# Controllability of discrete-time linear systems on solvable Lie groups

Thiago M. Cavalheiro
Departamento de Matemática, Universidade Estadual de Maringá
Maringá, Brazil  João A. N. Cossich
Departamento de Matemática, Universidad de Tarapacá - Arica, Chile  Alexandre
J. Santana
Departamento de Matemática, Universidade Estadual de Maringá
Maringá, Brazil

###### Abstract

The aim of this paper is to study controllability of discrete-time linear
control systems on solvable Lie groups. In special case of nilpotent Lie
groups, it is established a necessary and sufficient condition for
controllability. Moreover, all discrete-time linear systems on bidimensional
affine Lie group are constructed and it is stated a condition for
controllability of these systems on this group.

##  1 Introduction

Continuous-time linear control systems on $\mathbb{R}^{d}$ are given by a
family of differential equations of the form

$\Sigma_{c}:\ \dot{x}(t)=Ax(t)+Bu(t),\ t\in\mathbb{R},$

parametrized by control functions $u\in\mathcal{U}:=\\{u:\mathbb{R}\rightarrow
U\subset\mathbb{R}^{m};\ u\mbox{ is piecewise continuous}\\}$, where
$A\in\mathbb{R}^{d\times d}$ and $B\in\mathbb{R}^{d\times m}$. This class of
systems constitutes a widely known and applicable class of control systems
(see Ogata [16]). The study of controllability of this system can be found
e.g. in Sontag [20] where it is proved a classical result which establishes
that the necessary and sufficient conditions for controllability are
$\mbox{rank}[B\ AB\ \cdots A^{d-1}B]=d$ and $A$ admits only eigenvalues with
zero real part .

However, when it comes to digital signal processing modelling, for example,
the discrete-time version of the above system becomes more effective (see
Wilsky [21]). Such a class of systems is given by

$\Sigma_{d}:\ x_{k+1}=Ax_{k}+Bu_{k},\
k\in\mathbb{N}_{0},u=(u_{k})_{k\in\mathbb{N}_{0}}\in\mathcal{U}:=U^{\mathbb{N}_{0}}$

where $U\subset\mathbb{R}^{m}$ is non-empty, $A\in\text{Gl}(d,\mathbb{R})$ and
$B\in\mathbb{R}^{d\times d}$.

Regarding the controllability of ($\Sigma_{d}$), Colonius et al in [8] proved
that such a system is controllable if, and only if, $\mbox{rank}[B\ AB\ \cdots
A^{d-1}B]=d$ and all eigenvalues of $A$ have absolute value equal to $1$.
Moreover they characterized the control sets, that is, proper subsets of the
state space where approximate controllability property holds.

A natural extension of the above system $(\Sigma_{c})$ is known linear system
on Lie group $G$, that is, a family of differential equations
$\cdot{g}(t)=\mathcal{X}(g(t))+u(t)X(g(t))$, where $\mathcal{X}$ comes from
and automorphism of $G$, $X$ is a right invariant vector field and
$u\in\mathcal{U}$. This class of systems has been extensively studied and Lie
theory has been shown to be powerful in obtaining results about the
controllability, conjugacy and invariance entropy of this system (e.g. Ayala
and Da Silva [2, 3], Ayala, Da Silva and Zsigmond [4], Da Silva [10, 11],
Jouan [13] and Jouan and Dath [14]).

The discrete-time version of linear systems on Lie groups was introduced by
Colonius et al in [9]. Essentially, its dynamic is given by a family of
difference equations

$\Sigma\ :\ g_{k+1}=f(g_{k},u_{k}),\ k\in\mathbb{N}_{0},\ u_{k}\in U,$

on a connected Lie group $G$, where $0\in U\subset\mathbb{R}^{m}$,
$f_{0}:=f(\cdot,0)$ is an automorphism of $G$ and for each $u\in U$,
$f_{u}:=f(\cdot,u):G\rightarrow G$ satisfies

$f_{u}(g)=f_{u}(e)\cdot f_{0}(g)\text{ for all }g\in G,$

where “$\cdot$” denotes the product of $G$. The authors established a formula
for calculating the outer invariance entropy of admissible pairs in terms of
the eigenvalues of the Lie algebra automorphism
$d(f_{0})_{e}:\mathfrak{g}\rightarrow\mathfrak{g}$.

In this context, the subject of our paper is to study controllability of the
above system $(\Sigma)$. Our results were inspired in some results of Da Silva
[10] where continuous-time linear systems are considered. Specifically, we
show that if all eigenvalues of the automorphism $d(f_{0})_{e}$ associated to
a discrete-time linear systems on a solvable Lie groups have absolute value
equal to $1$ and that the reachable and the controllable set from the identity
are open, then the system is controllable. It is also proved that the
reciprocal of this result holds when $G$ is nilpotent.

This paper is structured as follows: Section 2 is dedicated to present Lie
theoretic notations, facts and some general properties of discrete-time linear
control systems as well. In Section 3 is shown a sufficient condition for
controllability of discrete-time linear systems on connected solvable Lie
groups. In addiction, the class of linear systems on the two dimensional Lie
group $\hbox{Aff}(2,\mathbb{R})$ is constructed and a condition for
controllability is derived. Section 4 deals with nilpotent Lie groups and it
is proved that the sufficient condition for the controllability obtained for
solvable Lie groups is also a necessary condition.

##  2 Preliminaries

This section establishes the notations, basic concepts and necessary results
for the development of this work. Besides, we present the definition of
discrete-time linear control systems on Lie groups as some of their properties
as well.

###  2.1 Decompositions of Lie algebras and Lie groups

In this subsection we show some dynamical decomposition on Lie algebras and
connected Lie groups introduced by Ayala, Román-Flores and Da Silva [1] for a
given automorphism.

For a Lie algebra $\mathfrak{g}$, the generalized eigenspaces of an
automorphism $\xi:\mathfrak{g}\rightarrow\mathfrak{g}$ associated with an
eigenvalue $\alpha$ is given by

$\mathfrak{g}_{\alpha}=\\{X\in\mathfrak{g}:(\xi-\alpha)^{n}X=0,\hbox{ for some
}n\in\mathbb{N}\\}.$

Due to [1, Proposition 2.1], the following subspaces

$\mathfrak{g}^{+}=\bigoplus_{|\alpha|>1}\mathfrak{g}_{\alpha},\
\mathfrak{g}^{-}=\bigoplus_{|\alpha|<1}\mathfrak{g}_{\alpha},\
\mathfrak{g}^{0}=\bigoplus_{|\alpha|=1}\mathfrak{g}_{\alpha}$ (1)

are Lie subalgebras of $\mathfrak{g}$, called unstable, stable and center Lie
subalgebras of $\mathfrak{g}$ in relation to $\xi$, respectively. Moreover,
the decomposition

$\mathfrak{g}=\mathfrak{g}^{+}\oplus\mathfrak{g}^{0}\oplus\mathfrak{g}^{-}$
(2)

holds. It can be also defined the center-unstable and center-stable Lie
subalgebras

$\displaystyle\mathfrak{g}^{+,0}=\mathfrak{g}^{+}\oplus\mathfrak{g}^{0}\
\text{ and }\ \mathfrak{g}^{-,0}=\mathfrak{g}^{-}\oplus\mathfrak{g}^{0},$ (3)

resp.

If $G$ is a connected Lie group with Lie algebra $\mathfrak{g}$, we denote by
$G^{+}$, $G^{-}$, $G^{0}$, $G^{+,0}$ and $G^{-,0}$ the connected Lie subgroups
of $G$ corresponding to $\mathfrak{g}^{+}$, $\mathfrak{g}^{-}$,
$\mathfrak{g}^{0}$, $\mathfrak{g}^{+,0}$ and $\mathfrak{g}^{-,0}$,
respectively. We refer to the subgroups above as unstable, stable, center,
center-unstable and the center-stable subgroups, resp.

###### Remark 1

  *   * 1)

It follows from [1, Proposition 2.1] that the stable and unstable subalgebras
are nilpotent. Moreover, since
$[\mathfrak{g}^{+},\mathfrak{g}^{0}]\subset\mathfrak{g}^{+}$, then
$\mathfrak{g}^{+}$ is an ideal of $\mathfrak{g}^{+,0}$. Consequently, $G^{+}$
is a normal subgroup of $G^{+,0}$. The same holds for
$\mathfrak{g}^{-}\subset\mathfrak{g}^{-,0}$ and $G^{-}\subset G^{-,0}$.

  * 2)

Item 1 above implies that $G^{+,0}=G^{+}G^{0}=G^{0}G^{+}$ and
$G^{-,0}=G^{-}G^{0}=G^{0}G^{-}$.

Given a homomorphism $\xi:\mathfrak{g}\rightarrow\mathfrak{g}$, a Lie
subalgebra is $\xi$-invariant if $\xi(\mathfrak{h})\subset\mathfrak{h}$. In
case when $\xi$ is an automorphism and $\mathfrak{h}$ is $\xi$-invariant, it
is clear that $\xi^{k}(\mathfrak{h})=\mathfrak{h}$, for all $k\in\mathbb{Z}$.

On the Lie group level, given a homomorphism $\phi$ of $G$, we say that a Lie
subgroup $H$ of $G$ is $\phi$-invariant if $\phi(H)\subset H$. Analogously, if
$\phi$ is an automorphism and $H$ is $\phi$-invariant, it also holds that
$\phi^{k}(H)=H$, for any $k\in\mathbb{Z}$. Note also that if $H$ is connected
and $\mathfrak{h}$ is its Lie algebra, $H$ is $\phi$-invariant iff
$\mathfrak{h}$ is $d\phi_{e}$-invariant, where $e$ denotes the identity of
$G$.

Next lemma shows that the above decompositions are preserved by a surjective
Lie algebra homomorphism since this homomorphism commutes with two
automorphisms.

###### Lemma 2

Let $\eta:\mathfrak{g}\rightarrow\mathfrak{h}$ a surjective Lie algebra
homomorphism, $\xi_{1}$ and $\xi_{2}$ automorphisms of $\mathfrak{g}$ and
$\mathfrak{h}$, respectively. If $\eta\circ\xi_{1}=\xi_{2}\circ\eta$, then

$\eta(\mathfrak{g}^{+})=\mathfrak{h}^{+},\
\eta(\mathfrak{g}^{-})=\mathfrak{h}^{-}\mbox{ and
}\eta(\mathfrak{g}^{0})=\mathfrak{h}^{0}.$

In addition, if $G$ and $H$ are connected Lie groups associated with
$\mathfrak{g}$ and $\mathfrak{h}$, respectively, and $\pi:G\rightarrow H$ is a
surjective homomorphism such that $d\pi_{e}\circ\xi_{1}=\xi_{2}\circ
d\pi_{e}$, then

$\pi(G^{+})=H^{+},\ \pi(G^{-})=H^{-}\mbox{ and }\pi(G^{0})=H^{0}.$

Proof. For an eigenvalue $\alpha$ of $\xi_{1}$, there exists $n\in\mathbb{N}$
with $(\xi_{1}-\alpha)^{n}X=0$, for all $X\in\mathfrak{g}$. By hypothesis,

$(\xi_{2}-\alpha)^{n}d\pi_{e}(X)=d\pi_{e}(\xi_{1}-\alpha)^{n}(X)=0.$

Since $\eta$ is surjective, $\alpha$ is also an eigenvalue of $\xi_{2}$, hence
$\eta(\mathfrak{g}_{\alpha})\subset\mathfrak{h}_{\alpha}$ which shows that
$\eta(\mathfrak{g}^{+})\subset\mathfrak{h}^{+}$,
$\eta(\mathfrak{g}^{-})\subset\mathfrak{h}^{-}$ and
$\eta(\mathfrak{g}^{0})\subset\mathfrak{h}^{0}$. The equality follows from
subjectivity of $\eta$ and decomposition (2) applied to $\mathfrak{g}$ and
$\mathfrak{h}$.

The equalities at group level holds due subjectivity of $\pi$ and the equality
$\pi(\exp_{G}(X))=\exp_{H}(d\pi_{e}X)$. $\square$

In sequence we get that solvable groups can be decomposable as a product of
subgroups $G^{+}$, $G^{0}$ and $G^{-}$.

###### Proposition 3

If $G$ is a connected solvable Lie group with Lie algebra $\mathfrak{g}$ and
$\phi$ an automorphism of $G$, then the Lie subgroups associated to the
decomposition (2) of $d\phi_{e}$ satisfy $G=G^{+,0}G^{-}=G^{-,0}G^{+}$.

Proof.

In order to prove that $G=G^{+,0}G^{-}$, we proceed by induction on $\dim G$.
If $G$ is unidimensional, the group is abelian and the result follows.

Now, suppose the result is true for all connected solvable Lie group with
dimension less than $d$. Let $G$ be a Lie group with $\dim G=d$ and consider
the derivative series of its Lie algebra $\mathfrak{g}$

$\mathfrak{g}=\mathfrak{g}^{(0)}\supset\mathfrak{g}^{(1)}\supset\cdots\supset\mathfrak{g}^{(k)}\supset\mathfrak{g}^{(k+1)}=\\{0\\},$

where $\mathfrak{g}^{(i)}=[\mathfrak{g}^{(i-1)},\mathfrak{g}^{(i-1)}]$ for
$i\in\\{1,\ldots,k\\}$. Each $\mathfrak{g}^{(i)}$ is an ideal of
$\mathfrak{g}$, hence each Lie subgroup $G^{(i)}$ associated to
$\mathfrak{g}^{(i)}$ is normal in $G$. The $d\phi_{e}$-invariance of
$\mathfrak{g}^{(i)}$ implies the $\phi$-invariance of $G^{(i)}$. In
particular, $G^{(k)}$ is $\phi$-invariant, abelian and normal, so its closure
$\overline{G^{(k)}}$ is a closed Lie subgroup with the same properties. The
Lie group $H:=G/\overline{G^{(k)}}$ is solvable with

$\dim H=\dim G-\dim\overline{G^{(k)}}\leq\dim G-\dim G^{(k)}<\dim G,$

because $\dim G^{(k)}>0$. Denote by $\pi$ the natural projection of $G$ on $H$
and note that

$H=H^{+,0}H^{-}=\pi(G^{+,0})\pi(G^{-})=\pi(G^{+,0}G^{-})$

by Lemma 2 and induction hypothesis. Then
$G=G^{+,0}G^{-}\overline{G^{(k)}}=G^{+,0}\overline{G^{(k)}}G^{-}$.

If $\overline{\mathfrak{g}^{(k)}}$ denotes the Lie algebra of
$\overline{G^{(k)}}$, one has that
$\overline{\mathfrak{g}^{(k)}}^{+,0}=\overline{\mathfrak{g}^{(k)}}\cap\mathfrak{g}^{+,0}$
and
$\overline{\mathfrak{g}^{(k)}}^{-}=\overline{\mathfrak{g}^{(k)}}\cap\mathfrak{g}^{-}$
by $d\phi_{e}$-invariance of $\overline{\mathfrak{g}^{(k)}}$. Moreover, since
it is abelian, it holds that
$\overline{G^{(k)}}=\overline{G^{(k)}}^{+,0}\overline{G^{(k)}}^{-}$, which
shows that $\overline{G^{(k)}}^{+,0}\subset G^{+,0}$ and
$\overline{G^{(k)}}^{-}\subset G^{-}$, therefore

$G=G^{+,0}\overline{G^{(k)}}G^{-}=G^{+,0}\overline{G^{(k)}}^{+,0}\overline{G^{(k)}}^{-}G^{-}\subset
G^{+,0}G^{-}\subset G,$

and the statement holds. The equality $G=G^{-,0}G^{+}$ follows analogously.
$\square$

###### Proposition 4

Every compact, connected and $\phi$-invariant Lie subgroup of $G$ is contained
in $G^{0}$.

Proof. Let $H$ be a Lie subgroup of $G$ satisfying the conditions above. If
$\mathfrak{h}$ denotes the Lie algebra of $H$, then [12, Corollary 4.25]
implies that
$\mathfrak{g}=\mathfrak{z}(\mathfrak{h})\oplus[\mathfrak{h},\mathfrak{h}]$,
where $\mathfrak{z}(\mathfrak{h})$ is the center of h and
$[\mathfrak{h},\mathfrak{h}]$ is semisimple. Note that the $\phi$-invariance
of $H$ implies the $d\phi_{e}$ invariance of $\mathfrak{h}$.

The connected Lie group associated to $\mathfrak{z}(\mathfrak{h})$ is the
connected component of $Z(G)$, denoted by $Z(G)_{0}$. Since $Z(G)_{0}$ is
compact and abelian, the subset $S\subset Z(G)_{0}$ of all elements with
finite order is dense in $Z(G)_{0}$. In this case, given
$X\in\mathfrak{z}(\mathfrak{h})$ such that $\exp X\in S$, there is
$k\in\mathbb{N}$ with $(\exp X)^{k}=e$, hence

$\displaystyle e=\phi((\exp X)^{k})=(\phi(\exp X))^{k}=\exp(d\phi_{e})^{k}X.$

Since the exponential map of an abelian Lie group is a diffeomorphism and
$(d\phi_{e})^{k}X=0$, then $X=0$. Hence $\mathfrak{z}(\mathfrak{h})$ is
trivial.

In addition, by compactness of $H$ and semisimplicity of
$[\mathfrak{h},\mathfrak{h}]$ it follows that the Cartan-Killing form
restricted to $[\mathfrak{h},\mathfrak{h}]$ is non-degenerated and negative
defined (see [12, Theorem 1.45 and Corollary 4.26]). Since $d\phi_{e}$ is an
automorphism, it is an isometry by Cartan-Killing form, hence all eigenvalues
of $d\phi_{e}|_{[\mathfrak{h},\mathfrak{h}]}$ has absolute value equal to $1$,
which yields
$\mathfrak{h}=[\mathfrak{h},\mathfrak{h}]\subset\mathfrak{g}^{0}$. $\square$

By Proposition 4 the following result is immediate.

###### Corollary 5

If $\phi$ is an automorphism of a compact Lie group $G$, then $d\phi_{e}$ has
only eigenvalues with absolute value equal to $1$.

The next lemma will often be used and can be found in [20, Lemma 3.1].

###### Lemma 6

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and $N$ a normal Lie
subgroup of $G$ with Lie algebra $\mathfrak{n}$. Then for every
$X\in\mathfrak{g}$, we have that

$\exp{(X+\mathfrak{n})}\subset\exp{(X)}N.$

###  2.2 Linear control systems on Lie groups

This subsection presents some general properties of discrete-time control
systems. We start by recalling that a discrete-time control system on a
topological space $M$ is given by a difference equation

$x_{k+1}=f(x_{k},u_{k}),\ k\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\},\ u_{k}\in
U,$ (4)

where $U$ is a non-empty set and $f:M\times U\rightarrow M$ is a map such that
for each $u\in U$, $f_{u}(\cdot):=f(\cdot,u):M\rightarrow M$ is continuous
function. The set $M$ is called state space of (4) and $U$ is the control
range. Moreover, the shift space is defined as the set of all sequences in
$U$, that is, $\mathcal{U}:=\displaystyle\prod_{i=0}^{\infty}U$. The elements
in $\mathcal{U}$ are called controls. In this case, the following solutions of
(4) is well-defined

$\varphi(k,x_{0},u)=\left\\{\begin{array}[c]{lll}x_{0},&\mbox{if}&k=0\\\
f_{u_{k-1}}\circ\cdots\circ f_{u_{1}}\circ f_{u_{0}}(x_{0}),&\mbox{if}&k\geq
1\end{array},\right.$

for all $k\in\mathbb{N}_{0}$, $x_{0}\in M$ and
$u=(u_{i})_{i\in\mathbb{N}_{0}}\in\mathcal{U}$. Note that
$\varphi(k,\cdot,u):M\rightarrow M$ is continuous for each
$k\in\mathbb{N}_{0}$ and $u=(u_{i})_{i\in\mathbb{N}_{0}}\in\mathcal{U}$.

The shift map $\Theta:\mathbb{N}_{0}\times\mathcal{U}\rightarrow\mathcal{U}$
given by $\Theta(k,(u_{j}))=\Theta_{k}((u_{j})):=(u_{j+k})$ defines a
dynamical system on $\mathcal{U}$. It is well known that $\varphi$ satisfies
the cocycle property, that is,

$\varphi(k+l,x,u)=\varphi(k,\varphi(l,x,u),\Theta_{l}(u)),\ \forall\
k,l\in\mathbb{N}_{0},\ \forall\ x\in M,\ \forall\ u\in\mathcal{U}.$

###### Definition 7

For $x\in M$, the reachable and the controllable set from x at time k are
given by

$\displaystyle\mathcal{R}_{k}(x)=\\{y\in M:\hbox{ there is
}u\in\mathcal{U}\hbox{ with }\varphi(k,x,u)=y\\}$

and

$\displaystyle\mathcal{C}_{k}(x)=\\{y\in M:\hbox{ there is
}u\in\mathcal{U}\hbox{ with }\varphi(k,y,u)=x\\},$

respectively. Moreover, the reachable and the controllable set from x up to
time k are given by $\mathcal{R}_{\leq k}(x)=\displaystyle\bigcup_{t\leq
k}\mathcal{R}_{t}(x)$ and $\mathcal{C}(x)=\displaystyle\bigcup_{t\leq
k}\mathcal{C}_{t}(x)$, resp. The sets
$\mathcal{R}(x)=\displaystyle\bigcup_{k\in\mathbb{N}}\mathcal{R}_{k}(x)$ and
$\mathcal{C}(x)=\displaystyle\bigcup_{k\in\mathbb{N}}\mathcal{C}_{k}(x)$
denote the reachable and the controllable set from x, respectively.

The system (4) is forward accessible (resp. backward accessible) if
$\hbox{int}\mathcal{R}(x)\neq\emptyset$ (resp.
$\hbox{int}\mathcal{C}(x)\neq\emptyset$), for all $x\in M$ and it is called
accessible if both conditions are satisfied.

According to Wirth [22], for any $k\in\mathbb{N}$ and $g\in G$, define the
smooth map $G_{k}:G\times U^{k}\rightarrow G$ given by

$G_{k}(g,u)=\varphi(k,g,u).$

A pair $(g,u)\in G\times\hbox{int}U^{k}$ such that
$\hbox{rank}\left[\frac{\partial}{\partial u}G_{k}(g,u)\right]=\dim G$ is
called regular. The regular reachable set at time k is defined by

$\hat{\mathcal{R}}_{k}(g)=\\{\varphi(k,g,u):(x,u)\hbox{ is regular}\\}$

and the regular reachable set
$\hat{\mathcal{R}}(g)=\cup_{k\in\mathbb{N}}\hat{\mathcal{R}}_{k}(g)$. One can
prove that $\hat{\mathcal{R}}(g)$ is open, for every $g\in G$.

Now, let us define the main object of this work.

###### Definition 8

Consider a discrete-time control system

$g_{k+1}=f(g_{k},u_{k}),u_{k}\in U,$ (5)

on a connected Lie group $G$ with $U\subset\mathbb{R}^{m}$ a compact
neighborhood of $0$. The system (5) is called linear if $f_{0}$ is an
automorphism of $G$ and for each $g\in G$

$f_{u}(g)=f_{u}(e)\cdot f_{0}(g).$ (6)

where $"\cdot"$ denotes de product on $G$.

The group product will be omitted when it is clear. Moreover, the equation (6)
can be write as

$f_{u}(g)=f_{u}(e)f_{0}(g)=L_{f_{u}(e)}\circ f_{0}(g),$

where $L_{g}$ is the left translation by $g\in G$. Considering the expression
above, we can see that $f_{u}$ is a diffeomorphism of $G$, for each $u\in U$,
with inverse

$f_{u}^{-1}(g)=f_{0}^{-1}\circ
L_{(f_{u}(e))^{-1}}(g)=f_{0}^{-1}((f_{u}(e))^{-1}\cdot g).$

###### Remark 9

It follows from Definition 8 that:

  * 1)

$\mathcal{U}$ endowed with the product topology is a compact space.

  * 2)

for all $k\in\mathbb{N}_{0}$ and $u\in\mathcal{U}$, the map
$\varphi(k,\cdot,u)$ is a diffeomorphism of $G$.

  * 3)

Since $f_{0}$ is an automorphism of $G$, its differential
$d(f_{0})_{e}:\mathfrak{g}\rightarrow\mathfrak{g}$ is an automorphism and one
has the Lie subalgebras (1) and (3) related to $d(f_{0})_{e}$.

###### Example 10

The standard example of discrete-time linear system is the difference equation

$x_{k+1}=Ax_{k}+Bu_{k},\ \ u_{k}\in U,$

where $G$ is the additive euclidean space $\mathbb{R}^{d}$, $A\in
GL(d,\mathbb{R}),B\in\mathbb{R}^{d\times m}$, and $0\in
U\subset\mathbb{R}^{m}$. The map $f:\mathbb{R}^{d}\times
U\rightarrow\mathbb{R}^{d}$ is given by $f(x,u)=Ax+Bu$ and satisfies

  * •

$f_{0}(x)=Ax$ is an automorphism of $\mathbb{R}^{d}$;

  * •

$f_{u}(x)=Ax+Bu=f_{0}(x)+f_{u}(0)=f_{u}(0)+f_{0}(x)$, since $f_{u}(0)=Bu$

In this case, the solutions are given by

$\varphi(k,x,u)=A^{k}x+\sum_{j=0}^{k-1}A^{k-1-j}Bu_{j}.$

The next proposition shows that the solutions of 5, from an element $g\in G$
can be given by a translation of the solution from $e\in G$ (see [9,
Proposition 3])

###### Proposition 11

Consider the discrete-time linear system (5) on a Lie group $G$. Then, for all
$g\in G$ and $u\in\mathcal{U}$ it holds that

$\varphi(k,g,u)=\varphi(k,e,u)f_{0}^{k}(g).$

In case of linear systems, the identity $e$ of $G$ satisfies
$e\in\mathcal{R}_{k}(e)$ for all $k\in\mathbb{N}$, because it is a fixed point
of $f_{0}$. In addition, using the notation $\mathcal{R}(e)=\mathcal{R}$,
$\mathcal{R}_{k}(e)=\mathcal{R}_{k}$ and $\mathcal{R}_{\leq
k}=\mathcal{R}_{\leq k}(e)$, we get the following proposition.

###### Proposition 12

For all $k,k_{1},k_{2}\in\mathbb{N}$ $g\in G$ and $u\in\mathcal{U}$, it holds
that

  * 1)

$\mathcal{R}_{k}=\mathcal{R}_{\leq k}$;

  * 2)

If $k_{1}\leq k_{2}$, then $\mathcal{R}_{k_{1}}\subset\mathcal{R}_{k_{2}}$;

  * 3)

$\mathcal{R}_{k}(g)=\mathcal{R}_{k}f_{0}^{k}(g)$;

  * 4)

If $k_{1},k_{2}\in\mathbb{N}$, then
$\mathcal{R}_{k_{1}+k_{2}}=\mathcal{R}_{k_{1}}f_{0}^{k_{1}}(\mathcal{R}_{k_{2}})=\mathcal{R}_{k_{2}}f_{0}^{k_{2}}(\mathcal{R}_{k_{1}})$;

  * 5)

For any $u\in\mathcal{U}$, $g\in G$ and $k\in\mathbb{N}$, then

$\varphi(k,\mathcal{R}(g),u)\subset\mathcal{R}(g);$

  * 6)

$e\in\hbox{int}\mathcal{R}$ if and only if $\mathcal{R}$ is open.

Proof.

  * 1)

It is clear that $\mathcal{R}_{k}\subset\mathcal{R}_{\leq k}.$ On the other
hand, taking $t\in[1,k)\cap\mathbb{N}$, an arbitrary $u\in\mathcal{U}$ and
considering the control

$v=\left\\{\begin{array}[]{lcl}0,&\text{for}&j<k-t\\\
u_{j-k+t},&\text{for}&j\geq k-t\end{array},\right.$

we have

$\varphi(t,e,u)=\varphi(t,\varphi(k-t,e,0),u)=\varphi(t,\varphi(k-t,e,v),\Theta_{k-t}(v))=\varphi(k,e,v),$

which implies that $\mathcal{R}_{k}\subset\mathcal{R}_{t}$, proving the
statement.

  * 2)

Consequence of (1).

  * 3)

It follows from Proposition 11.

  * 4)

Note that

$\displaystyle\varphi(k_{1}+k_{2},e,u)$ $\displaystyle=$
$\displaystyle\varphi(k_{1},\varphi(k_{2},e,u),\Theta_{k_{2}}(u))$
$\displaystyle=$
$\displaystyle\varphi(k_{1},e,\Theta_{k_{2}}(u))f_{0}^{k_{1}}(\varphi(k_{2},e,u))\in\mathcal{R}_{k_{1}}f_{0}^{k_{1}}(\mathcal{R}_{k_{2}})$

which means that
$\mathcal{R}_{k_{1}+k_{2}}\subset\mathcal{R}_{k_{1}}f_{0}^{k_{1}}(\mathcal{R}_{k_{2}})$.
Now, given $u,v\in\mathcal{U}$, we have

$\displaystyle\varphi(k_{1},e,u)f_{0}^{k_{1}}(\varphi(k_{2},e,v))$
$\displaystyle=$ $\displaystyle\varphi(k_{1},\varphi(k_{2},e,v),u)$
$\displaystyle=$ $\displaystyle\varphi(k_{1}+k_{2},e,w),$

with

$w=\left\\{\begin{array}[]{lcc}v_{j},&j<k_{2}\\\ u_{j-k_{2}},&j\geq
k_{2}\end{array}.\right.$

The inclusion
$\mathcal{R}_{k_{2}}f_{0}^{k_{2}}(\mathcal{R}_{k_{1}})\subset\mathcal{R}_{k_{1}+k_{2}}$
follows by the same arguments above.

  * 5)

Take $k\in\mathbb{N}$, $g\in G$ and $u\in\mathcal{U}$. If
$h\in\mathcal{R}(g)$, then $h=\varphi(t,g,v)$ for some $t\in\mathbb{N}$ and
$v\in\mathcal{U}$. Thus

$\varphi(k,h,u)=\varphi(k,\varphi(t,g,v),u)=\varphi(k+t,g,w)\in\mathcal{R}(g),$

with

$w=\left\\{\begin{array}[]{lcc}v_{j},&j<t\\\ u_{j-t},&j\geq
t\end{array}.\right.$

  * 6)

As $e\in\mathcal{R}$ if $\mathcal{R}$ is open then
$e\in\hbox{int}\mathcal{R}$. Suppose that $e\in\hbox{int}\mathcal{R}$ and take
$g\in\mathcal{R}$. Then there are $t\in\mathbb{N}$ and $u\in\mathcal{U}$ such
that

$\varphi(t,e,u)=g.$

Take $V_{e}\subset\mathcal{R}$ a neighborhood of $e$. As $f_{u}$ is
diffeomorphism, the set $V_{g}=\varphi(t,V_{e},u)$ is a neighborhood of $g$
and $\varphi(t,V_{e},u)\subset\varphi(t,\mathcal{R},u)\subset\mathcal{R}$.

$\square$

Since the map $f_{u}$ of the linear system (5) is a diffeomorphism of $G$ for
each $u\in U$ we can define its reversed counterpart by

$\displaystyle h_{k+1}=\tilde{f}_{u_{k}}(h_{k}),\ u\in\mathcal{U},$ (7)

where $\tilde{f}_{u}(h)=f_{u}^{-1}(e)f_{0}^{-1}(h)$ for all $h\in G$. Note
that (7) is also a linear system on $G$ with $\tilde{f}_{0}(x)=f_{0}^{-1}(x)$,
hence $d\tilde{f}_{0}=df_{0}^{-1}$. In this case, if $\alpha$ is an eigenvalue
of $df_{0}$, $\alpha^{-1}$ is an eigenvalue of $d\tilde{f}_{0}$.

Therefore, we can consider the generalized eigenspaces

$\mathfrak{g}_{*}^{+}=\sum_{|\alpha|>1}\mathfrak{g}^{*}_{\alpha},\
\mathfrak{g}_{*}^{-}=\sum_{|\alpha|<1}\mathfrak{g}^{*}_{\alpha},\
\mathfrak{g}_{*}^{0}=\sum_{|\alpha|=1}\mathfrak{g}^{*}_{\alpha}.$

where $\mathfrak{g}_{\alpha}^{*}$ is the generalized eigenspace associated
with the eigenvalue $\alpha$ of $d\tilde{f}_{0}$. One can easily see that
$\mathfrak{g}^{-}_{*}=\mathfrak{g}^{+},\mathfrak{g}^{+}_{*}=\mathfrak{g}^{-}$
and $\mathfrak{g}^{0}_{*}=\mathfrak{g}^{0}$. Analogously, $G_{*}^{+}=G^{-}$,
$G_{*}^{-}=G^{+}$ and $G_{*}^{0}=G^{0}$.

###### Example 13

Consider the linear system presented in Example 10. In this case, the reverse
counterpart of this system is given by

$x_{k+1}=A^{-1}x_{k}-A^{-1}Bu_{k},\ \ u_{k}\in U\subset\mathbb{R}^{m}.$

Let us denote by $\mathcal{R}_{k}^{*}$ and $\mathcal{C}_{k}^{*}$ the reachable
and the controllable set from $e$ up to time $k$ of (7), respectively. Then
the following lemma holds.

###### Lemma 14

If holds that ’$\mathcal{R}_{k}^{*}=\mathcal{C}_{k}$ and
$\mathcal{R}_{k}=\mathcal{C}_{k}^{*}$, for all $k\in\mathbb{N}$.

Proof. Note that for any $k\in\mathbb{N}$, $g\in G$ and $u\in\mathcal{U}$, we
have

$\displaystyle\tilde{f}_{u_{k-1}}\circ\cdots\circ\tilde{f}_{u_{0}}(g)=e\Leftrightarrow\tilde{f}_{u_{0}}^{-1}\circ\cdots\circ\tilde{f}_{u_{k-1}}^{-1}(e)=g\Leftrightarrow
f_{u_{0}}\circ\cdots\circ f_{u_{k-1}}(e)=g,$

because $\tilde{f}_{u}^{-1}=f_{u}$, for all $u\in U$. If $\varphi^{*}$ denote
the solution of (7), then

$\varphi^{*}(k,g,u)=e\Leftrightarrow\varphi(k,e,\tilde{u})=g,$

where

$\tilde{u}=\left\\{\begin{array}[]{lcc}u_{k-1-j},&j<k\\\ 0,&j\geq
k\end{array},\right.$

This shows that $\mathcal{R}_{k}=\mathcal{C}_{k}^{*}$. The other equality
follows analogously. $\square$

In sequence we present a lemma that will be widely used throughout the work.

###### Lemma 15

Let $g\in\mathcal{R}$ such that $f_{0}^{k}(g)\in\mathcal{R}$, for all
$k\in\mathbb{Z}$. Then $\mathcal{R}\cdot g\subset\mathcal{R}$.

Proof. In fact, given $k\in\mathbb{N}$ and $u\in\mathcal{U}$ one has

$\displaystyle\varphi(k,e,u)g=\varphi(k,e,u)f_{0}^{k}(f_{0}^{-k}(g))=\varphi(k,f_{0}^{-k}(g),u),$

for all $g\in G$. Hence, if we assume that $g$ is an element in $\mathcal{R}$
satisfying the required assumptions, then there are $l\in\mathbb{N}_{0}$ and
$v\in\mathcal{U}$ such that $f_{0}^{-k}(g)=\varphi(l,e,v)$. Then

$\varphi(k,e,u)g=\varphi(k,f_{0}^{-k}(g),u)=\varphi(k,\varphi(l,e,v),u)=\varphi(k+l,e,w),$

with

$w=\left\\{\begin{array}[]{lcc}v_{j},&j<l\\\ u_{j-l},&j\geq
l\end{array},\right.$

that is $\mathcal{R}\cdot g\subset\mathcal{R}$. $\square$

To simplify, we denote $d(f_{0})_{e}$ by $df_{0}$. Since $f_{0}$ and $df_{0}$
are automorphisms on $G$ and $\mathfrak{g}$, respectively, one has

$f_{0}^{n}(\exp X)=\exp df_{0}^{n}X,$

for all $n\in\mathbb{Z}$ and all $X\in\mathfrak{g}$. This fact is useful in
the next result.

###### Corollary 16

Suppose that $H$ is a connected $f_{0}$-invariant Lie subgroup of $G$ with Lie
algebra $\mathfrak{h}$. If $\exp{X}\in\mathcal{R}$, for any $X\in\mathfrak{h}$
then $H\subset\mathcal{R}$.

Proof. Since $H$ is $f_{0}$-invariant, then $\mathfrak{h}$ is
$df_{0}$-invariant. Hence

$f_{0}^{k}(\exp{X})=\exp{df_{0}^{k}(X)}\in\mathcal{R},k\in\mathbb{Z}.$

The result follows by connectedness of $H$ and Lemma 15. $\square$

###### Corollary 17

Suppose that $H$ is a connected Lie subgroup of $G$ and there is a
neighborhood $B$ of $e$ in $H\cap\mathcal{R}$ which is invariant by $f_{0}$
and $f_{0}^{-1}$, then $H$ is $f_{0}$-invariant and $H\subset\mathcal{R}$.

Proof. The invariance of $B\subset H\cap\mathcal{R}$ by $f_{0}$ and
$f_{0}^{-1}$ yields that $B^{n}\subset\mathcal{R}$ due Lemma 15. Since $H$ is
connected, then $H=\bigcup_{n\in\mathbb{N}}B^{n}\subset\mathcal{R}$. $\square$

Following [9, Section 3], we finish this section by recalling that given a
linear system (5) on a connected Lie group $G$ and $H$ is a closed
$f_{0}$-invariant subgroup of $G$, one can induce the following control system
on the homogeneous space $G/H$

$x_{k+1}=\bar{f}(x_{k},u_{k}),\ x_{k}\in G/H,\
u=(u_{i})_{i\in\mathbb{N}_{0}}\in\mathcal{U},$ (8)

where $\bar{f}_{u}$ is a diffeomorphism of $G/H$ given by
$\bar{f}_{u}(gH)=f_{u}(g)H$ for each $u\in U$, which satisfies
$\bar{f}_{u}^{k}\circ\pi=\pi\circ f_{u}^{k}$ for all $k\in\mathbb{Z}$, where
$\pi:G\rightarrow G/H$ is the standard projection. If $\varphi$ denotes the
solution of (5) and $\bar{\varphi}$ is the solution of (8), it holds that
$\pi(\varphi(k,g,u))=\bar{\varphi}(k,\pi(g),u)$ and
$\bar{\varphi}(k,gH,u)=\mathcal{L}_{\varphi_{k,u}(e)}(\bar{f}_{0}^{k}(gH))$,
where $\mathcal{L}_{g}$ denotes the left translation in $G/H$ given by
$xH\mapsto gxH$ for each $g\in G$. The system (8) is called induced linear
system on $G/H$.

##  3 Controllability on solvable Lie groups

The aim of this section is to present a sufficient condition for the
controllability of discrete-time linear systems on solvable connected Lie
groups. If $\mathfrak{g}$ is the Lie algebra of $G$, it is known from the
theory of smooth manifolds that there is a neigborhood $U\subset\mathfrak{g}$
of $0$ such that the function $\exp:U\longrightarrow V$ is a diffeomorphism.
Given $X,Y\in U$, then we can write

$\exp{(X)}\exp{(Y)}=\exp{(C(X,Y))}.$

The Baker-Campbell-Hausdorff series gives an expression for the element
$C(X,Y)$, which is construct as

$C(X,Y)=X+Y+\sum_{n\geq 2}C_{n}(X,Y),$ (9)

where

$\begin{array}[]{ccl}C_{2}(X,Y)&=&-\frac{1}{2}[X,Y],\\\
C_{3}(X,Y)&=&\frac{1}{12}\left([X,[X,Y]]-[Y,[Y,X]]\right),\\\
C_{4}(X,Y)&=&-\frac{1}{24}([Y,[X,[X,Y]]]),\end{array}$

and so on. The others elemenets of $C_{n}$ are very difficult to express. In
[17, Proposition 8.5, chapter 8] is proved that the expression 9 is always
valid under the condition $|X|,|Y|\leq\rho$, for some $\rho>0$ small enough.

Consider $N\subset G^{0}$ a solvable Lie subgroup of $G$. If $\mathfrak{n}$ is
the Lie subalgebra of $N$, there exists $l\in\mathbb{N}$ such that its derived
series satisfies

$\mathfrak{n}_{1}\supset\mathfrak{n}_{2}\supset\cdots\supset\mathfrak{n}_{l+1}=\\{0\\}.$

where $\mathfrak{n}_{1}=\mathfrak{n}$ and
$\mathfrak{n}_{i+1}=[\mathfrak{n}_{i},\mathfrak{n}_{i}]$. Each set
$\mathfrak{n}_{i}$ can be considered as

$\mathfrak{n}_{i}=\bigoplus_{|\alpha|=1}\mathfrak{n}_{i,\alpha},$ (10)

where $\mathfrak{n}_{i,\alpha}=\mathfrak{g}_{\alpha}\cap\mathfrak{n}_{i}$. We
also can consider

$\mathfrak{n}_{i,\alpha}=\bigcup_{j\in\mathbb{N}_{0}}\mathfrak{n}_{i,\alpha}^{j},\mathfrak{n}_{i,\alpha}^{j}=\\{X\in\mathfrak{n}_{i,\alpha}:(df_{0}-\alpha)^{j}X=0\\}.$

In this section, we will suppose that $\mathcal{R}$ is open. The next lemma is
the most important in this section.

###### Lemma 18

Let $N\subset G^{0}$ be a solvable connected $f_{0}$-invariant Lie subgroup of
$G^{0}$. Then $N\subset\mathcal{R}.$

Proof. As $N$ is $f_{0}$-invariant, then $\mathfrak{n}$ is $df_{0}$-invariant,
which implies in $df_{0}(\mathfrak{n})=\mathfrak{n}$. Consequently
$df_{0}(\mathfrak{n}_{i})=\mathfrak{n}_{i}$, for any $i=1,\ldots,l+1.$ Let us
denote by $N_{i}=\langle\exp{\mathfrak{n}_{i}}\rangle$ the connected subgroup
associated to the ideal $\mathfrak{n}_{i}$, as in the expression (10). Each
$N_{i}$ is a normal subgroup of $N$ and $f_{0}$-invariant, since $f_{0}$ is a
automorphism and each $\mathfrak{n}_{i}$ is a ideal.

Note that if for every $X\in\mathfrak{n}_{i,\alpha}$ we have
$\exp{X}\in\mathcal{R}$, then $N_{i}\in\mathcal{R}$. In fact, as
$\mathfrak{n}_{i}=\bigoplus_{|\alpha|=1}\mathfrak{n}_{i,\alpha}$, the set

$B=\prod_{|\alpha|=1}\exp{\mathfrak{n}_{i,\alpha}}$

is a neighborhood of $e$ in $N_{i}$ (see remark (19)). Besides, as
$df_{0}^{k}(\mathfrak{n}_{i,\alpha})=\mathfrak{n}_{i,\alpha}$ for every
$k\in\mathbb{Z}$ we have

$f_{0}^{k}(\exp{\mathfrak{n}_{i,\alpha}})=\exp{(df_{0}^{k}(\mathfrak{n}_{i,\alpha}))}=\exp{\mathfrak{n}_{i,\alpha}},$

for every $k\in\mathbb{Z}$. It is clear that $B$ is $f_{0}$ and
$f_{0}^{-1}$-invariant and as $\exp{X}\in\mathcal{R}$ for every
$X\in\mathfrak{n}_{i,\alpha}$, the Lemma 15 ensures that
$B\subset\mathcal{R}$. By Corollary 17 we get $N_{i}\subset\mathcal{R}$.

Now, we claim that if $N_{i+1}\subset\mathcal{R}$ then
$N_{i}\subset\mathcal{R}$, for every $i=0,\ldots,l$. In fact, note that we
just need to prove that $\exp{X}\in\mathcal{R}$, for every
$X\in\mathfrak{n}_{i,\alpha}$ with $|\alpha|=1$. Using the decomposition (10)
it is enough to prove that for every $j\in\mathbb{N}_{0}$ and
$X\in\mathfrak{n}^{j}_{i,\alpha}$ we have $\exp{X}\in\mathcal{R}$. We prove
this by induction. It is clear for $j=0$ since
$\mathfrak{n}_{i,\alpha}^{0}=\\{0\\}$ and $e\in\hbox{int}\mathcal{R}$. If it
is true for $j>0$ then $\exp{Z}\in\mathcal{R}$ for every
$Z\in\mathfrak{n}^{j-1}_{i,\alpha}$. By [8, Lemma 13], there is a sequence of
natural numbers $S_{\alpha}=\\{n_{k}\\}_{k\in\mathbb{N}}$ such that
$n_{k}\rightarrow\infty$ and $\alpha^{n_{k}}\rightarrow 1$, given that
$|\alpha|=1$. Let us take $X\in\mathfrak{n}^{j}_{i,\alpha}$ and consider
$U\subset\mathfrak{g}$ such that $\exp:U\rightarrow\exp{U}$ is a
diffeomorphism with $\exp{U}\subset\mathcal{R}$. There is a $m\in\mathbb{N}$
such that $v=\frac{X}{m}\in U$. Then, there is a $\tau\in\mathbb{N}$ such that
$\exp{v}\in\mathcal{R}_{\tau}$. By Proposition 12, item 2, we have
$\mathcal{R}_{\tau}\subset\mathcal{R}_{\tau n}$ for any $n\in\mathbb{N}$.
Using the polar form of a complex number, we get $\alpha^{\tau n}=e^{i(\tau
n\theta)}=\cos{(\tau n\theta)}+i\sin{(\tau n\theta)}$ for some
$\theta\in(0,2\pi]$. Without loss of generality, let us suppose that
$\theta\in 2\pi\mathbb{Q}$ (see remark (19)). Then $n\theta=2\pi N$, for some
$n\in\mathbb{N}$ and $N\in\mathbb{Z}\setminus\\{0\\}$. Consider the sequence
$S_{\alpha}$ such that every element of it satisfies $n<n_{k}$. Hence

$\alpha^{n}=\cos{(n\theta)}+i\sin{(n\theta)}=1.$ (11)

Also, for the sequence $n_{k}$, we get $n_{k}\theta=2\pi N_{k}$ for every
$k\in\mathbb{N}$ and

$\alpha^{n_{k}}v=\left(\cos{(n_{k}\tau\theta)}+i\sin{(n_{k}\tau\theta)}\right)\frac{X}{m}=\cos{(2N_{k}\pi)}\frac{X}{m}=\frac{X}{m},\forall
n_{k}\in S_{\alpha}.$ (12)

such that $n<n_{k}$. Taking a subsequence $\\{n_{s_{j}}\\}\subset S_{\alpha}$,
we have $n_{k_{2}}-n_{k_{1}}\in\mathbb{N}$, then
$\exp{v}\in\mathcal{R}_{\tau(n_{k_{2}}-n_{k_{1}})}$ for any
$n_{k_{1}},n_{k_{2}}\in S_{\alpha}$. As $v\in\mathfrak{n}_{i,\alpha}^{j}$,
given any $k\in\mathbb{N}$ satisfying $n<k$ we obtain

$(df_{0})^{k}v=(df_{0}-\alpha+\alpha)^{k}v=\sum_{i=0}^{k}\binom{k}{i}(df_{0}-\alpha)^{k-i}\alpha^{i}v=\alpha^{k}v+v_{k}$

with $\alpha^{k}v\in\mathfrak{n}_{i,\alpha}^{j}$ and

$v_{k}=\sum_{i=0}^{k-1}\binom{k}{i}(df_{0}-\alpha)^{k-i}\alpha^{i}v\in\mathfrak{n}_{j,\alpha}^{j-1}.$

for some $k\geq j$. Hence we can consider

$\alpha^{k}v=df_{0}^{k}v+u_{k},\forall k\in\mathbb{N}$ (13)

with $u_{k}=-v_{k}$. Let us set conditions for the convergence of BCH series.
Consider the proposition [18, Corollary 2.19], which states that
$[\mathfrak{n},\mathfrak{n}]\subset\mathfrak{rn}(\mathfrak{g}_{0})$, where
$\mathfrak{rn}(\mathfrak{g}_{0})$ is the nilradical of $\mathfrak{g}_{0}$.
Then every element in the form $[Y,Z]$ is nilpotent. We can conclude that
$C_{n}(Y,Z)=0$ for some $n\in\mathbb{N}$ which depends on $Y$ and $Z$ and also
$C(Y,Z)$ is a finite sum, hence, convergent.

Now, consider $k\in\mathbb{N}$ such that $\alpha^{k}v=v$, for any $n\leq k$.
The expression (13) holds for every $k\in\mathbb{N}$. Using BCH series, we
have

$\exp{(df_{0}^{\tau n_{k_{1}}}v)}\exp{(df_{0}^{\tau
n_{k_{2}}}v)}=\exp{(df_{0}^{\tau n_{k_{1}}}v+df_{0}^{\tau
n_{k_{2}}}v+O_{1})},$

where $O_{1}$ is given by the Brackets between $df_{0}^{\tau n_{k_{1}}}v$ and
$df_{0}^{\tau n_{k_{2}}}v$, which is in
$[\mathfrak{n}^{j}_{i,\alpha},\mathfrak{n}^{j}_{i,\alpha}]\subset[\mathfrak{n}_{i},\mathfrak{n}_{i}]=\mathfrak{n}_{i+1}$
and we have $O_{1}\in\mathfrak{n}_{i+1}$. By the Lemma 6, we obtain

$\exp{(df_{0}^{\tau n_{k_{1}}}v+df_{0}^{\tau
n_{k_{2}}}v+O_{1})}=\exp{(df_{0}^{\tau n_{k_{1}}}v+df_{0}^{\tau
n_{k_{2}}}v)}g_{1}$

for some $g_{1}\in N_{i+1}\subset\mathcal{R}$. By item $(4)$ in the
Proposition 12 we get

$\exp{(df_{0}^{\tau n_{k_{1}}}v)}\exp{(df_{0}^{\tau n_{k_{2}}}v)}=f_{0}^{\tau
n_{k_{1}}}(\exp{v})f_{0}^{\tau n_{k_{2}}}(\exp{v})\in\mathcal{R}_{2(\tau
n_{k_{2}}-\tau n_{k_{1}})}\subset\mathcal{R}.$

Once $g_{1},g_{1}^{-1}\in N_{i+1}\subset\mathcal{R}$, we obtain

$\exp{(df_{0}^{\tau n_{k_{1}}}v+df_{0}^{\tau n_{k_{2}}}v)}=\exp{(df_{0}^{\tau
n_{k_{1}}}v+df_{0}^{\tau n_{k_{2}}}v+O_{1})}g_{1}^{-1}\in\mathcal{R}\cdot
g_{1}^{-1}\subset\mathcal{R}.$

By Proposition 12 item $2$, we can choose
$n_{k_{3}}\in\\{n_{s_{j}}\\}_{j\in\mathbb{N}}$ such that

$\exp{(df_{0}^{\tau n_{k_{1}}}v+df_{0}^{\tau n_{k_{2}}}v)}\in\mathcal{R}_{\tau
n_{k_{3}}}.$

Using BCH series we have

$\exp{(df_{0}^{\tau n_{k_{1}}}v+df_{0}^{\tau n_{k_{1}}}v)}\exp{(df_{0}^{\tau
n_{k_{3}}}v)}=\exp{(\sum_{i=1}^{3}df_{0}^{\tau
n_{k_{i}}}v+O_{2})},O_{2}\in\mathfrak{n}_{i+1}.$ (14)

where the convergence is discussed in the remark (19). Again by the Lemma 6 we
get

$\exp{(\sum_{i=1}^{3}df_{0}^{\tau
n_{k_{i}}}v+O_{2})}=\exp{(\sum_{i=1}^{3}df_{0}^{\tau
n_{k_{i}}}v)}g_{2},g_{2}\in N_{i+1}.$

Then

$\exp{(df_{0}^{\tau n_{k_{1}}}v+df_{0}^{\tau n_{k_{2}}}v)}\exp{(df_{0}^{\tau
n_{k_{3}}}v)}\in\mathcal{R}_{\tau(n_{k_{3}}+n_{k_{2}}-n_{k_{1}})}\subset\mathcal{R}.$

and again by the $f_{0}$-invariance of $N_{i+1}$ we have

$\exp{(\sum_{i=1}^{3}df_{0}^{\tau
n_{k_{i}}}v)}=\exp{(\sum_{i=1}^{3}df_{0}^{\tau
n_{k_{i}}}v+O_{2})}g_{2}^{-1}\in\mathcal{R}\cdot
g_{2}^{-1}\subset\mathcal{R}.$

Repeating the idea $m-2$ times, we get

$\exp{(\sum_{i=1}^{m}df_{0}^{\tau n_{k_{i}}}v)}\in\mathcal{R}.$

Summing up to $m$ in the expression $\alpha^{k}v=df_{0}^{k}v+u_{k}$ we get

$X=\sum_{i=1}^{m}\alpha^{\tau n_{k_{j}}}v=\sum_{i=1}^{m}df_{0}^{\tau
n_{k_{i}}}v+u^{\prime}$ (15)

from the fact of $\sum_{i=1}^{m}\alpha^{\tau n_{k_{j}}}=m$, for
$u^{\prime}\in\mathfrak{n}_{i,\alpha}^{j-1}$ defined by
$u^{\prime}=\sum_{j=1}^{m}u_{\tau n_{k_{j}}}$. By the $f_{0}$-invariance of
$\mathfrak{n}_{i,\alpha}^{j-1}$ and the fact of $u^{\prime}$ and
$\sum_{i=1}^{m}df_{0}^{\tau n_{k_{i}}}v$ comutes, we get

$\exp{(X)}=\exp{\left(\sum_{i=1}^{m}df_{0}^{\tau
n_{k_{i}}}v\right)}\exp{u^{\prime}}\in\mathcal{R}\cdot\exp{u^{\prime}}$

By the induction hypothesis

$f_{0}^{k}(\exp{u^{\prime}})=\exp{(df_{0}^{k}(u^{\prime}))}\in\mathcal{R},\forall
k\in\mathbb{Z}.$

The Lemma 6 guarantees that

$\exp{(X)}=\exp{\left(\sum_{i=1}^{m}df_{0}^{\tau
n_{k_{i}}}v\right)}\exp{u^{\prime}}\in\mathcal{R}\cdot\exp{u^{\prime}}\subset\mathcal{R}$

as requested.

To prove that $N\subset\mathcal{R}$, as $N_{l+1}=\\{e\\}\subset\mathcal{R}$,
then $N_{l}\subset\mathcal{R}$ by the previous affirmation. Using this
reasoning $l$ times, we get $N\subset\mathcal{R}$. $\square$

###### Remark 19

Here we have some refinements about the proof above. To prove that
$B=\prod_{|\alpha|=1}\exp{\mathfrak{n}_{i,\alpha}}$ is a neighborhood of $e$,
consider without loss of generality, that
$\mathfrak{n}_{i}=\mathfrak{n}_{i,\alpha_{1}}\oplus\mathfrak{n}_{i,\alpha_{2}}$.
If $\\{X_{1},...,X_{k_{1}}\\}$ and $\\{Y_{1},...,Y_{k_{2}}\\}$ are basis of
$\mathfrak{n}_{i,\alpha_{1}}$ and $\mathfrak{n}_{i,\alpha_{2}}$ respectivelly,
consider the function
$\psi:\mathbb{R}^{k_{1}}\times\mathbb{R}^{k_{2}}\longrightarrow N_{i}$ given
by

$\psi(t_{1},...,t_{k_{1}},s_{1},...,s_{k_{2}})=\exp{(t_{1}X_{1}+...+t_{k_{1}}X_{k_{1}})}\exp{(s_{1}Y_{1}+...+s_{k_{2}}Y_{2})}.$

Then $\psi(0,0)=e$, $d(\psi)_{(0,0)}(e_{j},0)=X_{j}$ and
$d(\psi)_{(0,0)}(0,e_{k})=Y_{k}$ with $e_{k}$ the $k^{th}$ vector of the
canonical basis of $\mathbb{R}^{d}$. Then $d(\psi)_{(0,0)}$ takes basis to
basis and then, is a linear isomorphism. By the inverse theorem function,
$\psi$ is a local diffeomorphism. In this case, there is a open set
$U_{1}\times U_{2}\subset\mathbb{R}^{k_{1}}\times\mathbb{R}^{k_{2}}$
containing $(0,0)$ such that $\psi(U_{1}\times U_{2})=\exp{V_{1}}\exp{V_{2}}$
is a open neighborhood of $e\in N_{i}$, where
$V_{1}=\\{t_{1}X_{1}+...+t_{k_{1}}X_{k_{1}}:(t_{1},...,t_{k_{1}})\in U_{1}\\}$
and $V_{2}=\\{s_{1}Y_{1}+...+s_{k_{2}}Y_{k_{1}}:(s_{1},...,s_{k_{2}})\in
U_{2}\\}$. Then
$\exp{(\mathfrak{n}_{i,\alpha_{1}})}\exp{(\mathfrak{n}_{i,\alpha_{2}})}$ is a
neighborhood of $e$ in $N_{i}$. The reasoning above can be generalized for any
sum $\mathfrak{n}_{i}=\oplus_{|\alpha|=1}\mathfrak{n}_{i,\alpha}$.

Now, in the expression (11) we suppose wlog that $\theta\in 2\pi\mathbb{Q}$.
If we suppose otherwise, one can consider a sequence
$\\{n_{k}\\}\subset\mathbb{N}$ such that
$\alpha^{n_{k}}=\cos{n_{k}\theta}+i\sin{n_{k}\theta}\longrightarrow 1$ [8,
Lemma 13]. Summing up to $m$ in the expression
$\alpha^{k}v=df_{0}^{k}v+u_{k}$, we would have

$X+\varepsilon_{k}X=\left(\sum_{i=1}^{m}df_{0}^{\tau
n_{k_{i}}}v\right)+u^{\prime}.$

with $\varepsilon_{k}\longrightarrow 0$. Thus, it is enough to consider the
limit when $k\longrightarrow\infty$ in the expression above to get the
equality (15).

From now on, we consider the dynamical Lie algebras $\mathfrak{g}^{*}$,
$*\in\\{+,-,0\\}$ associated to the Lie algebra automorphism $df_{0}$, and the
corresponding connected Lie groups $G^{*}$, $*\in\\{+,-,0\\}$, unless stated
otherwise. Moreover, we often assume that $\mathcal{R}$ is an open set of $G$.

With the above lemma we can prove the following controllability results.

###### Theorem 20

Let $G$ be a connected solvable Lie group and consider the system (5) defined
on $G$. Then $G^{+,0}\subset\mathcal{R}$.

Proof. The space $\mathfrak{g}^{+}$ is the unstable subspace associated with
the differential $df_{0}$. Since $G^{+}=\exp{\mathfrak{g}^{+}}$ is nilpotent
and taking $g\in G^{+}$, then exists a $X\in\mathfrak{g}^{+}$ such that
$g=\exp{X}$. As $0\in\mathfrak{g}^{+}$ is stable in negative time there is a
$k\in\mathbb{N}$ such that $df_{0}^{-k}X$ is as close as necessary of $0$ for
$k$ large enough. By continuity,

$f_{0}^{-k}(\exp{X})=\exp{(df_{0}^{-k}X)}\in\mathcal{R}.$

Then $g=\exp{X}\in f_{0}^{-k}(\mathcal{R})\subset\mathcal{R}$, that is
$G^{+}\subset\mathcal{R}$. Solvability of $G$ implies that $G^{0}$ is
solvable. Therefore $G^{0}\subset\mathcal{R}$ by Lemma 18. Thus
$G^{+,0}\subset\mathcal{R}$. $\square$

The final result of this section gives a sufficient condition for
controllability of $(\ref{linearcontrolsystem}).$ Remember that the
controllability of (5) can be seen by proving that
$G=\mathcal{R}\cap\mathcal{C}=\mathcal{R}\cap\mathcal{R}^{*}$. The following
result present us a sufficient condition for the controllability of discrete-
time linear systems on solvable Lie groups.

###### Theorem 21

The linear system (5) is controllable if $\mathcal{R}$ and $\mathcal{C}$ are
open sets and $G^{0}=G$.

Proof. By Theorem 20, it follows that $G^{0,+}\subset\mathcal{R}$ and
$G^{0,-}\subset\mathcal{R}^{*}=\mathcal{C}$. As
$\mathfrak{g}=\mathfrak{g}^{0}$ then $G=\mathcal{R}\cap\mathcal{R}^{*}$.
Following the ideas above the system (5) is controllable. $\square$

###  3.1 Controllability and accessibility of affine Lie group

In this subsection, as application of the previous results we study
controllability of linear systems on two dimensional affine Lie group. The
real abelian solvable Lie groups of dimension $2$ are $\mathbb{R}^{2}$,
$\mathbb{T}\times\mathbb{R}$ and $\mathbb{T}^{2}$ (see e.g. [15]). In the non-
abelian case, the unique (up to an isomorphism) real solvable lie group is the
open half plane $G=\mathbb{R}^{+}\ltimes\mathbb{R}$, endowed with the product

$(x_{1},y_{1})\cdot(x_{2},y_{2})=(x_{1}x_{2},y_{2}+x_{2}y_{1}).$

The Lie group $(G,\cdot)$ is called affine group and denoted by
$\hbox{Aff}(2,\mathbb{R})$. In particular, the automorphisms of
$\hbox{Aff}(2,\mathbb{R})$ are given by

$\phi(x,y)=(x,a(x-1)+dy),$

with $d\in\mathbb{R}^{*}$ and $a\in\mathbb{R}$. Then, the linear systems on
$\hbox{Aff}(2,\mathbb{R})$ can be defined by the functions

$f((x,y),u):=f_{u}(x,y)=(h(u)x,a(x-1)+dy+g(u)x),$ (16)

where $h:\mathbb{R}^{m}\rightarrow\mathbb{R}^{+}$ and
$g:\mathbb{R}^{m}\rightarrow\mathbb{R}$ are $\mathcal{C}^{\infty}$ functions
satisfying $h(0)=1$ and $g(0)=0$. In fact, note that
$f_{0}(x,y)=(x,a(x-1)+dy)$ is an automorphism of $\hbox{Aff}(2,\mathbb{R})$.
Note that

$f_{0}^{-1}(x,y)=\left(x,\frac{-a}{d}(x-1)+\frac{y}{d}\right)$

and

$\displaystyle f_{u}(x,y)$ $\displaystyle=$
$\displaystyle(h(u)x,a(x-1)+dy+g(u)x)$ $\displaystyle=$
$\displaystyle(h(u),g(u))(x,a(x-1)+dy)$ $\displaystyle=$ $\displaystyle
f_{u}(1,0)f_{0}(x,y),$

for each $u\in U$ and $(x,y)\in\hbox{Aff}(2,\mathbb{R})$. Hence $f$ defines a
linear system on $\hbox{Aff}(2,\mathbb{R})$ given by

$x_{k+1}=f(x_{k},u_{k}),\ k\in\mathbb{N},\ u\in U,$ (17)

where we assume that $U$ is a compact and convex neighborhood of
$0\in\mathbb{R}^{m}$. Note also that

$f_{0}^{k}(x,y)=(x,a(x-1)(\sum_{j=0}^{k-1}d^{j})+d^{k}y),\forall k\geq 1.$

From this previous equality and the Proposition 11, one can define the
solution $\varphi(k,(x,y),u)$ for every $(x,y)\in\hbox{Aff}(2,\mathbb{R})$ and
$u\in\mathcal{U}$. In the identity we have that

$\varphi(k,(1,0),u)=\left\\{\begin{array}[]{ccc}(1,0),&if&k=0\\\
(h(u_{0}),g(u_{0})),&if&k=1\\\ \hat{f}_{k}(u),&if&k\geq 2\end{array},\right.$

with

$\hat{f}_{k}(u)=\left(\prod_{j=0}^{k-1}h(u_{k-1-j}),-a\left(\sum_{j=0}^{k-2}d^{j}\right)+d^{k-1}g(u_{0})+\sum_{j=0}^{k-2}\left(d^{k-2-j}(a+g(u_{j+1}))\prod_{i=0}^{j}h(u_{i})\right)\right).$

Then by Proposition 11, we get

$\displaystyle\varphi(k,(x,y),u)$ $\displaystyle=$
$\displaystyle(\prod_{j=0}^{k-1}h(u_{k-1-j})x,d^{k-1}(a+g(u_{0}))x+\sum_{j=0}^{k-2}d^{k-2-j}(a+g(u_{j+1}))\prod_{i=0}^{j}h(u_{i})x$
$\displaystyle+d^{k}y-a(\sum_{j=0}^{k-1}d^{k-1-j})).$

for every $(x,y)\in\hbox{Aff}(2,\mathbb{R})$ and $u=(u_{i})\in\mathcal{U}$.

###### Remark 22

As we can see above, it is difficult to work with the solution of (17).
However, by considering the case $h(u)=1$, for every
$u=(u_{i})\in\mathcal{U}$, the solution are given by

$\varphi(k,(x,y),u)=\left(x,\sum_{j=0}^{k-1}d^{k-1-j}((a+g(u_{j}))x-a)+d^{k}y\right)$

which means that for each $(x,y)\in\hbox{Aff}(2,\mathbb{R})$, the solutions
are contained in the set $\\{x\\}\times\mathbb{R}$. Then
$\hbox{int}\mathcal{R}(x,y)=\emptyset$, for every pair
$(x,y)\in\hbox{Aff}(2,\mathbb{R})$. Therefore, the results of the previous
sections does not apply in this case.

Consider the linear system (17) and define the vector fields

$\displaystyle X^{+}_{u}(x)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial v}\bigg{|}_{v=0}f_{u}^{-1}\circ
f_{u+v}(x)$ $\displaystyle X^{-}_{u}(x)$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial v}\bigg{|}_{v=0}f_{u}\circ
f_{u+v}^{-1}(x)$ $\displaystyle\hbox{Ad}_{u_{k}\ldots u_{1}}X^{+}_{u_{0}}(x)$
$\displaystyle=$ $\displaystyle(df_{u_{k}}\circ\cdots\circ
f_{u_{1}})_{e}^{-1}X^{+}_{u_{0}}(f_{u_{k}}\circ\cdots\circ f_{u_{1}}(x)).$
$\displaystyle\hbox{Ad}^{-1\cdots-1}_{u_{k}\ldots u_{1}}X^{-}_{u_{0}}(x)$
$\displaystyle=$ $\displaystyle(df_{u_{k}}^{-1}\circ\cdots\circ
f_{u_{1}}^{-1})_{e}^{-1}X^{-}_{u_{0}}(f_{u_{k}}^{-1}\circ\cdots\circ
f_{u_{1}}^{-1}(x)).$

Moreover, defining the sets

$\displaystyle\Gamma^{+}=\\{\hbox{Ad}_{u_{k}\cdots
u_{1}}X^{+}_{u_{0}}:k\in\mathbb{N},u_{k},\ldots,u_{0}\in U\\}$
$\displaystyle\Gamma^{-}=\\{\hbox{Ad}^{-1\cdots-1}_{u_{k}\cdots
u_{1}}X^{-}_{u_{0}}:k\in\mathbb{N},u_{k},\ldots,u_{0}\in U\\}$

and considering $\Gamma^{+}(x)$ as the spanned vector subspace generated by
the vectors of $\Gamma^{+}$ on $x$, Jakubcyk and Sontag [19] proved the
following accessibility criteria.

###### Theorem 23

Consider a discrete-time control system of the form (4) on a $n$-dimensional
smooth manifold $M$, where the control range $U\subset\mathbb{R}^{m}$ is a
compact and convex neighborhood of $0$ in $\mathbb{R}^{m}$ and
$f_{u}:M\rightarrow M$ is a diffeomorphism for any $u\in U$.

  * 1-

The system is forward accessible if and only if

$\dim\Gamma^{+}(x)=n,\forall x\in M.$

  * 2-

The system is backward accessible if and only if

$\dim\Gamma^{-}(x)=n,\forall x\in M.$

  * 3-

The system is accessible if both conditions are satisfied.

For linear systems on $\hbox{Aff}(2,\mathbb{R})$, the previous result can be
applied in order to get a sufficient condition for the accessibility of (17).

###### Proposition 24

The system (17) is accessible if $-ah^{\prime}(0)\neq g^{\prime}(0)(d-1)$ and
$h^{\prime}(0)\neq 0$.

Proof. Given any $u\in U$ we have

$df_{u}=\begin{bmatrix}h(u)&0\\\ a+g(u)&d\end{bmatrix}\mbox{ and
}(df_{u})^{-1}=\begin{bmatrix}\dfrac{1}{h(u)}&0\\\
-\dfrac{a+g(u)}{dh(u)}&\dfrac{1}{d}\end{bmatrix}.$

The compositions $f_{u}^{-1}\circ f_{u+v}(x,y)$ and $f_{u}\circ
f_{u+v}^{-1}(x,y)$ are given by

$\displaystyle f^{-1}_{u}\circ f_{u+v}(x,y)$ $\displaystyle=$
$\displaystyle\left(\frac{h(u+v)}{h(u)}x,\frac{x}{d}\left(g(u+v)+a-\frac{h(u+v)}{h(u)}(a+g(u))\right)+y\right),$
$\displaystyle f_{u}\circ f_{u+v}^{-1}(x,y)$ $\displaystyle=$
$\displaystyle\left(\frac{h(u)}{h(u+v)}x,y+x\left(\frac{g(u)}{h(u+v)}-\frac{g(u+v)}{h(u+v)}\right)\right).$

Then

$\displaystyle X^{+}_{u}(x,y)$ $\displaystyle=$
$\displaystyle\left(\frac{h^{\prime}(u)}{h(u)}x,\frac{x}{d}\left(\frac{h^{\prime}(u)}{h(u)}(-a-g(u))+g^{\prime}(u)\right)\right),$
$\displaystyle X^{-}_{u}(x,y)$ $\displaystyle=$
$\displaystyle\left(-\frac{h^{\prime}(u)}{h(u)}x,-\frac{g^{\prime}(u)}{h(u)}x\right).$

Besides, we have

$\displaystyle\hbox{Ad}_{u_{1}}X^{+}_{u_{0}}(x,y)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\dfrac{h^{\prime}(u_{0})}{h(u_{0})}x\\\
-\dfrac{a+g(u_{1})}{dh(u_{0})}h^{\prime}(u_{0})x-\dfrac{(a+g(u_{0}))h^{\prime}(u_{0})h(u_{1})}{d^{2}h(u_{0})}x+\dfrac{g^{\prime}(u_{0})h(u_{1})}{d^{2}}x\end{bmatrix},$
$\displaystyle\hbox{Ad}^{-1}_{u_{1}}X_{u_{0}}^{-}(x,y)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\dfrac{h^{\prime}(u_{0})}{h(u_{0})}x\\\
-\dfrac{a+g(u_{1})}{h(u_{0})h(u_{1})}h^{\prime}(u_{0})x-\dfrac{dg^{\prime}(u_{0})}{h(u_{0})h(u_{1})}x\end{bmatrix},$
$\displaystyle\hbox{Ad}_{u_{2}u_{1}}X^{+}_{u_{0}}(x,y)=\begin{bmatrix}\dfrac{h^{\prime}(u_{0})}{h(u_{0})}x\\\
T_{21}(x,u_{0},u_{1},u_{2})\end{bmatrix}\mbox{ and
}\hbox{Ad}^{-1-1}_{u_{2}u_{1}}X^{+}_{u_{0}}(x,y)=\begin{bmatrix}\dfrac{h^{\prime}(u_{0})}{h(u_{0})}x\\\
S_{21}(x,u_{0},u_{1},u_{2})\end{bmatrix},$

with

$\displaystyle T_{21}(x,u_{0},u_{1},u_{2})$ $\displaystyle=$
$\displaystyle-\dfrac{a+g(u_{2})}{dh(u_{0})}h^{\prime}(u_{0})x-\dfrac{(a+g(u_{1}))h^{\prime}(u_{0})h(u_{2})}{d^{2}h(u_{0})}x$
$\displaystyle-\dfrac{(a+g(u_{0}))h^{\prime}(u_{0})h(u_{1})h(u_{2})}{d^{3}h(u_{0})}x+\dfrac{g^{\prime}(u_{0})h(u_{2})h(u_{1})}{d^{3}}x,$
$\displaystyle S_{21}(x,u_{0},u_{1},u_{2})$ $\displaystyle=$
$\displaystyle-\frac{(a+g(u_{1}))h^{\prime}(u_{0})x}{h(u_{0})h(u_{1})}-\frac{d(a+g(u_{2}))h^{\prime}(u_{0})x}{h(u_{0})h(u_{1})h(u_{2})}-\frac{d^{2}g^{\prime}(u_{0})x}{h(u_{0})h(u_{1})h(u_{2})}.$

By taking $u_{0}=u_{1}=u_{2}=0$ in the above vector fields above we get

$\displaystyle\hbox{Ad}_{u_{2}u_{1}}X^{+}_{u_{0}}(x,y)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}h^{\prime}(0)x\\\
-\dfrac{ah^{\prime}(0)x}{d}\left(\dfrac{1}{d^{2}}+\dfrac{1}{d}+1\right)+\dfrac{g^{\prime}(0)}{d^{3}}x\end{bmatrix},$
$\displaystyle\hbox{Ad}_{u_{1}}X^{+}_{u_{0}}(x,y)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}h^{\prime}(0)x\\\
-\dfrac{ah^{\prime}(0)x}{d}\left(\dfrac{1}{d}+1\right)+\dfrac{g^{\prime}(0)}{d^{2}}x\end{bmatrix},$
$\displaystyle\hbox{Ad}^{-1-1}_{u_{2}u_{1}}X^{-}_{u_{0}}(x,y)$
$\displaystyle=$ $\displaystyle\begin{bmatrix}h^{\prime}(0)x\\\
-ah^{\prime}(0)x-dag^{\prime}(0)x-d^{2}g^{\prime}(0)x\end{bmatrix},$

and

$\displaystyle\hbox{Ad}^{-1}_{u_{1}}X^{-}_{u_{0}}(x,y)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}h^{\prime}(0)x\\\
-ah^{\prime}(0)x-dg^{\prime}(0)x\end{bmatrix}.$

If we consider the sets

$\displaystyle\alpha=\\{\hbox{Ad}^{-1-1}_{u_{2}u_{1}}X^{-}_{u_{0}}(x,y),\hbox{Ad}^{-1}_{u_{1}}X^{-}_{u_{0}}(x,y)\\}\hbox{
and
}\beta=\\{\hbox{Ad}_{u_{2}u_{1}}X^{+}_{u_{0}}(x,y),\hbox{Ad}_{u_{1}}X^{+}_{u_{0}}(x,y)\\}$

with $u_{0}=u_{1}=u_{2}=0$, one can prove that $\alpha$ and $\beta$ are linear
independent sets if, and only if, $h^{\prime}(0)\neq 0$ and
$-ah^{\prime}(0)\neq g^{\prime}(0)(d-1)$ as required. $\square$

Now, we prove that such condition guarantees that $\mathcal{R}$ and
$\mathcal{C}$ are open sets.

###### Proposition 25

If the system $(\ref{generalsystem})$ is accessible then it is reachable and
controllable sets are open.

Proof. Initially, note that

$\frac{\partial}{\partial(u,v)}f_{u,v}(e)=\begin{bmatrix}h^{\prime}(u)h(v)&h^{\prime}(v)h(u)\\\
g^{\prime}(u)h(v)&ah^{\prime}(v)+dg^{\prime}(v)+h^{\prime}(v)g(u)\end{bmatrix}$

and for $u=v=0$ one get

$\frac{\partial}{\partial(u,v)}f_{u,v}(e)=\begin{bmatrix}h^{\prime}(0)&h^{\prime}(0)\\\
g^{\prime}(0)&ah^{\prime}(0)+dg^{\prime}(0)\end{bmatrix}$ (18)

since $g(0)=0$ and $h(0)=1$. Then the matrix above has rank $2$ if, and only
if, $-ah^{\prime}(0)\neq g^{\prime}(0)(d-1)$ and $h^{\prime}(0)\neq 0$. The
matrix in (18) guarantees that $e\in\hat{\mathcal{R}}$, which is open.
Therefore, as $\hat{\mathcal{R}}\subset\mathcal{R}$ we have
$e\in\hbox{int}\mathcal{R}$. By the Proposition 12 item $6$, $\mathcal{R}$ is
open. Analogously we show that

$g_{k+1}=f_{u_{k}}^{-1}(e)f_{0}^{-1}(g_{k})$

then $\mathcal{C}=\mathcal{R}^{*}$ is also open. $\square$

The next theorem present a characterization of accessibility which is an
immediate statement from the above results.

###### Theorem 26

The reachable and controllable sets of the system (17) are open if and only if
(17) is accessible.

We conclude this section presenting a sufficient condition for the
controllability of (17).

###### Theorem 27

Consider the discrete-time linear system (17). If $h^{\prime}(0)\neq 0$,
$-ah^{\prime}(0)\neq g^{\prime}(0)(d-1)$ and $d=1$ then (17) is controllable.

Proof. In fact, the conditions $h^{\prime}(0)\neq 0$ and $-ah^{\prime}(0)\neq
g^{\prime}(0)(d-1)$ implies by the Theorem 26 that the sets $\mathcal{R}$ and
$\mathcal{C}$ are open. Furthermore, since $d$ and $1$ are the only
eigenvalues of $f_{0}$, by Theorem 21 the assumption $d=1$ implies that (17)
is controllable. $\square$

##  4 Controllability of nilpotent Lie groups

In this section we present and kind of converse of Theorem 21. It is assumed
that $G$ is connected, simply connected and nilpotent Lie group. Recall the
following well known fact will be useful. For connected, simply connected and
nilpotent Lie groups, each connected subgroups are closed and simply connected
(see e.g. Knapp [12, Corollary 1.134] or San Martin [17, Corollary 10.10]).

To achieve this goal we need the following lemma.

###### Lemma 28

Suppose that $df_{0}$ has no eigenvalue with absolute value greater than $1$
and $\mathcal{R}$ is open. If $M:=G/G^{0}$ admits a $G$-invariant metric, then
$\mathcal{R}_{G^{-}}=\mathcal{R}\cap G^{-}$ is a relative compact set.

Proof. Consider the induced linear system

$x_{k+1}=\bar{f}(x_{k},u_{k}),\ x_{k}\in M,\
u=(u_{i})_{i\in\mathbb{N}_{0}}\in\mathcal{U},$

on $M:=G/G^{0}$ with solutions denoted by $\bar{\varphi}$ which satisfies
$\pi(\varphi(k,g,u))=\bar{\varphi}(k,\pi(g),u)$, for all $k\in\mathbb{N}_{0}$,
$g\in G$ and $u\in\mathcal{U}$, where $\pi:G\rightarrow G/G^{0}$ denotes the
standard projection (see end of Section 2).

If $\varrho$ is the distance induced by the $G$-invariant Riemannian metric on
$M$, then given $x,y\in M$ and a smooth curve $\gamma:[0,1]\rightarrow M$ with
$\gamma(0)=x$ and $\gamma(1)=y$ we have that $\bar{f}_{0}^{k}\circ\gamma$ is a
smooth curve with $\bar{f}_{0}^{k}\circ\gamma(0)=\bar{f}_{0}^{k}(x)$ and
$\bar{f}_{0}^{k}\circ\gamma(1)=\bar{f}_{0}^{k}(y)$. Hence

$\varrho(\bar{f}_{0}^{k}(x),\bar{f}_{0}^{k}(y))\leq\displaystyle\int_{0}^{1}|(d\bar{f}_{0}^{k})_{\gamma(t)}\gamma^{\prime}(t)|dt.$

Note that $\|(d\bar{f}_{0}^{k})_{gG^{0}}\|=\|(d\bar{f}_{0}^{k})_{eG^{0}}\|$
holds for all $x\in M$ since the metric is $G$-invariant. From [9, Remark 9
and Proposition 10] there exist a $c\geq 1$ and $\sigma\in(0,1)$ such that

$\displaystyle\varrho(\bar{f}_{0}^{k}(x),\bar{f}_{0}^{k}(y))\leq\displaystyle\int_{0}^{1}\|(d\bar{f}_{0}^{k})_{eG^{0}}\||\gamma^{\prime}(t)|dt\leq
c^{-1}\sigma^{k}\displaystyle\int_{0}^{1}|\gamma^{\prime}(t)|dt,$

that is,

$\displaystyle\varrho(\bar{f}_{0}^{k}(x),\bar{f}_{0}^{k}(y))\leq
c^{-1}\sigma^{k}\varrho(x,y),\ \forall\ k\in\mathbb{N}.$ (19)

Moreover, the solutions $\bar{\varphi}$ satisfy

$\displaystyle\bar{\varphi}(k+l,eG^{0},u)=\mathcal{L}_{\varphi(k,e,\Theta_{k}(u))}(\bar{f}_{0}^{k}(\bar{\varphi}(l,eG^{0},u))).$
(20)

Now we prove by induction on $k$ that

$\varrho(\bar{\varphi}(k,eG^{0},u),eG^{0})\leq
c^{-1}a\displaystyle\sum_{i=0}^{k-1}\sigma^{i},$

where $a:=\displaystyle\max_{u\in U}\varrho(\bar{f}_{u}(eG^{0}),eG^{0})$.
Assume without loss of generality that $c^{-1}\geq 1$. For $k=1$, it follows
from (19) that

$\displaystyle\varrho(\bar{\varphi}(1,eG^{0},u),eG^{0})=\varrho(\bar{f}_{u}(eG^{0}),eG^{0})\leq
c^{-1}a.$

Suppose that the assertion holds for $n\geq 1$. Equality (20) yields

$\bar{\varphi}(k+1,eG^{0},u)=\mathcal{L}_{\varphi(k,e,\Theta_{k}(u))}(\bar{f}_{0}^{k}(\bar{f}_{u}(eG^{0}))),$

consequently,

$\displaystyle\varrho(\bar{\varphi}(k+1,eG^{0},u),eG^{0})$ $\displaystyle\leq$
$\displaystyle\varrho(\mathcal{L}_{\varphi(k,e,\Theta_{k}(u))}(\bar{f}_{0}^{k}(\bar{f}_{u}(eG^{0}))),\mathcal{L}_{\varphi(k,e,\Theta_{k}(u))}(eG^{0}))$
$\displaystyle+$
$\displaystyle\varrho(\mathcal{L}_{\varphi(k,e,\Theta_{k}(u))}(eG^{0}),eG^{0})$
$\displaystyle=$
$\displaystyle\varrho(\bar{f}_{0}^{k}(\bar{f}_{u}(eG^{0})),eG^{0})+\varrho(\bar{\varphi}(k,eG^{0},u),eG^{0}),$

because $\mathcal{L}_{g}$ is an isometry and $\mathcal{L}_{g}(eG^{0})=\pi(g)$
for all $g\in G$. By (19) and the inductive hypothesis we have

$\displaystyle\varrho(\bar{\varphi}(k+1,eG^{0},u),eG^{0})$ $\displaystyle\leq$
$\displaystyle\varrho(\bar{f}_{0}^{k}(\bar{f}_{u}(eG^{0})),eG^{0})+\varrho(\bar{\varphi}(k,eG^{0},u),eG^{0})$
$\displaystyle\leq$ $\displaystyle
c^{-1}\sigma^{k}\varrho(\bar{f}_{u}(eG^{0}),eG^{0})+c^{-1}a\displaystyle\sum_{i=0}^{k-1}\sigma^{i}$
$\displaystyle\leq$ $\displaystyle
c^{-1}a\sigma^{k}+c^{-1}a\displaystyle\sum_{i=0}^{k-1}\sigma^{i}=c^{-1}a\displaystyle\sum_{i=0}^{k}\sigma^{i}.$

Since $\sigma\in(0,1)$, one has

$\pi(\mathcal{R})=\pi\left(\displaystyle\bigcup_{\tau\in\mathbb{N}}\mathcal{R}_{\tau}\right)=\displaystyle\bigcup_{\tau\in\mathbb{N}}\pi(\mathcal{R}_{\tau})\subset\overline{B_{\delta}(eG^{0})},$

where $\delta=c^{-1}a\displaystyle\sum_{i=0}^{\infty}\sigma^{i}<\infty$. Hence
$\pi(\mathcal{R})$ is relatively compact in $M$. Note also that
$\pi|_{G^{-}}:G^{-}\rightarrow M$ is a homeomorphism because
$G^{-,0}=G^{-}G^{0}$ and $G^{-}\cap G^{0}=\\{e\\}$ (c.f. [1, Lemma 3.6]).
Therefore, $\mathcal{R}_{G^{-}}$ is relatively compact in $G^{-}$. The result
follows since $G^{-}$ is closed in $G$. $\square$

The next proposition establishes a necessary and sufficient condition for the
controllability from the identity of $G$.

###### Proposition 29

Assume that $\mathcal{R}$ is open. Then $\mathcal{R}=G$ if, and only if,
$G=G^{+,0}$.

Proof. If $G=G^{+,0}$, then Theorem 20 implies $\mathcal{R}=G$. Reciprocally,
assume $\mathcal{R}=G$. The proof that $G=G^{+,0}$ will proceed by induction
on $\dim G$. If $\dim G=1$, then $G$ is abelian and we can conjugate the
system to its linearization by exponential map, consequently the result
follows from [8, Corollary 16 (i)]. Now, suppose that the assertion holds for
any nilpotent connected and simply connected Lie group with dimension smaller
than $d$. Consider a nilpotent connected and simply connected Lie group $G$
with $\dim G=n$. Since the center of a nilpotent Lie group is non-trivial, the
Lie group $H:=G/Z(G)$ is nilpotent, connected and simply connected (see [17,
Proposition 9.11]) with $\dim H<d$.

The $f_{0}$-invariance of $Z(G)$ implies that the induced linear system on $H$
is well-defined. By hypothesis, $\mathcal{R}=G$ yields $\pi(\mathcal{R})=H$
where $\pi:G\rightarrow H$ is the standard projection and by induction one has
$H=H^{+,0}$. Lemma 2 implies that $G^{-}\subset Z(G)$ and by Proposition 3 we
get that $G^{+}$ is a normal subgroup of $G$. If $\mathcal{R}^{-,0}$ denotes
the reachable set of the induced linear system on $G^{-,0}=G/G^{+}$, it holds
that $\mathcal{R}^{-,0}=G^{-,0}$. Moreover, $G^{-,0}/G^{0}$ admits a
$G$-invariant Riemannian metric because $G^{-}\subset Z(G)$. Then
$\text{Ad}(G^{0})|_{\mathfrak{g}^{-}}=\\{\text{id}_{\mathfrak{g}^{-}}\\}$ is
compact in $\text{Gl}(\mathfrak{g}^{-})$ (see [7, Proposition 3.16]). By Lemma
28, $\mathcal{R}^{-,0}\cap G^{-}=G^{-}$ is relatively compact in $G$. As
$G^{-}$ is closed then it is compact. Hence by Proposition 4 $G^{-}=\\{e\\}$,
therefore $G=G^{+,0}$. $\square$

###### Remark 30

In the proof of Proposition 29, the hypothesis that $G$ is nilpotent is
important to get $Z(G)$ non-trivial.

Finally, the main result of this section can be stated.

###### Theorem 31

Consider the linear system (5) on a nilpotent connected and simply connected
Lie group $G$. Then the system is controllable if, and only if, $\mathcal{R}$
and $\mathcal{C}$ are open and $G=G^{0}$.

Proof. If we assume that $\mathcal{R}$ and $\mathcal{C}$ are open and
$G=G^{0}$, the result follows from Theorem 21. On the other hand, if the
system is controllable, then $\mathcal{R}$ and $\mathcal{C}$ are clearly open.
Moreover, since $G=\mathcal{R}\cap\mathcal{C}$, we can apply Proposition 29 to
$\mathcal{R}$ and $\mathcal{R}^{*}=\mathcal{C}$ in order to conclude that
$G=G^{0}$. $\square$

###### Remark 32

Theorem 31 generalizes [8, Corollary 16 (iii)] about controllability of linear
systems on euclidean spaces.

We finish this work with an application of the previous result in the case of
Heisenberg group.

###### Example 33

Consider the Heisenberg group

$\mathbb{H}=\left\\{\begin{bmatrix}1&x_{2}&x_{1}\\\ 0&1&x_{3}\\\
0&0&1\end{bmatrix};\ x_{1},x_{2},x_{3}\in\mathbb{R}\right\\},$

which is diffeomorphic to $\mathbb{R}^{3}$ with the product

$(x_{1},x_{2},x_{3})\cdot(y_{1},y_{2},y_{3})=(x_{1}+y_{1}+x_{2}y_{3},x_{2}+y_{2},x_{3}+y_{3}).$

Let $U$ be a compact and connected neighborhood of $0\in\mathbb{R}$ and
$f:\mathbb{H}\times U\rightarrow\mathbb{H}$ given by

$f_{u}(x_{1},x_{2},x_{3})=\left(x_{1}+x_{2}+\dfrac{x_{2}^{2}}{2}+ux_{2}+ux_{3}-\dfrac{u}{2}-\dfrac{u^{2}}{3},x_{2}+u,x_{2}+x_{3}-\dfrac{u}{2}\right).$

It is not difficult to see that

$\displaystyle g_{k+1}=f_{u_{k}}(g_{k}),\ u_{k}\in U$ (21)

is a linear system on $\mathbb{H}$ (see [9, Example 6]). Note that

$f_{u}^{-1}(x_{1},x_{2},x_{3})=\left(x_{1}-x_{2}-\frac{x_{2}^{2}}{2}+ux_{2}-ux_{3}+\frac{3u}{2}-\frac{2u^{2}}{2},x_{2}-u,-x_{2}+x_{3}-\frac{3u}{2}\right).$

Since each coordinate of both $f_{u}(0,0,0)$ and $f_{u}^{-1}(0,0,0)$ are
polynomials in $u$ with root $u=0$ and non-null derivative in $u=0$, there is
$\varepsilon>0$ such that $B_{\varepsilon}(0,0,0)\subset
f_{\mathrm{int}U}(0,0,0)\cap f_{\mathrm{int}U}^{-1}(0,0,0)$ is a neighborhood
of $(0,0,0)$ in $\mathcal{R}$ and $\mathcal{R}^{*}$, respectively. By item (6)
of Proposition 12, $\mathcal{R}$ and $\mathcal{R}^{*}=\mathcal{C}$ are open.
Moreover, the unique eigenvalue of $df_{0}$ is $1$ (see [9, Example 20]),
hence $G=G^{0}$. By Theorem 31, the system (21) is controllable.

## References

  * [1] Ayala, V., Román-Flores H., Da Silva, A. _The dynamic of a Lie group endomorphism_ Open Math, Vol. 15. (2017), pp. 1477 - 1486.
  * [2] Ayala, V., Da Silva, A. _A semigroup associated to a linear control system on a Lie group._ System & control letters. Vol. 98. (2016), pp. 33-36.
  * [3] Ayala, V., Da Silva, A. _The control set of a linear control system on the two dimensional solvable Lie group._ Journal of Differential Equations, 268, (2020), pp. 6683 - 6701.
  * [4] Ayala, V., Da Silva, A., Zsigmond G. _Control sets of linear systems on Lie groups_ , Nonlinear Differential Equations and Applications 24(8) (2017), pp. 7-15.
  * [5] Ayala V., San Martin L.A.B. _Controllability properties of a class of control systems on Lie groups_ , Lecture Notes in Control and Inform. Sci. 258, Springer, Berlin, (2001), pp. 83–92.
  * [6] Ayala V., Tirao J. _Linear control systems on Lie groups and controllability,_ Proceedings of Symposia in Pure Mathematics, Vol 64, American Mathematical Society, (1999), pp. 47-64.
  * [7] Cheeger J., Ebin D. G., _Comparison Theorems in Riemannian Geometry_ , North-Holland, Amsterdam, 1975.
  * [8] Colonius, F., Cossich, J. A. N., Santana, A. J. _Controllability Properties and Invariance Pressure for Linear Discrete-Time Systems._ J Dyn Diff Equat 34, (2022), pp 5–28.
  * [9] Colonius, F., Cossich, J. A. N., Santana, A. J. _Outer invariance entropy for discrete-time linear systems on Lie groups._ ESAIM: Control, Optimisation and Calculus of Variations, vol. 27, (2021).
  * [10] Da Silva, A.J. _Controllability of linear systems on solvable Lie groups._ SIAM: Journal on Control and Optimization, Vol. 54, (2016), pp. 372-390.
  * [11] Da Silva A.J. _Outer invariance entropy for linear systems on Lie groups_ , J. Dyn. Control Syst., 52 (2014), pp. 3917–3934.
  * [12] Knapp A.,_Lie Groups Beyond an Introduction,_ Second Edition, Birkhäuser, (2002).
  * [13] Jouan P. _Equivalence of control systems with linear systems on Lie groups and homogeneous spaces_ , ESAIM Control Optim. Calc. Var., 16 (2010), pp. 956–973.
  * [14] Jouan, P., Dath, M. _Controllability of Linear Systems on Low Dimensional Nilpotent and Solvable Lie Groups._ Journal of dynamical and control systems. Vol. 22(2). (2014), pp. 207–225.
  * [15] Onishchik, A.L., Vinberg, E.B. _Lie groups and Lie algebras_ Springer. Vol. 41. (1993).
  * [16] Ogata T. _Modern Control Engineering_ , 3rd Edition, Prentice Hall, (1997).
  * [17] San Martin, L.A.B., _Lie groups_. Latin American Mathematics Series. Springer, Berlin
  * [18] San Martin, L.A.B., _Algebras de Lie,_ Second Edition, Editora Unicamp. (2010).
  * [19] Jakubczyk, B., Sontag, E._Controllability of nonlinear discrete-time systems: a Lie-algebraic approach_ SIAM J. Control and Optimization. Vol. 28. (1990), pp. 1-33.
  * [20] Sontag, E.D., _Mathematical Control Theory. Deterministic finite- dimensional systems,_ Springer-Verlag, New York, (1998)
  * [21] Willsky A.S., _Digital Signal Processing and Control and Estimation Theory: Points of Tangency, Areas of Intersection, and Parallel Directions_ , The MIT Press, 1979.
  * [22] Wirth, F., _Dynamics and controllability of nonlinear discrete-time control systems._ IFAC Proc. Vol. 31, (1998), pp. 267–272.
  * [23] Wyss, W. _Two non-commutative binomial theorems._ arXiv preprint 1707.03861v2 (2017)


consecutive buffet cycles, however, the amplitude of the peak anti-correlation
varies from cycle-to-cycle. The magnitude of the anti-correlations is stronger
than those observed at $\AR={2}$ (Figure 20), suggesting more pronounced
large-scale 3D flow structures at this wider aspect ratio for a fixed angle of
attack. This is consistent with the instantaneous flow-field visualisations
shown in Figure 11. At $\AR={3}$, the correlation threshold detects both
single and double separation cell patterns, associated with an undulation of
the shock front in this region ($0.25<x<0.6)$. Multiple structures are also
observed in the region downstream of the shock-wave. Comparing the red
correlation curves in Figures 32 (b,c) associated with the upstream
($0.25<x<0.6$) and downstream regions ($0.6<x<1.0$) of the aerofoil, we can
now clearly see a time lag between anti-correlation peaks of $\Delta t\approx
2-3$ for $\AR={3}$, as the buffet-cell structures are convected downstream.

## References

  * Aihara & Kawai (2023) Aihara, Aya & Kawai, Soshi 2023 Effects of Spanwise Domain Size on LES-Predicted Aerodynamics of Stalled Airfoil. AIAA Journal 61 (3), 1440–1446.
  * Bhagatwala & Lele (2009) Bhagatwala, Ankit & Lele, Sanjiva K 2009 A modified artificial viscosity approach for compressible turbulence simulations. Journal of Computational Physics 228 (14), 4965–4969.
  * Bogey & Bailly (2004) Bogey, Christophe & Bailly, Christophe 2004 A family of low dispersive and low dissipative explicit schemes for flow and noise computations. Journal of Computational Physics 194 (1), 194–214.
  * Borges et al. (2008) Borges, Rafael, Carmona, Monique, Costa, Bruno & Don, Wai Sun 2008 An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics 227 (6), 3191–3211.
  * Carpenter & Kennedy (1994) Carpenter, Mark H & Kennedy, Christopher A 1994 Fourth-order 2N-storage Runge-Kutta schemes. NASA Langley Research Center .
  * Chapelier et al. (2024) Chapelier, Jean-Baptiste, Lusher, David J., Van Noordt, William, Wenzel, Christoph, Gibis, Tobias, Mossier, Pascal, Beck, Andrea, Lodato, Guido, Brehm, Christoph, Ruggeri, Matteo, Scalo, Carlo & Sandham, Neil 2024 Comparison of high-order numerical methodologies for the simulation of the supersonic Taylor–Green vortex flow. Physics of Fluids 36 (5), 055146\.
  * Coppola et al. (2019) Coppola, G., Capuano, F., Pirozzoli, S. & de Luca, L. 2019 Numerically stable formulations of convective terms for turbulent compressible flows. Journal of Computational Physics 382, 86–104.
  * Crouch et al. (2009) Crouch, J. D., Garbaruk, A., Magidov, D. & Travin, A. 2009 Origin of transonic buffet on aerofoils. Journal of Fluid Mechanics 628, 357–369.
  * Crouch et al. (2019) Crouch, J. D., Garbaruk, A. & Strelets, M. 2019 Global instability in the onset of transonic-wing buffet. Journal of Fluid Mechanics 881, 3–22.
  * Dandois et al. (2018) Dandois, Julien, Mary, Ivan & Brion, Vincent 2018 Large-eddy simulation of laminar transonic buffet. Journal of Fluid Mechanics 850, 156–178.
  * Deck (2005) Deck, S. 2005 Numerical simulation of transonic buffet over a supercritical airfoil. AIAA Journal 43(7), 1556–1566.
  * Dolling (2001) Dolling, D. 2001 Fifty years of shock-wave/boundary-layer interaction research: what next? AIAA Journal 39(8), 1517–1531.
  * Ducros et al. (1999) Ducros, F, Ferrand, V, Nicoud, F, Weber, C, Darracq, D, Gacherieu, C & Poinsot, T 1999 Large-eddy simulation of the shock/turbulence interaction. Journal of Computational Physics 152 (2), 517 – 549.
  * Eagle & Driscoll (2014) Eagle, W Ethan & Driscoll, James F 2014 Shock wave–boundary layer interactions in rectangular inlets: three-dimensional separation topology and critical points. Journal of Fluid Mechanics 756, 328–353.
  * Fu (2023) Fu, Lin 2023 Review of the high-order TENO schemes for compressible gas dynamics and turbulence. Archives of Computational Methods in Engineering 30 (4), 2493–2526.
  * Fukushima & Kawai (2018) Fukushima, Y. & Kawai, S. 2018 Wall-modeled large-eddy simulation of transonic airfoil buffet at high reynolds number. AIAA Journal 56(6), 1–18.
  * Garnier & Deck (2013) Garnier, E. & Deck, S. 2013 Large-Eddy Simulation of transonic buffet over a supercritical airfoil. Turbulence and Interactions (ed. M. Deville, T.-H. Le and P. Sagaut). Springer, Berlin, Heidelberg.
  * Garnier et al. (1999) Garnier, Eric, Mossi, Michele, Sagaut, Pierre, Comte, Pierre & Deville, Michel 1999 On the use of shock-capturing schemes for large-eddy simulation. Journal of Computational Physics 153 (2), 273–311.
  * Giannelis et al. (2017) Giannelis, Nicholas F., Vio, Gareth A. & Levinski, Oleg 2017 A review of recent developments in the understanding of transonic shock buffet. Progress in Aerospace Sciences 92, 39–84.
  * Grinstein et al. (2007) Grinstein, Fernando F, Margolin, Len G & Rider, William J 2007 Implicit large eddy simulation, , vol. 10. Cambridge university press Cambridge.
  * Hamzehloo et al. (2021) Hamzehloo, Arash, Lusher, David J., Laizet, Sylvain & Sandham, Neil D. 2021 On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high-speed compressible flows. International Journal for Numerical Methods in Fluids 93 (1), 176–196.
  * Hamzehloo et al. (2023) Hamzehloo, Arash, Lusher, David J. & Sandham, Neil D. 2023 Direct numerical simulations and spectral proper orthogonal decomposition analysis of shocklet-containing turbulent channel counter-flows. International Journal of Heat and Fluid Flow 104, 109229.
  * Hartmann et al. (2013) Hartmann, A, Feldhusen, A & Shroder, W. 2013 On the interaction of shock waves and sound waves in transonic buffet flow. AIAA Journal 28 (942).
  * Hashimoto et al. (2018) Hashimoto, Atsushi, Ishida, Takashi, Aoyama, Takashi, Ohmichi, Yuya, Yamamoto, Takahiro & Hayashi, Kenji 2018 Current progress in unsteady transonic buffet simulation with unstructured grid cfd code. In 2018 AIAA Aerospace Sciences Meeting, p. 0788\.
  * Hashimoto et al. (2012) Hashimoto, Atsushi, Murakami, Keiichi, Aoyama, Takashi, Ishiko, Keiichi, Hishida, Manabu, Sakashita, Masahide & Lahur, Paulus 2012 Toward the fastest unstructured cfd code ’fastar’. 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition (AIAA 2012-1075).
  * He & Timme (2021) He, Wei & Timme, Sebastian 2021 Triglobal infinite-wing shock-buffet study. Journal of Fluid Mechanics 925, A27.
  * Houtman et al. (2023) Houtman, Jelle, Timme, Sebastian & Sharma, Ati 2023 Resolvent analysis of a finite wing in transonic flow. Flow 3, E14.
  * Iovnovich & Raveh (2015) Iovnovich, M. & Raveh, D. E. 2015 Numerical study of shock buffet on three-dimensional wings. AIAA Journal 53(2), 449–463.
  * Ishida et al. (2016) Ishida, T., Ishiko, K., Hashimoto, A., Aoyama, T. & Takekawa, K. 2016 Transonic buffet simulation over supercritical airfoil by unsteady-FaSTAR code. AIAA Paper 2016-1310 .
  * Iwatani et al. (2023) Iwatani, Yuta, Asada, Hiroyuki, Yeh, Chi-An, Taira, Kunihiko & Kawai, Soshi 2023 Identifying the self-sustaining mechanisms of transonic airfoil buffet with resolvent analysis. AIAA Journal 61 (6), 2400–2411.
  * Jacquin et al. (2009) Jacquin, L., Molton, P., Deck, S., Maury, B. & Soulevant, D. 2009 Experimental study of shock oscillation over a transonic supercritical profile. AIAA Journal 47, 1985–1994.
  * Jones et al. (2008) Jones, LE, Sandberg, RD & Sandham, ND 2008 Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence. Journal of Fluid Mechanics 602, 175–207.
  * Lee (1990) Lee, B. H. K. 1990 Oscillatory shock motion caused by transonic shock boundary-layer interaction. AIAA Journal 28(5), 942–944.
  * Lee (2001) Lee, B. H. K. 2001 Self-sustained shock oscillations on airfoils at transonic speeds. Progress in Aerospace Sciences 37, 147–196.
  * Lusher & Coleman (2022) Lusher, David J. & Coleman, Gary N. 2022 Numerical study of compressible wall-bounded turbulence – the effect of thermal wall conditions on the turbulent Prandtl number in the low-supersonic regime. International Journal of Computational Fluid Dynamics 36 (9), 797–815.
  * Lusher et al. (2018) Lusher, David J., Jammy, Satya P. & Sandham, Neil D. 2018 Shock-wave/boundary-layer interactions in the automatic source-code generation framework OpenSBLI. Computers & Fluids 173, 17 – 21.
  * Lusher et al. (2021) Lusher, David J., Jammy, Satya P. & Sandham, Neil D. 2021 OpenSBLI: Automated code-generation for heterogeneous computing architectures applied to compressible fluid dynamics on structured grids. Computer Physics Communications 267, 108063.
  * Lusher & Sandham (2020a) Lusher, David J. & Sandham, Neil D. 2020a The effect of flow confinement on laminar shock-wave/boundary-layer interactions. Journal of Fluid Mechanics 897, A18.
  * Lusher & Sandham (2020b) Lusher, David J & Sandham, Neil D 2020b Shock-Wave/Boundary-Layer Interactions in Transitional Rectangular Duct Flows. Flow, Turbulence and Combustion .
  * Lusher & Sandham (2021) Lusher, David J & Sandham, Neil D 2021 Assessment of low-dissipative shock-capturing schemes for the compressible taylor–green vortex. AIAA Journal 59 (2), 533–545.
  * Lusher et al. (2024) Lusher, David J., Sansica, Andrea & Hashimoto., Atsushi 2024 Effect of Tripping and Domain-Width on Transonic Buffet on Periodic NASA-CRM Airfoils, arXiv: 2401.14793.
  * Lusher et al. (2023) Lusher, D. J., Zauner, M., Sansica, A. & Hashimoto, A. 2023 Automatic Code-Generation to Enable High-Fidelity Simulations of Multi-Block Airfoils on GPUs. In AIAA Scitech 2023 Forum, AIAA SciTech Forum.
  * Masini et al. (2020) Masini, L., Timme, S. & Peace, A. J. 2020 Analysis of a civil aircraft wing transonic shock buffet experiment. Journal of Fluid Mechanics 884.
  * Mavriplis (2003) Mavriplis, D. J. 2003 Revisiting the least-squares procedure for gradient reconstruction on unstructured. In 16th AIAA Computational Fluid Dynamics Conference, AIAA paper 2003-3986.
  * Memmolo et al. (2018) Memmolo, A., Bernardini, M. & Pirozzoli, S. 2018 Scrutiny of buffet mechanisms in transonic flow. International Journal of Numerical Methods for Heat and Fluid Flow 28, 1031–1046.
  * Mengaldo & Maulik (2021) Mengaldo, Gianmarco & Maulik, Romit 2021 PySPOD: A Python package for Spectral Proper Orthogonal Decomposition (SPOD). Journal of Open Source Software 6 (60), 2862.
  * Moise et al. (2022) Moise, Pradeep, Zauner, Markus & Sandham, Neil D. 2022 Large-eddy simulations and modal reconstruction of laminar transonic buffet. Journal of Fluid Mechanics 944, A16.
  * Moise et al. (2023) Moise, Pradeep, Zauner, Markus, Sandham, Neil D., Timme, Sebastian & He, Wei 2023 Transonic buffet characteristics under conditions of free and forced transition. AIAA Journal 61 (3), 1061–1076.
  * NASA-LaRC (2012) NASA-LaRC 2012 Crm.65.airfoil sections [online]. Available:https://commonresearchmodel.larc.nasa. gov/crm-65-airfoil-sections .
  * Nguyen et al. (2022) Nguyen, Ngoc Cuong, Terrana, Sebastien & Peraire, Jaime 2022 Large-eddy simulation of transonic buffet using matrix-free discontinuous galerkin method. AIAA Journal 60 (5), 3060–3077.
  * Obayashi & Guruswamy (1995) Obayashi, S. & Guruswamy, G. P. 1995 Convergence acceleration of an aeroelastic Navier-Stokes solver. AIAA Journal 33, 1134–1141.
  * Ohmichi et al. (2018) Ohmichi, Y., Ishida, T. & Hashimoto, A. 2018 Modal decomposition analysis of three-dimensional transonic buffet phenomenon on a swept wing. AIAA Journal 56 (10), 3938–3950.
  * Paladini et al. (2019) Paladini, Edorado, Beneddine, Samir, Dandois, Julien, Sipp, Denis & Robinet, Jean-Christophe 2019 Transonic buffet instability: From two-dimensional airfoils to three-dimensional swept wings. Phys. Rev. Fluids 4, 103906.
  * Plante (2020) Plante, Frédéric 2020 Towards understanding stall cells and transonic buffet cells. PhD thesis, Ecole Polytechnique, Montreal (Canada).
  * Plante et al. (2021) Plante, F., Dandois, J., Beneddine, S., Laurendeau, E. & Sipp, D. 2021 Link between subsonic stall and transonic buffet on swept and unswept wings: from global stability analysis to nonlinear dynamics. Journal of Fluid Mechanics 908.
  * Plante et al. (2020) Plante, F., Dandois, J. & Laurendeau, E. 2020 Similarities between cellular patterns occurring in transonic buffet and subsonic stall. AIAA Journal 58 (1), 71–84.
  * Poplinger et al. (2019) Poplinger, L., Raveh, D. E. & Dowell, E. H. 2019 Modal analysis of transonic shock buffet on 2d airfoil. AIAA Journal 57 (7).
  * Reguly et al. (2018) Reguly, István Z., Mudalige, Gihan R. & Giles, Michael B. 2018 Loop Tiling in Large-Scale Stencil Codes at Run-Time with OPS. IEEE Transactions on Parallel and Distributed Systems 29 (4), 873–886.
  * Reguly et al. (2014) Reguly, István Z., Mudalige, Gihan R., Giles, Michael B., Curran, Dan & McIntosh-Smith, Simon 2014 The OPS Domain Specific Abstraction for Multi-block Structured Grid Computations. WOLFHPC ’14 10, pp. 58–67. IEEE Press.
  * Rodríguez & Theofilis (2011) Rodríguez, Daniel & Theofilis, Vassilis 2011 On the birth of stall cells on airfoils. Theoretical and Computational Fluid Dynamics 25, 105–117.
  * Sansica & Hashimoto (2023) Sansica, Andrea & Hashimoto, Atsushi 2023 Global stability analysis of full-aircraft transonic buffet at flight reynolds numbers. AIAA Journal 61 (10), 4437–4455.
  * Sansica et al. (2022) Sansica, A., Loiseau, J.-Ch., Kanamori, M. Hashimoto, A. & Robinet, J.-C. 2022 System identification of two-dimensional transonic buffet. AIAA Journal 60 (5).
  * Sartor et al. (2015) Sartor, F., Mettot, C. & Sipp, D. 2015 Stability, receptivity, and sensitivity analyses of buffeting transonic flow over a profile. AIAA Journal 53(7), 1980–1933.
  * Sartor & Timme (2017) Sartor, F. & Timme, S. 2017 Delayed detached–eddy simulation of shock buffet on half wing–body configuration. AIAA Journal 55, 1230–1240.
  * Sharov & Nakahashi (1998) Sharov, D. & Nakahashi, K. 1998 Reordering of hybrid unstructured grids for lower-upper symmetric Gauss-Seidel computations. AIAA Journal 36, 484–486.
  * Shur et al. (2000) Shur, Michael L., Strelets, Michael K., Travin, Andrey K. & Spalart, Philippe R. 2000 Turbulence modeling in rotating and curved channels: Assessing the spalart-shur correction. AIAA Journal 38 (5), 784–792.
  * Song et al. (2024) Song, Hang, Wong, Man Long, Ghate, Aditya S. & Lele, Sanjiva K. 2024 Numerical study of transonic laminar shock buffet on the OALT25 airfoil, chap. AIAA 2024-2148.
  * Spalart & Allmaras (1992) Spalart, P. R. & Allmaras, S. R. 1992 A one-equation turbulence model for aerodynamic flows. In 30th Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings.
  * Sugioka et al. (2018) Sugioka, Y., Koike, S., Nakakita, K., Numata, D., Nonomura, T. & Asai, K. 2018 Experimental analysis of transonic buffet on a 3D swept wing using fast-response pressure-sensitive paint. Experiments in Fluids 59 (108).
  * Sugioka et al. (2022) Sugioka, Y., Kouchi, T. & Koike, S. 2022 Experimental comparison of shock buffet on unswept and 10-deg swept wings. Experiments in Fluids 63 (132).
  * Sugioka et al. (2021) Sugioka, Y., Nakakita, K., Koike, S., Nakajima, T., Nonomura, T. & Asai, K. 2021 Characteristic unsteady pressure field on a civil aircraft wing related to the onset of transonic buffet. Experiments in Fluids (62:20).
  * Taira et al. (2017) Taira, Kunihiko, Brunton, Steven L., Dawson, Scott T. M., Rowley, Clarence W., Colonius, Tim, McKeon, Beverley J., Schmidt, Oliver T., Gordeyev, Stanislav, Theofilis, Vassilios & Ukeiley, Lawrence S. 2017 Modal analysis of fluid flows: An overview. AIAA Journal 55 (12), 4013–4041.
  * Tamaki & Kawai (2024) Tamaki, Yoshiharu & Kawai, Soshi 2024 Wall-Modeled Large-Eddy Simulation of Transonic Buffet over NASA-CRM Using FFVHC-ACE. AIAA Journal 0 (0), 1–16.
  * Thiery & Coustols (2006) Thiery, M. & Coustols, E. 2006 Numerical prediction of shock induced oscillations over a 2D airfoil: Influence of turbulence modelling and test section walls. International Journal of Heat and Fluid Flow 27, 661–670.
  * Timme (2020) Timme, S 2020 Global instability of wing shock-buffet onset. Journal of Fluid Mechanics 885.
  * Tinoco (2019) Tinoco, Edward N. 2019 An Evaluation and Recommendations for Further CFD Research Based on the NASA Common Research Model (CRM) Analysis from the AIAA Drag Prediction Workshop (DPW) Series. NASA/CR-2019-220284 pp. 13–17.
  * Tinoco et al. (2018) Tinoco, Edward N., Brodersen, Olaf P., Keye, Stefan, Laflin, Kelly R., Feltrop, Edward, Wahls, Richard A., Morrison, Joseph H., Vassberg, John C., Mani, Mori, Rider, Ben, Hue, David, Roy, Christopher J., Mavriplis, Dimitri J. & Murayama, Mitsuhiro 2018 Summary of Data from the Sixth AIAA CFD Drag Prediction Workshop: CRM Cases 2 to 5. Journal of Aircraft 55 (4), 1352–1379.
  * Tobak & Peake (1982) Tobak, Murray & Peake, David J 1982 Topology of three-dimensional separated flows. Annual review of fluid mechanics 14 (1), 61–85.
  * Towne et al. (2018) Towne, Aaron, Schmidt, Oliver T. & Colonius, Tim 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. Journal of Fluid Mechanics 847, 821–867.
  * Visbal & Gordnier (2000) Visbal, M. R. & Gordnier, R. 2000 A high-order flow solver for deforming and moving meshes. In Fluids 2000 Conference and Exhibit, AIAA Paper 2000-2619.
  * Yee et al. (1999) Yee, H.C, Sandham, N.D & Djomehri, M.J 1999 Low-dissipative high-order shock-capturing methods using characteristic-based filters. Journal of Computational Physics 150 (1), 199–238.
  * Yee & Sjögreen (2018) Yee, H C & Sjögreen, Björn 2018 Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows. Computers & Fluids 169, 331–348.
  * Zauner et al. (2019) Zauner, Markus, De Tullio, Nicola & Sandham, Neil D 2019 Direct numerical simulations of transonic flow around an airfoil at moderate reynolds numbers. AIAA Journal 57 (2), 597–607.
  * Zauner et al. (2022) Zauner, Markus, Moise, Pradeep & Sandham, Neil D. 2022 On the co-existence of transonic buffet and separation-bubble modes for the oalt25 laminar-flow wing section. Flow, Turbulence and Combustion 110, 1023–1057.
  * Zauner & Sandham (2020) Zauner, Markus & Sandham, Neil D. 2020 Wide domain simulations of flow over an unswept laminar wing section undergoing transonic buffet. Phys. Rev. Fluids 5, 083903.