# Global Controllability for General Nonlinear Systems

Yuanyuan Liu<EMAIL_ADDRESS>Wei Zhang<EMAIL_ADDRESS>Northwestern Polytechnical University, Xi’an 710072, China.

###### Abstract

This note studies the global controllability of a general nonlinear system by
extending it to affine one. The state space of the obtained affine system
admits a nature foliation, each leaf of which is diffeomorphic to the state
space of the original system. Through this foliation, the global
controllability of these two kinds of systems are closely related and we prove
that they are indeed equivalent. The result is then extended to the case with
bounded inputs to make it practically more useful. To demonstrate the power of
our approach, several examples are presented.

###### keywords:

Nonlinear systems, global controllability, foliation, Lie algebra.

††journal: arXiv

##  1 Introduction

Controllability is one of the central issue in modern control theory. For
linear systems, it has been fully studied and it turns out that the
controllability of a linear system is completely characterized by the rank of
its controllability matrix. As for nonlinear systems, things are very
different. First, the reachability of linear systems is symmetric (which means
that $x_{0}$ is reachable by $x_{1}$ if and only if $x_{1}$ is reachable by
$x_{0}$) while in general it is not the case for nonlinear systems. Second,
there is a difference between local and global controllability for nonlinear
systems.

For global controllability, existing researches mainly focus on a special
class of nonlinear systems known as affine systems, that is, control systems
of the following form

$\dot{x}=f(x)+g(x)u,$ (1)

where $x$ is the state and $u$ is the control. (Rigorous definitions will be
given in Section 2.) Although, as far as we know, a complete characterization
of global controllability for system (1) is still lacking, many insightful
results have been achieved. For example, the controllability problem of
switched linear systems has been completely solved in [1] and the criterion is
very similar to that of linear ones. For codimension-$1$ affine systems, i.e.,
that with $n$ state variables and $n-1$ independent inputs, necessary and
sufficient conditions are given (separately) in [2]. However, for the
$2$-dimensional case, a simple and easily verifiable condition which is both
necessary and sufficient is obtained in [3]. The author proved an assertion
which is essential for the sufficiency of the condition, that is, any control
curve separates the plane into two disjoint components. The result was then
extended to $3$-dimensional case under some mild assumptions (see [4]). To our
best knowledge, the closest result to a complete solution to this problem was
achieved in [5] which, to some extent, covers all the aforementioned results.

In the study of controllability of nonlinear systems, Chow’s theorem (see [6])
plays a key role. It gives a sufficient condition for distributions to be
globally controllable, but the condition needs not to be necessary. Indeed,
counterexamples are given in [7] along with a condition which is both
sufficient and necessary. The ideal is to extend the given distribution to a
new and “bigger” one. This ideal is essential. However, the controllability of
system (1) is different from that of a distribution. Since the reachability of
the latter is symmetric while generally it is not the case for the former. To
overcome this difficulty, the concept of foliation was introduced (see [8,
9]). It decomposes the controllability of a nonlinear system into two
directions, that is, directions along and traverse the leaves of a foliation.
This concept turns out to be powerful in analysis of nonlinear systems, see
[10, 11, 12, 13] for some applications of it. For a comprehensive discussion
in the global controllability of nonlinear systems, we refer to the books [14,
15, 16].

This note concerns the global controllability of a general nonlinear system,
i.e. control system of the following form,

$\dot{x}=f(x,u),$ (2)

where $x$ is the state and $u$ is the control. (Rigorous definitions will be
given in Section 2.) We are going to prove the equivalence between the global
controllability of system (2) and a specific affine system using some of the
aforementioned techniques.

The organization of the note is as follows. In Section 2, we review some basic
preliminaries and fix some notations for statement ease. Our main result along
with some illustrative examples are presented in Section 3. The proof of the
main result is given in Section 4. Finally, conclusions are given in Section
5.

##  2 Preliminaries and Notations

By a general nonlinear control system, we mean system of the form (2), where
the state $x$ belongs to a paracompact manifold $M$, called the state space of
the system. Throughout the note the word “manifold ” means smooth, finite-
dimensional, second countable, connected and Hausdorff. We use $n$ to denote
the dimension of $M$. The control $u$ is a function valued in $\mathcal{C}$,
called the control space and, each component $u_{i}$ of $u$ belongs to
$\mathcal{U}$, called the set of feasible controls. In the note, $\mathcal{C}$
is always assumed to be the Euclidean space $\mathbb{R}^{m}$ (except in
Corollary 2 and its proof) and $\mathcal{U}$ the set of piecewise constant
functions. (In the literature, $\mathcal{U}$ is usually assumed to be the set
of measurable or piecewise continuous functions, etc. But the choice of
$\mathcal{U}$ doesn’t affect the controllability much, see [17, 18].)
Moreover, $f:M\times\mathbb{R}^{m}\rightarrow TM$ is a smooth mapping which
assigns, to each $x\in M$ and $u\in\mathbb{R}^{m}$ a tangent vector $f(x,u)$
in $T_{x}M$. Here, $TM$ denotes the tangent bundle of $M$ and $T_{x}M$ the
tangent space of $M$ at $x$.

Affine control system is a special nonlinear system which is of the form (1),
where $x$ and $u$ are as previously. In addition, $f(\cdot)$ and
$g_{1}(\cdot),\cdots,g_{m}(\cdot)$ are smooth vector fields on $M$. Here,
$g_{i},i=1,\cdots,m$ denotes the $i$-th component of $g$.

An absolutely continuous curve $x:[0,T]\rightarrow M$ is said to be a
trajectory of system (2), if there exists a feasible control
$u:[0,T]\rightarrow\mathbb{R}^{m}$ such that $\dot{x}(t)=f(x(t),u(t))$ for
almost every $t\in[0,T]$. In this case, we call it the trajectory of system
(2) associated with control $u$. Given two points $x_{0},x_{1}\in M$, we say
that $x_{1}$ is reachable by $x_{0}$, if there exists a feasible control
$u:[0,T]\rightarrow\mathbb{R}^{m}$ such that the associated trajectory
$x:[0,T]\rightarrow M$ satisfies $x(0)=x_{0}$ and $x(T)=x_{1}$. The set of all
points in $M$ that are reachable by $x_{0}$ is called the reachable set of
$x_{0}$ and is denoted by $R(x_{0})$. The system (2) is said to be globally
controllable, if for each $x\in M$, $R(x)=M$. In the rest of the note, the
word “controllability” always means “global controllability” and
“controllable” always means “globally controllable”, unless otherwise
specified.

In the note, we shall use $\mathbb{R}_{+}$ to denote the subset of
$\mathbb{R}$ that contains all the positive real numbers and
$\mathbb{R}^{l}_{+}$ to denote its $l$-fold Cartesian product
$\underbrace{\mathbb{R}_{+}\times\cdots\times\mathbb{R}_{+}}_{l}$. We shall
use $I_{\epsilon}^{l}$ to denote the open subset
$\underbrace{(-\epsilon,\epsilon)\times\cdots\times(-\epsilon,\epsilon)}_{l}$
of $\mathbb{R}^{l}$. We simply use $I^{l}$ when the specific value of
$\epsilon>0$ is of no great importance.

Let $F$ be a family of vector fields on $M$, then $X$ belongs to $F^{l}$,
denoted by $X\in F^{l}$, if there exist $l$ elements $X_{1},\cdots,X_{l}\in F$
such that $X=(X_{1},\cdots,X_{l})$. Let
$t=(t_{1},\cdots,t_{l})\in\mathbb{R}^{l}$, we use $e^{tX}$ to denote
$e^{t_{1}X_{1}}\circ\cdots\circ e^{t_{l}X_{l}}$ and $e^{tX}_{\ast}$ to denote
its tangent map. A mapping $e^{tX}:M\times\mathbb{R}\rightarrow M$ is called
physically realizable by system (2) if $t\geq 0$ and

$X\in\left\\{f(x,u)|u\in\mathcal{U}^{m}\right\\}.$

##  3 Main Results and Examples

To formulate our main result, we extend system (2) to the following affine
system by adding an integrator,

$\begin{split}\dot{x}&=f(x,y)\\\ \dot{y}&=v.\end{split}$ (3)

Here $(x,y)^{T}\in M\times\mathbb{R}^{m}$ is regarded as the state and
$v\in\mathcal{U}^{m}$ the control of the extended system (3). We shall prove,
in the next section, the equivalence between controllability of system (2) and
(3), that is,

######  Theorem 1.

Nonlinear system (2) is globally controllable if and only if affine system (3)
is globally controllable.

We shall also prove that if $M$ and $\mathcal{C}$ are compact, then

######  Corollary 2.

Nonlinear system (2) is globally controllable with input $u$ valued in the
bounded set $\in\mathcal{C}$ if and only if system (3) with $(x,y)^{T}\in
M\times\mathcal{C}$ is globally controllable with bounded input $v$.

Here, we give examples to illustrate some applications of our main result.

######  Example 1.

Consider the following linear system

$\dot{x}=Ax+Bu,\\\ $ (4)

where $x=(x_{1},x_{2},x_{3})^{T},A=(a_{ij}),1\leq i,j\leq 3$ and
$B=(0,0,1)^{T}$.

According to Theorem 1, system (4) is controllable if and only if the
following system

$\begin{pmatrix}\dot{x}_{1}\\\ \dot{x}_{2}\\\
\end{pmatrix}=\hat{A}\begin{pmatrix}x_{1}\\\ x_{2}\\\
\end{pmatrix}+\begin{pmatrix}a_{13}\\\ a_{23}\\\ \end{pmatrix}v,$ (5)

where $\hat{A}=(a_{ij}),1\leq i,j\leq 2$, is controllable. For the above
system to be controllable, $a_{13}^{2}+a_{23}^{2}$ has to be positive.
Therefore, we can apply the change of coordinates

$\begin{pmatrix}y_{1}\\\ y_{2}\\\ \end{pmatrix}=P\begin{pmatrix}x_{1}\\\
x_{2}\\\ \end{pmatrix}$

to (5) and obtain

$\begin{pmatrix}\dot{y}_{1}\\\
\dot{y}_{2}\end{pmatrix}=\bar{A}\begin{pmatrix}y_{1}\\\ y_{2}\\\
\end{pmatrix}+\begin{pmatrix}0\\\ a_{13}^{2}+a_{23}^{2}\\\ \end{pmatrix}v,$

where $\bar{A}=P\hat{A}P^{-1}$ and

$P=\begin{pmatrix}a_{23}&-a_{13}\\\ a_{13}&a_{23}\\\ \end{pmatrix}.$

Using Theorem 1 again, we conclude that system (4) is controllable if and only
if

$\bar{A}_{12}\neq 0.$ (6)

It can be seen that criterion (6) is equivalent to the controllability matrix
criterion $\mathrm{dim}(B,AB,A^{2}B)=3$.

######  Example 2.

Consider the following affine system

$\begin{split}&\dot{x}_{1}=\sin{x_{3}},\\\ &\dot{x}_{2}=\cos{x_{3}},\\\
&\dot{x}_{3}=x_{4},\\\ &\vdots\\\ &\dot{x}_{n-1}=x_{n},\\\
&\dot{x}_{n}=u.\end{split}$ (7)

It is clear that system (7) is controllable (see Theorem 6.2 in [14]). Using
Theorem 1, this claim can be verified easily. Indeed, repeatedly use Theorem 1
$n-3$ times we conclude that the controllability of system (7) is equivalent
to that of

$\begin{split}&\dot{x}_{1}=\sin{v},\\\ &\dot{x}_{2}=\cos{v},\\\ \end{split}$

which is obviously controllable. Hence, system (7) itself is controllable.

The previous two examples show the necessary of Theorem 1, the following
example which is given in [9] will show the sufficiency of Theorem 1.

######  Example 3.

Consider the nonlinear system

$\begin{split}\dot{x}_{1}&=u,\\\ \dot{x}_{2}&=x_{3}^{3},\\\
\dot{x}_{3}&=u^{3},\end{split}$ (8)

which is not affine. We use Theorem 1 to verify the controllability of system
(8). Extending it to the affine system

$\begin{split}\dot{x}_{1}&=x_{4},\\\ \dot{x}_{2}&=x_{3}^{3},\\\
\dot{x}_{3}&=x_{4}^{3},\\\ \dot{x}_{4}&=v,\end{split}$

of which the controllability can be verified by Theorem 6.2 in [14], then the
controllability of system (8) can be established by applying Theorem 1.

##  4 The Proofs of Main results

To prove Theorem 1, we study the controllability of system (3). Note that the
state space $M\times\mathbb{R}^{m}$ of it admits a nature foliation
$\mathcal{F}=\left\\{M\times\left\\{y\right\\}|y\in\mathbb{R}^{m}\right\\}$,
and the vector field $(f(\cdot,\cdot),0)$ is tangent to the leaf
$L_{y}\coloneqq M\times\left\\{y\right\\}$ for any $y\in\mathbb{R}^{m}$.
Therefore, we can restrict the vector field $(f(\cdot,\cdot),0)$ to the
submanifold $L_{y}$ of $M\times\mathbb{R}^{m}$ to obtain a vector field
$(f(\cdot,y),0)$ on $L_{y}$. Moreover, let $g_{i},i=1,\cdots,m$ be the vector
field associated with the $i$-th input $v_{i}$, that is
$g_{i}=\partial_{y_{i}}$, and $G=\left\\{g_{1},\cdots,g_{m}\right\\}$, then
clearly the mapping $\varphi:L_{y}\rightarrow L_{y_{0}}$ defined by

$\varphi(p)=e^{tg}(p),\quad\forall p\in L_{y}$

where $g=(g_{1},\cdots,g_{m})\in G^{m}$ and $t=y_{0}-y\in\mathbb{R}^{m}$, is a
diffeomorphism. Hence, we can push-forward the vector fields $(f(\cdot,y),0)$
on all leaves $L_{y}$ of $\mathcal{F}$ to a single leaf $L_{y_{0}}$, resulting
in a family of vector fields

$\bar{F}\coloneqq\left\\{\varphi_{\ast}(f(\cdot,y),0)\mid
y\in\mathbb{R}^{m}\right\\}=\left\\{(f(\cdot,y),0)|y\in\mathbb{R}^{m}\right\\}$

on $L_{y_{0}}$ for each $y_{0}\in\mathbb{R}^{m}$.

Note that the family of vector fields $\bar{F}$ on $L_{y_{0}}$ can be
identified with the family of vector fields
$F\coloneqq\left\\{f(\cdot,u)|u=\text{const}\right\\}$ on $M$. Indeed, let
$\pi$ be the projection mapping from $L_{y_{0}}$ to $M$, i.e.,

$\pi:(x,y_{0})\mapsto x.$

It is clear that $\pi$ is a diffeomorphism. Moreover, the tangent mapping

$\pi_{\ast}:(f(x,y),0)\mapsto f(x,y),$

induced by $\pi$ defines a one-to-one correspondence between $\bar{F}$ and
$F$.

In addition, we need the following lemma which follows directly from the proof
of Lemma 1 in [19].

######  Lemma 3.

If system (2) is controllable, then for any point $x\in M$, there exist a
integer $l>0$, a subset $U$ of $\mathbb{R}_{+}^{l}$ and $X\in F^{l}$, such
that the mapping $\phi:U\rightarrow M$ defined by

$\phi(t)\coloneqq e^{tX}(x)$

satisfies that the image $\phi(U)$ of $U$ under $\phi$ contains a non-empty
open subset of $M$.

Figure 1:

###### Proof.

(Proof of Theorem 1) The sufficiency of Theorem 1 is obvious. Indeed, if
system (3) is controllable, then system (2) is controllable by piecewise
continuous control and consequently by piecewise constant control in view of
Theorem 3 in [19].

The necessity is proved in two steps.

Step 1. We prove that for arbitrary given two points $p_{0}=(x_{0},y_{0})$ and
$p_{1}=(x_{1},y_{1})\in M\times\mathbb{R}^{m}$, there exists a mapping
$\varphi:U\times I^{m}\rightarrow M\times\mathbb{R}^{m}$, where $U$ is a
subset of $\mathbb{R}_{+}^{l}$, such that the image $\varphi(U\times I^{m})$
is a neighborhood of $p_{1}$.

Indeed, since system (2) is controllable, there exist $X\in F^{l}$ and a
subset $U$ of $\mathbb{R}_{+}^{l}$, thanks to Lemma 3, such that the image of
$U$ under the mapping $\phi:U\rightarrow M$ defined by

$\phi(t)\coloneqq e^{tX}(x_{0}),\quad\forall t\in U$

contains a non-empty open subset of $M$. Let $x_{2}$ be a point in this open
subset, then $\phi(U)$ is a neighborhood of $x_{2}$. Now, we lift $\phi$ to
the mapping $\varphi_{1}:U\rightarrow L_{y_{0}}$ by

$\varphi_{1}(t)\coloneqq\pi^{-1}\circ\phi(t),\quad\forall t\in U,$

where $\pi^{-1}$ is the inverse of the projection mapping $\pi$ defined
previously. It can be verified that

$\varphi_{1}(t)=e^{t\bar{X}}(p_{0}),$

where
$\bar{X}=(\pi_{\ast}^{-1}X_{1},\cdots,\pi_{\ast}^{-1}X_{l})\in\bar{F}^{l}$.
Clearly, $\varphi_{1}(U)$ is a neighborhood (with respect to the subspace
topology of $L_{y_{0}}$) of $p_{2}\coloneqq\pi^{-1}(x_{2})=(x_{2},y_{0})$ (see
Fig. 1).

Let $\eta=y_{1}-y_{0}\in\mathbb{R}^{m}$ then the mapping $\varphi_{2}$ defined
by

$\varphi_{2}(p)\coloneqq e^{\eta g}(p),\quad\forall p\in L_{y_{0}}.$

is a diffeomorphism between $L_{y_{0}}$ and $L_{y_{1}}$.

Next, let $p_{3}\coloneqq\varphi_{2}(p_{2})$, then $p_{3}\in L_{y_{1}}$ and
hence can be written as $p_{3}=(x_{3},y_{1})$ where $x_{3}\in M$ and
$y_{1}\in\mathbb{R}^{m}$. By the definition of controllability of system (2),
there exists a feasible control $u:[0,T]\rightarrow\mathbb{R}^{m}$ such that
the associated trajectory $x(t)$ satisfies $x(0)=x_{3}$ and $x(T)=x_{1}$.
Since $u$ is piecewise constant, there exists a partition
$0=T_{0}<T_{1}<\cdots<T_{k}=T$ of $[0,T]$ such that $u$ is constant on each
interval $[T_{i-1},T_{i}),i=1,\cdots,k$. Let
$u(i)=u(T_{i-1}),Y_{i}=f(x,u(i))\in F$ and $\tau_{i}=T_{i}-T_{i-1}$ for
$i=1,\cdots,k$, then we have

$x_{1}=e^{\tau Y}(x_{3}),$

where $Y=(Y_{k},\cdots,Y_{1})\in F^{k}$ and
$\tau=(\tau_{k},\cdots,\tau_{1})\in\mathbb{R}_{+}^{k}$. Analogously, we can
lift $e^{\tau Y}:M\rightarrow M$ to the mapping
$\varphi_{3}:L_{y_{1}}\rightarrow L_{y_{1}}$ by

$\varphi_{3}(p)\coloneqq e^{\tau\bar{Y}}(p),\quad\forall p\in L_{y_{1}},$

where
$\bar{Y}=(\tilde{\pi}_{\ast}^{-1}Y_{k},\cdots,\tilde{\pi}_{\ast}^{-1}Y_{1})\in\bar{F}^{k}$and
$\tilde{\pi}$ is the projection mapping from $L_{y_{1}}$ to $M$. Clearly, the
mapping $\varphi_{3}$ is a diffeomorphism and satisfies
$\varphi_{3}(p_{3})=p_{1}$.

Since $\varphi_{2}$ and $\varphi_{3}$ are diffeomorphisms, the image
$\varphi_{3}\circ\varphi_{2}\circ\varphi_{1}(U)$ of $U$ is a neighborhood of
$p_{1}$ with respect to the subspace topology of $L_{y_{1}}$. Hence, if we
define the mapping $\varphi:U\times I^{m}\rightarrow M\times\mathbb{R}^{m}$ by

$\varphi(t,s)\coloneqq
e^{sg}\circ\varphi_{3}\circ\varphi_{2}\circ\varphi_{1}(t),\quad\forall t\in
U,s\in I^{m},$

then the image $\varphi(U\times I^{m})$ of $U\times I^{m}$ under $\varphi$ is
a neighborhood of $p_{1}$ with respect to the topology of
$M\times\mathbb{R}^{m}$.

Step 2. We prove in this step that the mapping $\varphi$ can be modified in a
manner so that it is realizable by system (3) and the image $\varphi(U\times
I^{m})$ remains a neighborhood of $p_{1}$, thereby proving that $p_{1}$ is
reachable by $p_{0}$ and consequently the controllability of system (3) in
view of the arbitrariness of $p_{0}$ and $p_{1}$. To this end, note that the
vector fields $V$ appear in $\varphi$ falling into the following two
categories.

Case 1. $V\in G$, that is $V=g_{i}$ for some
$i\in\left\\{1,\cdots,m\right\\}$. In this case, we have

$e^{\sigma V}=e^{\sigma/{v_{i}}(((f,0)+g_{i}v_{i})-(f,0))}.$

Choosing $v_{i}$ so that $\sigma/v_{i}\geq 0$ and defining

$\rho(\sigma)\coloneqq e^{\sigma/{v_{i}}((f,0)+g_{i}v_{i})},$

then $\rho(\sigma)$ is physically realizable by system (3). Replacing
$e^{\sigma V}$ in $\varphi$ by $\rho(\sigma)$ as long as $V\in G$, we will
obtain another mapping $\varphi^{\prime}:U\times I^{m}\rightarrow
M\times\mathbb{R}^{m}$. In general, the image $\varphi^{\prime}(U\times
I^{m})$ of $U\times I^{m}$ under $\varphi^{\prime}$ will be different from
that under $\varphi$. But as long as $|v_{i}|$ is large enough, the image
$\varphi^{\prime}(U\times I^{m})$ will remain a neighborhood of $p_{1}$.

Case 2. $V\in\bar{F}$, that is $V=e^{\beta h}_{\ast}(f,0)$, where
$\beta=(\beta_{1},\cdots,\beta_{j})\in\mathbb{R}^{j}$ and
$h=(h_{1},\cdots,h_{j})\in G^{j}$. In this case, we have (see Lemma 2.20 in
[20])

$e^{\sigma V}=e^{\beta h}\circ e^{\sigma(f,0)}\circ e^{-\hat{\beta}\hat{h}},$

where $\hat{\beta}=(\beta_{j},\cdots,\beta_{1})$ and
$\hat{h}=(h_{j},\cdots,h_{1})$. Note that $e^{\sigma(f,0)}$ is physically
realizable since $\sigma$ is positive, thus no modification is required. The
proof is complete once we notice that the vector fields $\pm
h_{i},i=1,\cdots,j$ falling into Case 1. ∎

###### Proof of Corollary 2.

The “if” part is obvious. To prove the “only if” part, we suppose that system
(2) is globally controllable with input $u$ valued in the bounded set
$\mathcal{C}$ and let $p_{0}\in M\times\mathcal{C}$ be fixed. By the proof of
Theorem 1, for any $p\in M\times\mathcal{C}$, $p_{0}$ can reach all the points
in an open neighborhood $W_{p}$ of $p$ with input $v$ valued in a bounded set
$\mathcal{B}_{p}$. Since $\\{W_{p}\\}_{p\in M\times\mathcal{C}}$ is an open
cover of the compact set $M\times\mathcal{C}$, there exist a finte number of
points $p_{1},\cdots,p_{k}$ such that $\bigcup_{i=1}^{k}W_{p_{i}}$ covers
$M\times\mathcal{C}$ and consequently, $p_{0}$ can reach any point in
$M\times\mathcal{C}$ with input $v$ valued in the bounded set
$\bigcup_{i=1}^{k}\mathcal{B}_{p_{i}}$. Applying the same argument to the
time-reversed systems of (2) and (3), one can see that $p_{0}$ can also be
reached by any point in $M\times\mathcal{C}$ with bounded input. Put these two
facts together, we conclude that system (3) is globally controllable with
bounded input $v$. ∎

######  Remark 1.

Theorem 1 was proved in the situation where $\mathcal{U}$ is the set of
piecewise constant functions, but it can be extended to other situations where
$\mathcal{U}$ is more general. Since controllability using arbitrary controls
is equivalent to controllability using just piecewise constant controls (see
[19, 17, 21]for details).

######  Remark 2.

Compared with the approach of [14] in their proofs of Theorem 6.1 and 6.2, the
proposed one is novel in the following way. In the approach of [14], the state
space is foliated by submanifolds which are tangent to $G$ (i.e. maximal
integral submanifolds of $G$), therefore the “controllability” within the
leaves is guaranteed while that traverses the leaves is not. Hence, the main
task is to prove the flows can traverse the leaves in all directions. But in
our approach, the state space is foliated by submanifolds perpendicular to
$G$, therefore the motions traverse the leaves can be in all directions and
the problem is to prove the “controllability” within the leaves.

##  5 Conclusion

In this note, we have proved that the global controllability of a general
nonlinear system is preserved when an integrator is added. Also, we have
extended this result to the case with bounded inputs. By making use of our
result, the controllability problem of a general nonlinear system can be
converted to that of an affine system which has been studied more thoroughly.
The power of our result has been demonstrated by several examples. In
addition, the differences between our approach and existing ones have been
illustrated.

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