3) with
$\partial_{\infty}(D)\cap e(_{\mathbb{P}}E_{j}^{\hom})\not=\varnothing$ for
some $D$.

###### Proof.

By Proposition 9(i) there is a unique $\sigma$-periodic solution of the affine
system with $g(u)x^{0}=x^{0}$. Proposition 12 implies that $x^{0}$ is in the
interior of a control set $D_{0}$. Now let $D_{1}$ be any control set in
$\mathbb{R}^{n}$ with nonvoid interior. It suffices to show that there is a
chain control set ${}_{\mathbb{P}}E$ in $\mathbb{P}^{n}$ containing
$\pi_{\mathbb{P}}D_{0}^{1}$ and $\pi_{\mathbb{P}}D_{1}^{1}$ and that its
boundary at infinity $\partial_{\infty}(_{\mathbb{P}}E)$ contains all chain
control sets $e(_{\mathbb{P}}E_{j}^{\hom})$ with $\partial_{\infty}(D_{1})\cap
e(_{\mathbb{P}}E_{j}^{\hom})\not=\varnothing$.

Pick $x^{1}\in\mathrm{int}(D_{1})$. Then Proposition 12(i) implies that there
are $\tau_{1}>0$ and
$g_{1}=g(u^{1})\in\mathcal{S}_{\tau_{1}}\cap\mathrm{int}(\mathcal{S})$ with
$x^{1}=g(u^{1})x^{1}$. Proposition 16(i) yields $\tau_{\alpha}$-periodic
controls $w^{\alpha}\in\mathcal{U}_{pc}$ and continuous paths
$p_{1}:[0,1]\rightarrow\mathrm{int}(\mathcal{S})$ with
$p_{1}(0)=g(u),\,p_{1}(\alpha)=g(w^{\alpha})$ for $\alpha\in[0,1]$ and
$p_{1}^{\hom}:[0,1]\rightarrow\,_{\mathbb{R}}\mathcal{S}^{\hom}$ with
$p_{1}^{\hom}(0)=\Phi_{u}(\sigma,0),\,p_{1}^{\hom}(\alpha)=\Phi_{w^{\alpha}}(\tau_{\alpha},0)$
such that

$\left\|p_{1}^{\hom}(1)-\Phi_{u^{1}}(\tau_{1},0)\right\|<\varepsilon,\,\left\|p_{1}(1)-g(u^{1})\right\|<\varepsilon$

and $1\not\in\mathrm{spec}(\Phi_{w^{\alpha}}(\tau_{\alpha},0)$ for all but at
most finitely many $\alpha\in[0,1]$. Denote the $\alpha$-values with
$1\in\mathrm{spec}(\Phi_{w^{\alpha}}(\tau_{\alpha},0))$ by
$0<\alpha_{1}<\cdots<\alpha_{r}<1$, where the last inequality holds without
loss of generality. The continuity and smoothness properties from Proposition
16(i) hold for
$w^{\alpha},\tau_{\alpha},\Phi_{w^{\alpha}}(\tau_{\alpha},0)\in\,_{\mathbb{R}}\mathcal{S}^{\hom}$,
and $g(w^{\alpha})\in\mathcal{S}$. For $\alpha$ with
$1\not\in\mathrm{spec}(\Phi_{w^{\alpha}}(\tau_{\alpha},0)$ Proposition 16(ii)
shows that there are unique $\tau_{\alpha}$-periodic solutions with initial
values $x^{\alpha}$ depending continuously on $\alpha$. Define

$A:=\\{\alpha\in[0,1]\left|\text{
}1\not\in\mathrm{spec}(\Phi_{w^{\alpha}}(\tau_{\alpha},0))\right.\\}.$

The set $A$ consists of $r+1$ intervals. For $\alpha\in A$, Proposition 12(ii)
implies that $x^{\alpha}\in\mathrm{int}(D^{\alpha})$ for some control set
$D^{\alpha}$ since $g(w^{\alpha})\in\mathrm{int}(\mathcal{S})$. For each
$\alpha$ in an interval contained in $A$, the $x^{\alpha}$ depend continuously
on $\alpha$, hence they are contained in the interior of a single control set.
By construction, $x^{0}\in\mathrm{int}(D_{0})$ and, for $\varepsilon>0$ small
enough, $x^{1}\in\mathrm{int}(D_{1})$. Denote the other control sets
containing the $x^{\alpha}$ by $D_{i},i\geq 2$. By Theorem 33 the control sets
$D_{i}$ are unbounded, hence their boundary at infinity
$\partial_{\infty}(D_{i})$ is nonvoid.

The control sets $\pi_{\mathbb{P}}D_{i}^{1}$ in $\mathbb{P}^{n,1}$ are
contained in chain control sets ${}_{\mathbb{P}}E_{i}\subset\mathbb{P}^{n}$
and it follows that
$\partial_{\infty}(_{\mathbb{P}}E_{i})\supset\partial_{\infty}(D_{i})$ is
nonvoid. Thus Lemma 40(ii) implies that
$\partial_{\infty}(_{\mathbb{P}}E_{i})$ contains a chain control set
$e(_{\mathbb{P}}E_{j}^{\hom})$ where ${}_{\mathbb{P}}E_{j}^{\hom}$ is a chain
control set of the homogeneous part in $\mathbb{P}^{n-1}$. By Lemma 40(i),
$\partial_{\infty}(_{\mathbb{P}}E_{i})$ contains every chain control set of
the homogeneous part that it intersects. The theorem follows from the next
claim.

Claim. All chain control sets ${}_{\mathbb{P}}E_{i},i\geq 2$, in
$\mathbb{P}^{n}$ coincide.

For every point $\alpha_{i}\in(0,1)\setminus A$ there are control sets which
we denote by $D_{i}$ and $D_{i+1}$ such that all $\alpha$ in a neighborhood of
$\alpha_{i}$ satisfy $x^{\alpha}\in\mathrm{int}(D_{i})$ for
$\alpha<\alpha_{i}$ and $x^{\alpha}\in\mathrm{int}(D_{i+1})$ for
$\alpha>\alpha_{i}$.

We have to show that the chain control sets ${}_{\mathbb{P}}E_{i}$ for
$\alpha<\alpha_{i}$ and ${}_{\mathbb{P}}E_{i+1}$ for $\alpha>\alpha_{i}$
coincide. The projected eigenspace
$\pi_{\mathbb{P}}\mathbf{E}(\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},0);1))$
consists of $\tau_{\alpha_{i}}$-periodic solutions for the
$\tau_{\alpha_{i}}$-periodic control $u^{\alpha_{i}}$ and hence is contained
in a chain control set ${}_{\mathbb{P}}E_{j}^{\hom}$ in $\mathbb{P}^{n-1}$. We
will show that

$e(_{\mathbb{P}}E_{j}^{\hom})\subset\partial_{\infty}(_{\mathbb{P}}E_{i})\cap\partial_{\infty}(_{\mathbb{P}}E_{i+1}),$

which implies that ${}_{\mathbb{P}}E_{i}$ and ${}_{\mathbb{P}}E_{i+1}$ have
nonvoid intersection and hence coincide.

First consider parameters
$\beta_{k}\rightarrow\alpha_{i},\beta_{k}<\alpha_{i}$. As stated above, the
points $x^{\beta_{k}}$ with $x^{\beta_{k}}=g(u^{\beta_{k}})x^{\beta_{k}}$
satisfy $x^{\beta_{k}}\in\mathrm{int}(D_{i})$.

Case 1.
$\int_{0}^{\tau_{\alpha_{i}}}\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},s)\left[Cu^{\alpha_{i}}(s)+d\right]ds\not\in\operatorname{Im}(I-\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},0))$.

Lemma 17 implies that the $x^{\beta_{k}}\in D_{i}$ satisfy

$\left\|x^{\beta_{k}}\right\|\rightarrow\infty\text{ and
}\frac{x^{\beta_{k}}}{\left\|x^{\beta_{k}}\right\|}\rightarrow\mathbf{E}(\Phi^{\alpha_{i}}(\tau_{\alpha_{i}},0);1),$
(32)

and by Lemma 40(i)
$e(_{\mathbb{P}}E_{j}^{\hom})\subset\partial_{\infty}(_{\mathbb{P}}E_{i})$
since

$\varnothing\not=\partial_{\infty}(D_{i})\cap
e\left(\pi_{\mathbb{P}}\mathbf{E}(\Phi^{\alpha_{i}}(\tau_{\alpha_{i}},0);1)\right)\subset\partial_{\infty}(_{\mathbb{P}}E_{i})\cap\,e(_{\mathbb{P}}E_{j}^{\hom}).$
(33)

Case 2.
$\int_{0}^{\tau_{\alpha_{i}}}\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},s)\left[Cu^{\alpha_{i}}(s)+d\right]ds\in\operatorname{Im}(I-\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},0))$.

Let, for $k=1,2,\ldots$,

$A_{k}:=I-\Phi_{u^{\beta_{k}}}(\tau_{\beta_{k}},0),\quad
b_{k}:=\int_{0}^{\tau_{\beta_{k}}}\Phi_{u^{\beta_{k}}}(\tau_{\beta_{k}},s)\left(Cu^{\beta_{k}}(s)+d\right)ds.$

Then $x^{\beta_{k}}=g(u^{\beta_{k}})x^{\beta_{k}}$ implies
$A_{k}x^{\beta_{k}}=b_{k}$ and by Proposition 9(iii), (iv)

$A_{k}\rightarrow
A_{0}:=I-\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},0),\,b_{k}\rightarrow
b_{0}:=\int_{0}^{\tau_{\alpha_{i}}}\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},s)\left(Cu^{\alpha_{i}}(s)+d\right)ds.$

If $x^{\beta_{k}}$ remains bounded, we may assume that
$x^{\beta_{k}}\rightarrow y^{0}$ for some
$y^{0}\in\overline{D_{i}}\subset\mathbb{R}^{n}$ and hence $A_{0}y^{0}=b_{0}$.
By Lemma 18 there are $x^{k}\in
y^{0}+\mathbf{E}(\Phi_{u^{\alpha_{i}}}(\tau_{\alpha_{i}},0);1)$ with
$\left\|x^{k}\right\|\rightarrow\infty$ and
$\frac{x^{k}}{\left\|x^{k}\right\|}\rightarrow\mathbf{E}(\Phi^{\alpha_{i}}(\tau_{\alpha_{i}},0);1)$
and again (32) follows implying
$e(_{\mathbb{P}}E_{j}^{\hom})\subset\partial_{\infty}(_{\mathbb{P}}E_{i})$.

If $x^{\beta_{k}}$ becomes unbounded, we obtain

$\left\|A_{0}\frac{x^{\beta_{k}}}{\left\|x^{\beta_{k}}\right\|}\right\|\leq\left\|A_{0}-A_{k}\right\|+\left\|A_{k}\frac{x^{\beta_{k}}}{\left\|x^{\beta_{k}}\right\|}\right\|=\left\|A_{0}-A_{k}\right\|+\frac{\left\|b_{k}\right\|}{\left\|x^{\beta_{k}}\right\|}\rightarrow
0,$

and (32) follows implying
$e(_{\mathbb{P}}E_{j}^{\hom})\subset\partial_{\infty}(_{\mathbb{P}}E_{i})$.

We have shown this inclusion using
$\beta_{k}\rightarrow\alpha_{i},\beta_{k}<\alpha_{i}$. The same arguments can
be applied to parameters
$\beta_{k}\rightarrow\alpha_{i},\beta_{k}>\alpha_{i}$, showing that also for
the chain control set ${}_{\mathbb{P}}E_{i+1}$ the boundary at infinity
$\partial_{\infty}(_{\mathbb{P}}E_{i+1})$ contains the chain control set
$e(_{\mathbb{P}}E_{j}^{\hom})$. This proves the claim and concludes the proof
of the theorem. ∎

The following controlled linear oscillator illustrates Theorem 41.

######  Example 42.

Consider the affine control system (cf. [9, Example 5.17])

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

where $\rho\in\left(1,\frac{5}{4}\right)$ and $d<1$. The equilibria are, for
$u\in[-\rho,-1)$ and $u\in(-1,\rho]$,

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

resp. The equilibria in $\mathcal{C}_{1}$ are hyperbolic, since here the
eigenvalues of $A(u)$ are $\lambda_{1}(u)<0<\lambda_{2}(u)$. The equilibria in
$\mathcal{C}_{2}$ are stable nodes since here
$\lambda_{1}(u)<\lambda_{2}(u)<0$. For $u^{0}=-1$ the matrix
$A(-1)=\left(\begin{array}[c]{cc}0&1\\\ 0&-3\end{array}\right)$ has the
eigenvalue $\lambda_{2}(-1)=0$ with eigenspace $\mathbb{R}\times\\{0\\}$,
hence $e^{A(-1)t}$ has the eigenvalue $1$. There are control sets
$D_{1}\not=D_{2}$ containing the equilibria in $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$, resp., in the interior. For $u^{k}\nearrow u^{0}=-1$, the
equilibria in $D_{1}$ satisfy $(x_{u^{k}},0)\rightarrow(\infty,0)$ and for
$u^{k}\searrow u^{0}=-1$, the equilibria in $D_{2}$ satisfy
$(x_{u^{k}},0)\rightarrow(-\infty,0)$ for $k\rightarrow\infty$. There is a
single chain control set ${}_{\mathbb{P}}E$ in $\mathbb{P}^{2}$ containing the
images of $D_{1}$ and $D_{2}$ since the eigenspace
$\mathbf{E}(e^{A(-1)\tau};1)=\mathbb{R}\times\\{0\\}$ satisfies

$e(\pi_{\mathbb{P}}\mathbf{E}(e^{A(-1)\tau};1))\subset\partial_{\infty}(D_{1})\cap\partial_{\infty}(D_{2})\quad\text{for
any }\tau>0.$

Concerning the homogeneous part in $\mathbb{P}^{1}$ the projectivized
eigenspace $\pi_{\mathbb{P}}\mathbf{E}(e^{A(-1)\tau};1)$ is contained in the
invariant control set
${}_{\mathbb{P}}D_{2}^{\hom}=\pi_{\mathbb{P}}\\{\left.(x,\lambda_{2}(u)x)^{\top}\right|x\not=0,u\in[-\rho,\rho]\\}$
and ${}_{\mathbb{P}}D_{2}^{\hom}$ is the projection of a control set
${}_{\mathbb{R}}D_{2}^{\hom}$ in $\mathbb{R}^{2}$, since
$0\in\mathrm{int}(\Sigma_{Fl}(_{\mathbb{P}}D_{2}^{\hom}))$.

##  7 Appendix

This appendix presents the proof of Proposition 9. Assertion (i) follows from
the variation-of-parameters formula and (ii) is a consequence of (i).

(iii) The principal fundamental solutions $\Phi^{k}(t,s)$ satisfy for
$t,s\in[0,\tau_{0}+1]$

$\left\|\Phi^{k}(t,s)\right\|\leq
1+\left|\int_{s}^{t}\left\|P^{k}(\sigma)\right\|\left\|\Phi^{k}(\sigma,s)\right\|d\sigma\right|.$

By the generalized Gronwall inequality (cf. Amann [1, Lemma 6.1]) it follows
that
$c_{1}=\sup_{k\in\mathbb{N},\,\,t,s\in[0,\tau_{0}+1]}\left\|\Phi^{k}(\,t,s)\right\|<\infty$.
For all $\,t,s\in[0,\tau_{0}+1]$

$\displaystyle\left\|\Phi^{k}(t,s)-\Phi^{0}(t,s)\right\|=\left\|\int_{s}^{t}\left[P^{k}(\sigma)\Phi^{k}(\sigma,s)-P^{0}(\sigma)\Phi^{0}(\sigma,s)\right]d\sigma\right\|$
$\displaystyle\leq\left\|\int_{s}^{\,t}\left[P^{k}(\sigma)-P^{0}(\sigma)\right]\Phi^{k}(\sigma,s)d\sigma\right\|+\left|\int_{s}^{\,t}\left\|P^{0}(\sigma)\right\|\left\|\Phi^{k}(\sigma,s)-\Phi^{0}(\sigma,s)\right\|d\sigma\right|.$

The first term is bounded by
$c_{k}:=c_{1}\int_{0}^{\tau_{0}+1}\left\|P^{k}(\sigma)-P^{0}(\sigma)\right\|d\sigma\rightarrow
0$ for $k\rightarrow\infty$. Again by [1, Lemma 6.1] it follows that for
$\,t,s\in[0,\tau_{0}+1]$

$\left\|\Phi^{k}(t,s)-\Phi^{0}(t,s)\right\|\leq
c_{k}+\left|\int_{s}^{\,t}c_{k}\left\|P^{0}(r)\right\|\exp\left|\int_{r}^{\,t}\left\|P^{0}(\sigma)\right\|d\sigma\right|dr\right|.$

The right hand side converges to $0$ uniformly in $\,t,s$ for
$k\rightarrow\infty$ since $c_{k}\rightarrow 0$.

(iv) The assumption implies that there are unique $\tau_{k}$-periodic
solutions given by

$x^{k}:=(I-\Phi^{k}(\tau_{k},0))^{-1}\int_{0}^{\tau_{k}}\Phi^{k}(\tau_{k},s)z^{k}(s)ds.$

Then
$\left\|\int_{\tau_{0}}^{\tau_{k}}\Phi^{k}(\tau_{k},s)z^{k}(s)ds\right\|\rightarrow
0$ since $\tau_{k}\rightarrow\tau_{0}$ and the integrands are uniformly
bounded, and

$\displaystyle\left\|\int_{0}^{\tau_{0}}\left[\Phi^{k}(\tau_{k},s)z^{k}(s)-\Phi^{0}(\tau_{0},s)z^{0}(s)\right]ds\right\|$
$\displaystyle\leq\left\|\int_{0}^{\tau_{0}}\left[\Phi^{k}(\tau_{k},s)-\Phi^{0}(\tau_{0},s)\right]z^{k}(s)ds\right\|+\left\|\int_{0}^{\tau_{0}}\Phi^{0}(\tau_{0},s)\left[z^{k}(s)-z^{0}(s)\right]ds\right\|$
$\displaystyle\leq\left[\sup_{s\in[0,\tau_{0}]}\left\|\Phi^{k}(\tau_{k},s)-\Phi^{0}(\tau_{k},s)\right\|+\sup_{s\in[0,\tau_{0}]}\left\|\Phi^{0}(\tau_{k},s)-\Phi^{0}(\tau_{0},s)\right\|\right]\int_{0}^{\tau_{0}}\left\|z^{k}(s)\right\|ds$
$\displaystyle\qquad+\sup_{s\in[0,\tau_{0}]}\left\|\Phi^{0}(\tau_{0},s)\right\|\int_{0}^{\tau_{0}}\left\|z^{k}(s)-z^{0}(s)\right\|ds.$

This converges to $0$ and it follows that $x^{k}\rightarrow x^{0}$.

Acknowledgements.  We thank Luiz A.B. San Martin and Adriano Da Silva for
valuable suggestions and comments. We are also indebted to anonymous reviewers
of an earlier version whose significant queries helped to improve the paper.

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