{\TheTitle
Saber Jafarpour\thanks{Department of Mechanical Engineering and Center for Control
Dynamical Systems
and Computations at University of California
Santa Barbara (\email{saber.jafarpour@engineering.ucsb.edu
In this paper we study small-time local controllability of real analytic control systems under small perturbations of their vector fields. Consider a real analytic control system $\mathcal{X$ which is small-time locally controllable and whose reachable sets grow with the polynomial rate of order $N$ with respect to time. We will prove a general theorem which states that any real analytic control system whose vector fields are perturbations of the vector fields of $\mathcal{X$ with terms of order higher than $N$ is again small-time locally controllable. In particular, we show that this result connects two long-standing open conjectures about small-time local controllability of systems.
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\author{
  Saber Jafarpour\thanks{Department of Mechanical Engineering and
    Center for Control, Dynamical Systems, and Computations at 
    University of California, Santa Barbara
    (\email{saber.jafarpour@engineering.ucsb.edu}).}}

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\begin{document}

\maketitle

% REQUIRED
\begin{abstract}
In this paper we study small-time local controllability of real
analytic control systems under small perturbations of their vector fields. Consider a real analytic control
system $\mathcal{X}$ which is small-time locally controllable and whose reachable sets grow with the polynomial rate of
order $N$ with respect to time. We will prove a general theorem which
states that any real analytic control system whose vector fields are perturbations of the vector fields
of $\mathcal{X}$ with terms of order higher than $N$ is again
small-time locally controllable. In particular, we show that this result
connects two long-standing open conjectures about small-time local
controllability of systems. 
\end{abstract}

% REQUIRED
\begin{keywords}
Small-time local controllability, control variations, reachable sets,
real analytic systems. 
\end{keywords}

% REQUIRED
\begin{AMS}
93B03, 93B05, 93C10
\end{AMS}


\section{Introduction}

Controllability of systems is one of the central concepts
in mathematical control theory. For linear control systems, the notion of
controllability has been first studied by Kalman and his
coworkers~\cite{kalman1960contributions}. Using the state-space
approach, it can be shown that the well-known Kalman rank condition completely
characterizes controllability of linear systems~\cite{REK-YCH-SKN:1963}. For nonlinear
system, various notions of controllability have been defined and studied in
the literature~\cite{HJS:1983}. While different setting have been
proposed for studying nonlinear control systems~\cite{JCW:1979},~\cite{JPA-ACL:1984},~\cite{jurdjevicgeometric1996}, it turns out
that the geometric control theory is one of the most suitable frameworks
for studying local controllability of systems. In geometric control
theory, a control system is defined as a \emph{parametrized} family of
vector fields on a manifold, where the parameters are the controls and
the manifold is the state space of the system~\cite{jurdjevicgeometric1996}. For studying
the local properties of a control system, without loss of
generality one can assume that the state space of the system is an Euclidean
space $\R^n$. Also, for the sake of simplicity, we assume that the
control set of the system is the compact convex set $[-1,1]^m$. Therefore,
a control system can be considered as a family $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ of vector
fields on $\R^n$. A trajectory for the control system $\mathcal{X}$ is
an absolutely continuous curve $x:[0,T]\to \R^n$ such that 
\begin{align*}
\dot{x}(t)= X_0(x(t))+\sum_{i=1}^{m}u_{i}(t)X_i(x(t)),\quad \mbox{
  for almost every } t\in [0,T],
\end{align*}
for some measurable controls $u_1,u_2,\ldots,u_m:[0,T]\to [-1,1]$. For $x_0\in\R^n$ and time $t\in \R_{\ge 0}$, the reachable set of $\mathcal{X}$
form $x_0$, for time less than $t$, which is denoted by
$\mathrm{R}_{\mathcal{X}}(< t,x_0)$, is the set of points in state space
$\R^n$ which can be reached by traveling along the trajectories of the
vector fields in $\mathcal{X}$ for positive times less than $t$. More
precisely, we have 
\begin{align*}
\mathrm{R}_{\mathcal{X}}(< t,x_0)=\{x(T)\mid x:[0,T]\to \R^n \mbox{
                                      is a trajectory of }
                                      \mathcal{X}, \ x(0)=x_0, \ T< t\}.
\end{align*}
While this definition of reachable sets using measurable controls is
widely used and studied in the literature (cf.~\cite{sussmannlie1983},~\cite{sus1987},~\cite{MK:1990}), some authors consider other classes of
controls such as piecewise-constant controls~\cite{KAG:1992} and bang-bang
controls~\cite{AK:1974} for
studying control systems and their reachable sets. One can show that, in general,
the reachable sets defined using these different classes of controls are not the same. The
connections between some of these reachable sets have been studied in the literature~\cite{KAG:1992},~\cite{sus1987},~\cite{AK:1974}. 

Among different notions of controllability proposed in the literature,
small-time local controllability is arguably the most fundamental
one. A control system is small-time locally controllable form a point
$x_0$ if, for all times $t>0$, the reachable set
$\mathrm{R}_{\mathcal{X}}(< t,x_0)$ contains a neighbourhood of $x_0$ (a more rigorous definition will be given
later in the paper). It is clear from the definition that, by having the reachable sets, one can completely characterize small-time local
controllability of the system. However, a complete analytic description of reachable sets of
a control system requires solving a family of nonlinear differential equations,
which is generally very difficult, if not impossible. In the past few
decades, many different approaches have been developed for
studying reachable sets of a control system. The essence of most of
these approaches can be explained using the fundamental result of Nagano~\cite{TN:1966}, which connects
the diffeomorphism invariant properties of a system to the Lie
algebra of the vector fields of the of the system (cf.~\cite{YK-TO:16} for a beautiful alternative approach to
study small-time local controllability). Using these approaches, small-time local controllability has been studied throughly in the literature and many sufficient conditions  (cf.~\cite{AK:1974}, \cite{sussmanna1978},
\cite{sussmannlie1983}, \cite{sus1987}, \cite{RH:ADL:2004}) as well as
some necessary conditions (cf.~\cite{GS:1986}, \cite{sus1987},
\cite{MK:1987}, \cite{MIK:1998}) have been developed. However, to
our knowledge, many basic questions about properties of small-time locally controllable systems are still
unanswered. Moreover, except for some specific classes of systems
(cf.~\cite{sussmancontrollability1972},~\cite{NNP:1976},~\cite{COA-ADL:2012}), the characterization of small-time local controllability for nonlinear systems is far from complete. 

Another notion of controllability, which has a close connection with
small-time local controllability, is local accessibility. A control system is locally accessible from $x_0$ if
the reachable sets of $\mathcal{X}$ starting from $x_0$ has nonempty
interiors for all positive times. It is clear that if a system is small-time locally controllable
from $x_0$, then it is accesible from $x_0$. However, the converse may
not be true~\cite[Example 7.1]{bullogeometric2004}. In 1972, Sussmann
and Jurdjevic characterized the local 
accessibility of a real analytic control system using the Lie brackets
of the vector fields of the system at the point $x_0$~\cite[Corollary
4.7]{sussmancontrollability1972}. Sussmann and Jurdjevic's condition
implies that, for a given real analytic
control system, locally accessibility of the control system at the point $x_0$ can be checked using only ``finite'' number of
differentiations of vector fields of the system at $x_0$. This
interesting observation raises the following important question about the nature
of small-time local controllability: Is it possible to characterize small-time local
controllability of a given real analytic system using finite number of differentiations of vector fields of
the system at the point $x_0$? More precisely, this question can be 
formulated as the following conjecture (see~\cite{AA:1999}).

\begin{conjecture}\label{conj:2}
Given a real analytic control system $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ on $\R^n$ which is small-time locally
controllable from $x_0$, there exists $N\in\N$ such
that, if $\mathcal{Y}=\{Y_0,Y_1,\ldots,Y_m\}$ is another real analytic
system with the property that, for every $i\in \{0,1,2,\ldots,m\}$ we have
\begin{align*}
D^{\mathbf{r}}X_i(x_0)=D^{\mathbf{r}}Y_i(x_0),\qquad\forall \mathbf{r}=(r_1,\ldots,r_n)\in
  \mathbb{Z}^{n}_{\ge 0},\ \sum_{i=0}^{m}r_i\le N,
\end{align*}
then $\mathcal{Y}$ is small-time locally controllable from $x_0$.
\end{conjecture}
The above conjecture has the following nice interpretation based on the
perturbation of vector fields of controllable system: For a 
real analytic control system $\mathcal{X}$ which is small-time locally
controllable from $x_0$, there exists $N\in \N$
such that, if one perturbs the vector fields of $\mathcal{X}$ around
$x_0$ by terms of order higher than $N$, the resulting perturbed system is again small-time locally
controllable from $x_0$. 


% In this paper, for the sake of
% simplicity, we consider the space of parameters to be a discrete
% finite set $\{1,2,\ldots,m\}$, the state space manifold to be $\R^n$,
% and a control system is considered as a finite family of 
% the vector fields $\mathcal{X}=\{X_1,X_2,\ldots,X_m\}$. We assume that
% at least one of the vector fields in $\mathcal{X}$ has an equilibrium
% point at $x_0$. Since that small-time local controllability is a local
% property of the control system $\mathcal{X}$. Therefore, without loss of
% generality, we can assume that all the vector fields in $\mathcal{X}$
% are complete. Starting from a point $x_0$ in the state space $\R^n$,
% the reachable set of $\mathcal{X}$ at times less than $T$, which is
% denoted by $\mathrm{R}_{\mathcal{X}}(< T, x_0)$, is defined as the set
% of all points that can be reached by starting from $x_0$ and
% traveling, in positive times, on the trajectories of these vector fields for times less than $T$. More specifically, we have
% \begin{multline*}
% \mathrm{R}_{\mathcal{X}}(< T, x_0)=\{\ex(t_1X_{i_1})\scirc
%   \ex(t_2X_{i_2})\scirc \ldots\scirc \ex(t_sX_{i_s})(x_0)\\\suchthat s\in \N,\  t_i\ge 0,\ X_{i_j}\in
%   \mathcal{X},\ \sum_{i=1}^{s} t_i< T \},
% \end{multline*}
% where $\ex(tX)$ is the flow of the vector field $X$ at time $t$. Using the reachable sets of a control system, one can
% define small-time local controllability of $\mathcal{X}$. A
% control system $\mathcal{X}$ is small-time locally controllable from $x_0$ if, for all times $t$,
% $\mathrm{R}_{\mathcal{X}}(< t, x_0)$ contains an open neighbourhood of
% $x_0$. Another notion of controllability which has a close connection
% with small-time local controllability is local accessibility. A control system $\mathcal{X}$
% is called locally accessible from $x_0$, if, for all times $t$, $\mathrm{R}_{\mathcal{X}}(< t, x_0)$ has a nonempty interior.

One of the useful notions for studying small-time local
controllability is a control variation. Control variations can be
used as a suitable tool for approximating the reachable sets of control
systems (a rigorous defintion of control variations is given in Section~\ref{sec:variations}). Control variations can be considered as 
higher-order tangent vectors which shows the admissible directions in
the reachable set of a control system, i.e., for small times,
one can travel in these directions~\cite[Theorem 2.4]{MK:1990}. Therefore,
by constructing a suitable family of control variations which generates all
the directions in $\R^n$ and using a generalized open mapping theorem,
one can show that the system is small-time locally controllable~\cite[Corollary 2.5]{MK:1990}.  In the
control literature, many different families of control variations have been
introduced for studying small-time local controllability of systems (cf.~\cite{AJK:1977}, \cite{sus1987}, \cite{HF:1989}, \cite{MK:1990},
\cite{bianchinicontrollability1993}, \cite{COA-ADL:2012}). Control
variations can also be used for studying the rate of growth of the
reachable sets of a control system with respect to time. In fact, the
order of a control variation gives us some information about how fast one can travel in the
reachable set in that direction. More specifically, if one can get all
direction in $\R^n$ using families of control variations of order less
than equal to $N$ then, there exists a positive constant $C>0$ such that
the closed ball $\overline{\mathrm{B}}(x_0,Ct^N)$ is contained in the
reachable set  $\mathrm{R}_{\mathcal{X}}(< t,x_0)$ for small $t>0$~\cite[Corollary
2.5]{MK:1990}.

It turns out that this polynomial growth condition for reachable sets of a
system has a close connection with the regularity of
the time-optimal map of the system~\cite{RMB-GS:90}. One can show that, if the
control system $\mathcal{X}$ is small-time locally controllable, then
the time-optimal map of $\mathcal{X}$ is locally
continuous~\cite{NNP:1977} (cf.~\cite[Theorem 2.2]{RMB-GS:90}, where
this local result has been extended to
a larger domain called escape domain). Similarly, one can show that the polynomial growth
condition for a control system $\mathcal{X}$ is equivalent to local
H\"{o}lder continuity of the time-optimal map of $\mathcal{X}$~\cite{NNP:1971},~\cite[Theorem 2.5]{RMB-GS:90}.

As mentioned above, finding suitable families of control variations
which generate the space $\R^n$ guarantees small-time local
controllability of a control system from $x_0$. This raises the
following question: Given a small-time locally controllable system, does there exists a family of control
variations which can be used to prove small-time local controllability
of the system. It is not surprising
to see that this question has a close connection with the rate of
growth of reachable sets of the system with respect to time. Motivated
by the above question, one can propose the the following conjecture (see~\cite{AA:1999}).

\begin{conjecture}\label{conj:1}
Let $\mathcal{X}$ be a real analytic control system which is small-time
locally controllable from $x_0$. Then there exist $N\in \N$ and
$T,C>0$ such that 
\begin{equation*}
\overline{\mathrm{B}}(x_0,Ct^N)\subseteq \mathrm{R}_{\mathcal{X}}(< t,
x_0),\qquad\forall t\le T.
\end{equation*}
\end{conjecture}


One can easily check that, if the Conjecture
\ref{conj:1} is true then, for every small-time locally controllable system, one can find a
family of control variations for the control system $\mathcal{X}$
which generates all the directions in $\R^n$. As mentioned in~\cite{AA:1999}, the results in~\cite{NNP:1976} show that both Conjecture
\ref{conj:2} and Conjecture \ref{conj:1} hold for the case $n=2$. However, to the best of our knowledge, these two conjectures are still open for the general case, where
$n\ge 3$. 


Suppose that one can prove small-time local controllability of a
control system $\mathcal{X}$ using a family of control variations. Then, by~\cite[Corollary 2.5]{MK:1990}, there exists an
$N\in \N$ and $C,T>0$ such that we have
\begin{equation*}
\overline{\mathrm{B}}(x_0,Ct^N)\subseteq \mathrm{R}_{\mathcal{X}}(< t,
x_0),\qquad\forall t\le T.
\end{equation*}
It is interesting to investigate the validity of Conjecture~\ref{conj:2} for this control system. 
One can easily show that, if the family of control variations used for
proving small-time local controllability of $\mathcal{X}$ are the
classical ones constructed using the iterated
Lie brackets of vector fields of the system (e.g.,~the control
variations in~\cite{sus1987}), then
Conjecture~\ref{conj:2} holds for the control system
$\mathcal{X}$. However, there exist control systems which are small-time
locally controllable from $x_0$, but one cannot check their small-time local
controllability using classical control variations constructed
by the iterated family of Lie brackets~\cite{MK:88, MK:1990}. Example~\ref{ex:1}, adapted from~\cite{MK:88}, shows that one may need a more complicated family of
variations to prove small-time local controllability of a control system.

\begin{expl}\label{ex:1}
Consider the control system
$\mathcal{X}$ on $\R^4$, defined by
\begin{eqnarray*}
\dot{x}_1&=&u(t),\\
\dot{x}_2&=&x_1,\\
\dot{x}_3&=&x_1^3,\\
\dot{x}_4&=&x_3^2-x_2^7,
\end{eqnarray*}
where $u:\mathbb{R}\to [-1,1]$ is measurable. We want to study small-time local controllability of $\mathcal{X}$
from $\mathbf{0}\in \R^4$. Using suitable families of control
variations with finite number of switching, one can show that
$\left\{\pm\frac{\partial}{\partial x_1}, \pm\frac{\partial}{\partial
    x_2}, \pm\frac{\partial}{\partial x_3}, \frac{\partial}{\partial
    x_4}\right\}$ are the admissible directions in the reachable set
of the systems $\mathcal{X}$~\cite{MK:88},~\cite{sus1987}. In order to prove small-time local controllability of $\mathcal{X}$, one needs to find a control variation which generates the direction $-\frac{\partial}{\partial x_4}$.
It can be shown that there is no family of control variations with finite number of
switching which generates the direction $-\frac{\partial}{\partial x_4}$~\cite[Claim 2]{MK:88}. However, by defining families of control
variations with an increasing number of switching, which are called
fast switching variations, one can show that $\mathcal{X}$ is
small-time locally controllable from $\mathbf{0}\in \R^4$~\cite[Claim
1]{MK:88}~(see~\cite{MK:88} and \cite{MK:1990} for the beautiful and
detailed construction of these control variations). Now consider the
control system $\mathcal{Y}$ on $\R^4$ defined by
\begin{eqnarray*}
\dot{y}_1&=&u(t)+y_1^{58},\\
\dot{y}_2&=&y_1,\\
\dot{y}_3&=&y_1^3,\\
\dot{y}_4&=&y_3^2-y_2^7,
\end{eqnarray*}
where $u:\mathbb{R}\to [-1,1]$ is measurable. Note that $\mathcal{X}$ and $\mathcal{Y}$ have the same Taylor polynomial of order $57$ at
$\mathbf{0}\in \R^4$. Using the classical finite switching control
variations, it is easy to show that
$\left\{\pm\frac{\partial}{\partial y_1}, \pm\frac{\partial}{\partial
    y_2}, \pm\frac{\partial}{\partial y_3}, \frac{\partial}{\partial
    y_4}\right\}$ are the admissible directions in the reachable set
$\mathcal{Y}$. However, using the same fast switching variations as for
the system $\mathcal{X}$, it is very complicated to study whether $-\frac{\partial}{\partial
    y_4}$ is an admissible direction for the reachable sets of the control system
$\mathcal{Y}$ at point $\mathbf{0}\in \R^4$.
\end{expl}

In general, the families of control variations that are used to prove small-time local
controllability of a system might be even more complicated than the fast
switching control variations in Example~\ref{ex:1}. Therefore, for
small-time locally controllable systems, studying Conjecture
\ref{conj:2} using the form of families of control variations does not seem to be conclusive. 


Consider a real analytic system $\mathcal{X}$ which is
small-time locally controllable from $x_0$ and suppose that there
exist $C,T>0$ such that $\overline{\mathrm{B}}(x_0,Ct^N)\subseteq
\mathrm{R}_{\mathcal{X}}(< t,x_0)$, for every $t\le T$. Let
$\mathcal{Y}$ be another real analytic system with the property that
its vector fields have the same Taylor polynomial of order $N$ around $x_0$ as the vector fields of
$\mathcal{X}$. Thus, the control system $\mathcal{Y}$ can be
considered as an $N$th order perturbation of the control system
$\mathcal{X}$ around $x_0$. Our main result in this paper proves that, not only
is $\mathcal{Y}$ small-time locally controllable from $x_0$,
but also its reachable sets grow with polynomial rate of order
$N$ or higher with respect to time. More precisely, we show that there exist $\overline{T},\alpha>0$
such that we have 
\begin{align*}
\overline{\mathrm{B}}(x_0,\alpha t^N)\subseteq
\mathrm{R}_{\mathcal{Y}}(<t,x_0),\qquad\forall t\le \overline{T}.
\end{align*}
In particular, using the above result, we can show that if 
Conjecture \ref{conj:1} holds then Conjecture \ref{conj:2} should also hold. The proof of the main theorem is based on using suitable
class of control variations
  for studying small-time local controllability of systems. This
  general class of control variations is defined in~\cite{MK:1990}
  and~\cite{HF:1989}. One can show that this class of control
variations completely characterizes the control systems whose reachable
sets grow with polynomial rate. Using this characterization, we prove
the main result of the paper in three steps. 

% The first step is to characterize the the polynomial growth condition for reachable
% sets using the control variations define in
% Definition~\ref{def:control_variations}. This is done in
% Lemma~\ref{lem:characterization_growth_rate}, where we show 
% that for a control system $\mathcal{X}$, the cone generated by the
% control variations of order $k$ for $\mathcal{X}$ is the whole space
% $\R^n$ if and only if the reachable sets of the control system shrink
% with polynomial rate of order $k$ or higher with respect to time. 

In the first step, we study the effect of perturbations of vector fields of the control system $\mathcal{X}$ on
its reachable sets. This is done in section~\ref{sec:perturbations} by
constructing a suitable multi-valued map $F^t_{\mathcal{X},\mathcal{Y}}$ from
$\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$ (which is inside
$\mathrm{int}(\mathrm{R}_{\mathcal{X}}(< t,x_0))$) to the set
$\mathrm{R}_{\mathcal{Y}}(< t,x_0)$. Using the normal reachability of
real analytic control system $\mathcal{X}$, one can show that every point
in the set $\mathrm{int}(\mathrm{R}_{\mathcal{X}}(<
t,x_0))$ is reachable from $x_0$ by switching between constant vector
fields of $\mathcal{X}$~\cite[Theorem 5.5]{KAG:1992}. The idea is to use these switching times
to do the transition between the reachable sets of control system
$\mathcal{X}$ and $\mathcal{Y}$. More precisely, the map
$F^t_{\mathcal{X},\mathcal{Y}}$ can be defined as composition of two
mappings $\eta^t_{\mathcal{X}}$ and $\xi^t_{\mathcal{Y}}$. The first
mapping $\eta^t_{\mathcal{X}}$ maps a point in
$\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$ to its associated
switching times for the control system $\mathcal{X}$ and the second mapping
$\xi^t_{\mathcal{Y}}$ maps the switching times to its associated point
in the reachable set of the system $\mathcal{Y}$. The definition of 
the map $\xi^t_{\mathcal{Y}}$ is straightforward using the flows of
vector fields of $\mathcal{Y}$. However, since a point in the
reachable set of $\mathcal{X}$ might have more than one set of
switching times, there are different ways for defining the map
$\eta^t_{\mathcal{X}}$. In section~\ref{sec:perturbations}, using the
properties of the real analytic system $\mathcal{X}$, the map
$\eta^t_{\mathcal{X}}$ is defined such that it is finite-valued and satisfies nice regularity conditions. 



%  The idea is that
% the map $F^t_{\mathcal{X},\mathcal{Y}}$ consists of two parts $\eta$
% and $\xi$. The first part maps a point in a suitable domain in the
% reachable set of $\mathcal{X}$ to its associated swithching times for
% the system $\mathcal{X}$. The second part maps the swithcing times to
% the associed points for the system $\mathcal{Y}$. These two maps 


%  It is important to note
% that a point in the reachable set of $\mathcal{X}$ might have more
% than one set of switching times. Using the properties of normal
% reachability for real analytic control systems~\cite{KAG:1992}, we construct the mapping
% $x\mapsto \eta^t_{\mathcal{X}}(x)$ in such a way that it is
% finite-valued and has nice regularity properties. 



% that this map captures how points in the reachable set changes,
% when the vector fields of the system is perturbed from $\mathcal{X}$ to
% $\mathcal{Y}$. The mapping
% $F^{t}_{\mathcal{X},\mathcal{Y}}$ is defined by the composition of two
% other maps $\eta^t_{\mathcal{X}}$ and $\xi^t_{\mathcal{Y}}$ such that $F^t_{\mathcal{X},\mathcal{Y}}=\xi^t_{\mathcal{Y}}\scirc
% \eta^t_{\mathcal{X}}$. The mapping $\eta^t_{\mathcal{X}}$ maps a point in the reachable set of the
% system $\mathcal{X}$ into its switching times. It is important to note
% that a point in the reachable set of $\mathcal{X}$ might have more
% than one set of switching times. Use the properties of normal
% reachability for real analytic control systems~\cite{KAG:1992}, we construct the mapping
% $x\mapsto \eta^t_{\mathcal{X}}(x)$ in such a way that it is
% finite-valued and has nice regularity properties. The second part of $F^t_{\mathcal{X},\mathcal{Y}}$ is a single-valued
% mapping $\xi^t_{\mathcal{Y}}$ which maps a switching time to the
% associated point in reachable set of $\mathcal{Y}$. Therefore, by composing $\eta^t_{\mathcal{X}}$
% and $\xi^t_{\mathcal{Y}}$, we get the multi-valued mapping $F^t_{\mathcal{X},\mathcal{Y}}$
% which sends a point in the reachable set of $\mathcal{X}$ to the
% associated points in the reachable set of $\mathcal{Y}$. 

In the second step, we prove some regularity properties of the set-valued mapping
$t\rightrightarrows F^t_{\mathcal{X},\mathcal{Y}}$. This is done in Theorem~\ref{lem:multi-valued_map} where it has been shown that, for small $t$, the distance
between a point in the domain of $F^t_{\mathcal{X},\mathcal{Y}}$ and its image grows with polynomial rate of
order $N+1$ or higher with respect to time. The heart of the proof is the estimates
introduced in Definition~\ref{eq:sequence_iterations} for the flows of
time-varying vector fields and the Corollary~\ref{cor:estimate_picard_iterations}. It is worth mentioning that the role of the real
analyticity of vector fields is crucial in the proof of this result. 

% While
% it is clear that these estimates for the flows of time-varying vector
% fields can be defined for both smooth and real analytic vector
% fields, the convergence of these iterative estimates to the flow of the vector
% field only holds in the real analytic case~\cite[\S 2.4.4]{agrachevcontrol2004}. 

In the final step, we construct a family of control variations for the
control system $\mathcal{Y}$. We start by considering a specific family of control
variations for the control system $\mathcal{X}$ which generates the space $\R^n$. Then, we apply a single-valued selection
of the mapping $F^t_{\mathcal{X},\mathcal{Y}}$ to this family of 
of control variations to get a family of control variations for the
control system $\mathcal{Y}$. Using
the regularity of the map $t\rightrightarrows
F^t_{\mathcal{X},\mathcal{Y}}$, we show that this family of control
variations for the control system $\mathcal{Y}$ generates $\R^n$. This
implies that the control system $\mathcal{Y}$ is small-time locally controllable from $x_0$. 

This paper is organized as follows. In
section~\ref{sec:functions_vector_fields} we study functions and
vector fields on $\R^n$. In section~\ref{sec:chronological} we review
an operator approach for studying time-varying vector fields and their
flows, known as chronological calculus. This operator approach
has been first introduced in~\cite{agrachev1978exponential} using the Whitney compact-open topology on the
space $\mathrm{C}^{\infty}(\R^n)$. In this paper, we present an
extension of this framework to the space of real
analytic functions with $\mathrm{C}^{\omega}$-topology~\cite{SJ:2016}.
In section~\ref{sec:control_systems} we define $C^{\nu}$-control
systems and their reachable sets. We also define the notions of small-time
local controllability, growth rate condition, and normal reachability,
and we state a classical result connecting the normal reachability and
small-time local controllability for real analytic control systems. In
section~\ref{sec:variations} the notion of 
control variations is defined and a characterization of the growth rate condition is presented based on
the control variations of the system. In
section~\ref{sec:perturbations} we consider two real analytic control systems
$\mathcal{X}$ and $\mathcal{Y}$ whose vector fields have the same Taylor polynomials
of order $N$ around point $x_0$. We construct a
mapping $F^t_{\mathcal{X},\mathcal{Y}}$ between the reachable sets of
the control system $\mathcal{X}$ and $\mathcal{Y}$. In
Theorem~\ref{lem:multi-valued_map}, we study the regularity of this
map. Finally, in section~\ref{sec:main_result}, the main result of this paper is stated and proved. 


% For a single-valued differentiable mapping, one can study local
% openness of the range of the mapping using the derivatives of that
% mapping \cite[Chapter 1, \S4]{VG-AP:1974}. In the case that the mapping is only Lipschitz,
% one can use a topological argument, based on the Brouwer fixed-point
% theorem, to show the local openness of the range of the mapping
% \cite[Lemma 12.4]{agrachevcontrol2004}. This topological 
% argument plays an essential role in studying small-time local
% controllability of systems using a family of control variations (cf. \cite{bianchinicontrollability1993}, \cite{COA-ADL:2012}).

% Brouwer fixed-point theorem is
% one of the most fundamental theorems in topology. Roughly speaking,
% the Brouwer fix point theorem says that a continuous map, from a compact convex subset of $\R^n$
% into itself, has a fixed-point \cite[Chapter 2, \S 2]{VG-AP:1974}. There are many different generalization
% of the Brouwer fixed-point theorem in the literature. In \cite{PB:2006}, the Brouwer fixed-point
% theorem has been generalized for discontinuous mappings which
% satisfy a condition called ``half-continuity''. 

% In order to study local openness of the range of the map
% $F^t_{\mathcal{X},\mathcal{Y}}$, we focus on its single-valued selections. While, finding a continuous
% single-valued selection of the 
% multi-valued mapping $F^t_{\mathcal{X},\mathcal{Y}}$ seems elusive, one can show
% that $F^{t}_{\mathcal{X},\mathcal{Y}}$ admits a ``half-continuous'' single-valued selection. 
% Thus, using the generalized version of the Brouwer fixed-point
% theorem, it can be shown that the range of the mapping $F^t_{\mathcal{X},\mathcal{Y}}$
% contains an open neighbourhood of $x_0$ for all small enough $t$. 



\subsection{Notations and conventions}
In this paper, the set of integers is denoted by $\Z$ and the
set of positive integers is denoted by $\N$. We denote the $n$-dimensional Euclidean space by
$\R^{n}$ and the Euclidean norm of a vector $\mathbf{v}$ in $\R^n$ is
denoted by $\|\mathbf{v}\|$. The $n$-sphere $\mathbb{S}^n$ is defined as
\begin{align*}
\mathbb{S}^n=\{(x_1,x_2,\ldots,x_n)\in\R^{n+1}\mid
x^2_1+x_2^2+\ldots+x_{n+1}^2=1\}
\end{align*}
and we use the notation 
\begin{align*}
\R_{\ge 0}^{n}=\{(x_1,x_2,\ldots,x_n)\in\R^n\mid x_i\ge
0,\ \forall i\in \{1,2,\ldots,n\}\}.
\end{align*}
The interval $[-1,1]$ is defined by $[-1,1]=\{x\in \R\mid -1\le x\le 1\}$ and we denote
\begin{align*}
[-1,1]^n=\underbrace{[-1,1]\times [-1,1]\times \ldots\times [-1,1]}_{n
  \mbox{ times }}.
\end{align*}
For a nonempty subset $S\subseteq \R^n$, the interior of $S$ in $\R^n$ is
denoted by $\mathrm{int}(S)$ and the closure of $S$ in $\R^n$ is denoted by
$\overline{S}$. A multi-index of order $m$ is an element
$\mathbf{r}=(r_1,r_2,\ldots,r_m)\in \Z_{\ge 0}^m$. For all multi-indices $\mathbf{r}$ and $\mathbf{s}$ of order $m$, every
$\mathbf{x}=(x_1,x_2,\ldots,x_m)\in \R^m$, and $f:\R^m\to\R^n$, we define
\begin{eqnarray*}
|\mathbf{r}|&=&r_1+r_2+\ldots+r_m,\\
\mathbf{r}!&=&(r_1!)(r_2!)\ldots(r_m!),\\
\mathbf{x}^{\mathbf{r}}&=&x_1^{r_1}x_2^{r_2}\ldots x_m^{r_m},\\
D^{\mathbf{r}}f(x)&=&\frac{\partial^{|\mathbf{r}|} f}{\partial
               x_1^{r_1}\partial x_2^{r_2}\ldots \partial x_m^{r_m}}.
\end{eqnarray*}
The space of all decreasing sequences $\{a_i\}_{i\in\N}$ such that
$a_i\in \R_{>0}$ and $\lim_{n\to\infty} a_n=0$ is denoted by
$c^{\downarrow}_{0}$. Let $x\in \R^n$ and $r\in
\R^{>0}$. Then the open ball centered at $x$ with radius $r$ is
denoted by $\mathrm{B}(x,r)$. i.e.,  we have
\begin{align*}
\mathrm{B}(x,r)=\left\{y\in\R^n\suchthat \|y-x\|<r \right\}.
\end{align*} 
The closed ball centered at $x$ with radius $r$ is denoted by
$\overline{\mathrm{B}}(x,r)$. i.e., we have
\begin{align*}
\overline{\mathrm{B}}(x,r)=\left\{y\in\R^n\suchthat \|y-x\|\le r
\right\}.
\end{align*}
In this paper, whenever we use the letter $\nu$, we mean that $\nu\in
\N\cup\{\infty,\omega\}$. Let $U\subseteq\R^m$ be an open set. The
mapping $f:U\to \R^{n}$ is a $\mathrm{C}^{\nu}$-mapping if, for every multi-index
$\mathbf{r}\in \Z_{\ge 0}^m$ with property that $|\mathbf{r}|\le \nu$,
the mapping $D^{\mathbf{r}}f$ is continuous. Let $k\in \N$, $(V,\|.\|_{V})$ be a normed vector space, and $f:\R\to V$ and
$g:\R\to V$ be two curves on $V$. Then we write
\begin{align*}
f(x)=g(x)+\mathcal{O}(x^k)
\end{align*}
if there exists $\alpha\in \R$ such that we have $\lim_{x\to 0}\frac{\|f(x)-g(x)\|_{V}}{|x|^{k}}=\alpha$. Let $U$ and $V$ be
two sets and $F:U \rightrightarrows V$ be a multi-valued mapping. Then
a selection of $F$ is the single-valued mapping $f:U \to
V$ with the property that
\begin{align*}
f(x)\in F(x),\qquad\forall x\in U.
\end{align*}


\section{Functions and vector fields}\label{sec:functions_vector_fields}

In this section, we study functions and vector fields on the
Euclidean space $\R^n$. We first define the class of real analytic
mappings.
\begin{definition}[\textbf{Taylor series and real analytic mappings}]
Let $\Omega\subseteq \R^n$ be an open set, $x_0\in \Omega$, and
$f:\Omega\to \R^l$ be a $\mathrm{C}^{\infty}$-mapping at $x_0$. Then the Taylor
  series of $f$ at $x_0$ is the power series
\begin{equation}\label{eq:3}
\sum_{\mathbf{r}\in \Z_{\ge 0}^m} \frac{1}{\mathbf{r}!}\left[D^{\mathbf{r}}f(x_0)\right] (x-x_0)^{\mathbf{r}}.
\end{equation}
A $C^{\infty}$-mapping $f:\Omega\to \R^l$ is
real analytic or of class $\mathrm{C}^{\omega}$ if, for every
$x_0\in \Omega$, there exists $\rho>0$ such that the Taylor series \eqref{eq:3} of
$f$ at $x_0$ converges to $f(x)$ for all $\|x-x_0\|<\rho$. A mapping $f:\Omega\to \R^l$ is real analytic on $\Omega$ if, for every $x\in \Omega$, it is real analytic at $x$.
\end{definition}



\begin{definition}[\textbf{Functions and vector fields}]\label{def:vector_fields}
A $C^{\nu}$-vector field on $\R^n$ is a $\mathrm{C}^{\nu}$-mapping $V:\R^n\to \R^n\times \R^n$ such
that $\mathrm{pr}_1(V(x))=x$.\footnote{Note that this is a special case
of the more general definition of the vector fields on a manifold
$M$. Let $M$ be a manifold with the tangent bundle $\pi:TM\to M$. Then a
$\mathrm{C}^{\nu}$-mapping $V:M\to TM$ is a vector field of class $\mathrm{C}^{\nu}$ on $M$, if we have $\pi\scirc
V(x)=x$, for every $x\in M$. Using
the identifications $T\R^n\simeq\R^n\times \R^n$, it is easy to see that
this definition of vector fields is equivalent to
Definition~\ref{def:vector_fields}}
A $C^{\nu}$-function on $\R^n$ is a $C^{\nu}$-mapping $f:\R^n\to \R$. 
\end{definition}
Since, for a vector field $V:\R^n\to\R^n\times \R^n$, we have
$\mathrm{pr}_1(V(x))=x$, it is common to use the identification
$V\simeq \mathrm{pr}_2\scirc V$. In the rest of this paper, for the sake
of simplicity, we consider a vector field as a
mapping $V:\R^n\to\R^n$. If $V:\R^n\to\R^n$ is a vector field on
$\R^n$, then in the standard coordinate chart on $\R^n$, we write
\begin{align*}
V=(V^1,V^2,\ldots,V^n)^{\top},
\end{align*}
where, for every $i\in\{1,2,\ldots,n\}$, the $C^{\nu}$-function
$V^i:\R^n\to \R$ is the $i$th component of the vector field $V$. 
The space of all $\mathrm{C}^{\nu}$-functions on $\R^n$ is denoted by
$\mathrm{C}^{\nu}(\R^n)$ and the space of all $\mathrm{C}^{\nu}$-vector fields on
$\R^n$ is denoted by $\Gamma^{\nu}(\R^n)$. It is easy to see that
both $\mathrm{C}^{\nu}(\R^n)$ and $\Gamma^{\nu}(\R^n)$ are vector
spaces over $\R$. Given $x_0\in \R^n$, we define the functional
$\mathrm{ev}_{x_0}:\mathrm{C}^{\nu}(\R^n)\to \R$ by $\mathrm{ev}_{x_0}(f)=f(x_0)$,
for every $f\in \mathrm{C}^{\nu}(\R^n)$. In mathematical control theory, it is common to work with
time-varying vector fields. Therefore, it is essential to give a
rigorous definition of them in this paper. 

\begin{definition}[\textbf{Time-varying vector fields}]
Let $\mathbb{T}\subseteq \R$ be an interval. The map $X:\mathbb{T}\times \R^n\to \R^n$ is a
time-varying vector field of class $\mathrm{C}^{\nu}$ if the
following hold:
\begin{itemize}
\item[(i)] For every $t\in \mathbb{T}$, the map $X_t:\R^n\to \R^n$ defined by
\begin{equation*}
X_t(x)=X(t,x),\qquad\forall x\in \R^n,
\end{equation*}
is a vector field of class $\mathrm{C}^{\nu}$.
\item[(ii)] For every $x\in \R^n$, the curve $X^x:\mathbb{T}\to \R^n$ defined
  by
\begin{equation*}
X^x(t)=X(t,x),\qquad\forall t\in \mathbb{T},
\end{equation*}
is essentially bounded. 
\end{itemize}
\end{definition}

Let $X:\R\times\R^n\to\R^n$ be a time-varying vector
field. Then, by the fundamental theorem of ordinary differential
equations~\cite[Theorem 2.3]{coddington1955theory}, for every $x_0\in \R^n$, there exist an interval
$\mathbb{T}_{x_0}$ and an absolutely continuous curve $t\mapsto \exp(tX)(x_0)$ such that the following initial value
problem holds.
\begin{align*}
\frac{d}{dt}(\exp(tX)(x_0))&=X(t,\exp(tX)(x_0)),\qquad\mbox{ for almost
                              every }
                              t\in \mathbb{T}_{x_0}\\
\exp(0 X)(x_0)&=x_0.
\end{align*}
The map $t\mapsto \exp(tX)(x_0)$ is called the integral curve of the time-varying vector field
$X$ passing through $x_0$.

\begin{definition}[\textbf{Complete vector fields}]
A vector field is complete if, for every $x_0\in \R$, the integral
field of $X$ passing through $x_0$ is defined for all $t\in\R$.
\end{definition}
In the rest of this paper, for the sake of simplicity, we assume that all the vector fields are complete.
Note that this assumption is not restrictive for our analysis in this paper. The reason is that we only study local properties of control systems
with compact control sets~\cite[Chapter 2, Theorem 1.1]{coddington1955theory}.

\begin{definition}[\textbf{Flows of time-varying vector fields}]
Let $X:\R\times\R^n\to\R^n$ be a complete time-varying $C^{\nu}$-vector
field. Then the flow of $X$ is the map $\exp(X):\R\times \R^n\to \R^n$
defined by
\begin{align*}
\exp(X)(t,x)=\exp(tX)(x),\qquad\forall (t,x)\in \R\times \R^n.
\end{align*}
\end{definition}


\section{Operator approach for time-varying vector fields}\label{sec:chronological}

In this section, we introduce an operator calculus
which allows us to translate the nonlinear finite-dimensional systems
into linear infinite-dimensional systems. This
operator approach, which is known as \emph{chronological calculus},
was originally developed by Agrachev and Gamkrelidze
in~\cite{agrachev1978exponential}. In this framework, a $\mathrm{C}^{\infty}$-vector field is
considered as a derivation of the algebra $\mathrm{C}^{\infty}(\R^n)$
and a $\mathrm{C}^{\infty}$-diffeomorphism is considered as a unital
algebra isomorphism on $\mathrm{C}^{\infty}(\R^n)$. These
characterizations lead to the identification of the nonlinear
dynamical system governing
the flow of a time-varying vector field with a linear differential
equation an infinite-dimensional space. Using this identification and the Whitney
compact-open topology on the space of $\mathrm{C}^{\infty}(\R^n)$, an asymptotic expansion for the flow of a time-varying real
analytic vector field is developed and its convergence has been
studied in~\cite{agrachev1978exponential}. In~\cite{SJ-ADL:2014}
and~\cite{SJ:2016}, this framework has been extended in two directions. First, the real analytic vector fields
are considered as derivations on $\mathrm{C}^{\omega}(\R^n)$
and real analytic diffeomorphisms are considered as unital algebra homomorphism on
$\mathrm{C}^{\omega}(\R^n)$. Moreover, the space
$\mathrm{C}^{\omega}(\R^n)$ is endowed with the $\mathrm{C}^{\omega}$-topology
and the convergence of the asymptotic expansion for the flow of a
time-varying real analytic vector field is
studied in this new topology~\cite{SJ:2016}. In the sequel, we adopt the approach
of~\cite{SJ-ADL:2014} for studying time-varying real analytic vector fields and their flows.  

\begin{definition}[\textbf{Vector fields are derivations}]\label{def:vector_field_derivation}
Let $V:\R^n\to \R^n$ be a real analytic vector field. Then we define
the derivation $\widehat{V}:\mathrm{C}^{\omega}(\R^n)\to \mathrm{C}^{\omega}(\R^n)$ by
\begin{equation*}
\widehat{V}(f)=\mathcal{L}_{V}f,
\end{equation*}
where $\mathcal{L}_{V}f$ is the Lie derivative of the function $f$ in
direction of the vector field $V$.
\end{definition}


\begin{definition}[\textbf{Diffeomorphisms are algebra homomorphism}]
Let $\phi:\R^n\to \R^n$ be a real analytic diffeomorphism. Then we define the unital algebra homomorphism $\widehat{\phi}:\mathrm{C}^{\omega}(\R^n)\to
\mathrm{C}^{\omega}(\R^n)$ by
\begin{equation*}
\widehat{\phi}(f)=f\scirc\phi.
\end{equation*}
\end{definition}
By the above definitions, one can consider real analytic vector fields and real analytic diffeomorphisms on
$\R^n$ as linear operators on $\mathrm{C}^{\omega}(\R^n)$. The space of linear
mappings from $\mathrm{C}^{\omega}(\R^n)$ to $\mathrm{C}^{\omega}(\R^n)$ is denoted by
$\mathrm{LC}^{\omega}(\R^n)$ and it is clear that we
have $\Gamma^{\nu}(\R^n)\subset \mathrm{LC}^{\omega}(\R^n)$. Thus, for every real
analytic vector field $X:\R^n\to \R^n$ and
every real analytic mapping $\phi:\R^n\to \R^n$, we have 
\begin{equation*}
\widehat{\phi},\widehat{V}\in \mathrm{LC}^{\omega}(\R^n).
\end{equation*}
Using the operator characterization of vector fields, a time-varying $\mathrm{C}^{\nu}$-vector field $X:\mathbb{T}\times
\R^n\to \R^n$ can be considered as a curve $t\mapsto \widehat{X}_t$ on the space
$\mathrm{LC}^{\omega}(\R^n)$. Therefore, for studying properties of time-varying vector fields in this
framework, we need to define a suitable topology on the vector space $\mathrm{LC}^{\omega}(\R^n)$.

\begin{definition}[\textbf{Topological vector space}]
Let $E$ be a vector space over $\R$ and $\tau$ be a topology on $E$ such that, with
respect to $\tau$, both addition and scalar
multiplication are continuous. The pair $(E,\tau)$ is called a
topological vector space. A subset $B\subseteq E$
is bounded if, for every neighbourhood $U$of $0$ in $E$,
there exists $\alpha\in \R$ such that $B\subset \alpha U$.
\end{definition}

A locally convex topological vector space $(E,\tau)$ is topological vector spaces whose topology $\tau$ is generated using a family of seminorms.

\begin{definition}[\textbf{Locally convex space}]
A topological vector space $(E,\tau)$ is locally convex if the
topology $\tau$ is generated by a family of seminorms $\{p_{i}\}_{i\in\Lambda}$ on $E$.
\end{definition}

It is interesting to note that, in the locally convex spaces, bounded sets have the following nice
characterization~\cite[Theorem 1.37]{rudinfunctional}. 
\begin{theorem}[\textbf{Seminorm characterization of bounded sets}]\label{thm:boundedset_locallyconvex}
Let $E$ be a locally convex space which is generated by the family of
seminorms $\{p_{i}\}_{i\in\Lambda}$. A set $B\subseteq E$ is bounded
if and only if, for every $i\in\Lambda$, there exists $N_i\in \R_{>0}$
such that
\begin{align*}
p_i(v)\le N_i,\qquad\forall v\in B.
\end{align*}
\end{theorem}

We are now ready to define a locally convex topology on the vector
spaces $\mathrm{C}^{\omega}(\R^n)$ and $\mathrm{LC}^{\omega}(\R^n)$ using families of seminorms. 

\begin{definition}[\textbf{Real analytic seminorms}]
Let $K\subset \R^n$ be a compact set and $\mathbf{a}\in
c^{\downarrow}_0$. 
\begin{enumerate}
\item[(i)] We define the seminorm $\rho^{\omega}_{K,\mathbf{a}}:\mathrm{C}^{\omega}(\R^n)\to \R_{\ge 0}$ as
\begin{equation*}
\rho^{\omega}_{K,\mathbf{a}}(f)=\sup\left\{\frac{a_0a_1\ldots a_{|r|}}{|r|!}\left\|
    D^{(r)}f(x)\right\|\suchthat x\in K, |r|\in \Z_{\ge 0}\right\}.
\end{equation*}
The topology on $\mathrm{C}^{\omega}(\R^n)$ generated by the family of seminorms
$\rho^{\omega}_{K,\mathbf{a}}$ is denoted by the $\mathrm{C}^{\omega}$-topology.
\item[(ii)] Let $f\in \mathrm{C}^{\omega}(\R^n)$. We define the seminorm
$\rho^{\omega}_{K,\mathbf{a},f}:\mathrm{LC}^{\omega}(\R^n)\to \R_{\ge 0}$ as
\begin{equation*}
\rho^{\omega}_{K,\mathbf{a},f}(X)=\sup\left\{\frac{a_0a_1\ldots a_{|r|}}{|r|!}\left\|
    D^{(r)}\left(Xf\right)(x)\right\|\suchthat x\in K, |r|\in \Z_{\ge 0}\right\}
\end{equation*}
The topology on $\mathrm{LC}^{\omega}(\R^n)$ generated by the family of seminorms
$\left\{\rho^{\omega}_{K,\mathbf{a},f}\right\}$ is denoted by $\mathrm{C}^{\omega}$-topology.
\end{enumerate}
\end{definition}

Note that one can define another locally convex topology on
$\mathrm{C}^{\omega}(\R^n)$ by inducing the Whitney compact-open
topology and using the subspace relation~\cite[\S 2.2]{agrachevcontrol2004},~\cite[\S
6]{kriegl1997convenient}. It turns out that the $\mathrm{C}^{\omega}$-topology on the space
$\mathrm{C}^{\omega}(\R^n)$ is finer than the subspace topology
induced from the Whitney topology on
$\mathrm{C}^{\infty}(\R^n)$~\cite[Chapter 5]{SJ-ADL:2014}. The $\mathrm{C}^{\omega}$-topology and its properties has been studied throughly
in~\cite{AM:1966},~\cite{PD:2010}, and~\cite{kriegl1997convenient}. The
$\mathrm{C}^{\omega}$-topology on the space of real analytic functions
has been first defined and studied using advanced tools in analysis in~\cite{AM:1966}. The above seminorm characterization
of the $\mathrm{C}^{\omega}$-topology has been introduced and proved in~\cite{DV:2013} (see~\cite{PD:2010}
for a detailed study of the $\mathrm{C}^{\omega}$-topology on the space $\mathrm{C}^{\omega}(\R^n)$ ). 

Using the $\mathrm{C}^{\omega}$-topology on the vector space
$\mathrm{LC}^{\omega}(\R^n)$, we can study properties of
time-varying vector fields as curves on $\mathrm{LC}^{\omega}(\R^n)$. 



\begin{definition}[\textbf{Essentially bounded curves}]
Let $\mathbb{T}\subseteq \R$ be an interval. 
A measurable curve $\lambda:\mathbb{T}\to \mathrm{LC}^{\omega}(\R^n)$ is
essentially bounded if, for every compact set
$K\subset \R^n$, every $\mathbf{a}\in
c^{\downarrow}_0$, and every $f\in \mathrm{C}^{\omega}(\R^n)$, there exists $M>0$ such that
\begin{equation*}
\rho^{\omega}_{K,\mathbf{a},f}(\lambda(t))<M,\qquad\mbox{for almost every } t\in \mathbb{T}.
\end{equation*}

The set of essentially bounded curves with domain $\mathbb{T}$ on $\mathrm{LC}^{\omega}(\R^n)$
is denoted by $\mathrm{L}^{\infty}(\mathbb{T};\mathrm{LC}^{\omega}(\R^n))$. Let $S\subseteq \mathrm{LC}^{\omega}(\R^n)$. We define the subset
$\mathrm{L}^{\infty}(\mathbb{T};S)\subseteq
\mathrm{L}^{\infty}(\mathbb{T};\mathrm{LC}^{\omega}(\R^n))$ as
\begin{equation*}
\mathrm{L}^{\infty}(\mathbb{T};S)=\left\{\lambda\in
  \mathrm{L}^{\infty}(\mathbb{T};\mathrm{LC}^{\omega}(\R^n))\suchthat \lambda(t)\in S, \
  \mbox{ for almost every } t\in \mathbb{T} \right\}.
\end{equation*} 
\end{definition}
Let $X:\mathbb{T}\times \R^n\to \R^n$ be a time-varying real analytic
vector field. Then we say that $X$ is essentially bounded if $t\mapsto
\widehat{X}_t$ is an essentially bounded curve on $\mathrm{LC}^{\omega}(\R^n)$. By considering $X$ as a curve $t\mapsto \widehat{X}_t$ on
the space $\mathrm{LC}^{\omega}(\R^n)$, one can also translate the nonlinear differential equation governing the flow of $X$:
\begin{align*}
\frac{d}{dt}\exp(tX)(x_0)&=X(t,\exp(tX)(x_0)),\quad \mbox{ for almost
                           every }t\in \R\\
\exp(0X)(x_0)&=x_0,
\end{align*}
into the following linear differential equation:
\begin{eqnarray}\label{eq:1}
\begin{split}
\frac{d}{dt}\widehat{\exp}(tX)&=\widehat{\exp}(tX)\scirc
\widehat{X}_t,\quad \mbox{ for almost every }t\in\R\\
\widehat{\exp}(0X)&=\mathrm{id},
\end{split}
\end{eqnarray}
where $\widehat{\exp}(tX)$ is the unital algebra homomorphism
associated to $\exp(tX)$ (see
Definition~\ref{def:vector_field_derivation}). Note that equation~\eqref{eq:1} is a linear differential equation on
the infinite dimensional locally convex space $\mathrm{LC}^{\omega}(\R^n)$. 
One can study the sequence of Picard's iterations for this infinite dimensional linear
differential equation~\eqref{eq:1}~\cite[Chapter 1, \S
3]{coddington1955theory},~\cite{SGL-GOS:1994}. 
\begin{definition}[\textbf{Sequence of iterations for flows}]\label{eq:sequence_iterations}
Let $X$ be an essentially bounded time-varying vector field of class
$\mathrm{C}^{\omega}$. We define the curve 
$t\mapsto \widehat{\exp}_0(tX)$ on $\mathrm{LC}^{\omega}(\R^n)$ as:
\begin{equation*}
\widehat{\exp}_0(tX)=\mathrm{id},\qquad\forall t\in [0,T'].
\end{equation*}
Then, for every $k\in \N$, we define the curve $t\mapsto
\widehat{\exp}_k (tX)$ on $\mathrm{LC}^{\omega}(\R^n)$ inductively as
\begin{equation}\label{eq:Picards_iterations}
\widehat{\exp}_k(tX)=\mathrm{id}+\int_{0}^{t}\widehat{\exp}_{k-1}(\tau
X)\scirc\widehat{X}(\tau)d\tau,\qquad\forall t\in [0,T'].
\end{equation}
\end{definition}
It is worth mentioning that, for linear differential equations on infinite dimensional locally convex
spaces, there does not exist a general result for convergence of the sequence of
Picard's iterations~\cite{SGL-GOS:1994}. However, for the differential
equation~\eqref{eq:1}, one can prove the following estimates for the seminorms of the sequence of 
iterations of the flows in
Definition~\ref{eq:sequence_iterations}~\cite[Theorem
3.8.1]{SJ:2016}. The following theorem can be considered as extension of the
estimates in~\cite[\S 2.4.4]{agrachevcontrol2004} to the $\mathrm{C}^{\omega}$-topology. 

\begin{theorem}[\textbf{Estimates for iterations}]\label{th:3}
Let $B$ be a bounded set in $\Gamma^{\omega}(\R^n)$. Then the
following statements hold:
\begin{enumerate}
\item[(i)] there exists $T_{B}$ such that, for every $X\in
  \mathrm{L}^{\infty}([0,T_B];B)$ and every $k\in \N$, the map $t\mapsto
  \widehat{\exp}_k(tX)$ is defined on $[0,T_B]$,
\item[(ii)] for every compact set $K\subseteq \R^n$, every $f\in \mathrm{C}^{\omega}(\R^n)$, and every
$\mathbf{a}\in c^{\downarrow}_0$, there exist positive constants $M, M_f>0$ such that, for every $X\in \mathrm{L}^{\infty}([0,T_B];
B)$, we have
\begin{equation*}
\rho^{\omega}_{K,\mathbf{a},f}(\widehat{\exp}_k(t X)-\widehat{\exp}_{k-1}(t X))\le
\left(Mt\right)^{k+1} M_f, \quad\forall t\in [0,T_B], \ \forall k\in \N.
\end{equation*}
\end{enumerate}
\end{theorem}
Using the estimate in Theorem~\ref{th:3}, one can get an estimate for the flow of a vector field $X$ using the sequence of
iterations in Definition~\ref{eq:sequence_iterations}.
\begin{corollary}\label{cor:estimate_picard_iterations}
Let $B$ be a bounded set in $\mathrm{LC}^{\omega}(\R^n)$. Then there
exist $M,L>0$ and $\overline{T}\le T_B$ such that, for every $X\in \mathrm{L}^{\infty}([0,\overline{T}];
B)$ and every $i\in \{1,2,\ldots,n\}$, we have
\begin{equation*}
\|\mathrm{ev}_{x_0}\scirc\widehat{\exp}(t
X)(x^i)-\mathrm{ev}_{x_0}\scirc\widehat{\exp}_{k}(t X)(x^i)\|\le
\frac{Mt^{k+1}}{1-Mt}L, \quad\forall t\in [0,\overline{T}],
\end{equation*}
where $x^i$ is the $i$th coordinate function on $\R^n$.
\end{corollary}

It is worth mentioning that there is also an alternative way for proving this corollary using
the estimates in ~\cite[\S 2.4.4]{agrachevcontrol2004}. Since this
corollary is essential for the proof of the main results of this paper,
we provide a proof for it using Theorem~\ref{th:3} in
Appendix~\ref{app:1}. Using the estimate in Theorem~\ref{th:3} and
choosing $\overline{T}<T_B$ such that $M\overline{T}<1$, one can show the following result on convergence of
the iterations~\eqref{eq:Picards_iterations}~\cite[Theorem 3.8.1]{SJ:2016}. 
\begin{theorem}[\textbf{Convergence of iterations}]\label{thm:Picards_iterations_converges}
Let $X$ be an essentially bounded time-varying real analytic vector
field. Then the sequence $\{\widehat{\exp}_k(tX)\}_{k\in \N}$
converges to $\widehat{\exp}(tX)$ in $\mathrm{C}^{\omega}$-topology uniformly
for every $t\in [0,\overline{T}]$.
\end{theorem}




% Using Theorem \ref{th:3}, one can show the following result. 

% \begin{theorem}\label{th:4}
% Let $B$ be a bounded set in $\Gamma^{\omega}(T\R^n)$. Then, for every
% $f\in \mathrm{C}^{\omega}(\R^n)$, there
% exist positive real numbers $M>0$ and $M_f>0$ such that
% \begin{equation*}
% \left\|\mathrm{ev}_{x_0}(\widehat{\phi}^{X^{i_1,i_2,\ldots,i_p}_{t_1,t_2,\ldots,t_p}}(t)(f))-\mathrm{ev}_{x_0}(\widehat{\phi}^{X^{i_1,i_2,\ldots,i_p}_{t_1,t_2,\ldots,t_p}}_k(t)(f))\right\|\le
% \frac{\left(Mt\right)^{k+1}}{1-Mt} M_f,
% \end{equation*}
% for every $k\in N$, every $t\in [0,T]$ such that $Mt<1$, every family of real analytic vector fields
% $\{X_1,X_2,\ldots,X_m\}\subseteq B$, every $p\in \N$, every
% $i_1,i_2,\ldots,i_p\in \{1,2,\ldots,n\}$, and every
% $t_1,t_2,\ldots,t_p\in \R_{>0}$ such that $\sum_{i=1}^{p} t_i\le t$.
% \end{theorem}


\section{Control systems}\label{sec:control_systems}

In this section, we define the notion of $\mathrm{C}^{\nu}$-control system. In the
geometric control literature, a control system is usually defined as a
\emph{parametrized} family of vector fields on $\R^n$. 



\begin{definition}[\textbf{$\mathrm{C}^{\nu}$-control system}]
A $\mathrm{C}^{\nu}$-control system on $\R^n$ is a triple
$\{F,\mathcal{X},\mathfrak{C}\}$, where
\begin{enumerate}
\item[(i)] The set $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ is
a family of vector fields such that $X_i\in \Gamma^{\nu}(\R^n)$, for every $i\in \{0,1,\ldots,m\}$, and
\item[(ii)] the set $\mathfrak{C}$ is a compact convex set in $\R^m$ which is
  called the control set, 
\item[(iii)] the mapping $F:\mathfrak{C}\to
  \Gamma^{\nu}(\R^n)$ is defined by 
\begin{align*}
F(u_1,u_2,\ldots,u_m)=X_0+\sum_{i=1}^{m}u_iX_i.
\end{align*}
\end{enumerate} 
\end{definition}

In the rest of this paper, for the sake of simplicity, we assume that
$\mathfrak{C}=[-1,1]^m$. Therefore, without any confusion, we denote a
$\mathrm{C}^{\nu}$-control system on $\R^n$ by a family of $\mathrm{C}^{\nu}$-vector
fields $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$.

\begin{definition}[\textbf{Trajectories, reachable sets and
    controllability notions}]\label{def:reachable}
Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ be a $\mathrm{C}^{\nu}$-control
system on $\R^n$ and $t\in \R_{>0}$, and $x_0\in \R^n$. 
\begin{enumerate}
\item[(i)] The absolutely continuous curve $x:[0,t]\to \R^n$ is a trajectory of
  $\mathcal{X}$ if there exist measurable controls
  $u_1,u_2,\ldots,u_m:[0,t]\to [-1,1]$ such that
\begin{align*}
\dot{x}(\tau)=X_0(x(\tau))+\sum_{i=1}^{m}u_i(\tau)X_i(x(\tau)),\quad
  \mbox{ for almost all } \tau\in [0,t].
\end{align*}
\item[(ii)] The reachable sets of the $\mathrm{C}^{\nu}$-control system $\mathcal{X}$ from $x_0$
in times less than $t$ is the set
$\mathrm{R}_{\mathcal{X}}(< t,x_0)$ defined by
\begin{equation*}
\mathrm{R}_{\mathcal{X}}(< t,x_0)=\{x(\tau)\mid x:[0,\tau]\to \R^n
\mbox{ is a trajectory of } \mathcal{X}, \ x(0)=x_0, \tau< t\}.
\end{equation*}
\item[(iii)]
The $\mathrm{C}^{\nu}$-control system $\mathcal{X}$ is small-time locally controllable from
$x_0$ if, for every $t\in \R_{>0}$, we have
\begin{equation*}
x_0\in \mathrm{int}\left(\mathrm{R}_{\mathcal{X}}(< t, x_0)\right).
\end{equation*} 
\item[(iv)] Let $N\in \Z_{>0}$ be a positive integer. Then the
  $\mathrm{C}^{\nu}$-control system $\mathcal{X}$ satisfies growth
  rate condition of order $N$
  at point $x_0$ if there exist $C, T>0$ such that, for every $t\in (0,T]$, we have
\begin{equation*}
\overline{\mathrm{B}}(x_0, Ct^N)\subset \mathrm{R}_{\mathcal{X}}(< t, x_0).
\end{equation*} 
\end{enumerate}
\end{definition}
It is clear form Definition~\eqref{def:reachable} that, if a control system $\mathcal{X}$
satisfies the growth rate condition of order $N$ at point $x_0$, then it is small-time
locally controllable from $x_0$. 


While we defined the trajectories of a control systems using measurable
controls $u_1,u_2,\ldots,u_m:[0,t]\to [-1,1]$, it is sometimes useful to work with
piecewise-constant controls and their associated vector fields. 
\begin{definition}[\textbf{Piecewise-constant vector fields}]
Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ be a
$\mathrm{C}^{\nu}$-control system and
$\mathbf{u}=(u_1,u_2,\ldots,u_m)\in [-1,1]^m$. Then the vector
field of the control system $\mathcal{X}$ associated to $\mathbf{u}$ is
the vector field $X_{\mathbf{u}}\in \Gamma^{\nu}(\R^n)$ defined by
\begin{align*}
X_{\mathbf{u}}=X_0+\sum_{i=1}^{m} u_iX_i.
\end{align*}
Let $p\in N$, $\mathbf{I}=(\mathbf{u}^1,\mathbf{u}^2,\ldots,\mathbf{u}^p)\in
([-1,1]^m)^p$ be a $p$-tuple of constant controls, 
$\mathbf{t}=(t_0,t_1,\ldots,t_p)\in \R_{>0}^p$ be a $p$-tuple of
switching times. We define the piecewise-constant vector field $X^{\mathbf{I},\mathbf{t}}$ by
\begin{equation*}
X^{\mathbf{I},\mathbf{t}}=
\begin{cases}
X_{\mathbf{u}^p}, & t\in [0,t_p],\\
X_{\mathbf{u}^{p-1}}, & t\in (t_{p},t_{p-1}+t_p],\\
\vdots & \vdots\\
X_{\mathbf{u}^1}, & t\in (t_2+\ldots,t_{p}, t_1+\ldots+t_p].
\end{cases}
\end{equation*}
\end{definition}
It clear that, for every $p$-tuple
$\mathbf{I}\in ([-1,1]^m)^p$ and every $p$-tuple 
$\mathbf{t}\in\R^n_{>0}$ the vector field $X^{\mathbf{I},\mathbf{t}}$
is essentially bounded and the following property holds:
\begin{equation*}
\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x)=\exp(t_1X_{\mathbf{u}^1})\scirc
\exp(t_2X_{\mathbf{u}^2})\scirc
\ldots\scirc\exp(t_pX_{\mathbf{u}^p})(x),\qquad\forall x\in \R^n.
\end{equation*}


Another important notion relevant to small-time local controllability
of systems is normal reachability \cite{KAG:1992}. Normal reachability
has been first introduced and studied by Sussmann in \cite{HJS:1976}.
\begin{definition}[\textbf{Normal reachability}]
Let $\mathcal{X}= \{X_0,X_1,\ldots,X_m\}$ be a $\mathrm{C}^{\nu}$-control system on $\R^n$ and $x_1,x_0\in \R^n$. Then the point $x_1$ is normally reachable in time less
  than $t$ from $x_0$, if the following conditions hold:
\begin{enumerate}
\item there exist $p\in \N$,
  $\mathbf{u}^1,\mathbf{u}^2,\ldots,\mathbf{u}^p\in [-1,1]^m$, and $(s_1,s_2,\ldots,s_p)\in \R^p_{>0}$ such
that $s_1+s_2+\ldots+s_p<t$ and
\begin{equation*}
\exp(s_1 X_{\mathbf{u}^1})\scirc \exp(s_2 X_{\mathbf{u}^2})\scirc\ldots\scirc \exp(s_p X_{\mathbf{u}^p})(x_0)=x_1,
\end{equation*}
and 
\item there exists an open neighbourhood of $(s_1,s_2,\ldots,s_p)$ in
$\R^p_{>0}$ such that the map
\begin{equation*}
(t_1,t_2,\ldots,t_p)\mapsto \exp(t_1 X_{\mathbf{u}^1})\scirc \exp(t_2 X_{\mathbf{u}^2})\scirc\ldots\scirc \exp(t_p X_{\mathbf{u}^p})(x_0)
\end{equation*}
is $\mathrm{C}^1$ and of rank $n$ on this neighbourhood.
\end{enumerate}
\end{definition}
It is clear that, if the point $x_0$ is normally reachable from itself, the system is small-time locally
controllable from $x_0$. However, the converse is only true for
general control systems~\cite[Example 3.9]{KAG:1992}. For real analytic systems, the connection between small-time local controllability
and normal reachability has been studied in \cite{KAG:1992}. In fact,
in \cite{KAG:1992}, it has been shown that for real analytic control
systems, small-time local controllability from $x_0$ implies that, for every time $t$, every point in the interior of the reachable set from
$x_0$ in times less than $t$ is normally reachable from $x_0$ \cite[Theorem 5.5 and Corollary 4.15]{KAG:1992}.
\begin{theorem}\label{th:1}
Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ be a real analytic
control system. If $\mathcal{X}$ is small-time locally controllable
from $x_0$ then, for every $t>0$, every point in the set $\mathrm{int}\left(\mathrm{R}_{\mathcal{X}}(<t,x_0)\right)$ is normally
  reachable in time less than $t$ form $x_0$.
\end{theorem}
Suppose that $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ is a $\mathrm{C}^{\nu}$-control system. Then, in
order to consider perturbations of the control system $\mathcal{X}$, it is
reasonable to study the Taylor polynomials of the vector fields
$\{X_0,X_1,\ldots,X_m\}$. In order to capture the idea of a perturbation of
vector fields of a control system with Taylor polynomials of order
higher than $k$, we define the notion of $k$th contact. 
\begin{definition}[\textbf{$k$th contact at point $x_0$}]
Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ and
$\mathcal{Y}=\{Y_0,Y_1,\ldots,Y_m\}$ be two $\mathrm{C}^{\nu}$-control
systems on $\R^n$ and $k\le v$. Then $\mathcal{X}$ and $\mathcal{Y}$ have $k$th
contact at point $x_0$ if, for every multi-index $\mathbf{r}\in
\mathbb{Z}_{\ge 0}^n$ with $|\mathbf{r}|\le k$, we have
\begin{align*}
D^{\mathbf{r}}X_i(x_0)=D^{\mathbf{r}}Y_i(x_0),\qquad\forall i\in \{0,1,\ldots,m\}.
\end{align*}
\end{definition}
Roughly speaking, two control system have $k$th contact at point
$x_0$, if the Taylor polynomial of their vectors fields around $x_0$
agree up to order $k$. Using the notion of $k$th contact of
control systems at a point, we get the following theorem. The proof of this
theorem is given in Appendix~\ref{app:2}. 

\begin{theorem}\label{kth_contact_estimate_flow}
Suppose that $p\in \N$,
$\mathbf{I}=(\mathbf{u}^1,\mathbf{u}^2,\ldots,\mathbf{u}^p)\in
([-1,1]^m)^p$, $\mathbf{t}=(t_1,t_2,\ldots,t_p)\in \R^p_{\ge
  0}$, and $x_0\in \R^n$. Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ and
$\mathcal{Y}=\{Y_0,Y_1,\ldots,Y_m\}$ be two real analytic control
systems which have $k$th contact at point $x_0$. Then, for every $i\in
\{1,2,\ldots,n\}$ we have
\begin{align*}
\mathrm{ev}_{x_0}\scirc \widehat{\exp}_k(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)=\mathrm{ev}_{x_0}\scirc \widehat{\exp}_k(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i),
\end{align*}
where $x^i$ is the $i$th coordinate function on $\R^n$. 
\end{theorem}

\section{Control variations}\label{sec:variations}

The notion of control variation is one of the fundamental tools in studying
reachable sets of control systems. Roughly speaking control variations
can be considered as the directions constructed using the trajectories
of the system, along which one can steer the
control system. By constructing appropriate control variations and
using a suitable open mapping theorem, one can show that a control system
is small-time locally controllable~\cite[Theorem 2.4]{MK:1990}. The first use of the notion of control variations for approximating reachable sets of control systems
can be traced back to the original work of Pontryagin and his
coworkers for studying the boundary of reachable sets~\cite{LSP:1964}. Since then, many different and technical notions of variations with
various properties have been proposed in the control
literature~\cite{AJK:1977,bianchinicontrollability1993,MK:1990,
  HF:1989}. In this section we study a general class of control
variations introduced in \cite{HF:1989} and \cite{MK:1990}. 
\begin{definition}[\textbf{Control Variations}]\label{def:control_variations}
Let $k\in \N$ and $\mathcal{X}$ be a $\mathrm{C}^{\nu}$ control system on $\R^n$. Then a vector $\mathbf{v}\in \R^n$ is called a variation
of $k$th order for the control system $\mathcal{X}$ at point $x_0$ if there exists a
parametrized family of points $\gamma:\R_{\ge 0}\to \R^{n}$ such that,
for every $t\in \R_{\ge 0}$, we have $\gamma(t)\in \mathrm{R}_{\mathcal{X}}(<t,x_0)$ and 
\begin{equation}\label{eq:2}
\gamma(t)=x_0+t^k\mathbf{v}+\mathcal{O}(t^{k+1}). 
\end{equation}
The set of all variations of $k$th order for the system $\mathcal{X}$
at point $x_0$ is denoted by $\mathcal{K}^k_{\mathcal{X}}(x_0)$. We also
define the cone $\widehat{\mathcal{K}}^k_{\mathcal{X}}(x_0)$ by
\begin{align*}
\widehat{\mathcal{K}}^k_{\mathcal{X}}(x_0)=\bigcup_{\alpha\ge
  0}\alpha \left(\mathcal{K}^k_{\mathcal{X}}(x_0)\right).
\end{align*}
\end{definition}
Note that in Definition~\ref{def:control_variations} we do not assume
any regularity of the parametrized family of points $\gamma$. The next theorem shows how
these control variations can be used to deduce the small-time local
controllability of systems. The proof can be found
in~\cite[Corollary 2.5]{MK:1990}.
\begin{theorem}[\textbf{High-order Open Mapping Theorem}]\label{thm:control_variations_STLC}
Suppose that $\mathcal{X}$ is a $C^{\nu}$-control system. If we have
$\widehat{\mathcal{K}}^k_{\mathcal{X}}(x_0)=\R^n$, then the control system $\mathcal{X}$ satisfies the growth rate condition of order $k$. 
\end{theorem}


We first show that the control variations in
Definition~\ref{def:control_variations} can be used to characterize
the growth rate condition of
order $N$. 
\begin{lemma}[\textbf{Characterization of growth rate condition}]\label{lem:characterization_growth_rate}
Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ be a $\mathrm{C}^{\nu}$ control
system on $\R^n$, $x_0\in \R^n$ be a point, and $N\in \N$. Then the following conditions are equivalent:
\begin{enumerate}
\item[(i)] There exists $C,T>0$ such that
\begin{align*}
\overline{\mathrm{B}}(x_0,Ct^N)\subseteq \mathrm{R}_{\mathcal{X}}(<t,x_0),\qquad\forall t\in [0,T],
\end{align*}
\item[(ii)] we have $\widehat{\mathcal{K}}^N_{\mathcal{X}}(x_0)=\R^n$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i)$\Rightarrow$ (ii): Since the control system $\mathcal{X}$
  satisfies the growth rate condition of order $N$, there exists $T>0$ and $C>0$ such
  that, for every $t\in [0,T]$, we have
  $\overline{\mathrm{B}}(x_0,Ct^N)\subseteq
\mathrm{R}_{\mathcal{X}}(<t,x_0)$. For every $\mathbf{v}\in \mathbb{S}^{n-1}$, we define the curve $\gamma_{\mathbf{v}}:[0,T]\to \overline{\mathrm{B}}(x_0,Ct^N)$ as
  follows:
\begin{equation*}
\gamma_{\mathbf{v}}(t)=x_0+Ct^N\mathbf{v}.
\end{equation*}
Since, for every $t\in [0,T]$, we have $\overline{\mathrm{B}}(x_0,Ct^N)\subseteq
  \mathrm{R}_{\mathcal{X}}(<t,x_0)$, we get 
\begin{equation*}
\gamma_{\mathbf{v}}(t)\in
\mathrm{R}_{\mathcal{X}}(<t,x_0),\qquad\forall t\in [0,T].
\end{equation*}
Therefore, by Definition~\ref{def:control_variations}, the vector
$\mathbf{v}$ is control
variations of order $N$ for the system $\mathcal{X}$ at point
$x_0$. This means that we have
$\mathcal{K}^N_{\mathcal{X}}(x_0)=\mathbb{S}^{n-1}$ and this implies that
\begin{align*}
\widehat{\mathcal{K}}^N_{\mathcal{X}}(x_0)=\bigcup_{\alpha\ge 0}\alpha \mathcal{K}^N_{\mathcal{X}}(x_0)=\R^n
\end{align*}
Therefore $(ii)$ holds. 

(ii)$\Rightarrow$ (i): The proof of this part follows from Theorem~\ref{thm:control_variations_STLC}.
\end{proof}
Roughly speaking, Lemma~\ref{lem:characterization_growth_rate} states that small-time local
controllability of a system is checkable using variations of order
$N$ if and only if the system satisfies the growth rate condition of order
$N$. Note that this lemma holds for control systems in any
regularity class $\mathrm{C}^{\nu}$. 



\section{Perturbations of real analytic control systems}\label{sec:perturbations}

Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ be a
real analytic control system which satisfies the growth rate condition of
order $N$ at the point $x_0$ and let $\mathcal{Y}=\{Y_0,Y_1,\ldots,Y_m\}$ be another real
analytic control system which has $N$th contact at point $x_0$ with control system
$\mathcal{X}$. Note that, since vector fields of $\mathcal{Y}$ and $\mathcal{X}$ have
the same $N$th order Taylor polynomials at point $x_0$, the control
system $\mathcal{Y}$ can be considered as an $N$th order
perturbation of $\mathcal{X}$ around $x_0$. In this section, we study the effect of perturbation of
vector fields of the system from $\mathcal{X}$ to $\mathcal{Y}$ on the
reachable sets of the system. Since
$\mathcal{X}$ satisfies the growth rate condition of order $N$, there exists $C,T>0$ such that we have
\begin{align*}
\overline{\mathrm{B}}(x_0,Ct^N)\subseteq \mathrm{R}_{\mathcal{X}}(< t,x_0),\qquad\forall t\le T.
\end{align*}
For every $t\in [0,T]$, we define a perturbation mapping $F^t_{\mathcal{X},\mathcal{Y}}:\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)
\rightrightarrows \mathrm{R}_{\mathcal{Y}}(<t,x_0)$ which captures the
transition from reachable sets of $\mathcal{X}$ to the reachable sets
of $\mathcal{Y}$. The multi-valued mapping
$F^t_{\mathcal{X},\mathcal{Y}}$ is defined as composition of two mappings
$\xi^t_{\mathcal{Y}}$ and $\eta^t_{\mathcal{X}}$ as follows:
\begin{equation*}
F^t_{\mathcal{X},\mathcal{Y}}(x)=\xi^t_{\mathcal{Y}}\scirc
\eta^t_{\mathcal{X}}(x),\qquad\forall x\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N).
\end{equation*}
The mapping $\eta^t_{\mathcal{X}}$ applies to a point in the closed
ball $\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$ (which is a subset of the
reachable set $\mathrm{int}(\mathrm{R}_{\mathcal{X}}(< t,x_0))$) and gives us the
switching times associated to that point in the $\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$. Note that, for a point $x\in
\mathrm{int}(\mathrm{R}_{\mathcal{X}}(<t,x_0))$, there might exist more than one set
of switching times. This implies that there exist many different ways to
define the mapping $\eta^t_{\mathcal{X}}$. We use the normal
reachability of real analytic control system $\mathcal{X}$ to assign the switching
times to the points in $\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$ in
such a way that the mapping $\eta^t_{\mathcal{X}}$ has some nice
regularity properties. The other mapping $\xi^t_{\mathcal{Y}}$ applies
to the set of switching times and gives us the associated point in the reachable set of the control
system $\mathcal{Y}$. It is clear that $\xi^{t}_{\mathcal{Y}}$ is a single-valued mapping from the
switching times to $\mathrm{R}_{\mathcal{Y}}(<t,x_0)$. As a result, by
composing these two mappings, we get the multi-valued mapping
$F^t_{\mathcal{X}}$ which maps points in
$\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$ (which is a subset of
reachable set $\mathrm{int}(\mathrm{R}_{\mathcal{X}}(<t,x_0))$) to the
points in the the reachable set $\mathrm{R}_{\mathcal{Y}}(<t,x_0)$ through their sets of
switching times. 

We start by constructing the multi-valued mapping
$\eta^{t}_{\mathcal{X}}$. Since $\mathcal{X}$ satisfies the growth rate condition of order $N$, we have
$\overline{\mathrm{B}}(x_0,Ct^N)\subseteq \mathrm{R}_{\mathcal{X}}(< t,x_0)$. This implies that 
\begin{align*}
\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\subset
  \mathrm{B}(x_0,Ct^N)\subseteq
  \mathrm{int}(\mathrm{R}_{\mathcal{X}}(< t,x_0))
\end{align*}
By Theorem~\ref{th:1}, for every $x\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$, there exist $p_x\in \N$,
$s_1,s_2,\ldots,s_{p_x}\in \R_{>0}$, and $\mathbf{w}^1,\mathbf{w}^2,\ldots,\mathbf{w}^{p_x}\in
[-1,1]^m$ such that $s_1+s_2+\ldots+s_{p_x}<t$ and
\begin{equation*}
\exp(s_1X_{\mathbf{w}^1})\scirc \exp(s_2X_{\mathbf{w}^2})\scirc\ldots\scirc \exp(s_{p_x} X_{\mathbf{w}^{p_x}})(x_0)=x.
\end{equation*}
Moreover, there exists an open neighbourhood $V$ of $(s_1,s_2,\ldots,s_{p_x})$ in
$\R^{p_x}_{>0}$ such that the map $\xi^x_{\mathcal{X}}:V\to \R^n$, defined by
\begin{equation*}
\xi^x_{\mathcal{X}}(t_1,t_2,\ldots,t_{p_x})=\exp(t_1X_{\mathbf{w}^1})\scirc \exp(t_2X_{\mathbf{w}^2})\scirc\ldots\scirc \exp(t_{p_x} X_{\mathbf{w}^{p_x}})(x_0)
\end{equation*}
is $\mathrm{C}^ 1$ and of rank $n$ on $V$. Without loss of generality we can assume that,
for every $(t_1,t_2,\ldots,t_{p_x})\in V$, we have
\begin{equation*}
t_1+t_2+\ldots+t_{p_x}<t.
\end{equation*}
By \cite[Lemma 2.2]{IK-PWM-JS:1993}, there exists a submanifold $M_x$
of $V$ containing $(s_1,s_2,\ldots,s_{p_x})$ such that $\xi^x_{\mathcal{X}}(M_x)$ is an open neighbourhood of
$x$ in $\R^n$ and $\xi^x_{\mathcal{X}}\mid_{M_x}$ is a
$\mathrm{C}^{1}$-diffeomorphism. Let $S_x$ be an open neighbourhood of
$(s_1,s_2,\ldots,s_{p_x})$ in $M_x$ such that $\overline{S}_{x}\subseteq
M_x$. Since $\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$ is compact and, for
every $x\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$, the set $\xi^{x}_{\mathcal{X}}(S_{x})$ is open in $\R^n$, there exists
$x_1,x_2,\ldots,x_r\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$ such that 
\begin{equation*}
\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\subseteq \bigcup_{i=1}^{r} \xi^{x_i}_{\mathcal{X}}(S_{x_i}).
\end{equation*} 
Now let us define $p=p_{x_1}+p_{x_2}+\ldots+p_{x_r}$ and let
$(\mathbf{u}^1,\mathbf{u}^2,\ldots,\mathbf{u}^p)$ be the ordered set obtained by concatenation
of the controls $\mathbf{w}^1,\mathbf{w}^2,\ldots,\mathbf{w}^{p_{x_k}}$ for $1\le k\le r$. Since,
$\R^{p_{x_i}}\subseteq \R^{p}$, for every $i\in\{1,2,\ldots,r\}$, one can consider $S_{x_i}$ as a submanfiold of $\R^p$. We define the multi-valued map
$\eta^t_{\mathcal{X}}:\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\rightrightarrows \bigcup_{i=1}^{r} \overline{S}_{x_i}$ as
\begin{equation*}
\eta^t_{\mathcal{X}}(x)=\bigcup_{i\in
  \{1,2,\ldots,r\}}\left\{\left(\xi^{x_i}_{\mathcal{X}}\right)^{-1}(x)\suchthat
x\in \xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})\right\},\qquad\forall
x\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N).
\end{equation*}
Note that, for every $x\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$, the number of
elements in $\eta^t_{\mathcal{X}}(x)$ is at most $r$.


The next step is to construct the single-valued mapping
$\xi^t_{\mathcal{Y}}:\R^p\to \mathrm{R}_{\mathcal{Y}}(<t,x_0)$. We
define the map $\xi^t_{\mathcal{Y}}:\bigcup_{i=1}^{r} \overline{S}_{x_i} \to \R^n$ by
\begin{equation*}
\xi^t_{\mathcal{Y}}(t_1,t_2,\ldots,t_p)=\exp(t_1Y_{j_1})\scirc \exp(t_2Y_{j_2})\scirc\ldots\scirc \exp(t_{p_x} Y_{j_{p_x}})(x_0).
\end{equation*}
Then the multi-valued map
$F^t_{\mathcal{X},\mathcal{Y}}:\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\rightrightarrows
\R^n$ is given by
\begin{equation*}
F^t_{\mathcal{X},\mathcal{Y}}(x)=\xi^t_{\mathcal{Y}}\scirc \eta^t_{\mathcal{X}}(x).
\end{equation*}
One can observe that the mapping $F^t_{\mathcal{X},\mathcal{Y}}$ is finite-valued and has the following regularity properties.
\begin{theorem}[\textbf{Regularity of the perturbation mapping}]\label{lem:multi-valued_map}
Let $\mathcal{X}$ be a real analytic control system and $x_0\in
\R^n$. Suppose that there exist $C,T>0$ and $N\in \N$ with the property that
\begin{align*}
\overline{B}(x_0,Ct^N)\subseteq
\mathrm{R}_{\mathcal{X}}(<t,x_0),\qquad\forall t\le T.
\end{align*}
Let $\mathcal{Y}$ be another real analytic control system which has
$N$th contact at point $x_0$ with the control system $\mathcal{X}$ and, for
every $t\in [0,T]$, the map $F^t_{\mathcal{X},\mathcal{Y}}:\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\to
\mathrm{R}_{\mathcal{Y}}(<t, x_0)$ is defined as above. Then the following statements hold:
\begin{enumerate}
\item\label{p1:reg_spatial}  For every $t\in [0,T]$ and every $x\in
  \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$, there exist a positive integer $l\in \N$, a neighbourhood $W$ containing $x$, and
continuous functions $f^1,f^2,\ldots,f^l:W\to \R^n$ such that
\begin{equation*}
\left\{f^1(y)\right\}\subseteq F^t_{\mathcal{X},\mathcal{Y}}(y)\subseteq\left\{f^1(y), f^2(y), \ldots,
  f^l(y)\right\},\qquad\forall y\in W.
\end{equation*}
\item\label{p2:reg_time} there exist $\alpha>0$ and $T_{\min}\in (0,T)$ such that, for every 
  $t\le T_{\min}$ and every $x\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$, we have
\begin{equation*}
\|y-x\|\le \alpha t^{N+1},\qquad \forall y\in F^t_{\mathcal{X},\mathcal{Y}}(x).
\end{equation*}
\end{enumerate}
\end{theorem}
\begin{proof}
Regarding part~\ref{p1:reg_spatial}, suppose that, for every $i\in
\{1,2,\ldots, r\}$, the map
$\xi^{x_i}_{\mathcal{X}}$ and the manifold $S_{x_i}$ are defined as
above. Since $\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\subseteq
\bigcup_{i=1}^{r} \xi^{x_i}_{\mathcal{X}}(S_{x_i})$, we have $x\in \bigcup_{i=1}^{r} \xi^{x_i}_{\mathcal{X}}(S_{x_i})$. Without loss of
generality, we can assume that
\begin{equation*}
x\in \xi^{x_1}_{\mathcal{X}}(S_{x_1}).
\end{equation*}
Note that $\xi^{x_1}_{\mathcal{X}}(S_{x_1})$ is open in $\R^n$. Therefore, there exists a
neighbourhood $U$ of $x$ such that $U\subseteq
\xi^{x_1}_{\mathcal{X}}(S_{x_1})$. On the other hand, since we have $\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\subseteq
\bigcup_{i=1}^{r} \xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})$, without loss of generality, we
can assume that there exists $l\in \{1,2,\ldots,r\}$ such that
\begin{eqnarray*}
x\in \xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i}), &\quad i\in\{1,2,\ldots,l\},\\
x\not\in \xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i}), &\quad  i\in\{l+1, l+2,\ldots,r\}.
\end{eqnarray*}
For every $i\in\{l+1,l+2,\ldots,r\}$, the set
$\xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})$ is closed in $\R^n$. Therefore,
\begin{equation*}
\bigcup_{i=l+1}^{r} \xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})
\end{equation*}
is closed in $\R^n$. Moreover, we know that $x\not\in \bigcup_{i=l+1}^{s}
\xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})$. This implies that there exists a
neighbourhood $V$ of $x$ such that
\begin{equation*}
V\cap \left(\bigcup_{i=l+1}^{r} \xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})\right)=\emptyset.
\end{equation*}
Note that, for every $i\in \{1,2,\ldots,l\}$,
$\overline{S}_{x_i}\subseteq M_{x_i}$. Therefore, for every $i\in \{1,2,\ldots,l\}$, we have
$\xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})\subseteq
\xi^{x_i}_{\mathcal{X}}(M_{x_i})$. Since $x\in
\xi^{x_i}_{\mathcal{X}}(M_{x_i})$, for every $i\in \{1,2,\ldots,l\}$, the
set $\bigcap_{i=1}^{l}\xi^{x_i}_{\mathcal{X}}(M_{x_i})$ is nonempty. We set 
\begin{equation*}
W=\left(\bigcap_{i=1}^{l}\xi^{x_i}_{\mathcal{X}}(M_{x_i})\right)\cap V\cap U.
\end{equation*} 
For every $i\in \{1,2,\ldots,l\}$, we define the function $f^i:W\to
\R^n$ as 
\begin{equation*}
f^i(y)=\xi^t_{\mathcal{Y}}\scirc \left(\xi^{x_i}_{\mathcal{X}}\right)^{-1}(y),
\qquad\forall y\in W.
\end{equation*}
Note that, for every $i\in \{1,2,\ldots,l\}$, the map $\xi^{x_i}_{\mathcal{X}}$
is a $\mathrm{C}^{1}$-diffeomorphism on $M_{x_i}$. Therefore, for every $i\in
\{1,2,\ldots,l\}$, the map $f^i:W\to \R^n$ is continuous. Now, it is
clear from the definition of $F^t_{\mathcal{X},\mathcal{Y}}$ that we have
\begin{equation*}
F^t_{\mathcal{X},\mathcal{Y}}(t,x)=\left\{f^1(x), f^2(x), \ldots, f^l(x)\right\}.
\end{equation*}
Since $W\subseteq V$, and $V$ is chosen such that
\begin{equation*}
V\cap \left(\bigcup_{i=l+1}^{r} \xi^{x_i}_{\mathcal{X}}(\overline{S}_{x_i})\right)=\emptyset,
\end{equation*}
for every $i\in\{l+1,l+2,\ldots,r\}$ and every
$y\in W$, we have
\begin{equation*}
\xi^t_{\mathcal{Y}}\scirc \left(\xi^{x_i}_{\mathcal{X}}\right)^{-1}(y)\not\in
F^t_{\mathcal{X}, \mathcal{Y}}(y). 
\end{equation*}
Thus, we have 
\begin{equation*}
F^t_{\mathcal{X},\mathcal{Y}}(y)\subseteq\left\{f^1(y), f^2(y), \ldots,
  f^l(y)\right\},\qquad\forall y\in W.
\end{equation*}
Finally, since $W\subseteq U$, and $U$ is chosen such that
\begin{equation*}
U\subseteq \xi^{x_1}_{\mathcal{X}}(S_{x_1}),
\end{equation*}
for every $y\in W$, we have 
\begin{equation*}
\xi^t_{\mathcal{Y}}\scirc \left(\xi^{x_1}_{\mathcal{X}}\right)^{-1}(y)\in
F^t_{\mathcal{X}, \mathcal{Y}}(y). 
\end{equation*}
Therefore, for every $y\in W$, we have $f^{1}(y)\in F^t_{\mathcal{X},
  \mathcal{Y}}(y)$. This completes the proof of the part~\ref{p1:reg_spatial}. 

Regarding part~\ref{p2:reg_time}, define the set $B\subseteq
\Gamma^{\omega}(\R^n)$ by
\begin{align*}
B = \left\{X_{\mathbf{u}}\mid \mathbf{u}\in [-1,1]^m\right\}\bigcup \left\{Y_{\mathbf{u}}\mid \mathbf{u}\in [-1,1]^m\right\}.
\end{align*}
Note that, for every $\mathbf{u}\in [-1,1]^m$, every compact set $K$,
every $C^{\omega}$-function $f$, and every $\mathbf{a}\in c^{\downarrow}_0$, we have 
\begin{align*}
\rho^{\omega}_{K,\mathbf{a},f}(X_{\mathbf{u}})=\rho^{\omega}_{K,\mathbf{a},f}(X_0+\sum_{i=1}^{m}u_iX_i)\le
  (m+1)\max\{\rho^{\omega}_{K,\mathbf{a},f}(X_i)\mid i\in \{0,1,\ldots,m\}\}.
\end{align*}
Similarly, for every $\mathbf{u}\in [-1,1]^m$, every compact set $K$,
every $C^{\omega}$-function $f$, and every $\mathbf{a}\in c^{\downarrow}_0$, we have 
\begin{align*}
\rho^{\omega}_{K,\mathbf{a},f}(Y_{\mathbf{u}})=\rho^{\omega}_{K,\mathbf{a},f}(Y_0+\sum_{i=1}^{m}u_iY_i)\le
  (m+1)\max\{\rho^{\omega}_{K,\mathbf{a},f}(Y_i)\mid i\in \{0,1,\ldots,m\}\}.
\end{align*}
For every $i\in \{0,1,\ldots,m\}$ we define
$L_i=\max\{\rho^{\omega}_{K,\mathbf{a},f}(X_i),\rho^{\omega}_{\mathbf{a},K,f}(Y_i)\}$. Thus,
if we define the constant $L\in \R_{>0}$ by
\begin{align*}
L=(m+1)\max\{L_i\mid i\in \{0,1,\ldots,m\}\},
\end{align*}
then we have 
\begin{align*}
\rho^{\omega}_{K,\mathbf{a},f}(v)\le L,\qquad\forall v\in B.
\end{align*}
By Theorem~\ref{thm:boundedset_locallyconvex}, this implies that the set $B$ is bounded in $\Gamma^{\omega}(\R^n)$. Using Corollary
\ref{cor:estimate_picard_iterations}, there exist $M,L>0$ and $\overline{T}<T_B$ such that, for every $t\in (0,T]$ and every real analytic
vector field $Z\in\mathrm{L}^{\infty}([0,\overline{T}] ;B)$, we have
\begin{equation*}
\left\|\mathrm{ev}_{x_0}\scirc\exp(tZ)(x^i)-\mathrm{ev}_{x_0}\scirc\exp_{N}(tZ)(x^i)\right\|\le
\frac{\left(Mt\right)^{N+1}}{1-Mt} L,
\end{equation*}
We set $\alpha=\sqrt{n}M^{N+1}L$ and $T_{\min}=\min\{\overline{T},T\}$
Let $t\in [0,T_{\min}]$ and $x\in
\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$. If $y\in F^t_{\mathcal{X},\mathcal{Y}}(x)$, then there exist
$\mathbf{I}=(\mathbf{u}^1,\mathbf{u}^2,\ldots,\mathbf{u}^p)\in ([-1,1]^m)^p$
and $\mathbf{t}=(t_1,t_2,\ldots,t_p)\in \R^p_{\ge 0}$ such that
$|\mathbf{t}|< t$ and
\begin{eqnarray*}
 x&=&\exp(t_1X_{\mathbf{u}^1})\scirc \exp(t_2X_{\mathbf{u}^2})\scirc \ldots \scirc \exp(t_pX_{\mathbf{u}^p})(x_0)=\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}}),\\
y&=&\exp(t_1Y_{\mathbf{u}^1})\scirc \exp(t_2Y_{\mathbf{u}^2})\scirc \ldots \scirc \exp(t_pY_{\mathbf{u}^p})(x_0)=\exp(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}}).
\end{eqnarray*}
Note that, for every $\mathbf{k}\in \mathbb{Z}^n_{\ge 0}$ with the
property that $|\mathbf{k}|\le N$, we have
\begin{equation*}
D^{\mathbf{k}} X_i(x_0)=D^{\mathbf{k}} Y_i(x_0),\qquad i\in \{1,\ldots,m\}.
\end{equation*}
By Theorem~\ref{kth_contact_estimate_flow}, this implies that we have
\begin{equation*}
\mathrm{ev}_{x_0}\scirc\exp_N(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)=\mathrm{ev}_{x_0}\scirc\exp_N(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i).
\end{equation*}
Thus we have 
\begin{multline*}
\left\|\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)-\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i)\right\|\\\le
\left\|\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)-\mathrm{ev}_{x_0}\scirc\exp_N(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)\right\|\\+\left\|\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i)-\mathrm{ev}_{x_0}\scirc\exp_N(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i)\right\|.
\end{multline*}
Since $X^{\mathbf{I},\mathbf{t}}, Y^{\mathbf{I},\mathbf{t}}\in
\mathrm{L}^{\infty}([0,|\mathbf{t}|]; B)$, we have
\begin{eqnarray*}
\left\|\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)-\mathrm{ev}_{x_0}\scirc\exp_N(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)\right\|\le
  \frac{\left(Mt\right)^{N+1}}{1-Mt} L,\\
\left\|\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i)-\mathrm{ev}_{x_0}\scirc\exp_{N}(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i)\right\|
\le \frac{\left(Mt\right)^{N+1}}{1-Mt} L.
\end{eqnarray*}
Thus, we have
\begin{multline*}
\left\|\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)-\mathrm{ev}_{x_0}\scirc\exp(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i)\right\|\\\le
2\frac{\left(Mt\right)^{N+1}}{1-Mt} L,\qquad\forall i\in \{1,2,\ldots,n\}.
\end{multline*}
Therefore, we have
\begin{equation*}
\left\|\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x_0)-\exp(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x_0)\right\|\le
2\sqrt{n}\frac{\left(Mt\right)^{N+1}}{1-Mt} L.
\end{equation*}
Note that, by our choice of $t_1,t_2,\ldots,t_p\in \R_{>0}$, we have
\begin{eqnarray*}
y&=&\exp(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x_0),\\
x&=&\exp(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x_0).
\end{eqnarray*}
Note that, by the defintion, we have $\alpha=\sqrt{n}M^{N+1}L$ and
$Mt\le M\overline{T}\le\frac{1}{2}$. This implies that 
\begin{equation*}
\left\|y-x\right\|\le \alpha t^{N+1}.
\end{equation*}
Since $\alpha$ does not depend on $t$, the above inequality holds for
every $t\in [0,T_{\min}]$. This completes the proof of the theorem.
\end{proof}

Part~\ref{p1:reg_spatial} of
Theorem~\ref{lem:multi-valued_map} can be considered as a 
regularity result for the finite-valued mapping $x\rightrightarrows
F^t_{\mathcal{X},\mathcal{Y}}(x)$ and part~\ref{p2:reg_time}
of Theorem~\ref{lem:multi-valued_map} can be considered as a regularity result for the
finite-valued mapping $t\rightrightarrows F^t_{\mathcal{X},\mathcal{Y}}(x)$. It is interesting to notice that the condition ``control systems $\mathcal{X}$ and $\mathcal{Y}$ have $N$th contact at point
$x_0$'' is not required for the construction of the mapping $F^t_{\mathcal{X},\mathcal{Y}}$ and also for the proof of
part~\ref{p1:reg_spatial} of Theorem~\ref{lem:multi-valued_map}. In
fact, one can construct $F^t_{\mathcal{X},\mathcal{Y}}$ and prove the
part~\ref{p1:reg_spatial} of Theorem~\ref{lem:multi-valued_map}, for any arbitrary real analytic control
system $\mathcal{Y}$. However, this condition is essential for the proof of
part~\ref{p2:reg_time} of Theorem~\ref{lem:multi-valued_map}.


\section{The main theorem}\label{sec:main_result}

In this section we prove the main result of this paper, which can be
considered as a robustness of the growth rate condition of
order $N$ with respect to perturbations of order higher than $N$. Roughly
speaking it states that, given a real analytic system $\mathcal{X}$ which
satisfies the growth rate condition of order $N$ at point $x_0$, if we
perturb the vector fields of $\mathcal{X}$ around $x_0$ by terms
of order higher than $N$, then the resulting system again satisfies the growth rate condition of order $N$. 

\begin{theorem}\label{thm:main_result}
Let $\mathcal{X}=\{X_0,X_1,\ldots, X_m\}$ and $\mathcal{Y}=\{Y_0,Y_1,\ldots,Y_m\}$ be two real analytic
control systems on $\R^n$. Suppose that the following conditions
hold:
\begin{enumerate}
\item[(i)] the control system $\mathcal{X}$ satisfies the growth rate condition of order
  $N$ at point $x_0\in \R^n$, and
\item[(ii)] control systems $\mathcal{X}$ and $\mathcal{Y}$ has $N$th
  contact at point $x_0$.
\end{enumerate} 
Then $\mathcal{Y}$ satisfies the growth rate condition of order $N$ at point $x_0\in \R^n$. 
\end{theorem}
\begin{proof}
Since $\mathcal{X}$ satisfies the growth rate condition of order $N$, there
exist $T,C>0$ such that, for every $t\in [0,T]$, we have 
\begin{equation*}
\overline{\mathrm{B}}(x_0,Ct^N)\in \mathrm{R}_{\mathcal{X}}(< t,x_0).
\end{equation*}
We first show that, for every $\mathbf{v}\in\mathbb{S}^{n-1}$, the vector
$\mathbf{v}$ is a control variation of $N$th order for the control
system $\mathcal{Y}$ at point $x_0$. For every $\mathbf{v}\in \mathbb{S}^n$, consider the
parametrized family of points $\gamma_{\mathbf{v}}:[0,T]\to \R^n$ defined by
\begin{equation*}
\gamma_{\mathbf{v}}(t)=x_0+\tfrac{C}{2}t^N\mathbf{v}.
\end{equation*}
It is clear that, for every $t\in [0,T]$, we have
$\gamma_{\mathbf{v}}(t)\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$. Suppose that, for every
$t\in [0,T]$, the single-valued mapping
$f^t_{\mathcal{X},\mathcal{Y}}:\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)\to
\mathrm{R}_{\mathcal{Y}}(< t,x_0)$ is a selection of the multi-valued mapping $F^t_{\mathcal{X},\mathcal{Y}}$. Then, for every
$\mathbf{v}\in \mathbb{S}^{n-1}$, we define the curves $\mu_{\mathbf{v}}:[0,T]\to \R^n$ by 
\begin{equation*}
\mu_{\mathbf{v}}(t)=f^t_{\mathcal{X},\mathcal{Y}}(\gamma_{\mathbf{v}}(t)).
\end{equation*}
Note that, for every $t\in [0,T]$, we have $\gamma_{\mathbf{v}}(t)\in
\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$. Therefore, the map $\mu_{\mathbf{v}}(t)$ is
well-defined. Moreover, by definition of
$f^t_{\mathcal{X},\mathcal{Y}}$, for every $t\in [0,T]$, we have
$\mu_{\mathbf{v}}(t)\in \mathrm{R}_{\mathcal{Y}}(< t,x_0)$. 
\begin{lemma*}\label{lem:standard_basis_are_variations}
For every $\mathbf{v}\in \mathbb{S}^{n-1}$, the map $\mu_{\mathbf{v}}:[0,T]\to
\mathrm{R}_{\mathcal{Y}}(< t,x_0)$ satisfies the following property:
\begin{equation*}
\mu_{\mathbf{v}}(t)=x_0+\tfrac{C}{2}t^N\mathbf{v}+\mathcal{O}(t^{N+1}).
\end{equation*}
\end{lemma*}
\begin{proof}
By Theorem~\ref{lem:multi-valued_map} part~\ref{p2:reg_time}, there
exist $\alpha>0$ and $0<T_{\min}<T$ such that, for every $t\le T_{\min}$ and every $x\in \overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$, we have 
\begin{align*}
\|f^t_{\mathcal{X},\mathcal{Y}}(x)-x\|\le \alpha t^{N+1}.
\end{align*} 
However, by definition, for every $t\in [0,T_{\min}]$, we have
$\gamma_{\mathbf{v}}(t)\in
\overline{\mathrm{B}}(x_0,\tfrac{C}{2}t^N)$. Therefore, for every $t\in
[0,T_{\min}]$, we get
\begin{align*}
\|f^t_{\mathcal{X},\mathcal{Y}}(\gamma_{\mathbf{v}}(t))-\gamma_{\mathbf{v}}(t)\|\le \alpha t^{N+1}.
\end{align*} 
This implies that 
\begin{align*}
f^t_{\mathcal{X},\mathcal{Y}}(\gamma_{\mathbf{v}}(t))=\gamma_{\mathbf{v}}(t)+\mathcal{O}(t^{N+1}),\qquad
  \forall t\in [0,T_{\min}],
\end{align*} 
which means that
\begin{equation*}
\mu_i(t)=x_0+\tfrac{C}{2}t^N\mathbf{v}+\mathcal{O}(t^{N+1}),\qquad\forall t \in [0,T_{\min}].
\end{equation*}
This completes the proof of the lemma.
\end{proof}
Thus, using Lemma~\ref{lem:standard_basis_are_variations} and
Definition~\ref{def:control_variations}, one can easily see
that, for every $\mathbf{v}\in \mathbb{S}^{n-1}$, the vector $\mathbf{v}$ is a
control variation of $N$th order for the control system
$\mathcal{Y}$. Thus, we have
$\mathcal{K}^N_{\mathcal{Y}}(x_0)=\mathbb{S}^{n-1}$. This implies that
$\widehat{\mathcal{K}}^N_{\mathcal{Y}}=\bigcup_{\alpha\ge 0}
\alpha\mathcal{K}^N_{\mathcal{Y}}(x_0)=\R^n$. Therefore, by Lemma~\ref{lem:characterization_growth_rate}, the control system $\mathcal{Y}$ satisfies
the growth rate condition of order $N$. This completes the proof of the
theorem. \end{proof}

As a direct consequence of Theorem~\ref{thm:main_result}, one can have the
following connection between Conjecture~\ref{conj:2} and Conjecture~\ref{conj:1}. 
\begin{corollary}\label{thm:connection_conjectures}
If Conjecture~\ref{conj:1} is true, then Conjecture~\ref{conj:2} holds.
\end{corollary}


\section*{Acknowledgments} The author would like to thank Andrew 
Lewis for reading this manuscript carefully and offering valuable suggestions for
improving the quality of the paper. The author is also grateful of the anonymous reviewers whose constructive
comments and suggestions have had a huge impact on both the quality and exposition of the paper. 


\appendix
\section{Proof of Corollary~\ref{cor:estimate_picard_iterations}}\label{app:1}

In this appendix, we present a proof of the Corollary~\ref{cor:estimate_picard_iterations} using the
$\mathrm{C}^{\omega}$-topology on the space
$\mathrm{LC}^{\omega}(\R^n)$. As mentioned in Section~\ref{sec:functions_vector_fields}, it is
also possible to give a proof of this corollary using the estimates in~\cite[\S 2.4.4]{agrachevcontrol2004} for
time-varying real analytic vector fields as curves on the space
$\mathrm{\Gamma}^{\infty}(\R^n)$ (see~\cite[\S 2.4.4 and Appendix A.2]{agrachevcontrol2004}). Note that, in~\cite[\S
2.4.4]{agrachevcontrol2004}, the space $\Gamma^{\infty}(\R^n)$ is equipped with the Whitney compact-open
topology~\cite[\S 2.2]{agrachevcontrol2004}. 
% \begin{corollary}
% Let $B$ be a bounded set in $\mathrm{LC}^{\omega}(\R^n)$. Then there
% exist $M,L>0$ and $\overline{T}\le T_{B}$ such that, for every $X\in \mathrm{L}^{\infty}([0,\overline{T}]; B)$, we have
% \begin{equation*}
% \|\mathrm{ev}_{x_0}\scirc\widehat{\exp}(t
% X)(x^i)-\mathrm{ev}_{x_0}\scirc\widehat{\exp}_{k}(t X)(x^i)\|\le
% \frac{Mt^{k+1}}{1-Mt}L, \quad\forall t\in [0,\overline{T}].
% \end{equation*}
% \end{corollary}
\begin{proof}
By Theorem~\ref{th:3}, for every $i\in \{1,2,\ldots,n\}$, every compact
set $K\subset \R^n$ containing $x_0$, and every $\mathbf{a}\in
c^{\downarrow}_{0}$, there exist $M, M_{x^i}>0$ such that
\begin{equation}\label{eq:estimate_picards}
\rho^{\omega}_{K,\mathbf{a},x^i}(\widehat{\exp}_{k}(t X)-\widehat{\exp}_{k-1}(t X))\le (Mt)^{k+1}M_{x^i}.
\end{equation} 
Note that, for every $\phi\in \mathrm{LC}^{\omega}(\R^n)$ and every
$i\in \{1,2,\ldots,n\}$, we have $\mathrm{ev}_{x_0}\scirc
\phi(x^i)=\phi(x^i)(x_0)$. Therefore, we get
\begin{equation*}
\|\mathrm{ev}_{x_0}\scirc\phi(x^i)\|\le \sup\{\phi(x^i)(y)\mid y\in
K\}\le \rho^{\omega}_{K,\mathbf{a},x^i}(\phi).
\end{equation*}
Now, by setting $\max_i\{M_{x^i}\}=L$ and using the estimate~\eqref{eq:estimate_picards}, we get 
\begin{equation*}
\|\mathrm{ev}_{x_0}\scirc\widehat{\exp}_{k+1}(t
X)(x^i)-\mathrm{ev}_{x_0}\scirc\widehat{\exp}_{k}(t X)(x^i)\|\le 
(Mt)^{k+1}L,\qquad\forall i\in\{1,2,\ldots,n\}.
\end{equation*}
Therefore, if we choose $\overline{T}\le T_{B}$ such that
$M\overline{T}<1$, we have 
\begin{align*}
\|\mathrm{ev}&_{x_0}\scirc\widehat{\exp}(t
X)(x^i)-\mathrm{ev}_{x_0}\scirc\widehat{\exp}_{k}(t X)(x^i)\|\\ &\le \sum_{r=k}^{\infty}\|\mathrm{ev}_{x_0}\scirc\widehat{\exp}_{r+1}(t
X)(x^i)-\mathrm{ev}_{x_0}\scirc\widehat{\exp}_{r}(t X)(x^i)\| \\ & \le
                                                                   \sum_{r=k}^{\infty}(Mt)^{r+1}L=\frac{(Mt)^{k+1}}{1-Mt}L,\qquad\forall
                                                                   t\in [0,\overline{T}].
\end{align*}
\end{proof}

\section{Proof of Theorem~\ref{kth_contact_estimate_flow}}\label{app:2}

% \begin{theorem}
% Suppose that $p\in \N$, $\mathbf{I}=(\mathbf{u}^1,\mathbf{u}^2,\ldots,\mathbf{u}^p)\in
% ([-1,1]^m)^p$, $\mathbf{t}=(t_1,t_2,\ldots,t_p)\in \R^p_{\ge
%   0}$, and $x_0\in \R^n$. Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ and
% $\mathcal{Y}=\{Y_0,Y_1,\ldots,Y_m\}$ be two real analytic control
% systems which have $k$th contact at point $x_0$. Then, for every $i\in
% \{1,2,\ldots,n\}$ we have
% \begin{align*}
% \mathrm{ev}_{x_0}\scirc \widehat{\exp}_k(|\mathbf{t}|X^{\mathbf{I},\mathbf{t}})(x^i)=\mathrm{ev}_{x_0}\scirc \widehat{\exp}_k(|\mathbf{t}|Y^{\mathbf{I},\mathbf{t}})(x^i),
% \end{align*}
% where $x^i$ is the $i$th coordinate function on $\R^n$. 
% \end{theorem}
\begin{proof}
By definition of $X^{\mathbf{I},\mathbf{t}}$ and
$Y^{\mathbf{I},\mathbf{t}}$, it suffices to show that, for every
$t\in\R_{>0}$, every $\mathbf{u}=(u_1,u_2,\ldots,u_m)\in [-1,1]^m$,
and every $i\in \{1,2,\ldots,n\}$, we have
\begin{align*}
\mathrm{ev}_{x_0}\scirc \widehat{\exp}_k(tX_{\mathbf{u}})(x^i)=\mathrm{ev}_{x_0}\scirc \widehat{\exp}_k(tY_{\mathbf{u}})(x^i).
\end{align*}
In order to prove the above equality, we first note that, by Definition~\ref{eq:sequence_iterations}, the following relation between the vector
field $X_{\mathbf{u}}$ and the linear mapping
$\widehat{\exp}_k(tX_{\mathbf{u}})$ holds:
\begin{align*}
\widehat{\exp}_k(tX_{\mathbf{u}})=\mathrm{id}+\int_{0}^{t}
  \widehat{X}_{\mathbf{u}} d\tau + \ldots +
  \int_{0}^{t}\int_{0}^{\tau_k}\ldots\int_{0}^{\tau_2}
  \underbrace{\widehat{X}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{X}_{\mathbf{u}}}_{k \mbox{ times}}
  d\tau_1 d\tau_2\ldots d\tau_k.
\end{align*}
By applying the coordinate function $x^i$ to the above equality, we
get 
\begin{align*}
&\widehat{\exp}_k(tX_{\mathbf{u}})(x^i)\\ & =x^i+\int_{0}^{t}
  \widehat{X}_{\mathbf{u}}(x^i) d\tau + \ldots +
  \int_{0}^{t}\int_{0}^{\tau_k}\ldots\int_{0}^{\tau_2}
  \underbrace{\widehat{X}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{X}_{\mathbf{u}}}_{k \mbox{ times}}(x^i)
  d\tau_1 d\tau_2\ldots d\tau_k.
\end{align*}
Note that, for the integrants in the right hand side of the above
equality, we have
\begin{align*}
&\widehat{X}_{\mathbf{u}}(x^i)=
  X^i_{\mathbf{u}}=X^i_0+\sum_{j=1}^{m}u_jX^i_{j},\\
&\widehat{X}_{\mathbf{u}}\scirc\widehat{X}_{\mathbf{u}}(x^i)=\sum_{j=1}^{n}
  X^j_{\mathbf{u}}\frac{\partial X^i_{\mathbf{u}}}{\partial x^j} =
  \sum_{j=1}^{n}(X^j_0+\sum_{l=1}^{m}u_lX^j_{l})\left(\frac{\partial
  X^i_0}{\partial x^j}+\sum_{l=1}^{m}u_l\frac{\partial X^i_{l}}{\partial x^j}\right), \\
&\qquad \vdots\\
&\underbrace{\widehat{X}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{X}_{\mathbf{u}}}_{k \mbox{ times}}(x^i)=
  \sum_{j_2=1}^{n}X^{j_2}_{\mathbf{u}}\frac{\partial}{\partial
  x^{j_2}}\left(\sum_{j_3=1}^{n}
  X^{j_3}_{\mathbf{u}}\frac{\partial}{\partial
  x^{j_3}}\left(\ldots\sum_{j_{k}=1}^{n}X^{j_k}_{\mathbf{u}}\frac{\partial X^i_{\mathbf{u}}}{\partial 
  x^{j_k}}\right)\right).
\end{align*}
From the above equalities, it is clear that, for every $l\in \{1,2,\ldots,k\}$, the term 
\begin{align*}
\underbrace{\widehat{X}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{X}_{\mathbf{u}}}_{l \mbox{ times}}(x^i)
\end{align*}
only contains partial derivatives of the vector fields
$\{X_0,X_1,\ldots,X_m\}$ of order less than or equal to $l$. Since real analytic control systems
$\mathcal{X}$ and $\mathcal{Y}$ have $k$th contact at point $x_0$, we
have 
\begin{align*}
D^{\mathbf{r}}X_j(x_0)=D^{\mathbf{r}}Y_j(x_0),\qquad\forall
  \mathbf{r}\in \Z_{\ge 0}^{n},\mbox{ such that } |\mathbf{r}|\le k \quad \forall j\in\{0,1,\ldots,m\}.
\end{align*}
This implies that, for every $l\in \{1,2,\ldots,k\}$ and every $i\in \{1,2,\ldots,n\}$, we have 
\begin{align*}
\mathrm{ev}_{x_0}\scirc \underbrace{\widehat{X}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{X}_{\mathbf{u}}}_{l \mbox{ times}}(x^i)=\mathrm{ev}_{x_0}\scirc \underbrace{\widehat{Y}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{Y}_{\mathbf{u}}}_{l \mbox{ times}}(x^i)
\end{align*}
Therefore, for every $i\in \{1,2,\ldots,n\}$, we get
\begin{align*}
& \mathrm{ev}_{x_0}\scirc\widehat{\exp}_k(tX_{\mathbf{u}})(x^i)\\ & =x^i_0+\int_{0}^{t}
  \mathrm{ev}_{x_0}\scirc\widehat{X}_{\mathbf{u}}(x^i) d\tau + \ldots +
  \int_{0}^{t}\int_{0}^{\tau_k}\ldots\int_{0}^{\tau_2}
  \mathrm{ev}_{x_0}\scirc\underbrace{\widehat{X}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{X}_{\mathbf{u}}}_{k \mbox{ times}}(x^i)
  d\tau_1 d\tau_2\ldots d\tau_k\\ & = x^i_0+\int_{0}^{t}
  \mathrm{ev}_{x_0}\scirc\widehat{Y}_{\mathbf{u}}(x^i) d\tau + \ldots +
  \int_{0}^{t}\int_{0}^{\tau_k}\ldots\int_{0}^{\tau_2}
  \mathrm{ev}_{x_0}\scirc\underbrace{\widehat{Y}_{\mathbf{u}}\scirc \ldots\scirc
  \widehat{Y}_{\mathbf{u}}}_{k \mbox{ times}}(x^i)
  d\tau_1 d\tau_2\ldots d\tau_k\\ & =
                                    \mathrm{ev}_{x_0}\scirc\widehat{\exp}_k(tY_{\mathbf{u}})(x^i). 
\end{align*}

This completes the proof of the theorem. 
\end{proof}


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