∎

11institutetext: Qianqian Xia 22institutetext: Nanjing University of
Information Science & Technology, Nanjing, 210044, China
22email<EMAIL_ADDRESS>
# ON CONTROLLED INVARIANCE OF REGULAR DISTRIBUTIONS

Qianqian Xia

(Received: date / Accepted: date)

###### Abstract

This paper considers the problem of controlled invariance of involutive
regular distribution, both for smooth and real analytic cases. After a review
of some existing work, a precise formulation of the problem of local and
global controlled invariance of involutive regular distributions for both
affine control systems and affine distributions is introduced. A complete
characterization for local controlled invariance of involutive regular
distributions for affine control systems is presented. A geometric
interpretation for this characterization is provided. A result on local
controlled invariance for real analytic affine distribution is given. Then we
investigate conditions that allow passages from local controlled invariance to
global controlled invariance, for both smooth and real analytic affine
distributions. We clarify existing results in the literature. Finally, for
manifolds with a symmetry Lie group action, the problem of global controlled
invariance is considered.

###### Keywords:

Controlled invarianceAffine control systemGeneralized distributionLocalGlobal
Real analyticity

##  1 Introduction

Let $r\in\\{\infty,\omega\\}$. Throughout the paper, all manifolds are assumed
to be paracompact Hausdorff $C^{r}$-manifolds if not specified. Let $L$ be a
$C^{r}$-distribution on a $C^{r}$-manifold $M$, i.e. a subbundle of the
tangent bundle $TM$, where the dimension of the the subspace $L_{p}$ is a
constant function of $p\in M$.

In control theory one encounters generalized distributions sussmann ; Stefan ;
Liu ; Bullo , where the subspace $L_{p}$ can have different dimensions at
different points. We call a distribution of a constant rank a regular
distribution. A generalized distribution $L$ is of class $C^{r}$, if for each
point $p\in M$, there exist local $C^{r}$-sections of $L$ whose values at $p$
span $L_{p}$. We refer to Section 2 for a detailed discussion on generalized
distributions.

Consider the affine control system $\Sigma$

$\dot{x}=f(x)+\sum_{i=1}^{m}g_{i}(x)u_{i}$ (1)

on $M$, where $f,g_{1},\cdots,g_{m}$ are $C^{r}$-vector fields on $M$.

Then associated to $\Sigma$ there is a $C^{r}$-generalized distribution $G$ of
$M$, defined by

$G_{p}=\text{span}_{\mathbb{R}}\\{g_{1}(p),\cdots,g_{m}(p)\\},$ (2)

for $p\in M$, where $g_{1},\cdots,g_{m}$ are said to be the $C^{r}$-generators
for $G$.

The following definition is taken from 1 .

###### Definition 1

Let $U$ be an open subset of $M$. A pair of $C^{r}$-functions
$(\alpha,\beta)$, where $\alpha:U\rightarrow\mathbb{R}^{m}$ and
$\beta:U\rightarrow GL(m,\mathbb{R})$, are called feedback functions on $U$.
When $U=M$ we refer to feedback functions as global feedback. If
$\alpha=(\alpha_{1}\cdots\alpha_{m})^{T}$ and $\beta=(\beta_{ij})_{m\times m}$
are feedback functions on $U$, then the new system obtained by feedback,

$\dot{x}=(f+\sum_{i=1}^{m}g_{i}\alpha_{i})+\sum_{i=1}^{m}(\sum_{j=1}^{m}\beta_{ij}g_{j})v_{i}=\hat{f}(x)+\sum_{i=1}^{m}\hat{g}_{i}(x)v_{i},$
(3)

is said to be pure feedback equivalent to $\Sigma$.

Let $D$ be an involutive regular $C^{r}$-distribution of $M$. In the previous
work there is the following definition of local controlled invariance 1 ;
Hirschorn ; Krener ; Nijmeijer ; Ikggm ; Byrnes ; Schaft ; schaft ; Isidory ;
Cheng .

###### Definition 2

An involutive regular $C^{r}$-distribution $D$ of $M$ is said to be locally
controlled invariant for $\Sigma$ at $p\in M$, if there exist a sufficiently
small neighborhood $U$ of $p$, and $C^{r}$-feedback functions $(\alpha,\beta)$
on $U$ such that

$\displaystyle[\hat{f},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D),$ (4a)
$\displaystyle[\hat{g}_{i},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D),i=1,\cdots,m,$
(4b)

where $\hat{f},\hat{g}_{i},i=1,\dots,m,$ are given by $(\ref{eq:1.2})$, and
$\Gamma^{r}(U;D)$ denotes the space of $C^{r}$-sections of $D$ over $U$.

Let $\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\}$ be the module over
$C^{r}(U)$ spanned by $g_{1},\cdots,g_{m}$, where $C^{r}(U)$ denotes the ring
of $C^{r}$-functions on the open subset $U$. From (4a) and (4b) it is obvious
that a necessary condition for local controlled invariance is that $D$
satisfies

$\displaystyle[f,\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\},$
(5a)
$\displaystyle[g_{j},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\},j=1,\cdots,m.$
(5b)

The notion of controlled invariant subspaces in the linear systems theory
dates back to the end of sixties Basile ; Wonham . A modern account of the use
of controlled and conditioned invariant subspaces is given in wonham . The
nonlinear generalization of the notion of controlled invariance, which plays a
crucial role in disturbance decoupling problems, has been initiated in
Hirschorn ; Krener . See also Isidory ; MW ; isidory ; Moog ; Krener2 ;
Respondek . There has been a lot of work on conditions that ensure local
controlled invariance for the $C^{\infty}$-affine control system $\Sigma$,
mainly focusing on the following conditions:

$\displaystyle[f,D](q)\subseteq(D+G)(q),$ (6a)
$\displaystyle[g_{j},D](q)\subseteq(D+G)(q),j=1,\cdots,m.$ (6b)

for each $q$ in a sufficiently small neighborhood $U$ of $p\in M$. In Schaft ,
Theorem 7.5 states that if $G$ and $D\cap G$ have constant rank, then (6a) and
(6b) are sufficient to ensure that $D$ is locally controlled invariant for
$\Sigma$. In 1 ; Hirschorn ; Krener ; Nijmeijer ; Ikggm ; Byrnes ; schaft ;
Isidory , the above conditions are replaced by only requiring that $G+D$ has
constant rank, or equivalently $(G+D)/D$ has constant rank, then the
sufficiency of (6a) and (6b) is ensured. This is known as Quaker Lemma. In
Cheng , a necessary and sufficient condition concerning the
$C^{\infty}(U)$-module spanned by $g_{1}/D,\cdots,g_{m}/D$ is presented for
the local controlled invariance of the involutive regular distribution $D$,
where no additional condition of constant rank is required. We refer to Remark
1 in Section 3 for a precise statement of this result and its equivalence with
our result. However, the proof provided in Cheng  is not a geometric one.

In this paper, by exploring the underlying geometric structure, we will show
that instead of (6a) and (6b), (5a) and (5b) are necessary and sufficient for
local controlled invariance for $C^{r}$-affine control system $\Sigma$. We
will provide an intrinsic and complete differential geometric proof for our
result. Differential geometric interpretations for the feedback functions
$(\alpha,\beta)$ required in Definition 2 will be given. A previous work using
geometric methods has been done in 1  on the proof of Quaker Lemma, where
connection and parallel translation are introduced. Although the result there
is not complete (we will give a detailed explanation for this in Section 3), a
geometric intuition is provided which inspires our consideration here.

One thing that should be pointed out is that it can not be assumed that any
$C^{r}$-local generators for a $C^{r}$-generalized distribution can span the
module of $C^{r}$-local sections of the $C^{r}$-generalized distribution Lewis
; Drager . Hence in general the conditions of (5a) and (5b) is stronger than
(6a) and (6b). This observation leads to the following definition of local
controlled invariance for generalized affine distribution, which is equivalent
with Definition 2 if $p$ is a regular point for $G$ and is a new one if $p$ is
a singular point for $G$.

###### Definition 3

An involutive regular $C^{r}$-distribution $D$ of $M$ is said to be locally
controlled invariant for a $C^{r}$-generalized affine distribution $\Delta$ at
$p$, if there exist a sufficiently small local neighborhood $U$ of $p$ and
$C^{r}$-vector fields $f,g_{i},i\in I$ on $U$, such that
$[f,\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)$ and
$[g_{i},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D),i\in I$, where
$\Delta_{q}=f(q)+\text{span}_{\mathbb{R}}\\{g_{i}(q)|i\in I\\}$ for each $q\in
U$, and $I$ is an index set.

We will provide a sufficient condition for local controlled invariance for
$C^{\omega}$-generalized affine distribution in Section 3.

Another main problem that we will investigate in this paper is the problem of
global controlled invariance. We have the following definitions for affine
distributions and affine control systems respectively.

###### Definition 4

An involutive regular $C^{r}$-distribution $D$ of $M$ is said to be globally
controlled invariant for a $C^{r}$-generalized affine distribution $\Delta$,
if there exist $C^{r}$-vector fields $f,g_{i},i\in I$ on $M$, such that

$\displaystyle[f,\Gamma^{r}(D)]\subseteq\Gamma^{r}(D),$ (7a)
$\displaystyle[g_{i},\Gamma^{r}(D)]\subseteq\Gamma^{r}(D),i\in I.$ (7b)

where $\Delta_{p}=f(p)+\text{span}_{\mathbb{R}}\\{g_{i}(p)|i\in I\\}$ for all
$p\in M$, and $I$ is an index set.

###### Definition 5

An involutive regular $C^{r}$-distribution $D$ of $M$ is said to be globally
controlled invariant for $\Sigma$, if there exist global feedback
$(\alpha,\beta)$ such that

$\displaystyle[\hat{f},\Gamma^{r}(D)]\subseteq\Gamma^{r}(D),$ (8a)
$\displaystyle[\hat{g}_{i},\Gamma^{r}(D)]\subseteq\Gamma^{r}(D),i=1,\cdots,m,$
(8b) where $\hat{f},\hat{g}_{i},i=1,\cdots,m$, are given by $(\ref{eq:1.2})$
with $U=M$.

Since $D$ has constant rank, there exist generators
$d_{1},\cdots,d_{k}\in\Gamma^{r}(U;D)$ for the $C^{r}(U)$-module of
$\Gamma^{r}(U;D)$ of $D$. Besides, $d_{1}|V,\cdots,d_{k}|V\in\Gamma^{r}(V;D)$
are generators for the $C^{r}(V)$-module of $\Gamma^{r}(V;D)$ of $D$, where
$V$ is any open subset of $U$. Since $M$ is paracompact and Hausdorff, then it
follows from partition of unity Abraham  in the smooth case and Cartan’s
Theorem A Cartan  in the real analytic case that, for each $p\in M$, there
exist generators for the module of local sections of $D$ on a neighborhood of
$p$ that are the restrictions of global sections of $D$ to this neighborhood.
Then by restrictions of globally defined vector fields in (7a), (7b), (8a),
and (8b) to the above neighborhood of $p$, it follows immediately that global
controlled invariance implies the corresponding local controlled invariance.
However, if $D$ does not have constant rank, complications may arise. A
previous work on controlled invariance of singular distribution can be found
in Rama .

It is obvious that in general we can not infer from local controlled
invariance at every point in $M$ that global controlled invariance holds. In
this paper, we will investigate cases and conditions that admit global
controlled invariance. Note that 1  studies a special case of global
controlled invariance for affine control systems, where $g_{1},\cdots,g_{m}$
are assumed to be globally defined linearly independent vector fields. In
Section 4 we will investigate the problem of global controlled invariance for
generalized affine distributions from the following two aspects: (1) special
geometric structure of the state space $M$ that admits global controlled
invariance; (2) conditions that allow passages from local controlled
invariance to global controlled invariance. Both require techniques different
from 1 . We will also review result in 1  and do some possible generalization.
We refer to 1  for detailed information about existing work on global
controlled invariance.

In summary, the main contribution of the paper is two fold: (1) We provide a
differential geometric interpretation for the problem of local controlled
invariance of involutive regular distribution for $C^{r}$-affine control
system. (2) We provide conditions that allow global controlled invariance for
(i) $C^{r}$-generalized affine distribution of general $C^{r}$-manifold; (ii)
smooth generalized affine distribution of smooth manifold with a symmetry Lie
group action.

The paper is organized as follows. Section 2 provides some basic definitions,
results and facts on generalized distributions. In Section 3 we study local
controlled invariance for both affine control system and affine distribution.
The corresponding problems of global controlled invariance are investigated in
Section 4, and the conclusions follow in Section 5.

##  2 Generalized distributions

In this section we introduce some basic notions and results on generalized
distributions and generators used in the paper. We refer to Lewis  for a
comprehensive and elegant treatment on this topic.

###### Definition 6

Let $M$ be a $C^{r}$-manifold. A generalized distribution of $M$ is a subset
$G\subseteq TM$ such that for each $p\in M$, the subset $G_{p}\subseteq
T_{p}M$ is a subspace of $T_{p}M$. A generalized affine distribution $\Delta$
of $M$ is an assignment of an affine subspace $\Delta_{p}\subseteq T_{p}M$ for
each $p\in M$. Associated with $\Delta_{p}$ is a subspace of $T_{p}M$. The
distribution which assigns to $p$ this subspace is the linear part of $\Delta$
and is denoted by $L(\Delta)$. We do not assume the subspaces $G_{p}$ or
$L(\Delta_{p})$ vary continuously with $p$ or have constant dimension.

A generalized distribution $G$ is of class $C^{r}$, if, for each $p\in M$,
there exist a neighbourhood $U$ of $p$ and a family $(g_{j})_{j\in J}$ of
$C^{r}$-sections, called local generators, of $TM|U$ such that

$G_{q}=\text{span}_{\mathbb{R}}\\{g_{j}(q)|j\in J\\},$ (9)

for each $q\in U$. A generalized affine distribution $\Delta$ is of class
$C^{r}$, if, for each $p\in M$, there exist a neighbourhood $U$ of $p$ and a
family $(h_{0},(h_{i})_{i\in I})$ of $C^{r}$-sections, called local
generators, of $TM|U$ such that

$\Delta_{q}=h_{0}(q)+\text{span}_{\mathbb{R}}\\{h_{i}(q)|i\in I\\},$ (10)

for each $q\in U$.

A generalized distribution $G$ of class $C^{r}$ is locally finitely generated
if, for each $p\in M$, there exist a neighbourhood $U$ of $p$ and a family
$(g_{1},\cdots,g_{k})$ of $C^{r}$-sections, called local generators, of $TM|U$
such that

$G_{q}=\text{span}_{\mathbb{R}}\\{g_{1}(q),\cdots,g_{k}(q)\\},$ (11)

for each $q\in U$. A generalized affine distribution $\Delta$ of class $C^{r}$
is locally finitely generated if the associated generalized distribution
$L(\Delta)$ of class $C^{r}$ is locally finitely generated.

A generalized distribution $G$ of class $C^{r}$, is globally finitely
generated if, there exists a family $(g_{1},\cdots,g_{k})$ of
$C^{r}$-sections, called global generators, of $TM$ such that for each $p\in
M$,

$G_{p}=\text{span}_{\mathbb{R}}\\{g_{1}(p),\cdots,g_{k}(p)\\}.$ (12)

The nonnegative integer $\dim(G_{p})$ at $p$ is called the rank of $G$ at $p$
and is sometimes denoted by $\text{rank}(G_{p})$. A generalized affine
distribution $\Delta$ of class $C^{r}$, is globally finitely generated if,
there exists a family $(h_{0},h_{1},\cdots,h_{l})$ of $C^{r}$-sections, called
global generators, of $TM$ such that for each $p\in M$,

$\Delta_{p}=h_{0}(p)+\text{span}_{\mathbb{R}}\\{h_{1}(p),\cdots,h_{l}(p)\\}.$
(13)

###### Definition 7

Let $M$ be a $C^{r}$-manifold. Let $G$ be a $C^{r}$-generalized distribution
of $M$ and $\Delta$ be a $C^{r}$-generalized affine distribution of $M$. If
$U\subseteq M$ is open, a local section of $G$ or $\Delta$ over $U$ is a
section $g:U\rightarrow TM$ such that $g(p)\in G_{p}$ or $g(p)\in\Delta_{p}$
for every $p\in U$. A local section $g$ of $G$ or $\Delta$ is of class
$C^{r}$, if it is of class $C^{r}$ as a local section of $TM$. The set of
local sections of $G$ or $\Delta$ over $U$ of class $C^{r}$ is denoted by
$\Gamma^{r}(U;G)$ or $\Gamma^{r}(U;\Delta)$, or simply by $\Gamma^{r}(G)$ or
$\Gamma^{r}(\Delta)$ when $U=M$.

It follows immediately that $\Gamma^{r}(U;G)$ is a module over $C^{r}(U)$. An
equivalent characterization of $C^{r}$-generalized distribution or
$C^{r}$-generalized affine distribution is given as follows: a generalized
distribution $G$ or a generalized affine distribution $\Delta$ of $M$ is of
class $C^{r}$, if and only if for every point $p$ and every $v\in G_{p}$ or
$\Delta_{p}$, there exist a neighbourhood $U$ of $p$ and a $C^{r}$-section
$s\in\Gamma(U;G)$ or $\Gamma(U;\Delta)$ such that $s_{p}=v$.

If $G$ is a smooth generalized distribution of a connected smooth manifold
$M$, the following result ensures that $G$ is globally finitely generated.

###### Theorem 2.1

Sussmann ; Drager  Let $M$ be a connected smooth manifold. Let $G$ be a smooth
generalized distribution of $M$. Then $G$ is globally finitely generated.

If $G$ is a $C^{\omega}$-generalized distribution of a $C^{\omega}$-manifold
$M$, the following results ensure that $G$ is locally finitely generated.
Besides, there exist sufficiently small neighborhood $U$ of $p\in M$, such
that the module $\Gamma^{\omega}(U;G)$ over $C^{\omega}(U)$ is finitely
generated.

###### Theorem 2.2

Lewis  Let $M$ be a real analytic manifold and $G$ be a generalized
distribution of $M$, then the following statements are equivalent:
1\. $G$ is real analytic.
2\. For each $p\in M$ and each $v_{p}\in G_{p}$, there exists a real analytic
section $g\in\Gamma^{\omega}(G)$ over $M$ such that $g_{p}=v_{p}$.
3\. For each $p\in M$ there exist a neighbourhood $\mathcal{N}$ of $p$ and
real analytic sections $g_{1},\cdots,g_{k}\in\Gamma^{\omega}(G)$ over $M$ such
that

$G_{q}=\text{span}_{\mathbb{R}}\\{g_{1}(q),\cdots,g_{k}(q)\\},$

for each $q\in\mathcal{N}$.

###### Proposition 1

Lewis  Let $M$ be a real analytic manifold and $G$ be a real analytic
generalized distribution of $M$. Then for each $p\in M$, there exist a
neighbourhood $U$ of $p$ and $s_{1},\cdots,s_{l}\in\Gamma^{\omega}(U;G)$ such
that $s_{1}|V,\cdots,s_{l}|V$ are generators for the module
$\Gamma^{\omega}(V;G)$ over $C^{\omega}(V)$, for every open set $V\subseteq
U$.

One of the complications ensuing from the notion of a generalized distribution
arises if the dimensions of the subspaces $G_{p},p\in M$, are not locally
constant. The following definition associates some language with this.

###### Definition 8

Let $M$ be a $C^{r}$-manifold and $p$ be point in $M$. Let $G$ be a
$C^{r}$-generalized distribution of $M$. $p$ is a regular point for $G$ if
there exists a neighbourhood $U$ of $p$ such that the dimension of the linear
subspace $G_{q}$ is equal to that of $G_{p}$, for every $q\in U$. $p$ is a
singular point for $G$ if it is not a regular point for $G$.

A generalized distribution $G$ is regular if every point in $M$ is a regular
point for $G$, and is singular otherwise.

Let $p$ be a regular point for the $C^{r}$-generalized distribution $G$ with
$\text{rank}(G_{p})=k$. From (10) we know that there exist local
$C^{r}$-sections $g_{1},\cdots,g_{k}$ of $G$ such that
$G_{p}=\text{span}_{\mathbb{R}}\\{g_{1}(p),\cdots,g_{k}(p)\\}$. Since $p$ is a
regular point, it follows that
$G_{q}=\text{span}_{\mathbb{R}}\\{g_{1}(q),\cdots,g_{k}(q)\\}$ for $q\in U$,
where $U$ is any sufficiently small neighborhood of $p$. That is, $G$ is
finitely generated over $U$. Besides, let $h_{1},\cdots,h_{l}$ be any local
generators of $G$ over $U$. $\Gamma^{r}(U;G)$ is a module over $C^{r}(U)$
generated by $h_{1},\cdots,h_{l}$. That is, we can identify a
$C^{r}$-distribution, the module spanned by any $C^{r}$-local generators, and
the space of its local $C^{r}$-sections around a regular point.

If $p$ is a singular point for a $C^{r}$-generalized distribution $G$, the
following two examples show that: (1) If $r=\infty$, there exists arbitrarily
small neighborhood $U$ of $p$ such that $\Gamma^{\infty}(U;G)$ is not finitely
generated. (2) If $r=\omega$, there exist local generators of $G$ which can
not span the module $\Gamma^{\omega}(U;G)$, where $U$ is any local
neighborhood of $p$.

###### Example 1

Drager  Let $M=\mathbb{R}$ and $TM=\mathbb{R}\times\mathbb{R}$. Consider the
generalized distribution $G$ defined by

$G_{p}=\begin{cases}\\{p\\}\times\mathbb{R},&p\textgreater 0,\\\
\\{p\\}\times\\{0\\},&p\leq 0.\end{cases}$

Consider the smooth section $g$ of $G$ defined by

$g_{p}=\begin{cases}(p,e^{-1/p}),&p\textgreater 0,\\\ (p,0),&p\leq
0.\end{cases}$

It follows that $G$ is a smooth distribution of $M$ since $g$ is a global
generator for $G$. However, Drager  shows that $\Gamma^{\infty}(U;G)$ is not
finitely generated, for any neighborhood $U=(-a,a),a\textgreater 0$ of the
singular point $0$ for $G$.

###### Example 2

Lewis  Let $M=\mathbb{R}$ and $TM=\mathbb{R}\times\mathbb{R}$. Consider the
$C^{\omega}$-generalized distribution $G$ defined by

$G_{p}=\begin{cases}\\{p\\}\times\mathbb{R},&p\neq 0,\\\
\\{p\\}\times\\{0\\},&p=0.\end{cases}$

It has a global generator $g(p)=(p,p^{2})$. However, consider the
$C^{\omega}$-section $h(p)=(p,p)$ of $G$. It is obvious that $h|U$ is not a
$C^{\omega}(U)$-multiple of $g|U$, where $U\subseteq M$ is any open subset
containing the singular point $0$ of $G$.

##  3 Local controlled invariance

We have the following result for local controlled invariance.

###### Theorem 3.1

Consider the $C^{r}$-affine control system (1) on $M$. Let $D$ be an
involutive regular $C^{r}$-distribution of $M$. $D$ is locally controlled
invariant for $\Sigma$ at $p\in M$, if and only if there exists a sufficiently
small neighborhood $U$ of $p$, such that

$\displaystyle[g_{i},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\},i=1,\cdots,m.$
(14a)
$\displaystyle[f,\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\},$
(14b)

where $\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\}$ denotes the
$C^{r}(U)$-module spanned by $g_{1},\cdots,g_{m}$.

We will propose a new proof based on the notion of connection. Actually, an
involutive regular distribution $D$ of $M$ defines a connection on $TM/D$ as
claimed in 1 . Consider the vector bundle $\rho:TM/D\rightarrow M$. Let
$\pi:TM\rightarrow TM/D$ be the projection map. Define a map $\nabla$ as
follows:

$\displaystyle\nabla:$
$\displaystyle\Gamma^{r}(D)\times\Gamma^{r}(TM/D)\rightarrow\Gamma^{r}(TM/D)$
(15) $\displaystyle\nabla_{X}\bar{Y}(p)=\pi([X,Y])(p)$

for each $p\in M$, where $Y$ is a $C^{r}$-vector field on a neighborhood of
$p$ in $M$ which projects to $\bar{Y}$. Since $TM/D$ has constant rank, then
the projections of any local generators for $TM$ span the module of local
section of $TM/D$. Hence the existence of $Y$ is ensured.

###### Proposition 2

The map $\nabla$ defined by (15) is well-defined and satisfies the following
conditions:
1\. $(X,\bar{Y})$ is $\mathbb{R}$-bilinear.
2\. $\nabla_{fX}\bar{Y}=f\nabla_{X}\bar{Y}$ for $f\in C^{r}(M)$.
3\. $\nabla_{X}(f\bar{Y})=f\nabla_{X}\bar{Y}+(L_{X}f)\bar{Y}$ for $f\in
C^{r}(M)$.

###### Proof

Let $p\in M$ and $Y_{1},Y_{2}$ be two $C^{r}$-sections defined on a local
neighborhood $U$ of $p$ such that $\bar{Y}|U=\pi(Y_{1})=\pi(Y_{2})$. It
follows that

$[X,Y_{1}-Y_{2}]\in[X,\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D),$

for $X\in\Gamma^{r}(U;D)$. That is,

$\pi([X,Y_{1}])=\pi([X,Y_{2}]).$ (16)

Hence $\nabla$ is well-defined. The proof for the three conditions is obvious,
so we omit it. ∎

Then from Bott ; Iliev  we know that the map $\nabla$ is a connection on
$TM/D$. Let $c:[0,1]\rightarrow M$ be a $C^{1}$ curve such that $\dot{c}(t)\in
D_{c(t)}$ for each $t\in[0,1]$. Recall that $\sigma:[0,1]\rightarrow TM/D$ is
said to be a parallel translation along $c$, if
$\nabla_{\dot{c}(t)}\sigma(t)=0$ for all $t\in[0,1]$. Since $\nabla$ satisfies
that
$\nabla_{X_{1}}\nabla_{X_{2}}\bar{Y}-\nabla_{X_{2}}\nabla_{X_{1}}\bar{Y}-\nabla_{[X_{1},X_{2}]}\bar{Y}=\pi([X_{1},[X_{2},Y]]-[X_{2},[X_{1},Y]]-[[X_{1},X_{2}],Y])=0$,
$\nabla$ is flat. Then from Bott ; Iliev  we know that for each $p\in M$,
there exists a neighborhood on the leaf of foliation of $D$ through $p$, such
that when restricted to this neighborhood, the parallel translation depends
only on $c(0)$ and $c(1)$.

Let $G$ be a $C^{r}$-generalized distribution of $M$. From local coordinates
representation of $\pi$, we know that $\pi(G+D)$ is a $C^{r}$-generalized
subbundle of $TM/D$. Let $U$ be a local neighborhood of $p$ and
$\phi:U\rightarrow\mathbb{R}^{n}$ be a local chart about $p$ satisfying that
$\phi(p)=0$ and
$D=\text{span}_{\mathbb{R}}\\{\frac{\partial}{\partial{x_{1}}},\cdots,\frac{\partial}{\partial{x_{k}}}\\}$.
Let $g_{1},\cdots,g_{m}$ be local generators for $G$ on a neighborhood $U$ of
$p$. We define the sections $\bar{Z}_{i}:U\rightarrow TM/D$ as follows:

For any $q\in U$ with $\phi(q)=(q_{1},\cdots,q_{n})$, consider the parallel
translation

$\sigma_{i}:[0,1]\rightarrow TM/D$ (17)

of

$\sigma_{i}(0)=\pi(g_{i}(\phi^{-1}(0,\cdots,0,q_{k+1},\cdots,q_{n})))$ (18)

along any curve $c:[0,1]\rightarrow U$ with $\dot{c}(t)\in D_{c(t)}$ such that

$\phi(c(0))=(0,\cdots,0,q_{k+1},\cdots,q_{n})$ (19)

and $c(1)=q$. We define $\bar{Z}_{i}(q)=\sigma_{i}(1),i=1,\cdots,m$.

Let $h_{i}$ be defined by

$h_{i}(q)=\sum_{j=k+1}^{n}g_{i}^{j}(0,\cdots,0,q_{k+1},\cdots,q_{n})\frac{\partial}{\partial
x^{j}},$ (20)

for $q\in U,i=1,\cdots,m$. Direct computation yields that
$\bar{Z}_{i}(q)=\pi(h_{i}(q)),i=1,\cdots,m$. Hence $\bar{Z}_{i}$ are
$C^{r}$-sections of $TM/D,i=1,\cdots,m$.

###### Proposition 3

Assume that $g_{1},\cdots,g_{m}$ satisfy that

$[g_{i},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\},i=1,\cdots,m.$
(21)

Then we have

$\pi(g_{i})\in\text{span}_{C^{r}(U)}\\{\bar{Z}_{1},\cdots,\bar{Z}_{m}\\},i=1,\cdots,m,$

where $\text{span}_{C^{r}(U)}\\{\bar{Z}_{1},\cdots,\bar{Z}_{m}\\}$ denotes the
$C^{r}(U)$-module spanned by $\bar{Z}_{1},\cdots,\bar{Z}_{m}$.

###### Proof

From (21) we know that there exist functions $\gamma^{l}_{ij}\in C^{r}(U)$
such that

$\nabla_{\frac{\partial}{\partial
x_{i}}}{\pi(g_{j})}=\sum_{l=1}^{m}\gamma^{l}_{ij}\pi(g_{l}),i=1,\cdots,k,j=1,\cdots,m.$
(22)

Let $\hat{\zeta}_{ii}(0,\cdots,0,q_{k+1},\cdots,q_{n})=1$,
$\hat{\zeta}_{ij}(0,\cdots,0,q_{k+1},\cdots,q_{n})=0,j\neq i,i,j=1,\cdots,m.$
Then it follows from the definition of $\bar{Z}_{j}$ that

$\displaystyle\sum_{j=1}^{m}\hat{\zeta}_{ij}(0,\cdots,0,q_{k+1},\cdots,q_{n})\bar{Z}_{j}(0,\cdots,0,q_{k+1},\cdots,q_{n})$
$\displaystyle=$ $\displaystyle\pi(g_{i})(0,\cdots,0,q_{k+1},\cdots,q_{n}).$

Assume that there exist
$\hat{\eta}_{ij}(q_{1},\cdots,q_{l},0,\cdots,0,q_{k+1},\cdots,q_{n})$
satisfying that

$\displaystyle\sum_{j=1}^{m}$
$\displaystyle\hat{\eta}_{ij}(q_{1},\cdots,q_{l},0,\cdots,0,q_{k+1},\cdots,q_{n})\bar{Z}_{j}(q_{1},\cdots,q_{l},0,\cdots,0,q_{k+1},\cdots,$
(23) $\displaystyle
q_{n})=\pi(g_{i})(q_{1},\cdots,q_{l},0,\cdots,0,q_{k+1},\cdots,q_{n}).$

We claim there exist
$\hat{\xi}_{ij}(q_{1},\cdots,q_{l},q_{l+1},0,\cdots,0,q_{k+1},\cdots,q_{n})$
satisfying that

$\displaystyle\sum_{j=1}^{m}$
$\displaystyle\hat{\xi}_{ij}(q_{1},\cdots,q_{l},q_{l+1},0,\cdots,0,q_{k+1},\cdots,q_{n})\bar{Z}_{j}(q_{1},\cdots,q_{l},q_{l+1},0,\cdots,0,$
$\displaystyle
q_{k+1},\cdots,q_{n})=\pi(g_{i})(q_{1},\cdots,q_{l},q_{l+1},0,\cdots,0,q_{k+1},\cdots,q_{n}).$

Consider the curve
$c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}:[-1,1]\rightarrow M$ with

$\phi(c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(0))=(q_{1},\cdots,q_{l},0,\cdots,0,q_{k+1},\cdots,q_{n})$
(24)

and

$\dot{c}_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t)=\frac{\partial}{\partial
x_{l+1}}.$ (25)

$c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t)$ lies in the leaf of
$\rho(K)$ passing $c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(0)$.

Let $\bar{Y}_{j}=\sum_{i=1}^{m}\xi_{ji}\bar{Z}_{i},j=1,\cdots,m$. We have

$\displaystyle\nabla_{\dot{c}_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t)}\bar{Y}_{j}=\sum_{r=1}^{m}\dot{\xi}_{jr}(c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t))\bar{Z}_{r}$
$\displaystyle+\sum_{r=1}^{m}{\xi}_{jr}({c}_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t))\nabla_{\dot{c}_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t)}\bar{Z}_{r}$
$\displaystyle=\sum_{r=1}^{m}\dot{\xi}_{jr}(c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t))\bar{Z}_{r}(c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t)).$
(26)

On the other hand,

$\sum_{r=1}^{m}\gamma^{r}_{{l+1}j}\bar{Y}_{r}=\sum_{r=1}^{m}\gamma^{r}_{{l+1}j}\sum_{a=1}^{m}\xi_{ra}\bar{Z}_{a}=\sum_{r=1}^{m}\sum_{a=1}^{m}\gamma^{r}_{{l+1}j}\xi_{ra}\bar{Z}_{a}.$
(27)

Set

$\dot{\sigma}_{jr}(t)=\sum_{a=1}^{m}\gamma^{a}_{{l+1}j}(c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t))\sigma_{ar}(t),j,r=1,\cdots,m.$

Then we get $m$-system of linear ordinary differential equations with
parameters $q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n}$ which has unique
solutions with prescribed initial value

$\sigma_{jr}(0,q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})=\hat{\eta}_{jr}(q_{1},\cdots,q_{l},0,\cdots,0,q_{k+1},\cdots,q_{n}),$
(28)

for $\hat{\eta}_{jr}$ given in (23) and $j,r=1,\cdots,m$. It follows that

$\nabla_{\dot{c}_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t)}{\sum_{r=1}^{m}\sigma_{jr}(t)\bar{Z}_{r}}=\sum_{r=1}^{m}\gamma^{r}_{{l+1}j}(c_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t))\sum_{a=1}^{m}\sigma_{ra}(t)\bar{Z}_{a},$
(29)

for $j=1,\cdots,m$.

Since there exists unique solution for system of equations

$\nabla_{\dot{c}_{(q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})}(t)}{Y_{i}}=\sum_{r=1}^{m}\gamma^{r}_{{l+1}i}Y_{r},i=1,\cdots,m,$

with prescribed initial value, it follows from (22), (23), (28) and (29) that

$\displaystyle\sum_{r=1}^{m}$
$\displaystyle\sigma_{jr}(q_{l},q_{1},\cdots,q_{l+1},q_{k+1},\cdots,q_{n})\bar{Z}_{r}(q_{1},\cdots,q_{l+1},0,\cdots,0,q_{k+1},$
$\displaystyle\cdots,q_{n})=\pi(g_{j})(q_{1},\cdots,q_{l+1},0,\cdots,0,q_{k+1},\cdots,q_{n}).$

Let
$\hat{\xi}_{ij}(q_{1},\cdots,q_{l},q_{l+1},0,\cdots,0,q_{k+1},\cdots,q_{n})=\sigma_{jr}(q_{l+1},q_{1},\cdots,q_{l},q_{k+1},\cdots,q_{n})$.
This completes the proof for the above claim. By induction, we know that there
exist $\hat{\lambda}_{ij}\in C^{r}(U),i,j=1,\cdots,m,$ such that

$\sum_{j=1}^{m}\hat{\lambda}_{ij}\bar{Z}_{j}=\pi(g_{i}),$ (30)

for $i=1,\cdots,m$. So the result follows immediately.∎

Proof of Theorem 3.1. The proof of the ”only if” part is obvious according to
Definition 1 and 2. We only need to prove the ”if” part. It follows from (14a)
and Proposition 3 that

$(\pi(g_{1})\cdots\pi(g_{m}))=(\bar{Z}_{1}\cdots\bar{Z}_{m})A,$ (31)

where $A=(A_{ij})_{m\times m}$ satisfies that $A_{ij}\in
C^{r}(U),i,j=1,\cdots,m$. From the definition of $\bar{Z}_{j}$ we know that
$A(p)=\text{Id}_{m\times m}$. Then by shrinking $U$ to $V\subseteq U$ if
necessary, we can ensure that $A(q)\in GL(m,\mathbb{R})$ for each $q\in V$.
Consider the vector fields $\hat{g}_{1},\cdots,\hat{g}_{m}$ on $V$ defined by

$(\hat{g}_{1}\cdots\hat{g}_{m})=(g_{1}\cdots g_{m})A^{-1}.$ (32)

It follows from (31) that $\pi(\hat{g}_{j})=\bar{Z}_{j},j=1,\cdots,m$, which
yield that

$\pi([X,\hat{g}_{j}])=\nabla_{X}\pi(\hat{g}_{j})=\nabla_{X}\bar{Z}_{j}=0,j=1,\cdots,m,$

for any $X\in\Gamma^{r}(V;D)$. That is,

$[\hat{g}_{j},\Gamma^{r}(V;D)]\subseteq\Gamma^{r}(V;D),j=1,\cdots,m.$ (33)

Hence (4b) follows immediately from (32) and (33), where $\beta=A^{-1}$.

It follows from (14b) that

$\nabla_{X}\pi(f)\in\text{span}_{C^{r}(U)}\\{\pi(g_{1}),\cdots,\pi(g_{m})\\}\subseteq\text{span}_{C^{r}(U)}\\{\bar{Z}_{1},\cdots,\bar{Z}_{m}\\},$

for any $X\in\Gamma^{r}(U;D)$. Hence there exist $\alpha_{ij}\in C^{r}(U)$,
such that

$\nabla_{\frac{\partial}{\partial
x_{i}}}\pi(f)=\sum_{j=1}^{m}\alpha_{ij}\bar{Z}_{j},$ (34)

for $i=1,\cdots,k,j=1,\cdots,m$. We claim that there exists
$\bar{g}=\sum_{j=1}^{m}\beta_{j}\bar{Z}_{j}$, such that

$\nabla_{\frac{\partial}{\partial
x_{i}}}\bar{g}=\sum_{j=1}^{m}\alpha_{ij}\bar{Z}_{j},i=1,\cdots,k.$ (35)

Let

$\displaystyle\beta_{j}$ $\displaystyle=$
$\displaystyle\int_{0}^{x_{k}}\alpha_{kj}(x_{1},\cdots,x_{k-1},\tau,x_{k+1},\cdots,x_{n})d\tau+\int_{0}^{x_{k-1}}\alpha_{(k-1)j}(x_{1},\cdots,x_{k-2},$
(36)
$\displaystyle\tau,0,x_{k+1},\cdots,x_{n})d\tau+\cdots+\int_{0}^{x_{1}}\alpha_{1j}(\tau,0,\cdots,0,x_{k+1},\cdots,x_{n})d\tau,$

for $j=1,\cdots,m$.

Since $\nabla$ satisfies that $\nabla_{\frac{\partial}{\partial
x_{i}}}\nabla_{\frac{\partial}{\partial
x_{j}}}\pi(f)-\nabla_{\frac{\partial}{\partial
x_{j}}}\nabla_{\frac{\partial}{\partial
x_{i}}}\pi(f)-\nabla_{[\frac{\partial}{\partial
x_{i}},\frac{\partial}{\partial x_{j}}]}\pi(f)=0$, we have

$\sum_{l=1}^{m}\frac{\partial\alpha_{il}}{\partial
x_{j}}\bar{Z}_{l}=\sum_{l=1}^{m}\frac{\partial\alpha_{jl}}{\partial
x_{i}}\bar{Z}_{l},i,j=1,\cdots,k.$ (37)

Hence for $i=1,\cdots,k$,

$\displaystyle\nabla_{\frac{\partial}{\partial
x_{i}}}\bar{g}=\nabla_{\frac{\partial}{\partial
x_{i}}}\sum_{j=1}^{m}\beta_{j}\bar{Z}_{j}=\sum_{j=1}^{m}{\frac{\partial}{\partial
x_{i}}}(\beta_{j})\bar{Z}_{j}$ (38) $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}(\int_{0}^{x_{k}}{\frac{\partial}{\partial
x_{i}}}\alpha_{kj}(x_{1},\cdots,x_{k-1},\tau,x_{k+1},\cdots,x_{n})d\tau+\int_{0}^{x_{k-1}}{\frac{\partial}{\partial{x_{i}}}}\alpha_{(k-1)j}$
$\displaystyle(x_{1},\cdots,x_{k-2},\tau,0,x_{k+1},\cdots,x_{n})d\tau+\cdots+$
$\displaystyle\alpha_{ij}(x_{1},\cdots,x_{i},0,\cdots,0,,x_{k+1},x_{n}))\bar{Z}_{j}.$

From (20) we know that in the local chart $(U,\phi)$, the section
$\bar{Z}_{j}$ only depends on $x_{k+1},\cdots,x_{n}$, for $j=1,\cdots,m$. It
follows that

$\displaystyle(35)$ $\displaystyle=$
$\displaystyle\int_{0}^{x_{k}}\sum_{j=1}^{m}{\frac{\partial\alpha_{kj}}{\partial
x_{i}}}\bar{Z}_{j}(x_{1},\cdots,x_{k-1},\tau,x_{k+1},\cdots,x_{n})d\tau+\int_{0}^{x_{k-1}}\sum_{j=1}^{m}{\frac{\partial\alpha_{(k-1)j}}{\partial
x_{i}}}\bar{Z}_{j}$ (39)
$\displaystyle(x_{1},\cdots,x_{k-2},\tau,0,x_{k+1},\cdots,x_{n})d\tau+\cdots+\sum_{j=1}^{m}\alpha_{ij}(x_{1},\cdots,x_{i},0,\cdots,0,,x_{k+1},x_{n})\bar{Z}_{j}$
$\displaystyle=$
$\displaystyle\int_{0}^{x_{k}}\sum_{j=1}^{m}{\frac{\partial\alpha_{ij}}{\partial
x_{k}}}\bar{Z}_{j}(x_{1},\cdots,x_{k-1},\tau,x_{k+1},\cdots,x_{n})d\tau+\int_{0}^{x_{k-1}}\sum_{j=1}^{m}{\frac{\partial\alpha_{ij}}{\partial
x_{k-1}}}\bar{Z}_{j}$
$\displaystyle(x_{1},\cdots,x_{k-2},\tau,0,x_{k+1},\cdots,x_{n})d\tau+\cdots+\sum_{j=1}^{m}\alpha_{ij}(x_{1},\cdots,x_{i},0,\cdots,0,,x_{k+1},x_{n})\bar{Z}_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{m}(\int_{0}^{x_{k}}{\frac{\partial}{\partial\tau}}\alpha_{ij}(x_{1},\cdots,x_{k-1},\tau,x_{k+1},\cdots,x_{n})d\tau+\int_{0}^{x_{k-1}}{\frac{\partial}{\partial\tau}}\alpha_{ij}$
$\displaystyle(x_{1},\cdots,x_{k-2},\tau,0,x_{k+1},\cdots,x_{n})d\tau+\cdots+\alpha_{ij}(x_{1},\cdots,x_{i},0,\cdots,0,,x_{k+1},x_{n}))\bar{Z}_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{m}(\alpha_{ij}(x_{1},\cdots,x_{k-1},x_{k},x_{k+1},\cdots,x_{n})-\alpha_{ij}(x_{1},\cdots,x_{k-1},0,x_{k+1},\cdots,x_{n})$
$\displaystyle+$
$\displaystyle\alpha_{ij}(x_{1},\cdots,x_{k-1},0,x_{k+1},\cdots,x_{n})-\alpha_{ij}(x_{1},\cdots,x_{k-2},0,0,x_{k+1},\cdots,x_{n})+$
$\displaystyle\cdots+\alpha_{ij}(x_{1},\cdots,x_{i},0,\cdots,0,,x_{k+1},x_{n}))\bar{Z}_{j}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\alpha_{ij}(x_{1},\cdots,x_{k-1},x_{k},x_{k+1},\cdots,x_{n})\bar{Z}_{j},$

where the second equality follows from (37). This completes the proof for the
above claim. From (31) we know that, by shrinking $U$ if necessary, there
exist $\hat{\alpha}_{1},\cdots,\hat{\alpha}_{m}\in C^{r}(U)$, such that
$\bar{g}=\sum_{j=1}^{m}\hat{\alpha}_{j}\pi(g_{j})$.

Consider the vector field $\hat{f}=f-\sum_{j=1}^{m}\hat{\alpha}_{j}g_{j}$. It
follows from 1 and 2 in Proposition 2 that

$\nabla_{X}(\pi(\hat{f}))=\nabla_{X}(\pi(f)-\bar{g})=0,$ (40)

for any $X\in\Gamma^{r}(U;D)$. Hence $\hat{f}$ satisfies that

$[\hat{f},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D).$

Hence (4a) follows immediately, where
$\alpha=(-\hat{\alpha}_{1}\cdots-\hat{\alpha}_{m})$. This completes the proof
for Theorem 3.1. ∎

###### Remark 1

In Cheng  Cheng and Tarn give the following sufficient and necessary
conditions for local controlled invariance of the smooth involutive regular
distribution $D$ for the smooth affine control system $\Sigma$:

$\displaystyle[f,D]/D\subseteq\text{span}_{C^{\infty}(U)}\\{g_{j}/D|j=1,\cdots,m\\},$
(41a)
$\displaystyle[g_{i},D]/D\subseteq\text{span}_{C^{\infty}(U)}\\{g_{j}/D|j=1,\cdots,m\\},i=1,\cdots,m.$
(41b)

(41a) and (41b) are equivalent with our result (14a) and (14b). However, the
proof provided by Cheng  is not a geometric one. Here we provide a
differential geometric proof for this result.

###### Corollary 1

Consider the $C^{r}$-affine control system (1) on $M$. Let $D$ be an
involutive regular $C^{r}$-distribution of $M$. Let $p$ be a regular point for
the distribution $G$. Then $D$ is locally controlled invariant for $\Sigma$ at
$p$, if and only if $D$ is locally controlled invariant for $\Delta$ at $p$,
where $\Delta$ is the $C^{r}$-generalized affine distribution defined by
$\Delta(p)=f(p)+\text{span}_{\mathbb{R}}\\{g_{1}(p),\cdots,g_{m}(p)\\}$, for
each $p\in M$, if and only if there exists a sufficiently small neighborhood
$U$ of $p$, such that

$[\Gamma^{r}(U;\Delta),\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\Gamma^{r}(U;G).$
(42)

###### Proof

Since $p$ is a regular point for the distribution $G$, if $U$ is a
sufficiently small neighborhood of $p$,
$\Gamma^{r}(U;G)=\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\}$. Assume (42)
holds, then we have

$[g_{i},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\},$

and

$[f,\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\text{span}_{C^{r}(U)}\\{g_{1},\cdots,g_{m}\\}.$

It follows from Theorem 3.1 that $D$ is locally controlled invariant for
$\Sigma$ at $p$. If $D$ is locally controlled invariant for $\Sigma$ at $p$,
it is obvious that $D$ is locally controlled invariant for $\Delta$ at $p$.
Assume that $D$ is locally controlled invariant for $\Delta$ at $p$, then
there exist generators $\hat{g}_{i}\in\Gamma^{r}(U;G),i\in I$ of $G$, such
that $[\hat{g}_{i},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)$. Since $p$ is a
regular point, let $U$ be sufficiently small such that
$\Gamma^{r}(U;G)=\text{span}_{C^{r}(U)}\\{\hat{g}_{i_{1}},\cdots,\hat{g}_{i_{k}}\\}$,
where $i_{1},\cdots,i_{k}\in I$. Then (42) follows immediately by direct
calculation. This completes the proof.∎

###### Corollary 2

Consider the $C^{r}$-affine control system (1) on $M$. Let $D$ be an
involutive regular $C^{r}$-distribution of $M$. Let $p$ be a regular point for
the generalized subbundle $(G+D)/D$. Then $D$ is locally controlled invariant
for $\Sigma$ at $p$, if and only if there exists a sufficiently small
neighborhood $U$ of $p$, such that

$\displaystyle[g_{i},\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\Gamma^{r}(U;G),i=1,\cdots,m,$
(43a)
$\displaystyle[f,\Gamma^{r}(U;D)]\subseteq\Gamma^{r}(U;D)+\Gamma^{r}(U;G).$
(43b)

###### Proof

To prove the ”if” part, since $p$ is a regular point for the generalized
subbundle $(G+D)/D$, we have

$\Gamma^{r}(U;\pi(G))=\text{span}_{C^{r}(U)}\\{\pi(g_{1}),\cdots,\pi(g_{m})\\},$
(44)

if $U$ is sufficiently small. It follows from (43b) that
$\pi([f,X])\in\Gamma^{r}(U;\pi(G))=\text{span}_{C^{r}(U)}\\{\pi(g_{1}),\cdots,\pi(g_{m})\\}$,
for any $X\in\Gamma^{r}(U;D)$. Hence (14b) holds. Similar arguments can be
applied to $g_{i}$. Hence (14a) holds. Then the ”if” part follows from Theorem
3.1. The proof for the ”only if” part is obvious.∎

###### Remark 2

As a classical result on controlled invariance, Corollary 2 is presented in
the classical books schaft ; Isidory  on geometric control theory. It also
appears as Lemma 1.5 in 1 . As mentioned in Section 1, the geometric
interpretation for Corollary 2 provided by Lemma 3.1 in 1  is not a complete
one. The detailed reason is given as follows: compared with our definition of
sections $\bar{Z}_{i}$ on the open subset $U$ of $M$ given by (17), (18) and
(19), the parallel vectors $\bar{Z}_{i}$ given in Lemma 3.1 in 1  are defined
only on each leaf of the foliation, not on any open subset of $M$. As a
consequence, consider the local chart $\phi:U\rightarrow\mathbb{R}^{n}$ such
that $D=\text{span}_{\mathbb{R}}\\{\frac{\partial}{\partial
x_{1}},\cdots,\frac{\partial}{\partial x_{k}}\\}$, and the projection map
$\psi(x_{1},\cdots,x_{n})=(x_{k+1},\cdots,x_{n})$. Then Lemma 3.1 is
equivalent with the following statement: $T_{x}\psi(G)=T_{y}\psi(G)$ if
$\psi(x)=\psi(y)$. But neither the smoothness of the distribution $T\psi(G)$
of the orbit space $\psi(U)$, nor the existence of $C^{r}$-feedback functions
$(\alpha,\beta)$ and the corresponding vector fields on any open subset of $M$
satisfying (4a) and (4b), are studied in Lemma 3.1. In fact, Lemma 3.1 is
weaker than Lemma 1.5 in 1 .

In the following we present a direct geometric proof for Corollary 2 as well
as Lemma 1.5 in 1 , which improves the proof given in 1  and is simpler than
the proof for the general case presented by Proposition 3.

###### Proof

Consider the curve $c:[0,1]\rightarrow U$ with $c(0)$ given by (19), $c(1)=q$
and $\dot{c}(t)\in D_{c(t)}$. Let
$\sigma_{i}(t)=\sum_{j=1}^{m}{f}_{ij}(c(t))\pi({g}_{j}(c(t)))$,
$i=1,\cdots,m$, where $f_{ij}\in C^{r}(U)$ satisfy that
$f_{ii}(0,0,\cdots,q_{k+1},\cdots,q_{n})=1$,
$f_{ij}(0,\cdots,0,q_{k+1},\cdots,q_{n})=0$, for $j\neq i$.

According to (43a) and (44), there exist $\gamma_{ij}\in C^{r}(U)$,
$j=1,\cdots,m$, such that

$\nabla_{\dot{c}(t)}\pi(g_{i}(c(t))=\sum_{j=1}^{m}\gamma_{ij}(c(t))\pi(g_{j}(c(t)).$

On the other hand, it follows from (43a) and (44) that
$\nabla_{\frac{\partial}{\partial
x^{l}}}\pi({g}_{j})=\sum_{r=1}^{m}\Gamma^{r}_{lj}\pi(g_{r}),$ where
$\Gamma^{k}_{lj}\in C^{r}(U),l=1,\cdots,k,j=1,\cdots,m$. Then we have

$\displaystyle\nabla_{\dot{c}(t)}\sigma_{i}(t)=\nabla_{\dot{c}(t)}\sum_{j=1}^{m}{f}_{ij}(c(t))\pi({g}_{j}(c(t)))=\sum_{j=1}^{m}\nabla_{\dot{c}(t)}{f}_{ij}(c(t))\pi({g}_{j}(c(t)))$
(45) $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\dot{f}_{ij}(c(t))\pi({g}_{j}(c(t)))+{f}_{ij}(c(t))\nabla_{\sum_{l=1}^{k}c^{l}(t)\frac{\partial}{\partial
x^{l}}}\pi({g}_{j}(c(t)))$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\dot{f}_{ij}(c(t))\pi({g}_{j}(c(t)))+{f}_{ij}(c(t))\sum_{l=1}^{k}c^{l}(t)\sum_{r=1}^{m}\Gamma_{lj}^{r}(c(t)))\pi(g_{r}(c(t)))$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{m}(\dot{f}_{ij}(c(t))+\sum_{l=1}^{k}\sum_{r=1}^{m}f_{ir}(c(t))c^{l}(t)\Gamma^{j}_{lr}(c(t)))\pi({g}_{j}(c(t))).$

Let
$\dot{f}_{ij}(c(t))+\sum_{l=1}^{k}\sum_{r=1}^{m}f_{ir}(c(t))c^{l}(t)\Gamma^{j}_{lr}(c(t))=0$,
for $j=1,\cdots,m$. Then we get $m$ systems of linear equations

$\dot{f}_{i}=A_{i}f_{i},i=1,\cdots,m,$

with $f_{i}=(f_{i1},\cdots,f_{im})^{T},A_{i}=(a_{i}^{jl})_{m\times m}$ where
$a_{i}^{jl}(t)=\sum_{r=1}^{k}c^{r}(t)\Gamma^{j}_{rl}(c(t))$. We know there
exist unique solutions $\hat{f}_{i}(t)$ defined on $[0,1]$ with prescribed
initial value $\hat{f}_{ii}(0)=1,\hat{f}_{ij}(0)=0$, for $j\neq
i,i=1,\cdots,m$.

Let $\hat{\sigma}_{i}(t)=\sum_{j=1}^{m}\hat{f}_{ij}(c(t))\pi({g}_{j}(c(t)))$.
We have $\nabla_{\dot{c}(t)}\hat{\sigma}_{i}(t)=0$. It follows from uniqueness
of the solution of the equation $\nabla_{\dot{c}(t)}\sigma_{i}(t)=0$ with
prescribed initial value that $\hat{\sigma}_{i}(t)$ is the parallel
translation along $c(t)$ with initial value defined by (18). Then according to
our previous definition of $\bar{Z}_{i}$, we have

$\bar{Z}_{i}(q)=\hat{\sigma}_{i}(1)=\sum_{j=1}^{m}\hat{f}_{ij}(c(1))\pi({g}_{j}(c(1)))\in\pi(G)_{q},$
(46)

for $i=1,\cdots,m$. Since $p$ is a regular point for $(G+D)/D$, by shrinking
$U$ if necessary we know there exist functions $\xi_{ij}\in C^{r}(U)$, such
that $\bar{Z}_{i}=\sum_{j=1}^{m}\xi_{ij}\pi(g_{j}),i,j=1,\cdots,m$. The
remaining proof follows the proof for Theorem 3.1. ∎

###### Corollary 3

Let $M$ be a $C^{\omega}$-manifold. Let $\Delta$ be a $C^{\omega}$-generalized
affine distribution of $M$ with associated generalized distribution
$G=L(\Delta)$. Let $D$ be an involutive regular $C^{\omega}$-distribution of
$M$. Then $D$ is locally controlled invariant for $\Delta$ at $p$, if there
exists a sufficiently small neighborhood $U$ of $p$, such that

$[\Gamma^{\omega}(U;\Delta),\Gamma^{\omega}(U;D)]\subseteq\Gamma^{\omega}(U;D)+\Gamma^{\omega}(U;G).$
(47)

###### Proof

Since $G$ is of class $C^{\omega}$, if $U$ is sufficiently small, then it
follows from Proposition 1 that there exist local generators
$g_{1},\cdots,g_{m}\in\Gamma^{\omega}(U;G)$ of $G$ such that

$\Gamma^{\omega}(U;G)=\text{span}_{C^{\omega}(U)}\\{g_{1},\cdots,g_{m}\\}.$
(48)

Then we have

$[g_{i},\Gamma^{\omega}(U;D)]\subseteq\Gamma^{\omega}(U;D)+\text{span}_{C^{\omega}(U)}\\{g_{1},\cdots,g_{m}\\},$

and

$[f,\Gamma^{\omega}(U;D)]\subseteq\Gamma^{\omega}(U;D)+\text{span}_{C^{\omega}(U)}\\{g_{1},\cdots,g_{m}\\}.$

Then the result follows immediately from Theorem 3.1.∎

##  4 Global controlled invariance

###  4.1 General cases

Let $D$ be an involutive regular $C^{r}$-distribution of $M$. Consider the
foliation $\mathfrak{F}$ defined by $D$. As claimed in 1 , $M/\mathfrak{F}$ is
assumed to be a $C^{r}$-manifold in this subsection.

###### Theorem 4.1

Let $M$ be a smooth manifold and $D$ be a smooth involutive regular
distribution of $M$ such that $M/\mathfrak{F}$ is a smooth manifold, where
$\mathfrak{F}$ is the foliation defined by $D$. Let $\Delta$ be a smooth
generalized affine distribution of $M$ with associated smooth generalized
distribution $L(\Delta)=G$. Assume that $(G+D)/D$ is a regular subbundle of
$TM/D$. Assume also that $G\cap D$ is a smooth generalized distribution of
$M$. Then $D$ is globally controlled invariant for $\Delta$ if and only if it
is locally controlled invariant for $\Delta$ at each point $p\in M$.

###### Proof

The ”only if ” part follows from the arguments given in Section 1. We only
need to prove the ”if” part. Consider the map $\Phi:M\rightarrow
M/\mathfrak{F}$. Since $D$ is locally controlled invariant for $\Delta$, it
follows that for each $p\in M$, there exists a local neighborhood $U$ of $p$,
such that $T\Phi(G|U)$ is a well-defined smooth generalized distribution of
$\Phi(U)$. Further, since $\Phi^{-1}(q)$ is a leaf of $\mathfrak{F}$, it is
connected, which yields that $T\Phi(G)$ is a well-defined smooth generalized
distribution of $M/\mathfrak{F}$. Then it follows from Theorem 2.1 that there
exist smooth vector fields $Y_{1},\cdots,Y_{l}$ on $M/\mathfrak{F}$ such that

$(T\Phi(G))_{\bar{p}}=\text{span}_{\mathbb{R}}\\{Y_{1}(\bar{p}),\cdots,Y_{l}(\bar{p})\\},$
(49)

for each $\bar{p}\in M/\mathfrak{F}$.

Given $Y\in\Gamma^{\infty}(T\Phi(G))$, we claim that there exists
$X\in\Gamma^{\infty}(G)$ such that $T\Phi(X)=Y$. Since $(G+D)/D$ is regular,
it follows from (44) in Corollary 2 that for each $p_{\alpha}\in M$, there
exist a local neighborhood $U_{\alpha}$ of $p$, and
$X_{\alpha}\in\Gamma^{\infty}(U_{\alpha};G)$ such that
$T\Phi(X_{\alpha})=Y|\Phi(U_{\alpha})$. Then we get an open covering
$\\{U_{\alpha}\\}$ of $M$. Since $M$ is a smooth, paracompact, Hausdorff
manifold, let $\\{(\mathcal{W}_{i},\lambda_{i})\\}$ be a partition of unity
Abraham  subordinated to the open covering $\\{U_{\alpha}\\}$ such that each
open set $\mathcal{W}_{i}$ is a subset of some open set
$\mathcal{U}_{\alpha(i)}$. Let $X$ be defined by

$X(p)=\sum_{i}\lambda_{i}X_{\alpha(i)}(p)$

a finite sum at each $p\in M$. Then $X\in\Gamma^{\infty}(G)$ satisfies that

$T_{p}\Phi(X)=\sum_{i}\lambda_{i}(p)T_{p}\Phi(X_{\alpha(i)}(p)))=\sum_{i}\lambda_{i}(p)Y(\Phi(p))=Y(\Phi(p)),$

for each $p\in M$. This completes the proof of the claim.

Hence we get $X_{1},\cdots,X_{l}\in\Gamma^{\infty}(G)$ such that
$T\Phi(X_{i})=Y_{i},i=1,\cdots,l$. On the other hand, since $G\cap D$ is a
smooth generalized distribution of $M$, it follows from Theorem 2.1 that there
exist global generators $X_{l+1},\cdots,X_{r}\in\Gamma^{\infty}(G\cap D)$ for
$G\cap D$. We claim that $X_{1},\cdots,X_{r}$ are global generators for $G$.
Let $v_{p}\in G_{p}$. From (49) we know that there exist constants
$c_{1},\cdots,c_{l}$ such that
$T_{p}\Phi(v_{p})=\sum_{i=1}^{l}c_{i}Y_{i}(\Phi(p))$. Then we have

$T\Phi(v_{p}-\sum_{i=1}^{l}c_{i}X_{i}(p))=T\Phi(v_{p})-\sum_{i=1}^{l}c_{i}Y_{i}(\Phi(p))=0.$

This yields that $v_{p}-\sum_{i=1}^{l}c_{i}X_{i}(p)\in D_{p}\cap G_{p}$. Then
there exist constants $c_{l+1},\cdots,c_{r}$ such that

$v_{p}-\sum_{i=1}^{l}c_{i}X_{i}(p)=\sum_{i=l+1}^{r}c_{i}X_{i}(p).$

Hence $v_{p}=\sum_{i=1}^{r}c_{i}X_{i}(p)$. That is, $X_{1},\cdots,X_{r}$ are
global generators for $G$. It is obvious that
$[X_{i},\Gamma^{\infty}(D)]\subseteq\Gamma^{\infty}(D)$, for $i=1,\cdots,r$.

It now remains to prove the existence of $f\in\Gamma^{\infty}(\Delta)$ such
that $[f,\Gamma^{\infty}(D)]\subseteq\Gamma^{\infty}(D)$. By using similar
arguments as above, we know that the affine distribution $T\Phi(\Delta)$ of
$M/\mathfrak{F}$ is well-defined and smooth. By using partition of unity we
get a smooth vector field $g$ on $M/\mathfrak{F}$ such that $g(\bar{p})\in
T\Phi(\Delta)_{\bar{p}}$ for each $\bar{p}\in M/\mathfrak{F}$. Since for each
$p\in M$, there exist a neighborhood $U$ of $p$ and
$f_{p}\in\Gamma^{\infty}(U;\Delta)$ such that
$T\Phi(f_{p})-g|U\in\Gamma^{\infty}(\Phi(U);T\Phi(G))$, where
$T\Phi(f_{p})\in\Gamma^{\infty}(\Phi(U);T\Phi(\Delta))$. Then it follows from
(44) in Corollary 2 that, by shrinking $U$ if necessary, there exists
$\hat{f}_{p}\in\Gamma^{\infty}(U;\Delta)$ such that
$T\Phi(\hat{f}_{p})=g|\Phi(U)$. By using partition of unity as above, we then
get a smooth vector field $\hat{f}=\sum_{i}\lambda_{i}\hat{f}_{p(i)}$ on $M$
such that $T\Phi(\hat{f})=g$. This yields that
$[\hat{f},\Gamma^{\infty}(D)]\subseteq\Gamma^{\infty}(D)$. We only need to
show that $\hat{f}(p)\in\Delta_{p}$ for each $p\in M$. Actually We have a
finite sum at $p$:

$\displaystyle\hat{f}(p)$
$\displaystyle=\sum_{i}\lambda_{i}(p)\hat{f}_{p(i)}(p)=\sum_{i}\lambda_{i}(p)(v_{0}+v_{p_{i}})$
$\displaystyle=(\sum_{i}\lambda_{i}(p))v_{0}+\sum_{i}\lambda_{i}(p)v_{p_{i}}=v_{0}+v\in\Delta_{p},$

where $v_{0}\in\Delta_{p},v_{p_{i}},v\in G_{p}$. This completes the proof.∎

###### Remark 3

It follows from the above proof that there exist finitely many global
generators $X_{1},\cdots,X_{r}\in\Gamma^{\infty}(G)$ of $G$, such that
$[X_{i},\Gamma^{\infty}(D)]\subseteq\Gamma^{\infty}(D)$, for $i=1,\cdots,r$.
That is, in Theorem 4.1 the index set $I$ given in Definition 4 can be finite.

###### Corollary 4

Let $M$ be a smooth manifold and $D$ be a smooth involutive regular
distribution of $M$ such that $M/\mathfrak{F}$ is a smooth manifold, where
$\mathfrak{F}$ is the foliation defined by $D$. Let $\Delta$ be a smooth
affine distribution of $M$ with associated smooth distribution $L(\Delta)=G$.
Assume that both $G$ and $G\cap D$ have constant rank on $M$. Then $D$ is
globally controlled invariant for $\Delta$ if and only if

$[\Gamma^{\infty}(\Delta),\Gamma^{\infty}(D)]\subseteq\Gamma^{\infty}(D+G).$
(50)

###### Proof

Since both $G$ and $G\cap D$ have constant rank on $M$, it follows that
$(G+D)/D$ has constant rank and $G\cap D$ is smooth. Then according to Theorem
4.1, $D$ is globally controlled invariant for $\Delta$ if and only if $D$ is
locally controlled invariant for $\Delta$. Since $G$ has constant rank, it
follows from Corollary 1 that $D$ is locally controlled invariant for $\Delta$
if and only if for each $p\in M$ there exists a local neighborhood $U$ of $p$,
such that (42) holds. Hence under the above assumptions, it suffices to show
the equivalence between (42) at every point in $M$ and (50).

Assume that (50) holds. Since $G$ and $D$ have constant rank, it follows that
for each $p\in U$, there exist $g_{1},\cdots,g_{m}$ and
$d_{1},\cdots,d_{k}\in\Gamma^{\infty}(U;D)$ which span the module
$\Gamma^{\infty}(U;G)$ and $\Gamma^{\infty}(U;D)$ over $C^{\infty}(U)$
respectively. Besides, $g_{1}|V,\cdots,g_{m}|V$ and
$d_{1}|V,\cdots,d_{k}|V\in\Gamma^{\infty}(V;D)$ span the module
$\Gamma^{\infty}(V;G)$ and $\Gamma^{\infty}(V;D)$ over $C^{\infty}(V)$
respectively, where $V$ is any open subset of $U$. Then by using partition of
unity we get that, for each $p\in M$, there exist generators for the module of
local sections of $G$ and $D$ on a sufficiently small neighborhood $W$ of $p$
that are the restrictions of global sections of $G$ and $D$ to $W$
respectively. Then by restrictions of globally defined sections in (50) to
$W$, we have

$[\Gamma^{\infty}(W;G),\Gamma^{\infty}(W;D)]\subseteq\Gamma^{\infty}(W;D+G).$

Since $G,D$ and $G\cap D$ have constant rank, it follows that
$\Gamma^{\infty}(W;D+G)=\Gamma^{\infty}(W;D)+\Gamma^{\infty}(W;G)$ if $W$ is
sufficiently small. Hence we have

$[\Gamma^{\infty}(W;G),\Gamma^{\infty}(W;D)]\subseteq\Gamma^{\infty}(W;D)+\Gamma^{\infty}(W;G).$

Similarly we can prove that
$[f,\Gamma^{\infty}(W;D)]\subseteq\Gamma^{\infty}(W;D)+\Gamma^{\infty}(W;G).$
Hence (42) holds.

Assume (42) holds at each point in $M$, it is obvious that (50) holds. This
completes the proof.∎

In what follows we will investigate the problem of global controlled
invariance for real analytic generalized affine distributions, We have the
following results where, instead of partition of unity used in the smooth
case, Cartan’s Theorem A and B play a central role.

###### Theorem 4.2

Let $M$ be a $C^{\omega}$-manifold and $D$ be an involutive regular
$C^{\omega}$-distribution of $M$ such that $M/\mathfrak{F}$ is a
$C^{\omega}$-manifold, where $\mathfrak{F}$ is the foliation defined by $D$.
Let $\Delta$ be a $C^{\omega}$-generalized affine distribution of $M$ with
associated $C^{\omega}$-generalized distribution $L(\Delta)=G$. Assume that
$(G+D)/D$ is a regular subbundle of $TM/D$ and $G\cap D$ is a
$C^{\omega}$-generalized distribution of $M$. Then $D$ is globally controlled
invariant for $\Delta$ if and only if it is locally controlled invariant for
$\Delta$ at each point $p\in M$.

We will first introduce some basic elements of sheaf theory and Cartan’s
Theorem A and B for coherent real analytic sheaf that we will use in the proof
of Theorem 4.2.

###### Definition 9

A presheaf of sets over $M$ is an assignment to each open set $U\subseteq M$ a
set $F(U)$ and, to each pair of open sets $V,U\subseteq M$ with $V\subseteq
U$, a map $r_{U,V}:F(U)\rightarrow F(V)$ called the restriction map, with
these assignments having the following properties:
1\. $r_{U,U}$ is the identity map.
2\. If $W,V,U\subseteq M$ are open with $W\subseteq V\subseteq U$, then
$r_{U,W}=r_{V,W}\circ r_{U,V}$.

A presheaf of rings over $M$ is a presheaf
$\mathscr{R}=(\mathscr{R}(U))_{U\text{open}}$ whose local sections are rings
and for which the restriction maps
$r_{U,V}:\mathscr{R}(U)\rightarrow\mathscr{R}(V)$, $U,V\subseteq M$ open,
$V\subseteq U$ are homomorphisms of rings.

Let $\mathscr{R}=(\mathscr{R}(U))_{U\text{open}}$ be a presheaf of rings over
$M$, a presheaf of $\mathscr{R}$-module is a presheaf
$\mathscr{F}=(\mathscr{F}(U))_{U\text{open}}$ of sets such that
$\mathscr{F}(U)$ is a module over $\mathscr{R}(U)$ and such that the
restriction maps $r_{U,V}^{\mathscr{R}}$ and $r_{U,V}^{\mathscr{F}}$ satisfy

$\displaystyle
r_{U,V}^{\mathscr{F}}(s+t)=r_{U,V}^{\mathscr{F}}(s)+r_{U,V}^{\mathscr{F}}(t),s,t\in\mathscr{F}(U),$
$\displaystyle
r_{U,V}^{\mathscr{F}}(fs)=r_{U,V}^{\mathscr{R}}(f)r_{U,V}^{\mathscr{F}}(s),f\in\mathscr{R}(U),s\in\mathscr{F}(U).$

###### Definition 10

A presheaf $\mathscr{F}=(\mathscr{F}(U))_{U\text{open}}$ is a sheaf if
1\. The presheaf $\mathscr{F}$ is separated: if $U\subseteq M$ is open, if
$(U_{a})_{a\in A}$ is an open covering of $U$, and if $s,t\in\mathscr{F}(U)$
satisfy $r_{U,U_{a}}(s)=r_{U,U_{a}}(t)$ for every $a\in A$. then $s=t$.
2\. The presheaf $\mathscr{F}$ has the gluing property when, if $U\subseteq M$
is open, if $(U_{a})_{a\in A}$ is an open covering of $U$, and if, for each
$a\in A$, there exists $s_{a}\in\mathscr{F}(U_{a})$ with the family
$(s_{a})_{a\in A}$ satisfying

$r_{U_{a_{1}},U_{a_{1}}\cap U_{a_{2}}}(s_{a_{1}})=r_{U_{a_{2}},U_{a_{1}}\cap
U_{a_{2}}}(s_{a_{2}}),$

for each $a_{1},a_{2}\in A$, then there exists $s\in\mathscr{F}(U)$ such that
$s_{a}=r_{U,U_{a}}(s)$ for each $a\in A$.

Let $\mathscr{F}$ be a presheaf of sets on $M$. Let $p\in M$ and let
$\mathcal{N}_{p}$ be the collection of open subsets of $M$ containing $p$. We
define an equivalence relation in $(\mathscr{F}(U))_{U\in\mathscr{N}_{p}}$ by
saying that $s_{1}\in\mathscr{F}(U_{1})$ and $s_{2}\in\mathscr{F}(U_{2})$ are
equivalent if there exists $\mathcal{V}\in\mathscr{N}_{p}$ such that
$V\subseteq U_{1},V\subseteq U_{2}$, and
$r_{U_{1},V}(s_{1})=r_{U_{2},V}(s_{2})$. The equivalence class of a section
$s\in\mathscr{F}(U)$ we denote by $[s]_{p}$.

###### Definition 11

Let $\mathscr{F}=(\mathscr{F}(U))_{U\text{open}}$ be a presheaf of sets over
$M$. For $p\in M$, the stalk of $\mathscr{F}$ at $p$ is the set of equivalence
classes under the equivalence relation defined above, and is denoted by
$\mathscr{F}_{p}$. The equivalence class $[s]_{p}$ of a section
$s\in\mathscr{F}(U)$ is called the germ of $s$ at $p$.

Then a natural subsheaf arising from a generalized subbundle is given as
follows.

###### Definition 12

Let $\rho:E\rightarrow M$ be a $C^{r}$-vector bundle. Let $F\subseteq E$ be a
$C^{r}$-generalized subbundle. The sheaf of sections of $F$ is the sheaf
$\mathscr{G}_{F}^{r}$ whose local sections over the open set $U\subseteq M$ is
the set of $C^{r}$-sections of $F|U$.

$\mathscr{G}_{F}^{r}$ is a sheaf of modules over the sheaf of rings
$\mathscr{C}_{M}^{r}$, where $\mathscr{C}_{M}^{r}$ is a sheaf of rings of
$C^{r}$-functions on $M$. The stalk of $\mathscr{G}_{F}^{r}$ at $p\in M$ is a
module over the ring $\mathscr{C}_{p,M}^{r}$.

In the holomorphic case, the big result proved by Cartan cartan  and known as
”Cartan’s Theorem A”, has as a consequence that the module of germs at $p$ of
sections of a holomorphic vector bundle over a Stein base is generated by
germs of global sections. In Cartan  these holomorphic results are extended to
the real analytic case. However, Cartan’s results extend far beyond sheaves of
sections of vector bundles to the setting of coherent analytic sheaves. We
refer to Lewis  for a detailed discussion on Cartan’s Theorem A and B for
coherent real analytic sheaf, and hence for sheaf of sections of real analytic
generalized distribution.

###### Definition 13

Let $M$ be a real analytic manifold. A coherent real analytic sheaf is a
coherent sheaf $\mathscr{F}$ of $\mathscr{C}_{M}^{\omega}$-modules.

###### Proposition 4

Lewis  Let $M$ be a real analytic manifold. Let $G$ be a real analytic
generalized distribution of $M$. Then $\mathscr{G}_{G}^{\omega}$ is a coherent
real analytic sheaf.

###### Theorem 4.3

Lewis (Cartan’s Theorem A) Let $M$ be a real analytic manifold and let
$\mathscr{F}$ be a coherent real analytic sheaf. Then, for $p\in M$, the
$\mathscr{C}_{p,M}^{\omega}$-module $\mathscr{F}_{p}$ is generated by germs of
global sections of $\mathscr{F}$.

Theorem 2.2 in Section 2 is a consequence of Proposition 4 and Theorem 4.3.

###### Theorem 4.4

Lewis (A consequence of Cartan’s Theorem B) If $M$ is a real analytic
manifold, if $\mathscr{F}$ is a coherent sheaf of
$\mathscr{C}_{M}^{\omega}$-modules, and if $\mathscr{U}=(U_{a})_{a\in A}$ is
an open cover of $M$, then $H^{1}(\mathscr{U};\mathscr{F})=0$, where
$H^{1}(\mathscr{U};\mathscr{F})$ is the first cohomology group of
$\mathscr{F}$ for the cover $\mathscr{U}$.

Proof of Theorem 4.2. We only need to prove the ”if” part. Since $D$ is
locally controlled invariant for $\Delta$, we know that $T\Phi(G)$ is a well-
defined $C^{\omega}$-generalized distribution of $M/\mathfrak{F}$ as claimed
in Theorem 4.1. Given $p\in M$ and $v\in G_{p}$, it follows from Theorem 2.2
that there exists $t\in\Gamma^{\omega}(T\Phi(G))$ such that
$t(\Phi(p))=T_{p}\Phi(v)$. We claim that there exists
$\hat{s}\in\Gamma^{\omega}(G)$ such that $T\Phi(\hat{s})=t$. By using similar
arguments as given in Theorem 4.1, we know that for each $q\in M$, there
exists $X_{q}\in\Gamma^{\omega}(U_{q};G)$ such that
$T\Phi(X_{q})=t|\Phi(U_{q})$, where $U_{q}$ is a local neighborhood of $q$.
Then we get an open covering $\\{U_{q}\\}$ of $M$. If $U_{x}\cap U_{y}\neq
0,x,y\in M$, consider

$X_{xy}=X_{x}|U_{x}\cap U_{y}-X_{y}|U_{x}\cap U_{y}.$

Since $T\Phi(X_{x}|U_{x}\cap U_{y}-X_{y}|U_{x}\cap U_{y})=0$, it follows that

$X_{x}|U_{x}\cap U_{y}-X_{y}|U_{x}\cap U_{y}\in\Gamma^{\omega}(U_{x}\cap
U_{y};D)\cap\Gamma^{\omega}(U_{x}\cap U_{y};G).$

On the other hand, since $\mathscr{G}^{\omega}_{D}$ and
$\mathscr{G}^{\omega}_{G}$ are both coherent real analytic sheaves, it follows
that $\mathscr{G}^{\omega}_{D}\cap\mathscr{G}^{\omega}_{G}$ is a coherent real
analytic sheaf. Then according to Theorem 4.4, we have

$\check{\mathrm{H}}^{1}(\mathscr{U};\mathscr{G}^{\omega}_{D}\cap\mathscr{G}^{\omega}_{G})=0,$

where
$\check{\mathrm{H}}^{1}(\mathscr{U};\mathscr{G}^{\omega}_{D}\cap\mathscr{G}^{\omega}_{G})$
is the first cohomology group of the sheaf
$\mathscr{G}^{\omega}_{D}\cap\mathscr{G}^{\omega}_{G}$ for the cover
$\\{U_{q}\\}$. Then it follows from $(X_{xy})_{x,y\in
M}\in\check{\mathrm{Z}}^{1}(\mathscr{U};\mathscr{G}^{\omega}_{D}\cap\mathscr{G}^{\omega}_{G})$
that there exist $(\hat{X}_{x})_{x\in M}$, where
$\hat{X}_{x}\in(\mathscr{G}^{\omega}_{D}\cap\mathscr{G}^{\omega}_{G})(U_{x})$
such that $X_{xy}=\hat{X}_{y}-\hat{X}_{x}$ for $x,y\in M,U_{x}\cap U_{y}\neq
0$.
Let

$\hat{s}_{x}=X_{x}+\hat{X}_{x},x\in M.$

We have $\hat{s}_{x}|U_{x}\cap U_{y}=\hat{s}_{y}|U_{x}\cap U_{y}$. Hence we
get a $C^{\omega}$-vector field $\hat{s}\in\Gamma^{\omega}(G)$ on $M$ such
that $T\Phi(\hat{s})=t$. This completes the proof of the claim.

Since $T_{p}\Phi(v-\hat{s}(p))=0$, we have $v-\hat{s}(p)\in D_{p}\cap G_{p}$.
Since $G\cap D$ is a $C^{\omega}$-generalized distribution of $M$, it follows
from Theorem 2.2 that there exists a vector field $Y\in\Gamma^{\omega}(G\cap
D)$, such that $Y(p)=v-\hat{s}(p)$. Now consider the vector field
$Y+\hat{s}\in\Gamma^{\omega}(G)$, we have $T\Phi(Y+\hat{s})=t$ which yields
that $[Y+\hat{s},\Gamma^{\omega}(D)]\subseteq\Gamma^{\omega}(D)$. Besides,
$(Y+\hat{s})(p)=v$. Hence (7b) follows immediately since $p\in M$ and $v\in
G_{p}$ are arbitrary.

To prove (7a), first we claim that there exists
$\bar{t}\in\Gamma^{\omega}(T\Phi(\Delta))$. Since for each $\bar{p}\in
M/\mathfrak{F}$, there exist a local neighborhood $W_{\bar{p}}$ of $\bar{p}$
and $t_{\bar{p}}\in\Gamma^{\omega}(W_{\bar{p}};T\Phi(\Delta))$, we get an open
covering $\\{W_{\bar{p}}\\}$ of $M/\mathfrak{F}$. If $W_{\bar{x}}\cap
W_{\bar{y}}\neq 0,\bar{x},\bar{y}\in M/\mathfrak{F}$, consider
$t_{\bar{x}\bar{y}}=t_{\bar{x}}|W_{\bar{x}}\cap
W_{\bar{y}}-t_{\bar{y}}|W_{\bar{x}}\cap W_{\bar{y}}$. It follows that
$t_{\bar{x}\bar{y}}\in\Gamma^{\omega}(W_{\bar{x}}\cap W_{\bar{y}};T\Phi(G))$.
On the other hand, since $\mathscr{G}^{\omega}_{T\Phi(G)}$ is a coherent real
analytic sheaf, according to Theorem 4.4 we have
$\check{\mathrm{H}}^{1}(\mathscr{W};\mathscr{G}^{\omega}_{T\Phi(G)})=0$. Then
it follows from $(t_{\bar{x}\bar{y}})_{\bar{x},\bar{y}\in
M/\mathfrak{F}}\in\check{\mathrm{Z}}^{1}(\mathscr{W};\mathscr{G}^{\omega}_{T\Phi(G)})$
that there exist $(\hat{t}_{\bar{x}})_{\bar{x}\in M/\mathfrak{F}}$, where
$\hat{t}_{\bar{x}}\in\mathscr{G}^{\omega}_{T\Phi(G)}(W_{\bar{x}})$ such that
$t_{\bar{x}\bar{y}}=\hat{t}_{\bar{y}}-\hat{t}_{\bar{x}}$ for
$\bar{x},\bar{y}\in M/\mathfrak{F},W_{\bar{x}}\cap W_{\bar{y}}\neq 0$. Let
$\bar{t}_{\bar{x}}=t_{\bar{x}}+\hat{t}_{\bar{x}},\bar{x}\in M/\mathfrak{F}$.
We have $\bar{t}_{\bar{x}}|W_{\bar{x}}\cap
W_{\bar{y}}=\bar{t}_{\bar{y}}|W_{\bar{x}}\cap W_{\bar{y}}$. Hence we get a
$C^{\omega}$-vector field $\bar{t}\in\Gamma^{\omega}(T\Phi(\Delta))$.

Since $(G+D)/D$ is regular, for each $p\in M$, there exist a neighborhood $U$
of $p$ and $f_{p}\in\Gamma^{\omega}(U;\Delta)$, such that
$T\Phi(f_{p})=\bar{t}|\Phi(U)$. By using similar arguments as given above, we
know there exists $f\in\Gamma^{\omega}(\Delta)$, such that $T\Phi(f)=\bar{t}$,
which yields that $[f,\Gamma^{\omega}(D)]\subseteq\Gamma^{\omega}(D)$. This
completes the proof. ∎

A special case that doesn’t require the condition of paracompact Hausdorff
manifold is the trivial case presented as a main result in 1 , where for each
$p\in M$, $T_{p}\Phi:G(p)\rightarrow T_{p}\Phi(G(p))$ is one-to-one.

###### Proposition 5

1  Consider the driftless control system

$\dot{x}=\sum_{i=1}^{m}g_{i}(x)u_{i},$ (51)

on a $C^{r}$-manifold $M$ (not necessary paracompact and Hausdorff), where the
vector fields $g_{1},\cdots,g_{m}$ are linearly independent on $M$. Let $D$ be
an involutive regular $C^{r}$-distribution of $M$ such that $M/\mathfrak{F}$
is a $C^{r}$-manifold (not necessary paracompact and Hausdorff). Assume that
$G\cap D=0$. Assume also $D$ is locally controlled invariant for (51) at each
$p\in M$. Then $D$ is globally controlled invariant for (51) if and only if
$T\Phi(G)$ is trivial, where $\Phi:M\rightarrow M/\mathfrak{F}$ is the
projection.

However, if there exists a drift term $f$ in (51), then the condition of
Hausdorff and paracompact manifold is still needed. This is because if
$M/\mathfrak{F}$ is not assumed to be Hausdorff and paracompact, then from
conditions given in Proposition 5 we can not ensure the existence of
$\bar{t}\in\Gamma^{r}(T\Phi(\Delta))$, which is necessary for (8a).

The $C^{\infty}$-part of the following theorem on global controlled invariance
for affine control systems is taken from 1 .

###### Theorem 4.5

Consider the affine control system

$\dot{x}=f(x)+\sum_{i=1}^{m}g_{i}(x)u_{i},$ (52)

on a $C^{r}$-manifold $M$, where $f,g_{1},\cdots,g_{m}$ are $C^{r}$-vector
fields on $M$ and $g_{1},\cdots,g_{m}$ are linearly independent. Let $D$ be an
involutive regular $C^{r}$-distribution of $M$ such that $M/\mathfrak{F}$ is a
$C^{r}$-manifold. Assume that $G\cap D=0$. Assume also $D$ is locally
controlled invariant for (52). Then $D$ is globally controlled invariant for
(52) if and only if $T\Phi(G)$ is trivial, where $\Phi:M\rightarrow
M/\mathfrak{F}$ is the projection.

###### Proof

The proof for $r=\infty$ is given in 1 . Let $r=\omega$. Since
$M/\mathfrak{F}$ is a paracompact, Hausdorff $C^{\omega}$ -manifold, it
follows from the Grauert-Morrey embedding theorem Grauert  that there exists
real analytic Riemannian metric on $M/\mathfrak{F}$. Hence the result follows
by replacing $r=\infty$ with $r=\omega$ in the proof of Proposition 5.1 and
Proposition 5.2 in 1 . ∎

###### Remark 4

It should be pointed out that the proof for (8a) in Proposition 5.1 in 1
relies on the assumptions that $G,T\Phi(G)$ are trivial and $G\cap D=0$.

###  4.2 Manifolds with a symmetry Lie group action

Let $P$ be a smooth manifold. Let

$\Phi:G\times P\rightarrow P:(g,p)\rightarrow\Phi_{g}(p)=gp$ (53)

be a proper action of a connected Lie group $G$ on $P$ (see Sniatycki ). The
isotropy group $G_{p}$ of a point $p\in P$ is $G_{p}=\\{g\in G|gp=p\\}$.

###### Definition 14

Sniatycki  A slice through $p\in P$ for an action of $G$ on $P$ is a
submanifold $S_{p}$ of $P$ containing $p$ such that:
1\. $S_{p}$ is transverse and complementary to the orbit $Gp$ of $G$ through
$p$. In other words,

$T_{p}P=T_{p}S_{p}\oplus T_{p}(Gp).$

2\. For every $p^{\prime}\in S_{p}$, the manifold $S_{p}$ is transverse to the
orbit $Gp^{\prime}$; that is,

$T_{p^{\prime}}P=T_{p^{\prime}}S_{p}+T_{p^{\prime}}(Gp^{\prime}).$

3\. $S_{p}$ is $G_{p}$-invariant.
4\. Let $p^{\prime}\in S_{p}$. If $gp^{\prime}\in S_{p}$, then $g\in G_{p}$.

The existence of a slice through $p\in P$ is ensured by the following result.

###### Proposition 6

Sniatycki  There is an open ball $B$ in $\text{hor}T_{p}P$ centred at $0$ such
that $S_{p}=Exp_{p}(B)$ is a slice through $p$ for the action of $G$ on $P$,
where $Exp_{p}(v)$ is the value at 1 of the geodesics of $G$-invariant
Riemannian metric originating from $p$ in the direction $v$. Further, the set
$GS_{p}=\\{gq|g\in G\,\text{and}\,q\in S_{p}\\}$ is a G-invariant open
neighbourhood of p in P.

A differential structure on a topological space $S$ is a family
$C^{\infty}(S)$ of real-valued functions on $S$ satisfying the following
conditions:
1\. The family

$\\{f^{-1}(I)|f\in C^{\infty}(S)\,\text{and}\,I\,\text{is an open interval
in}\,\mathbb{R}\\}$

is a subbasis for the topology of $S$.
2\. If $f_{1},\cdots,f_{n}\in C^{\infty}(S)$ and $F\in
C^{\infty}(\mathbb{R}^{n})$, then $F(f_{1},\cdots,f_{n})\in C^{\infty}(S)$.
3\. If $f:S\rightarrow\mathbb{R}$ is a function such that, for every $x\in S$,
there exist an open neighborhood $U$ of $x$, and a function $f_{x}\in
C^{\infty}(S)$ satisfying

$f_{x}|U=f|U,$

then $f\in C^{\infty}(S)$. Here, the subscript vertical bar $|$ denotes a
restriction. A topological space S endowed with a differential structure is
called a differential space. A differential space $S$ is subcartesian if it is
Hausdorff and every point $x\in S$ has a neighbourhood $U$ diffeomorphic to a
subset $V$ of $\mathbb{R}^{n}$.

For the orbit space $R=P/G$, let

$C^{\infty}(R)=\\{f:R\rightarrow\mathbb{R}|\rho^{*}f\in C^{\infty}(P)\\},$
(54)

where $\rho:P\rightarrow R$ is the canonical projection (the orbit map). We
have

###### Proposition 7

Sniatycki  The topology of $R$ induced by the differential structure
$C^{\infty}(R)$ coincides with the quotient topology. Further, $R$ is a
subcartesian space.

For locally compact, second countable Hausdorff differential spaces, there
exists a partition of unity.

###### Definition 15

A countable partition of unity on a differential space $S$ is a countable
family of functions $\\{f_{i}\\}\in C^{\infty}(S)$ such that:
(a) The collection of their supports is locally finite.
(b) $f_{i}(x)\geq 0$ for each $i$ and each $x\in S$.
(c) $\sum_{i=1}^{\infty}f_{i}(x)=1$ for each $x\in S$.

###### Theorem 4.6

Sniatycki  Let $S$ be a differential space with differential structure
$C^{\infty}(S)$, and let $\\{U_{\alpha}\\}$ be an open cover of $S$. If $S$ is
Hausdorff, locally compact and second countable, then there exists a countable
partition of unity $\\{f_{i}\\}\in C^{\infty}(S)$, such that, for each $i$,
there exists $\alpha$ such that the support of $f_{i}$ is contained in
$U_{\alpha}$ and the support of each $f_{i}$ is compact.

Since a subcartesian space is always Hausdorff and locally compact, to ensure
existence of partition of unity, we need to assume that the subcartesian space
is second countable.

Now consider the tangent distribution $\mathcal{V}$ spanned by the fundamental
vector fields of the action of $G$ on $P$. At every point $p\in P$ it is
defined by

$\mathcal{V}_{p}=\\{\xi_{P}(p)|\xi\in\mathfrak{g}\\}.$ (55)

$\mathcal{V}$ is a smooth involutive generalized distribution of $P$. We have

###### Theorem 4.7

Let $G$ be a connected Lie group acting in a proper way on a second countable
smooth manifold $P$. Let $\Delta$ be a smooth generalized affine distribution
of $P$ with associated smooth generalized distribution $L(\Delta)=\mathcal{J}$
satisfying that for each $p\in P$, vectors in $\mathcal{J}_{p}$ are
$G_{p}$-invariant. Assume that

$[\Gamma^{\infty}(\Delta),\Gamma^{\infty}(\mathcal{V})]\subseteq\Gamma^{\infty}(\mathcal{J}),$

where $\mathcal{V}$ is defined by (55). Then $\mathcal{V}$ is globally
controlled invariant for the generalized affine distribution $\Delta$.

###### Proof

Since $\mathcal{J}$ is smooth, then for each $p\in P$ and
$v_{p}\in\mathcal{J}_{p}$, there exists $X\in\Gamma^{\infty}(\mathcal{J})$
such that $X(p)=v_{p}$. By averaging $X$ over $G_{p}$, we get a
$G_{p}$-invariant vector field

$\tilde{X}=\int_{G_{p}}(\Phi_{g})_{*}Xd\mu(g),$

where $d\mu(g)$ is the Haar measure on $G_{p}$ normalized so that
$\text{vol}G_{p}=1$. Since
$[\Gamma^{\infty}(\mathcal{J}),\Gamma^{\infty}(\mathcal{V})]\subseteq\Gamma^{\infty}(\mathcal{J})$,
it follows that $(\Phi_{g})_{*}v_{q}\in\mathcal{J}_{gq}$ for each
$v_{q}\in\mathcal{J}_{q}$ and each $g\in G$. Hence we have
$\tilde{X}\in\Gamma^{\infty}(\mathcal{J})$. Besides, since vectors in
$\mathcal{J}_{p}$ are $G_{p}$-invariant, we have

$\tilde{X}(p)=\int_{G_{p}}(\Phi_{g})_{*}v_{p}d\mu(g)=\int_{G_{p}}v_{p}d\mu(g)=v_{p}.$

Now we can define a $G$-invariant vector field $\hat{X}$ on the $G$-invariant
open neighborhood $GS_{p}$ as follows. For each $p^{\prime\prime}\in GS_{p}$,
there exists $g\in G$ such that $p^{\prime\prime}=gp^{\prime}$ for
$p^{\prime}\in S_{p}$, and we set

$\hat{X}(p^{\prime\prime})=(\Phi_{g})_{*}\tilde{X}(p^{\prime}).$

Since $\tilde{X}$ is $G_{p}$-invariant, it follows that $\hat{X}$ is well-
defined and $\hat{X}\in\Gamma^{\infty}(GS_{p};\mathcal{J})$. On the other
hand, from the proof of Proposition 4.3.2 in Sniatycki  we know that there
exists a $G$-invariant smooth function $f$ on $P$ such that $f(p)=1$, and the
support of $f$ is contained in $GS_{p}$ such that $P\backslash\text{supp}{f}$
is an open subset of $P$. Now consider the $G$-invariant vector field
$f\hat{X}\in\Gamma^{\infty}(GS_{p};\mathcal{J})$. It can be extended to a
smooth $G$-invariant vector field $X^{\prime}$ on $P$, which vanishes on
$P\backslash\text{supp}{f}$. We have
$X^{\prime}\in\Gamma^{\infty}(\mathcal{J})$, and
$X^{\prime}(p)=f(p)\hat{X}(p)=\tilde{X}(p)=v_{p}$. Since $X^{\prime}$ is
$G$-invariant, it follows that
$[X^{\prime},\Gamma^{\infty}(\mathcal{V})]\subseteq\Gamma^{\infty}(\mathcal{V})$.
Hence $\mathcal{V}$ is globally controlled invariant for $\mathcal{J}$ since
$p\in P$ and $v_{p}\in\mathcal{J}_{p}$ are arbitrary.

Let $Y\in\Gamma^{\infty}(\Delta)$. Let $p\in P$. By averaging $Y$ over
$G_{p}$, we get a $G_{p}$-invariant vector field

$\tilde{Y}=\int_{G_{p}}(\Phi_{g})_{*}Yd\mu(g).$

Since $[Y,\Gamma^{\infty}(\mathcal{V})]\subseteq\Gamma^{\infty}(\mathcal{J})$,
it follows that $(\Phi_{g})_{*}v_{q}\in{\Delta}_{gq}$ for each
$v_{q}\in\Delta_{q}$ and each $g\in G$, which yields that
$\tilde{Y}\in\Gamma^{\infty}(\Delta)$, since $\text{vol}G_{p}=1$. By using
similar arguments as above, $\tilde{Y}$ can be extended to a $G$-invariant
vector field $\hat{Y}_{p}\in\Gamma^{\infty}(GS_{p};\Delta)$. Consider the open
cover $\\{GS_{p}|p\in P\\}$ of $P$. It follows that $\\{\pi(GS_{p})|p\in P\\}$
is an open cover of $P/G$, where $\pi:P\rightarrow P/G$ is the orbit map.
Since $P$ is second countable, it follows that $P/G$ with the quotient
topology is second countable. Then from Proposition 7 and Theorem 4.6 we know
that there exists a countable partition of unity $\\{\bar{f}_{i}\\}\in
C^{\infty}(P/G)$, subordinate to $\\{\pi(GS_{p})|p\in P\\}$ and such that the
support of each $\bar{f}_{i}$ is compact. Now we define the vector field
$Y^{\prime}$ on $P$ by

$Y^{\prime}=\sum_{i}\pi^{*}\bar{f}_{i}\hat{Y}_{\alpha(i)},$

where $\alpha(i)\in P$ is such that the support of $\bar{f}_{i}$ is contained
in $\pi(GS_{\alpha(i)})$. $Y^{\prime}$ is a smooth $G$-invariant vector field
on $P$. Hence
$[Y^{\prime},\Gamma^{\infty}(\mathcal{V})]\subseteq\Gamma^{\infty}(\mathcal{V})$.
It remains to show that $Y^{\prime}(p)\in\Delta(p)$ for each $p\in P$.
Actually we have a finite sum at each $p\in P$:

$\displaystyle Y^{\prime}(p)$
$\displaystyle=\sum_{i}(\pi^{*}\bar{f}_{i})(p)\hat{Y}_{\alpha(i)}(p)=\sum_{i}\bar{f}_{i}(\pi(p))(v_{0}+v_{p_{i}})$
$\displaystyle=(\sum_{i}\bar{f}_{i}(\pi(p)))v_{0}+\sum_{i}\bar{f}_{i}(\pi(p))v_{p_{i}}=v_{0}+v\in\Delta_{p},$

where $v_{0}\in\Delta_{p},v_{p_{i}},v\in\mathcal{J}_{p}$. This completes the
proof. ∎

If the action of $G$ on $P$ is free and proper, then the tangent distribution
$\mathcal{V}$ is regular. We have the following result for global controlled
invariance of the involutive regular distribution $\mathcal{V}$.

###### Theorem 4.8

Let $G$ be a connected Lie group acting in a free and proper way on a second
countable smooth manifold $P$. Let $\Delta$ be a smooth generalized affine
distribution of $P$ with associated smooth generalized distribution
$L(\Delta)=\mathcal{J}$. Assume that

$[\Gamma^{\infty}(\Delta),\Gamma^{\infty}(\mathcal{V})]\subseteq\Gamma^{\infty}(\mathcal{J}),$
(56)

where $\mathcal{V}$ is defined by (55). Then $\mathcal{V}$ is globally
controlled invariant for the generalized affine distribution $\Delta$.

###### Proof

Since the action of $G$ on $P$ is free, it follows that $G_{p}=e$ for each
$p\in P$. Hence vectors in $\mathcal{J}_{p}$ are $G_{p}$-invariant, for each
$p\in P$. Then the result follows immediately from Theorem 4.7.∎

###### Remark 5

In Theorem 4.1, it is required that $(G+D)/D$ is regular. From (56) we know
that $\mathcal{V}\subseteq\mathcal{J}$. So in the setting of Theorem 4.8, the
above condition is equivalent with requiring that $\mathcal{J}$ is regular,
which is not satisfied in Theorem 4.8. Despite this, the result for global
controlled invariance still holds as claimed in Theorem 4.8. This is due to
the special geometric structure of manifolds with a symmetry Lie group action.

##  5 Conclusions

In this paper, we have investigated the problem of controlled invariance of
involutive regular distributions for both affine control systems and
generalized affine distributions. A complete characterization of local
controlled invariance for both smooth and real analytic affine control systems
has been given. We have provided a differential geometric proof for this local
result. A sufficient condition for local controlled invariance for real
analytic generalized affine distribution has been presented. We have given
sufficient conditions that allow passages from local to global controlled
invariance for both smooth and real analytic generalized affine distribution
of general manifolds. Existing results in the literature have been reviewed
and clarified. Finally, for smooth manifolds with a proper Lie group action,
the problem of global controlled invariance has been investigated, where the
controlled invariant distribution is not required to be regular. As mentioned
in the introduction, the problem of controlled invariance of singular
distribution is more complicated and yet has not been solved well. It will be
addressed in our future work.

###### Acknowledgements.

This work was performed while the author was visiting the Department of
Mathematics and Statistics, Queen’s University. The author would like to thank
Professor Andrew Lewis for some enlightening talks, for example on the use of
Cartan’s Theorem B. The author would also like to thank the anonymous
reviewers for their careful reading of the manuscript and a lot of helpful and
constructive suggestions which help improve the presentation of the paper a
lot. This work was supported by CSC and the NSFC Grant 61703211.

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