%From rampazzo@math.rutgers.edu Sun Oct 26 08:16:53 2003
%Date: Sun, 26 Oct 2003 08:09:56 -0500 (EST)
%From: Franco Rampazzo <rampazzo@math.rutgers.edu>
%To: "Hector J. Sussmann" <sussmann@math.rutgers.edu>
%Subject: file of the paper as "text"
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% Pin No.: 06293
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% Paper ID: CDC01-REG1307.
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% Title: Set-valued differentials and a nonsmooth version of Chow's Theorem
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%            This is a LateX file, containing  the paper
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%                  SET-VALUED DIFFERENTIALS AND
%             A NONSMOOTH VERSION OF CHOW'S THEOREM
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%                 by F. Rampazzo and H. J. Sussmann
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%        This paper will be presented at the 40th IEEE Conference
%        on Decision and Control, to be held in Orlando, FL,
%        December 4-7, 2001,  and  will appear in the conference proceedings.
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%               Date of this version: August 30, 2001
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\begin{center}
A Maximum Principle for\\
hybrid optimal control problems\\

$\quad$\\

by\\

$\quad$\\
H\'ector J. Sussmann\\

\end{center}
}

\newpage

\null

\vspace{3.3cm}


{\Large

Paper ID:\\

$\,\,\,${\Huge CDC99-INV0007} \\

Session Title:\\

$\,\,\,$ Optimal Control II\\

Session Code: {\Huge TuM01-4} \\

Paper Category: REGULAR

\newpage
}}

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Franco Rampazzo\\
Dipartimento di Matematica pura ed applicata\\
Universit\`a degli Studi di Padova\\
Via G. Belzoni, 7\\
35131 PADOVA--ITALY\\
{\small\tt rampazzo@math.unipd.it}
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H\'ector J.\ Sussmann\\
Department of Mathematics\\
Rutgers, the State University of New Jersey\\
Hill Center---Busch Campus\\
%110 Frelinghuysen Road\\
Piscataway, NJ 08854-8019, USA\\
{\small\tt sussmann@hamilton.rutgers.edu}
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Hill Center---Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ
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\begin{document}
\title{  \bf Commutativity and higher order controllability \\for
nonsmooth vector fields }
\author{ Franco Rampazzo \\
Dipartimento di Matematica
\\
Pura ed Applicata \\
Universit\`a di Padova\\
Via Belzoni 7 ,
35131 Padova, Italy \\
 rampazzo@math.unipd.it\\
 \,\,\,\\
 %}
 %\author{
 Hector J. Sussmann \\
Department of Mathematics
\\
Rutgers, the State University of New Jersey\\
Hill Center---Busch Campus\\
Piscataway, NJ 08854-8019, USA \\
sussmann@hamilton.rutgers.edu}


\date{ }
\maketitle \abstract{bla bla}




\section{Introduction}
\label{sec-intro}
Among the classic results  concerning families of vector fields
one has:
\begin{itemize}
\item{} the characterization of commutative vector fields in terms of
vanishing of their Lie bracket;
 \item{} the several  versions of Chow's theorem on controllability,
including the one providing the degree of Holder-continuity of the
Optimal Time function near a target  point.
\end{itemize}
Here we shall give akin versions of such results
involving an extended notion of Lie bracket.


\section{Flows on manifolds: basic facts and notations}

\subsection{Locally Lipschitz continuous maps}
Unless otherwise specified, the term   {\it manifold} will mean
``finite-dimensional, second countable, Hausdorff, differentiable
manifold of class $C^{\infty}$.''

Let $M,N$  be manifolds of dimension $m$ and $n$, respectively, and
let $x\in M$.  A map $F:M\mapsto N$ is {\it locally Lipschitz
continuous near $x$} if it is continuous at $x$ and, for some pair
$(\varphi,\psi)$ of charts $\varphi:U\mapsto\R^{m}$,
$\psi:V\mapsto\R^{n}$ defined on open neighborhoods $U$ and $V$ of $x$
and $F(x)$, respectively, such that $F(U)\subseteq V$, the composite
map $\psi\circ F\circ\varphi^{{-1}}: \varphi(U)\mapsto \psi (V)$ is
Lipschitz continuous near $\varphi(x)$ in the usual sense. (Notice
that in this case  $\psi\circ F\circ\varphi^{{-1}}$ is Lipschitz
continuous near $\varphi(x)$ for {\em every} pair of charts
$(\varphi,\psi)$.) We call $F$ {\it locally Lipschitz continuous} if
it is Lipschitz continuous at $x$ for every $x\in M$.

\begin{remark} Since all Riemannian metrics are locally equivalent,
it is easy to check that a map $F:M\mapsto N$ is locally Lipschitz
continuous if and only if it is locally Lipschitz continuous in the
usual sense when $M$ and $N$ are endowed with  arbitrary Riemannian
metrics.
\end{remark}

The following version of Rademacher's Theorem on manifolds is an
obvious consequence of the standard version on Euclidean spaces. (Let
us recall that the concept of a ``null subset'' ---that is, a subset
of measure zero--- of a manifold $M$ is well defined: if $E\subseteq
M$, then $E$ is a null subset if $\varphi(E)$ is a null subset of
$\varphi(U)$ for every chart $\varphi:U\mapsto\R^m$ of $M$.)

\begin{theorem}\label{rademacher} Let $M, N$ be manifolds and let
$F$ be a locally Lipschitz map. Then there exists a null subset $E$ of
$M$ such that $F$ is differentiable at each point of $M\backslash E$.
\end{theorem}

\subsection{Notation on manifolds}\label{exp}
In  \cite {kawski}  the well-known fact that the set  ${\cal E}(M)$ of
$C^{\infty}$
 functions on $M$  completely determines $M$ has been exploited in order
to set up a
 notation and  formal rules which highly simplify some
computations involving vector fields, brackets, flows and their
derivatives, an so on. We shall use this notation, so, for the sake of
self-consistency,  let
 us recall some basic facts from \cite{kawski}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 1truecm {\bf HECTOR  here I have inserted  pages 2-3 of your
paper with Kawski} \vskip 1truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\medskip\noindent If $M$ is a manifold, we use ${\cal E}(M)$ to denote the
commutative algebra ---over $\rr$---
of smooth real-valued functions on $M$, topologized in
the usual way.  (A sequence $\{\varphi_j\}$ converges to a limit $\varphi$
in ${\cal E}(M)$ iff for every $k\ge 0$ and every $k$-tuple
$(X_1,\ldots,X_k)$ of smooth vector fields on $M$ the functions
$X_1X_2\ldots X_k\varphi_j$ converge to $X_1X_2\ldots X_k\varphi$
uniformly on compact sets.)  We let ${\cal E}'(M)$ denote the dual space
of
${\cal E}(M)$, i.e. the space of compactly supported Schwartz
distributions
on $M$.

It is well known that $M$ is completely determined by ${\cal E}(M)$, in
the
following sense.  A point $p$ of $M$ gives rise to a linear functional
$\delta_p\in {\cal E}'(M)$ ---the Dirac delta function at $p$--- defined
by
$\delta_p(\varphi)=\varphi(p)$.  This linear functional is {\sl
multiplicative}, i.e. is a homomorphism of $\rr$-algebras.  For any
commutative algebra $A$ over a field ${\bf k}$, let us use $\sigma(A)$ to
denote the {\sl
spectrum} of $A$, i.e. the set of all maps $\mu:A\to{\bf k}$ that are
algebra
homomorphisms and do not vanish identically. Then {\sl the map
$p\to\delta_p$ is a bijection from $M$ to $\sigma({\cal E}(M))$}.  So a
manifold can be canonically identified with the spectrum of its algebra of
smooth functions.

Via the map $p\to\delta_p$, we
regard $M$ as embedded in ${\cal E}'(M)$.  Moreover, many other objects
related to $M$ can also be naturally regarded as members of ${\cal
E}'(M)$. For example,  $TM$ is embedded in ${\cal E}'(M)$ as follows:
a tangent vector $v$ at a point is a
linear functional on ${\cal E}(M)$, which maps $\varphi\in{\cal E}(M)$
to $v\varphi$. Since this functional is clearly continuous, $v$ belongs to
${\cal E}'(M)$.

If $X$ and $Y$ are smooth vector fields, then the product $XY$, evaluated
at $p$, is a well defined member of ${\cal E}'(M)$, namely, the map
$\varphi\to (XY\varphi)(p)$.  So the product $XY$ ---i.e. the map $p\to
XY(p)$--- is a well defined map from $M$ to ${\cal E}'(M)$.  The
difference
$[X,Y]=XY-YX$ ---the {\sl Lie bracket} of $X$ and $Y$--- is also a map
from
$M$ to ${\cal E}'(M)$, that happens to take values in $TM$, and
is in fact a vector field.

On the other hand, ${\cal E}'(M)$ is a topological linear space, so linear
operations and limiting processes that in principle appear not to make
intrinsic sense on $M$ can be meaningful in ${\cal E}'(M)$. For example,
if $\gamma:[0,\eps]\to M$ is a curve, and $\gamma(0)=p$,
then we would like to define the tangent vector $\dot\gamma(0)$ by just
letting
\beq\label{islim}
\dot\gamma(0)\equaldef \lim_{h\to 0}h^{-1}(\gamma(h)-p)\,.
\eeq

\vspace{-0.2cm}
\noindent This may look unacceptable, since $M$ is not a linear space, and
$\gamma(h)-p$ does not make sense. However,
Formula (\ref{islim}) is perfectly meaningful, and gives the right answer,
if we regard it as an identity in ${\cal E}'(M)$,
since $\lim_{h\to 0}h^{-1}(\gamma(h)-p)$
is the distribution that maps $\varphi\in{\cal E}(M)$ to the number
$\lim_{h\to 0}h^{-1}(\varphi(\gamma(h))-\varphi(p))$, i.e.
directional differentiation at $p$ in the direction of $\dot\gamma(0)$.

To make the formalism work well, it is convenient to write
$\pi\varphi$ ---rather than $\pi(\varphi)$--- for the  value  of the
linear
functional $\pi\in{\cal E}'(M)$ at the function $\varphi\in{\cal E}(M)$.
With this notation, when $p\in M$ and $\varphi\in{\cal E}(M)$ the value
$\varphi(p)$ is now written $p\varphi$.  If $f$ is a smooth
vector field and  $\varphi\in{\cal E}(M)$, then $f\varphi$ is in
${\cal E}(M)$, and the number usually written as
$(f\varphi)(p)$ now becomes $p(f\varphi)$. On the other hand,
if we also write $pf$ rather than the more common $f(p)$,
the number $p(f\varphi)$ is also equal to $(pf)\varphi$,
so we can just write $pf\varphi$ without parentheses.

More generally, maps $\Phi$ from $M$ to any manifold $N$ should act on
points
of $M$ on the right, so we will write $p\Phi$ rather than $\Phi(p)$.
A smooth map $\Phi:M\to N$ gives rise to a map
$\tilde\Phi:{\cal E}'(M)\to{\cal E}'(N)$, defined by letting
$\tilde\Phi(D)$, for $D\in {\cal E}'(M)$, be the distribution
on $N$ given by $\tilde\Phi(D)(\psi)=D(\psi\compo\Phi)$, i.e., in our
formalism, $(D\tilde\Phi)\psi\equaldef D(\Phi\psi)$,
where, of course, $\Phi\psi$ is the function usually referred to
as the ``pullback'' $\Phi^*(\psi)$
of $\psi$ to $M$ via $\Phi$.  (Naturally,
since maps act on the right, $\Phi\psi$ is the composite map obtained by
applying $\Phi$ first, and then $\psi$.)  When $D$ is actually a
point $p\in M$, then $(D\tilde\Phi)\psi$ is what is normally written
as $\psi(\Phi(p))$, so $p\tilde\Phi=p\Phi$. So $\tilde\Phi$ is an
extension
to ${\cal E}'(M)$ of the map $\Phi$, originally defined on the subset
$M$ of ${\cal E}'(M)$.  It is convenient to just use
$\Phi$ for this extension.    With this notation,
if $p\in M$ and $v$ is a tangent vector at $p$, then
$v\Phi$ is the tangent vector at $p\Phi$ usually written as
$\Phi_*(v)$, or $D\Phi(p).v\,$.
\def\coma{(Indeed, if $\psi\in
{\cal E}(N)$, then $(v\Phi)\psi$ is, by definition, equal to
$v(\Phi\psi)$, i.e. to the directional derivative at $p$ in the direction
of $v$ of the pullback function $\Phi^*(\psi)$.) Notice that, if
$v=\lim_{h\to 0}\frac{\gamma(h)-p}{h}$, then
$v\Phi=\lim_{h\to 0}\frac{\Phi(\gamma(h))-\Phi(p)}{h}$
(or, using our notations,
$v\Phi=\lim_{h\to 0}\frac{h\gamma\Phi-p\Phi}{h}$).}

In particular, we will use $e^{tf}$ to denote the time $t$ flow map
arising
from a vector field $f$, and will write it as acting on the right. So
$t\to pe^{tf}$ is the integral curve $\xi_{f,p}$ of $f$ that goes
through $p$ at time $0$.  This curve satisfies the equation
\beq
(d/dt)(pe^{tf})=pe^{tf}f\,,
\eeq

\vspace{-0.2cm}
\noindent which is a much more elegant way of writing the formula
$(d/dt)(\xi_{f,p}(t))\!=\!f(\xi_{f,p}(t))$,
and amply justifies the use of the exponential notation.

With this formalism, several important formulas involving Lie brackets
become completely trivial formally, and the formal calculations can be
rigorously justified by working in ${\cal E}'(M)$.
For example, let $f,g$ be vector fields, and let $p\in M$.
Then

\vspace{-0.5cm}
\beqn
&\,\,&(d/dt)(pe^{tf}e^{tg}e^{-tf}e^{-tg})\nonumber\\
&=&pe^{tf}fe^{tg}e^{-tf}e^{-tg}+pe^{tf}e^{tg}ge^{-tf}e^{-tg}
-pe^{tf}e^{tg}e^{-tf}fe^{-tg}-pe^{tf}e^{tg}e^{-tf}e^{-tg}g\,,\nonumber
\eeqn

\vspace{-0.15cm}
\noindent whose value for $t=0$ is $pf+pg-pf-pg$, i.e. $0$.


\vskip 1truecm
\section{Lie brackets of locally Lipschitz vector \mbox{fields}}

The ordinary Lie bracket of two
 vector fields $f,g,$ will be denoted by
$[f,g]_{ord}$. We shall use both $[f,g]_{ord}(x)$ and $x[f,g]_{ord}$
to indicate its evaluation at $x$ That is, we set
$$[f,g]_{ord}(x)  = x[f,g]_{ord} \doteq
x(fg-gf)\,.
$$
 This means that, if we write
$x\sigma$, $xV$, rather than $\sigma(x)$, $V(x)$, for the value  at
$x$ of a function $\sigma$ or a vector field $V$,  then
$x[f,g]_{ord}\varphi = x(fg-gf)\varphi$ for every test function
$\varphi$.


If $f$ is a locally Lipschitz continuous vector field on a manifold
$M$, we let $DIFF(f)$ denote the set of points $x\in M$ such
that $f$ is \mbox{differentiable} at $x$.  Then Theorem \ref{rademacher}
\mbox{implies} that $M\backslash DIFF(f)$ is a null subset of $M$.

\begin{definition}
\label{DEFlie}If  $f,g$ are locally Lipschitz continuous vector
fields on a manifold $M$, we define the {\em Lie bracket} $[f,g]$ as follows.
For every $x\in M$ we define $[f,g](x)$---the Lie bracket of $f$ and $g$
at $x$---to be the convex hull of the set of all vectors
\begin{equation}\label{deflie}
v=\lim_{j\to\infty}[f,g]_{ord}(x_j),
\end{equation}
for all sequences $\{x_j\}_{j\in\nn}$ such that
\begin{itemize}
\item[1.] $x_j\in DIFF(f)\cap DIFF(g)$ for all $j$, \item[2.]
$\lim_{j\to\infty}x_j=x$, \item[3.] the limit $v$ of (\ref{deflie})
exists.
\end{itemize}
\end{definition}

For every $x\in M$, it is clear that $[f,g](x)$ is a {\it convex}, {\it
compact}, and {\it nonempty} subset of the tangent space $T_xM$
of $M$ at $x$. Moreover, the {\em skew-symmetry
identity}
\begin{equation}\label{skewsym}
[f,g](x)=-[g,f](x)
\end{equation}
clearly holds for all $x\in M$. (This  means that
$[f,g](x)=\{w:-w\in [g,f](x)$).
In addition, each locally Lipschitz continuous
vector field $g$ satisfies the identity
\begin{equation}\label{idemp}
[g,g](x) = \{0\}\,\,\,{\rm for\,\,\,every}\,\,\,x\in M\,.
\end{equation}
(Formula (\ref{idemp}) is {\em not} an immediate consequence of
(\ref{skewsym}), but follows easily from Definition \ref{DEFlie},
since $[g,g]_{ord}(x)=0$ for every $x\in DIFF(g)$.)


\begin{remark}If $x\in DIFF(f)\cap DIFF(g)$ the bracket $[f,g](x)$
we have just defined {\em does not coincide, in general, with the
singleton $\{[f,g]_{ord}(x)\}$}, although it always contains the
vector $[f,g]_{ord}(x)$. For example, let $M=\rr$ and let us
consider the vector fields
$$
f(x)={\partial\over\partial x}\text{ and}
\qquad g(x) = \left\{{\begin{array}{lr} x^2\sin(1/x){\partial\over\partial
x}
\quad & x\neq 0\\\,\,\,\,\,\, 0\qquad & x=0\end{array}}\right.
$$
Then, for $x=0$
$$
[f,g]_{ord}(x)=0,\qquad\text{whereas} [f,g](x) =
\Big\{a{\partial\over\partial x}:a\in [-1,1]\Big\}
$$
In the special case when $f$ and $g$ are of class $C^1$, the set
$[f,g](x)$ is indeed the singleton $\{[f,g]_{ord}(x)\}$ for each $x\in
M$.
\end{remark}


\subsection{Set-valued iterated brackets}\label{sectliemon}

\begin{definition} If  $F$ is a family of  vector fields, let us say that  its
elements are {\em (ordinary) Lie brackets of degree one}. If $h\in F$
and $k\in F$ are differentiable at any point, let us say that $[h,k]$
is a {\em a (ordinary) Lie bracket of degree 2}. Moreover, if $l$ and
$m$ are differentiable Lie brackets of degree $i$ and $j$,
respectively, let us say that $[h,k]_{ord}$ is {\it a Lie bracket of
degree $i+j$}. So,
 it remains inductively defined what a {\em (ordinary) Lie bracket of degree r } is,
for any
natural number $r$.
\end{definition}




\begin{definition}
For any natural number $q\geq 1$, we say that a vector field is
of class
  $C_{loc}^{q-1,1}$ if it is of class $C^{q-1}$ and the  $q-1$-th order
derivatives
  are locally Lipschitz continuous.
  \end{definition}
  \vs

\begin{definition} If $p,r,q$ are non-negative integers such that $p+r=q$,
 $F=\{h_1,\dots,h_{q+1}\}$
is a family of  $C^{q-1,1}_{loc}$ vector fields, and $k$ and $l$ are
(ordinary) brackets of degree $p$ and $r$, respectively, we say that
$[k,l]$  is a (set-valued) bracket of degree $q+1$. Notice that
Definition \ref{DEFlie} applies, in that  all brackets having a degree
less than $q+1$ turn out to be locally Lipschitz continuous.
\end{definition}

 For
instance, if $h_{1},\dots,h_{q+1}$  are of class  $C_{loc}^{2,1}$,
the bracket
$[[[h_1,h_2],h_3],h_4]$ is well defined, for both vector fields
$[[h_1,h_2],h_{3}]$ and
$h_4$ are locally Lipschitz continuous.

 Actually in some cases the regularity
assumptions on the vector fields $h_i$ can be weakened. For instance,
in order to define degree $2$ bracket $[[h_1,h_2],h_3]$ we need $h_1$
and
 $h_2$ to be of class $C^{1,1}_{loc}$, while it is sufficient for $h_3$
of being just
locally Lipschitz continuous. It also can happen that none of the $q+1$
 vector fields $h_i$ need to be of class $C^{q-1,1}_{loc}$,
 as in the case of the degree $4$ bracket
$[[h_1,h_2],[h_3.h_4]]$, where $h_1.\dots, h_4$
do not need to be of class $C^{2,1}_{loc}$, and can
 well
be of class $C^{1,1}_{loc}$.

{\bf Jacobi identity.}
 If $h_1,h_2,h_3$ are of class $C^{q-1,1}$, with $q\geq 2$, by exploiting
the standard Jacobi identity along sequences $\{x_j\}_{j\in \nn}$ where
the $h_i$'s are
 twice differentiable, one can trivially prove the following {\it extended
version
of the   Jacobi
identity}:
$$[[h_1,h_2],h_3](x)+[[h_3,h_1],h_2](x)+[[h_2,h_3],h_1](x)\supseteq
\{0\}
$$
for all $x\in M$. Of course this is nothing but the usual Jacobi
identity as soon as $q>2$, while in the case when $q=2$ it says that
for every $x$ there is a triple

$$
(a_1,a_2,a_3)\in  [[h_1,h_2],h_3](x)\times [[h_3,h_1],h_2](x) \times
[[h_2,h_3],h_1](x)$$
such that
$$
a_1+a_2+a_3 = 0.
$$
\vskip 0.8 truecm


{\bf Linear dependence.}
  Let us recall  a property of Lie brackets
(of smooth  vector fields )
which is an easy consequence of Jacobi's identity:

\begin{itproposition}\label{smooth-q-lie}
If $h_1,\dots,h_{q+1}$ are $q$ times differentiable, then any $(q+1)$-th
 degree Lie
bracket is a $\rr$-linear combinations of ($(q+1)$-th degree) Lie
brackets of the form
$$
[[\dots[h_{i_1},h_{i_2}],h_{i_3},\dots,]h_{i_{q+1}}]\quad
(i_1,\dots,i_{q+1})\in \{1,\dots,{q+1}\}^{q+1}.
$$
(Of course, in view of the skew-symmetry, some of these
 Lie brackets are identically equal to zero)
\end{itproposition}
As an example, it is straightforward to verify that, if
$h_1,h_2,h_3,h_4$ are of class $C^3$, then
$$
[[h_1,h_2],[h_3,h_4]]= -[[[h_3,h_4],h_1],h_2] + [[[h_3,h_4],h_2],h_1]
$$
 By arguing as in the case of the Jacobi's identity one easily obtains
the following {\it nonsmooth} extension of the previous proposition,
which will be exploited later.

\begin{itproposition}\label{nonsmooth-(q+1)-lie}
If $h_1,\dots,h_{q+1}$ are of class  $C^{q-1,1}_{loc}$,
then any (set-valued) $(q+1)$-th degree Lie
bracket {\em is contained}
in a $\rr$-linear combination of the (set-valued)  $(q+1)$-th degree Lie
brackets
$$
[[\dots[h_{i_1},h_{i_2}],h_{i_3},\dots,]h_{i_{q+1}}],\,
\quad (i_1,\dots,i_{q+1})\in \{1,\dots,{q+1}\}^{q+1}.
$$
 (Again, in view of the skew-symmetry, many of these
 Lie brackets are identically equal to singleton $\{0\}$).
\end{itproposition}
Notice that the nonsmooth version of the  example above reads as
follows: if $h_1,h_2,h_3,h_4$ are of class $C^{3,1}_{loc}$
$$
[[h_1,h_2],[h_3,h_4]](x) \subseteq -[[[h_3,h_4],h_1],h_2](x) +
 [[[h_3,h_4],h_2],h_1](x).
$$

\section{Commutativity of locally Lipschitz vector fields}
It is well known that if $f,g$ are vector fields of class $C^{1}$ then
the flows of $f$ and $g$ commute for small times if and only if
$[f,g]_{ord}=0$ identically. In this section we show that an analogous
result holds true in the case when $f$ and $g$ are just locally
Lipschitz vector fields.

To begin with, let us make precise what we mean by saying that the flows
of two vector fields {\it commute}.
\vskip 1truecm
\begin{definition}  Let $M$ be a manifold, and let $f,g$
be locally  Lipschitz continuous vector fields defined on $M$.
We say that {\em the flows of $f$ and $g$ commute} if
$xe^{sg}e^{tf}$ is defined and
\begin{equation}\label{fgcommute}
xe^{tf}e^{sg}=xe^{sg}e^{tf}
\end{equation}
whenever $x\in M$, $t,s\in\rr$ are such that
$xe^{\tau f}e^{\sigma g}$ is defined whenever $\min(t,0)\leq \tau
\leq \max(t ,0)$ and $\min(s ,0)\leq \sigma\leq\max(s ,0)$
\end{definition}

There is also a ``local'' version of the above concept, useful
to prove results on manifolds by reducing them to the case
when the manifold is $\rr^n$:

\begin{definition}  Let $M$ be a manifold, and let $f,g$
be locally  Lipschitz continuous vector fields defined on $M$. Let
$x\in M$. We say that {\em the flows of $f$ and $g$ commute for small
times near $x$} if there exist a neighborhood $U$ of $x$ and a
positive number $\varepsilon$ such that ($x'e^{tf}e^{sg}$ and
$x'e^{sg}e^{tf}$ are defined and)
\begin{equation}\label{fgcommutesmall}
x'e^{tf}e^{sg}=x'e^{sg}e^{tf}
\end{equation}
for all $x'\in M$, $t,s\in\rr$ such that $|t|\le\varepsilon$ and
$|s|\le\varepsilon$.
\end{definition}

\begin{theorem}\label{liezero1}
Let $M$ be a manifold, and let $f,g$ be locally Lipschitz continuous
vector fields on $M$.  Let ${\cal N}$ be a null subset of $M$. Then
the following conditions are equivalent:
\begin{itemize}
\item[i)] $[f,g]_{ord}(x)=0$
for every $x\in \Big(DIFF(f)\cap DIFF(g)\Big)\backslash{\cal N}$;
\item[ii)] $[f,g]_{ord}(x)=0$ for every $x\in DIFF(f)\cap DIFF(g)$;
\item[iii)] $[f,g](x)=\{0\}$ for every $x\in M$;
\item[iv)] the flows of $f$ and $g$ commute for small times near $x$ for
every $x\in M$;
\item[v)] the flows of $f$ and $g$ commute.
\end{itemize}
\end{theorem}

In order to prove Theorem \ref{liezero1}, let us first establish an
exact integral representation of the quantity
$x\,e^{tf}e^{sg}e^{-tf}e^{-sg}-x$ in the case when $f$ and $g$ are
$C^1$. A slightly different version of this representation have been
already proved in \cite{ramsus1}.

\begin{lemma}\label{lemmaintform}
Let $f$ and $g$ be vector fields of class $C^1$ on a manifold $M$.
Then, for all $x\in M$, $t,s\in\rr$ such that $ xe^{\tau f}e^{{\sigma
g}}e^{\tau f}e^{{-\sigma g}}$ is defined whenever $\min(t ,0)\leq \tau
\leq \max(t ,0)$ and $\min(s ,0)\leq \sigma \leq\max(s ,0)$, one has
\beq\label{integralformula} x\,e^{tf}e^{sg}e^{-tf}e^{-sg}-x
 = \int_0^{t}\int_0^s\Big(
x\,e^{\tau f}e^{(s-\sigma)g}[f,g]e^{\sigma g}e^{-\tau f}e^{-sg}\Big)\,d\tau\,d\sigma
\eeq
holds. Moreover, if $w$ is any vector field, then
\beq\label{integralformulavector}
xw\,e^{tf}e^{sg}e^{-tf}e^{-sg}-xw
 = \int_0^{t}\int_0^s\Big(xw\,e^{\tau f}e^{(s-\sigma)g}[f,g]
e^{\sigma g}e^{-\tau f}e^{-sg}\Big)\,d\tau\,d\sigma \eeq

(See Subsection \ref{exp} for the meaning of the subtraction on
manifolds.)
\end{lemma}

\begin{remark} Under the hypotheses of Lemma \ref{lemmaintform},
the vector field $[f,g]$ is continuous, so the integrand $x\,e^{\tau
f}e^{(s-\sigma)g}[f,g]e^{\sigma g}e^{-\tau f}e^{-sg}$ is
$[f,g](x)+o(1)$ as $(s,t)\to(0,0)$. Therefore (\ref{integralformula})
implies the usual  second order estimate
\beq\label{esti}
x\,e^{tf}e^{sg}e^{-tf}e^{-sg}-x = st[f,g](x) + o(st)\,,
\eeq
 from
which it follows, for instance, that $[f,g]$ vanishes if the flows of
$f$ and $g$ commute. Of course an {\em exact integral formula} is much
more interesting than (\ref{esti}). As an instance, the fact that the
condition $[f,g]=0$ implies that the flows of $f$ and $g$ commute is a
straightforward consequence of  (\ref{integralformula}), while the
standard proof using (\ref{esti}) is nothing but trivial.
\end{remark}
 \vskip 0.8 truecm

\begin{proof}
The proof of (\ref{integralformula}) reduces to the following
chain of equalities:

\begin{eqnarray*}
x\,e^{tf}e^{sg}e^{-tf}e^{-sg}-x&=&
\int_0^t\Big(x\,e^{\tau f}fe^{sg}e^{-\tau f}e^{-sg}
- x\,e^{\tau f}e^{sg}fe^{-\tau f}e^{-sg}\Big)\,d\tau\\
% \int_0^t\Big(x\,e^{\tau f}fe^{sg}e^{-\tau f}e^{-sg}
%
&=&\int_0^t x\,e^{\tau f}e^{sg}(e^{-sg}fe^{sg}-f) e^{-\tau f}e^{-sg}\,
d\tau\\
&=&\int_0^t\int_0^s x\,e^{\tau f}e^{sg}(e^{-\sigma g}[f,g]e^{\sigma g})
e^{-\tau f}e^{-sg}\,d\sigma d\tau \\
&=&
\int_0^t\int_0^s x\,e^{\tau f}e^{(s-\sigma)g}[f,g]e^{\sigma g}
e^{-\tau f}e^{-sg}\,d\sigma d\tau
\end{eqnarray*}
where we have used the identities:
$$
{d\over d\tau}\Big(x\,e^{\tau f}e^{sg}e^{-\tau f}e^{-sg}\Big)=
x\,e^{\tau f}fe^{sg}e^{-\tau f}e^{-sg}
- x\,e^{\tau f}e^{sg}fe^{-\tau f}e^{-sg}
$$
and
$$
{d\over d\sigma}\Big(y(e^{-\sigma g}fe^{\sigma g}-f)\Big)=
ye^{-\sigma g}[f,g] e^{\sigma g}\,.
$$
\vskip 0.8 truecm
As for (\ref{integralformulavector}), it follows straightforwardly from
(\ref{integralformula}). Indeed, one has
$$\begin{array}{c}
xw\,e^{tf}e^{sg}e^{-tf}e^{-sg}-xw\\
 =  \lim_{\epsilon\to 0} {1\over\epsilon}
\Big[\Big((x+\epsilon(xw))\,e^{tf}e^{sg}e^{-tf}e^{-sg}
 - x\,e^{tf}e^{sg}e^{-tf}e^{-sg}\Big)
-  \Big(x+\epsilon(xw) - x\Big)\Big]\\
= \lim_{\epsilon\to 0}
{1\over\epsilon}\int_0^{t}\int_0^s\Big[\Big(x+\epsilon(xw)(\,e^{\tau
f}e^{(s-\sigma)g}[f,g]
e^{\sigma g}e^{-\tau f}e^{-sg}\Big) - \\
\Big[\Big( x\,e^{\tau f}e^{(s-\sigma)g}[f,g]
e^{\sigma g}e^{-\tau f}e^{-sg}\Big)\Big] \,d\tau\,d\sigma\\
=  \int_0^{t}\int_0^s\Big(xw\,e^{\tau f}e^{(s-\sigma)g}[f,g]
e^{\sigma g}e^{-\tau f}e^{-sg}\Big)\,d\tau\,d\sigma.
\end{array}
$$
\end{proof}
\vskip 1.5truecm In order to prove Theorem \ref{liezero1} we need the
following result as well:
\begin{lemma}\label{lemmacommsmall}
Let $f$ and $g$ be locally Lipschitz continuous vector fields on a manifold
$M$.  Suppose that the flows of $f$ and $g$ commute for small times near
$x$ for every $x\in M$. Then the flows of $f$ and $g$ commute.
\end{lemma}

\begin{proof} Let $x\in M$, $t,s\in\rr$ be such that
$xe^{\tau f}e^{\sigma g}$ is defined whenever
$\min(t ,0)\leq \tau\leq \max(t ,0)$ and
$\min(s ,0)\leq \sigma\leq\max(s ,0)$. We want to prove that
$xe^{tf}e^{sg}=xe^{sg}e^{tf}$.

Let $I_t$, $I_s$ be the compact intervals
$[\min(t,0),\max(t,0)]$ $[\min(s,0),\max(s,0)]$, respectively.  Write
$I_t=[t_-,t_+]$, $I_s=[s_-,s_+]$.

Let $K=\{xe^{\tau f}e^{{\sigma g}}: \tau\in I_t,\,\sigma\in I_s\}$. Then
$K$ is a compact subset of $M$, because $xe^{\tau f}e^{{\sigma g}}$
exists whenever $\tau\in I_t$, $\sigma\in I_s$. So there exist
a positive $\varepsilon$ such that $x'e^{\tau f}e^{\sigma g}$
is defined and equal to $x'e^{\sigma g}e^{\tau f}$ whenever
$x'\in K$, $|\tau|\le\varepsilon$ and $|\sigma|\le\varepsilon$.

Let $N$ be a positive integer such that
$\frac{|t|}{N}<\varepsilon$ and $\frac{|s|}{N}<\varepsilon$.
Let $t_j=\frac{jt}{N}$, $s_k=\frac{kt}{N}$, for $j=0,\ldots,N$,
$k=0,\ldots,N$. We claim that
\begin{itemize}
\item[(*)] {\em If $j,k\in\{0,\ldots,N\}$,
then $xe^{t_j f}e^{s_k g}$ is defined and is equal to
$xe^{s_k g}e^{t_j f}$}
\end{itemize}
To prove (*), we first show that
\begin{itemize}
\item[(\#)] {\em If $j\in\{0,\ldots,N\}$, $x'\in K$, and
$x'e^{t_i f}\in K$ for all $i\in\{0,\ldots,j\}$,
then $x'e^{t_jf}e^{s_1g}$ is defined and equal to
$x'e^{s_1g}e^{t_jf}$.}
\end{itemize}
To prove (\#), we use induction with respect to $j$. We
first observe that the case when $j=0$ is trivial.
Assume that our conclusion is known to be true for a $j$ such
that $0\le j<N$, and let $x'\in K$ be such that
$x'e^{t_i f}\in K$ for all $i\in\{0,\ldots,j+1\}$.
Then in particular $x'e^{t_i f}\in K$ for all $i\in\{0,\ldots,j\}$,
so the inductive assumption implies that
$x'e^{t_jf}e^{s_1g}$ is defined and equal to
$x'e^{s_1g}e^{t_jf}$. Since $x'e^{t_jf}\in K$,
$|t_1|\le\varepsilon$, and $|s_1|\le\varepsilon$,
we can conclude that $x'e^{t_jf}e^{t_1f}e^{s_1g}$ (which is equal to
$x'e^{t_{j+1}f}e^{s_1g}$) is defined and equal to
$x'e^{t_jf}e^{s_1g}e^{t_1f}$. But
$x'e^{t_jf}e^{s_1g}=x'e^{s_1g}e^{t_jf}$. Therefore
$x'e^{t_jf}e^{s_1g}e^{t_1f}=x'e^{s_1g}e^{t_jf}e^{t_1f}
=x'e^{s_1g}e^{t_{j+1}f}$. It follows that
$x'e^{t_{j+1}f}e^{s_1f}$ is defined and equal to
$x'e^{s_1g}e^{t_{j+1}f}$, completing the proof of (\#).

We now prove (*) by (strong) induction on $j$. The case $j=0$ is trivial.
Suppose that $0\le j^*<N$, and the conclusion of (*)
has been proved for all $k$ and for all $j$ such that $0\le j\le j^*$.
We prove, by induction on $k$, that
then $xe^{t_{j^*+1}f}e^{s_k g}$ is defined and is equal to
$xe^{s_k g}e^{t_{j^*+1}f}$ for all $k\in\{0,\ldots,N\}$. Once again,
the conclusion is trivial if $k=0$. Suppose that $0\le k<N$,
and we know that $xe^{t_{j^*+1}f}e^{s_k g}$ is defined and is equal to
$xe^{s_k g}e^{t_{j^*+1}f}$. Then
$xe^{t_{j^*+1}f}e^{s_{k+1} g}$ is defined, because
$t_{j^*+1}\in I_t$ and $s_{k+1}\in I_s$. Furthermore,
\begin{eqnarray}
xe^{t_{j^*+1}f}e^{s_{k+1} g}&=&xe^{t_{j^*+1}f}e^{s_{k}g}e^{s_1g}
\label{triv-a}\\
&=&xe^{s_{k}g}e^{t_{j^*+1}f}e^{s_1g}\label{triv-b}\\
&=&xe^{s_{k}g}e^{t_{j^*}f}e^{t_{1}f}e^{s_1g}\label{triv-c}\\
&=&xe^{s_{k}g}e^{t_{j^*}f}e^{s_1g}e^{t_{1}f}\label{triv-d}\\
&=&xe^{s_{k}g}e^{s_1g}e^{t_{j^*}f}e^{t_{1}f}\label{triv-e}\\
&=&xe^{s_{k+1}g}e^{t_{j^*+1}f}\,,\label{triv-f}
\end{eqnarray}
where
\begin{itemize}
\item[1.] (\ref{triv-a}) follows because $s_{k+1}=s_k+s_1$ and the numbers
$s_k$ and $s_1$ have the same sign;
\item[2.] (\ref{triv-b}) follows from the inductive hypothesis of the
$k$-induction;
\item[3.] (\ref{triv-c}) follows because $t_{j^*+1}=t_{j^*}+t_1$
and the numbers $t_{j^*}$ and $t_1$ have the same sign;
\item[4.] (\ref{triv-d}) follows because
$xe^{s_{k}g}e^{t_{j^*}f}\in K$  (since
$xe^{s_{k}g}e^{t_{j^*}f}=xe^{t_{j^*}f}e^{s_{k}g}$ by the inductive
hypothesis of the $j$-induction), $|t_1|\le\varepsilon$, and
$|s_1|\le\varepsilon$;
\item[5.] (\ref{triv-e}) follows from (\#), which can be applied---with
$xe^{s_kg}$ in the role of $x'$---since
$xe^{s_kg}e^{t_if}\in K$ for $i=0,1,\ldots,j^*$, because
our inductive assumption tells us that
$xe^{s_kg}e^{t_if}=xe^{t_if}e^{s_kg}$ for $i=0,1,\ldots,j^*$,
and $xe^{t_if}e^{s_kg}\in K$ by the definition of $K$;
\item[6.] (\ref{triv-f}) follows because $t_{j^*+1}=t_{j^*}+t_1$
and $s_{k+1}=s_k+s_1$.
\end{itemize}
Having proved (*), we can take $k=N$, $j=N$, and conclude
that $xe^{tf}e^{sg}=xe^{sg}e^{tf}$. This completes our proof.
\end{proof}





Here and in the next sections we  will consider {\it regularizations} of
vector fields on $\R^n$:

\begin{definition}\label{regularization}
Let $\varphi$ is a nonnegative
real-valued function
on $\rr^n$ of class $C^\infty$ such that $\int_{\rr^n}\varphi=1$
and $\varphi(h)=0$ whenever $\|h\|>1$.
For any continuous vector field $k$ on $\R^n$ consider the {\em
regularizations} $k^\zeta$ by setting
\begin{equation}
k^\zeta(x)=\int_{\R^n}\varphi(h)k(x+\zeta h)\,dh\,,\qquad\zeta>0\,,
\end{equation}
where $\varphi$ is a nonnegative
real-valued function
on $\rr^n$ of class $C^\infty$ such that $\int_{\rr^n}\varphi=1$
and $\varphi(h)=0$ whenever $\|h\|>1$.
\end{definition}



 In order to prove  Theorem \ref{liezero1} we need the following technical
result:

\begin{lemma}\label{regbracket}
Let $f$ and $g$ be locally Lipschitz continuous vector fields  on $\R^n$
 and let
${\cal M}\subset \R^n$ be any subset  having  measure equal to zero.\,If
$$
x[f,g]_{ord}=0 \qquad \forall
x\in \Big(DIFF(f)\cap DIFF(g)\Big)\backslash{\cal M}\, ,
$$
 then
$$
[f^{\zeta},g^{\zeta}] \to 0
$$
uniformly on compact subsets of $\R^n$.
\end{lemma}
\begin{proof} Let us  fix a compact subset $K\subset \R^n$ and let
  $x\in K$. Since in any compact neighborhood of $K$
 the derivatives $Df$ and $Dg$ are bounded (in any metric),
 it is easy to see that
\begin{eqnarray*}
[f^{\zeta},g^{\zeta}](x)=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
\int_{M\backslash \cal N}\Big[\varphi(z)[Dg(x+\zeta z)
f(x+\zeta z)- Df(x+\zeta z)g(x+\zeta z)]\Big]\,dz +
\psi(\zeta)=\psi(\zeta)
\end{eqnarray*}
 where $\psi$ is infinitesimal as $\zeta$ tends to zero. This yields the
thesis.
\end{proof}

\vskip 0.8 truecm \noindent {\it Proof of Theorem \ref{liezero1}.} Let
us show that  $i)$ implies $iv)$. Let $\bar x\in\R^n$, let $U$ be a
neighborhood of $\bar x$, and let
$$
\phi : U\mapsto \phi(U)\subset\R^n
$$
be a coordinate chart near $\bar x$. Let $(y_1,\dots, y_n)= \phi(x)$ denote  the
 coordinates of a  generic point $x$. Let $f_{|U}$ and $g_{|U}$ denote the
restrictions of $f$ and $g$ to $U$, respectively, so that
\begin{equation}
\begin{array}{c}
f_{|U}(x) = \sum_{i=1}^n f_i(x){{\partial}\over{\partial y_i}}\\
g_{|U}(x) = \sum_{i=1}^n g_i(x){{\partial}\over{\partial y_i}}
\end{array}
\end{equation}
for suitable functions $f_i(x), g_i(x)$, $i=1,\dots,n$.

Let $K$ be a $C^{\infty}$  map on $\phi(U)$ such that $K=1$ in a
compact neighborhood $Q_1$  of $\phi(\bar x)$ and $K=0$ on
$\phi(U)\backslash Q_2$, where
 $Q_2\subset\phi(U)$ is a compact neighborhood
 of
$\phi(\bar x)$  containing
 $Q_1$ in its interior.
 Then the vector fields
\begin{equation}
\begin{array}{c}
\tilde f(y) =K(y)\Big(f_1(\phi^{-1}(y)),\dots,f_n(\phi^{-1}(y))\Big)\\
\tilde g(y) =K(y)\Big(g_1(\phi^{-1}(y)),\dots,g_n(\phi^{-1}(y))\Big)
\end{array}
\end{equation}
which are defined on the open subset $\phi(U)$, can be extended to vector fields $f'$ and $g'$ on $\R^n$, respectively, by setting
\begin{equation}\label{local}
f'(y) = \left\{{\begin{array}{l} \tilde f(y)\quad \forall y\in\phi(U)\\
0\qquad \forall y\in\R^n\backslash\phi(U)\end{array}}\right.
\qquad
g'(y) = \left\{{\begin{array}{l} \tilde g(y)\quad \forall y\in\phi(U)\\
0\qquad \forall y\in\R^n\backslash\phi(U)\end{array}}\right.
\end{equation}
 Notice that $f'$ and $g'$ are (globally) bounded and  Lipschitz
 continuous. Moreover, by i) they verify
$$
[f',g']_{ord} = 0\qquad \forall y\in \R^n\backslash\phi({\cal N}\cap U)
$$
In particular, denoting
the $\zeta-$regularizations of $f'$ and $g'$ by ${f'}^{\zeta}$ and ${g'}^{\zeta}$,
respectively,the flows
$(y,\tau)\mapsto ye^{\tau f'^{\zeta}}$ and $(z,\sigma)
\mapsto ze^{\sigma g'^{\zeta}}$ can be
defined for every $\tau,\sigma\in\R$ and $y,z\in \R^n$
  and for any
$\zeta\geq 0$. Moreover it is trivial to verify that there exist
$\epsilon>0$ and a compact neighborhood $Q_0\subset Q_1$ of $\phi(\bar
x)$ such that
\begin{equation}\label{local}
ye^{\tau {f'}^{\zeta}}e^{\sigma {g'}^{\zeta}},\,\,ye^{\sigma {g'}^{\zeta}}e^{\tau {f'}^{\zeta}},\,\,\,
ye^{\tau {f'}^{\zeta}}e^{\sigma {g'}^{\zeta}}e^{-\tau {f'}^{\zeta}}e^{-\sigma {g'}^{\zeta}}\in Q_1
\end{equation}
for all $y\in Q_0$ and $\tau,\sigma,\zeta$ such that $|\tau|\leq\epsilon$, $|\sigma|\leq\epsilon$,
and  $|\zeta|\leq\epsilon$

 By applying Lemma \ref{lemmaintform}
to the
regularizations of ${f'}^{\zeta}$ and ${g'}^{\zeta}$   we obtain
 \beq\label{intformreg}
ye^{tf'^{\zeta}}e^{sg'^{\zeta}}e^{-tf'^{\zeta}}e^{-sg'^{\zeta}}-y
 = \int_0^{t}\int_0^s\Big(y\,e^{\tau
f'^{\zeta}}e^{(s-\sigma)g'^{\zeta}}[f'^{\zeta},g'^{\zeta}]
e^{\sigma g'^{\zeta}}e^{-\tau
f'^{\zeta}}e^{-sg'^{\zeta}}\Big)\,d\tau\,d\sigma
\eeq
for every $y\in Q_0$ and  $|t|,\, |s|,\,|\zeta|\leq\epsilon$
Then, letting $\zeta$ tend to zero,
 from Lemma \ref{regbracket} and from the trivial fact that the left-hand
side of (\ref{intformreg}) tends to  $y\,e^{tf}e^{sg}e^{-tf}e^{-sg}-y$
we obtain
$$
ye^{tf'}e^{sg'}e^{-tf'}e^{-sg'} = y
$$
In turn, this implies
$$
ye^{tf'}e^{sg'} = ye^{sg'}e^{tf'}
$$
which, by (\ref{local}), yields
$$
xe^{tf}e^{sg} = xe^{sg}e^{tf}
$$
for all $x\in \phi^{-1}(Q_0).$

In order to show that $ii)$ follows from
 $iv)$,
let us interpret it  in local
coordinates ---which is made possible by the same kind
of arguments as in the previous point. Then, the fact that
    $ii)$ follows from
 $iv)$ is a straightforward consequence of
 the classic asymptotic formula
\beq\label{asymptform} x\,e^{tf}e^{tg}e^{-tf}e^{-tg}-x = t^2[f,g](x) +
o(t^2)\quad \text{as} t\downarrow 0 \eeq which is easily seen to hold
true at each point $x\in DIFF(f)\cap DIFF(g)$.

Of course $ii)$ implies  $i)$ and is equivalent to $iii)$.

Finally, by Lemma \ref{lemmacommsmall} iv) is equivalent to v), so the
 theorem is proved.








\section{A Chow's theorem with non-smooth vector fields}\label{sec-main}

Chow's theorem is perhaps the most classic higher order
controllability result. In this section we shall state a version of
Chow' s theorem which generalizes the
 classic one in two directions. The first, major generalization, consists
in
 considering (set-valued) $q$-th degree Lie brackets of vector fields of
class
$C^{q-1,1}_{loc}$. The second, minor, generalization consists in allowing
some of
the vector fields to be just {\it continuous}, so
that, in general,
 the corresponding flows turn out to be  set-valued. As is well-known the
proof  of Chow's theorem relies on the
application of an open mapping theorem to suitable compositions of the
involved flows.
 Of course, this requires the application of some
chain rule.
Similarly,  the version of Chow's theorem we are going to state are based
on an open mapping theorem.
 As a matter of fact we shall
 rely on  the theory of Generalized Differential Quotients, which has been
introduced by one
 of the authors in  \cite{susssmf2000}.
This  differentiation theory,
which will be here briefly recalled, apply to  non-smooth, set-valued
maps, so,
 in particular, set-valued flows of continuous vector fields can be
considered. Furthermore, the set-valued Lie brackets defined in the
sections above allow one to build
 Generalized Differential Quotients of suitable multi-flows and to apply an open
 mapping theorem.




%The
%result stated below involves
%the notion of a q-th degree ``Lie bracket''  of
% vector fields $g_1,...,g_q$ belonging to  $C^{q-2,1}_{loc}(\Omega)$
%In order to
%\mbox{understand}
%the statement, all the reader needs to know at this point
%is that for every $x$ \mbox{belonging} to
%the domain of  $g_1,...,g_q$ (the evaluation at $x$ of) a   q-th degree
%bracket of these vector fields is a  nonempty
%{\em set} of vectors, rather than a {\em single} vector as in the
%classical
%notion of Lie bracket of vector fields of class $C^{q-1}$.  The precise
%definition   is
%given in Definition \ref{DEFlie}  below, and is such that at each point
%%it is
%is always a compact and convex (nonempty)  subset of $\rr^n$.


     \vskip 0.7truecm
In order to make the statement of this controllability result clearer
 let us introduce some additional notation. Let $q$ be an integer greater
than $2$.
 Corresponding to
a set of
  vector fields $g_1,...,g_r$ on $M$ of class $C^{q-1,1}_{loc}$,
for every $p=2,...,q-1$ let   $V_{p}$ denote
the set of all
(classic) $p$-th degree Lie
brackets  of   $g_1,\ldots,g_r$. Let
 ${W}_q$
    denote the family of all the (set-valued) $q$-th degree Lie
brackets
(at $x$) of the vector fields $g_1,...,g_r$.

 Notice that the elements of $V_{p}$ are vector fields of class
$C^{q-p,1}_{loc}$, while   ${W}_q$ is a family of set-valued vector
fields.   Since  ${W}_q$
 is finite, say it contains $s$ elements, we can order it and
write
${W}_q=\{Z_1,\dots,Z_s\}$.

 For every $x\in M$ and every $p=2,...q$, let $V_{p}(x)\subset T_xM$ denote the set of the
evaluations of the elements of $V_{p}$ at $x$, and let
$Z_1(x),\dots,Z_s(x)$  denote  the (set-valued) images of $x$ of
 $Z_1,\dots,Z_s$, respectively. Finally let us set
$$
{W}_q(x)= \{Z_1(x),\dots,Z_s(x)\}
$$



\begin{theorem}\label{LCST}
Let $f_1,\ldots,f_m,g_1,\ldots,g_r$, $m,r\geq 0$,
vector fields on $M$. Let $f_1,\ldots,f_m$
be  continuous, and let
 $g_1,\ldots,g_r$ be of class  $C^{q-1,1}_{loc}$,
for an integer number  $q\geq 1$.

Let $\Sigma$ be the driftless control system on $M$
$$
\Sigma:\qquad \dot x=\sum_{i=1}^mu_if_i(x)+
\sum_{j=1}^rv_jg_j(x)\,,
$$
with control constraints $|u_i|\le 1$ for $i=1,\ldots,m$
and $|v_j|\le 1$ for $j=1,\ldots,r$. Let $x_*$ be a point of $M$
such that, {\em for all  choices}
$$
(z_1,...,z_s)\in
Z_1(x_*)\times\dots\times Z_s(x_*),
$$
 the set of vectors
$$
\{f_1(x_*),..,f_m(x_*)\}\cup
\{g_1(x_*),..,g_r(x_*)\}
\cup V_1(x_*)\cup..\cup V_{q-1}(x_*)\cup \{z_1,..,z_s\}
$$
linearly spans $T_{x_*}M$.

Then $\Sigma$ is locally controllable from
$x_*$ in small time. More precisely, there exists a
\mbox{positive} constant
$A$ having the property that for every \mbox{sufficiently} small
$\eps$ it is possible to reach every point within a {\it distance}
$\le\eps$ from $x_*$ in time not \mbox{exceeding} $A{\eps}^{1\over q}$
by means
of a piecewise constant bang-bang
 \mbox{controls} such that at each
time $t$ only one of the
 numbers
$$u_1(t),\ldots,u_m(t), v_1(t),\ldots,
v_r(t)
$$
 is nonzero.

\end{theorem}

\begin{remark} The statement might appear meaningless, for a notion of
distance is used without having specified which Riemannian structure
we are referring
 to. However the statement is meaningful because it has a local
character, and all Riemannian metrics are locally equivalent (actually,
only the constant $A$ will depend on the chosen  Riemannian metric).
\end{remark}
\vskip 0.7 truecm The remaining part of this paper will be devoted to
set up some tools to prove  Theorem \ref{LCST}. Actually, this proof,
which will be given in the last section,  will rely
 on a classic scheme: i) first, one constructs suitable
 compositions of local flows; ii) secondly one differentiates these
composed maps; iii)
 finally, one applies an open mapping theorem.
Though this scheme is standard, its pursue requires  new tools and
some technical,
 intermediate results. To begin with, in the next section we
briefly recall some basic facts  concerning  {\it Generalized Differential
Quotients}.
Next (see Sections \ref{GDQ of multiflow} and \ref{Proof of Theorem})  we
shall devote our attention to the construction of Generalized Differential
Quotients
 of suitable compositions of local flows. Also, we shall find Generalized
Differential Quotients
 for the flows of the continuous vector
fields $f_1,..f_m$.
By applying chain rules and a suitable open
 mapping theorem,  in the last section  we shall give the actual proof of
Theorem \ref{LCST}.

\section{Generalized Differential Quotients}\label{sec-gdq}

In \cite{} one of the authors has introduced the notion of Generalized
Differential Quotient. Let us recall some facts concerning this
concept.

 A {\em set-valued map} (abbr. SVM) is a triple $F=(A,B,G)$
such that $A$ and $B$ are sets and $G$ is a subset of $A\times B$. The
sets $A$, $B$, $G$ are, respectively, the {\em source}, {\em target},
and {\em graph} of $F$, and we write $A=\so(F)$, $B=\ta(F)$,
$G=\gr(F)$. If $x$ is any object, we write
\mbox{$F(x)=\{y:(x,y)\in\gr(F)\}$}. (Hence $F(x)=\emptyset$ unless
\mbox{$x\in\so(F)$}.)  The sets
$\dom(F)=\{x\in\so(F):F(x)\neq\emptyset\}$,
$\im(F)=\bigcup_{x\in\so(F)}F(x)$, are, respectively, the {\bem
domain} and {\em image} of $F$. If $F=(A,B,G)$ is an SVM, we say that
$F$ is an {\em SVM from $A$ to $B$}, and write $F:A\tto B$.  We use
$SVM(A,B)$ to denote the set of all SVMs {}from $A$ to $B$.


If $X$, $Y$ are metric spaces, then
$SVM_{comp}(X,Y)$ will denote the subset of
$SVM(X,Y)$ whose \mbox{members} are
the set-valued maps from $X$ to $Y$
that have a \mbox{compact} graph.
We say that a sequence  $\{F_j\}_{j\in\nn}$ of
\mbox{members} of $SVM_{comp}(X,Y)$
{\bem inward graph-converges} to an $F\in SVM_{comp}(X,Y)$---and
write $F_j\togr F$---if
for every open subset $\Omega$ of $X\times Y$ such that
$\gr(F)\subseteq\Omega$ there exists a
$j_\Omega\in\nn$ such that $\gr(F_j)\subseteq\Omega$
whenever $j\ge j_\Omega$.

\begin{definition}\label{defregu}
Assume that $X$, $Y$ are  metric spaces.
A {\bem regular set-valued map} from $X$ to $Y$ is a set-valued map
$F\in{}SVM(X,Y)$ such that
\begin{itemize}
\vspace{-6pt}
\item{} for every compact subset $K$ of $X$, the restriction
$F\restr K$ of $F$ to $K$ belongs to
$SVM_{comp}(K,Y)$  and is a limit---in the sense of inward
graph-convergence---of a sequence of continuous
single-valued maps from $K$ to $Y$.
\end{itemize}
\vspace{-6pt}\noindent We use $\regu{X}{Y}$ to denote the set of all
regular \mbox{set-valued}  maps from $X$ to $Y$.
\end{definition}
It is easy to see that if $F:X\mapsto Y$ is a single-valued and
everywhere defined map, then $F$ belongs to $\regu{X}{Y}$ if and only
if $F$ is continuous.

Moreover, one has
\begin{theorem}\label{comporeg-13}
Assume that $X$, $Y$, $Z$ are metric \mbox{spaces}. Let
\mbox{$F\in\regu{X}{Y}$}, $G\in\regu{Y}{Z}$. Then the \mbox{composite}
map $G\circ F$ belongs to $\regu{X}{Z}$.
\end{theorem}



\vspace{-8pt}
\begin{definition}\label{defquasi}
Let $m,n\in\zz_+$, let $F:\rr^m\tto\rr^n$ be a set-valued
map, and let $\Lambda$ be a nonempty compact subset of $\rr^{n\times m}$.
Let $S$ be a subset of  $\rr^m$.
We say that $\Lambda$ is a {\bem generalized differential quotient}
(abbreviated ``GDQ'') {\bem of
$F$ at $(0,0)$ in the direction of $S$}, and write
$\Lambda\in  GDQ(F;0,0;S)$,
if for every positive real number $\delta$ there exist $U$, $G$ such that
\ben
\vspace{-6pt}
\item[1.] $U$  is a  compact neighborhood of $0$ in $\rr^m$ and $U\cap S$
is compact;
\vspace{-8pt}
\item[2.] $G$  is a  regular set-valued map from $U\cap S$ to the
$\delta$-neighborhood  $\Lambda^\delta$ of $\Lambda$ in $\rr^{n\times m}$;

\vspace{-8pt} \item[3.] $G(x)\cdot x\subseteq F(x)$ for every $x\in
U\cap S$. \een
\end{definition}

\noindent If $X$, $Y$ are real linear spaces, we use ${\rm Lin}(X,Y)$
to \mbox{denote}
the set of all linear maps from $X$ to $Y$. A {\em linear multimap}
{}from $X$ to $Y$ is a subset of ${\rm Lin}(X,Y)$.
A {\em linear multimultimap}
{}from $X$ to $Y$ is a set of linear multimaps from $X$ to $Y$.

If $M$, $N$ are $C^1$ manifolds,
\mbox{$\bar x\!\in\! M$, $\bar y\!\in\! N$, $S\!\subseteq \!M$},
and \mbox{$F:M\tto N$}, then we can define a linear \mbox{multimultimap}
$GDQ(F;\bar x,\bar y;S)$ from $T_{\bar x}M$ to $T_{\bar y}N$
by picking coordinate charts \mbox{$M\ni x\to\xi(x)\in\rr^m$},
\mbox{$N\ni y\to\eta(y)\in\rr^n$}---where \mbox{$m={\rm dim}\,M$,
$n={\rm dim}\,N$}---defined near $\bar x$, $\bar y$ and such
that $\xi(\bar x)=0$, $\eta(\bar y)=0$, and
declaring a subset $\Lambda$ of ${\rm Lin}(T_{\bar x}M,T_{\bar y}N)$
to \mbox{belong} to $GDQ(F;\bar x,\bar y;S)$  if
the multimap
$D\eta(\bar y)\circ\Lambda\circ D\xi(\bar x)^{-1}$ is in $GDQ(\eta\circ
F\circ\xi^{-1};0,0;\xi(S))$.  It turns out that, with this definition,
{\it the set $GDQ(F;\bar x,\bar y;S)$  does not depend on the choice
of the charts $\xi$, $\eta$.} Moreover, the following  results hold true:

%%%%%%%%%%HECTOR%%%%%%%%%%%%%%%%
\vskip 1truecm {\bf HECTOR

WHICH ARE THE EXACT REFERENCES FOR THE FOLLOWING 3 THEOREMS? }\vskip
1truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{theorem}\label{contidif}
If $F:\rr^m\mapsto\rr^n$ is a continuous map, $x\in\rr^m$, and $F$ is
classically differentiable at $x$, then $\{DF(x)\}\in
GDQ(F;x,F(x);\rr^m)$.
\end{theorem}

\begin{theorem}\label{conticlark}
If $F:\rr^m\mapsto\rr^n$ is Lipschitz-continuous, and $x\in\rr^m$,
then the Clarke generalized Jacobian $\partial F(x)$ belongs to
$GDQ(F;x,F(x);\rr^m)$.
\end{theorem}

\begin{theorem}\label{hasomp}
GDQ has the strong directional open mapping property with linear rate
(SDOMPLR).
\end{theorem}
The meaning of the SDOMPLR is as follows:
\medbreak
\noindent{\bf (SDOMPLR)}\ {\em
Assume that $n,m$ are nonnegative \mbox{integers},
$F:\rr^m\tto\rr^n$,  $v\!\in\!\rr^n$,
\mbox{$C$ is a closed} \mbox{convex} cone in $\rr^m$,
$\Lambda$ \mbox{belongs} to $GDQ(F;0,0;C)$, and
\mbox{$v\!\in\!\bigcap_{L\in\Lambda}{\rm Int}(LC)$}. Then
there exist a closed convex cone $D$ in $\rr^n$
and a function}
$$
]\,0,+\infty[\,\ni\delta\!\mapsto\!\eps(\delta)\in\,]\,0,+\infty[
$$
{\em such that $v\in{\rm Int}(D)$, the \mbox{function}
\mbox{$\delta\!\mapsto\!\eps(\delta)$} is linear for \mbox{small}
enough $\delta$}, {\em and} \ben \item[(*)] {\em if $y\!\in\! D$ and
$\|y\|\!\le\!\eps(\delta)$ then there \mbox{exists} a compact
connected subset $Z_y$ of
\mbox{$\{x\!\in\!C\!:\!\|x\|\!\le\!\delta\}\!\times\![0,1]$} such that
\mbox{$(0,0)\!\in\!Z_y$}, $(x,1)\!\in\! Z_y$ for some $x\in C$ such
that $\|x\|\le\delta$, and $ry\!\in\!F(x)$ whenever
$0\!\le\!r\!\le\!1$ and $(x,r)\!\in\!Z_y\,$.} \een


\vskip 1truecm

\section{  Generalized Differential Quotients of
flows' compositions} \label{GDQ of multiflow}

\subsection{Continuous vector fields}\label{notprem}

The occurrence of the continuous vector fields $f_1,\dots,f_m$ in
Theorem \ref{LCST} poses the question of  building a GDQ for the
set-valued flow of a continuous vector field.

\noindent
Let us begin by introducing suitable linear maps, which will be utilized
in the next session as well:
\vskip 1 truecm

For a given $w\in\rr^n$, let $Q^w$ denote the linear map

\vspace{-4mm}
$$
\rr^{n+1}\sim\rr^n\times\rr\ni(v,r)\mapsto
v+rw\equaldef Q^w(v,r)\in\rr^n\,.
$$

If $W$ is a subset of $\rr^n$, write
$$
\mathcal{Q}^W\equaldef \{Q^w: w\in W\}
$$
\vskip 1 truecm

Let $f$ be  a continuous vector field
on $M$, and  let us consider   its (local) flow
\beq\label{flow}
(x,t)\to xe^{tf}
\eeq
 For $x$ in a compact neighborhood $K$
of a given point $x_*$, this flow  turns out to be defined on $K\times
]-\varepsilon,\varepsilon[$, for a suitable $\varepsilon>0$. Since $f$
is merely continuous, in general this flow is a {\it set-valued} map.
More precisely, we set
$$
xe^{tf} = \left\{ x=x(t) \,\,\,|\,\,\, x(\cdot)
\quad\hbox{solves}\quad \dot x=f\quad x(0) = x\right\}
$$


Let us notice that if $f$ were a $C^1$ map its flow would be continuously
 differentiable and the map $Q^{f(x_*)}$ would be nothing but its
differential
 at $(x,t)=(x_*,0)$. It is remarkable  that a similar fact occurs when $f$
is just continuous.
 In fact, the following theorem was proved in \cite{ramsus1}.

\begin{theorem}\label{diffocont}
 The singleton
${\cal Q}^{f(x_*)}$ is a Generalized Differential Quotient of the flow
$xe^{tf}$ at $((x_*,0),x_*)$ in the direction of $M$.
\end{theorem}



\begin{remark}\label{univalued}
Despite the fact that   at each point   the flow $(x,t)\mapsto xe^{tX}$
possesses   a
Generalized Differential Quotient
consisting of a (continuous) singleton, this map  is
  not differentiable,
 actually it is not even a single-valued map. On the other hand, it can be
proved that a
{\it  single-valued} map $F$ having a singleton $\{v\}$ as GDQ at a point
$(y,F(y)$ is in fact differentiable at $y$, and $F'(y)=v$.
\end{remark}

\subsection{Higher degree Lie brackets as GDQ's of flows' compositions}
We are now going to address an issue connected with  GDQ's of
compositions of the $g_i$'s flows.

Let $h_1,\dots,h_{q+1}$ be vector fields on $M$ of class $C^{q-1,1}$.
We wish to show that for every $x_*\in M$ and  for every Lie bracket
$L$ of the fields $h_1,\dots,h_{q+1}$   of degree $s\leq q+1$
 there is a  suitable composition of flows
having ${\mathcal{Q}}^{L(x_*)}$ as a Generalized Differential Quotient
at $((x_*,0),x_*)$. Notice that $L(x_*)$
 is a singleton as soon as  $s<q+1$.

In view of Proposition \ref{nonsmooth-(q+1)-lie}, we shall limit ourselves
to Lie  brackets of the form
$$
[[..[h_{i_1},h_{i_2}],..],h_{i_s}],
$$ for all $s\le q$, where $i_1,...i_s$ are positive integers less
than or equal to $q+1$. In fact, these are the only brackets we shall need in
the
proof of Theorem \ref{LCST} (see Section \ref{Proof-of-main})

\vskip 0.8 truecm

\begin{definition} Let $s\geq 2$ be an integer and let
 $h_1,\dots,h_s$ be
locally  Lipschitz continuous  vector fields on $M$. We recursively define
the map
$$
(x,t_1,\dots,t_s) \to x\psi^{h_1,\dots,h_{s}}(t_1,\dots,t_{s})
$$
as follows:

i) for $s=2$, let us set,for any $x\in\rr^n$ and for all $(t_1,t_2)$
in a sufficiently small neighborhood of $(0,0)$,
$$
x\psi^{h_1,h_2}(t_1,t_2) = xe^{-t_2h_2}e^{t_1h_1}e^{t_2h_2}e^{-t_1h_1}.
$$

ii) for $s>2$
  we set
\begin{eqnarray*}
& x\psi^{h_1,\dots,h_{s}}(t_1,\dots,t_{s})=\\
& xe^{-t_{s}h_{s}}
\psi^{h_1,..,h_{s-1}}(t_1,..,t_{s-1})e^{t_{s}h_{q-1}}
\left(\psi^{h_1,..,h_{s-1}}(t_1,..,t_{s-1})\right)^{-1},
\end{eqnarray*}
where
$$
y\mapsto y\left(\psi^{h_1,..,h_{s-1}}(t_1,..,t_{s-1})\right)^{-1}
$$
 denotes the inverse of the map
$$
x\mapsto  x\psi^{h_1,..,h_{s-1}}(t_1,..,t_{s-1}).
$$
\end{definition}
Let us also define the map $(x,\epsilon)\to
x\Xi^{h_1,\dots,h_s}(\epsilon)$
by letting
$$
x\Xi^{h_1,\dots,h_s}(\epsilon)=
\left\{\begin{array}{lll}
x\psi^{h_1,\dots,h_s}({\eps}^{1\over s},\dots,{\eps}^{1\over s})&{\rm
if}&\eps\ge 0\,,\\
x\Big(\psi^{h_1,\dots,h_s}((-\eps)^{1\over
s},\dots,(-\eps)^{1\over s})\Big)^{-1}&{\rm if}&\eps\le 0\,.
\end{array}\right.
$$
For every $x\in\Omega$ the domain of
 $\epsilon\to x\Xi^{h_1,\dots,h_s}(\epsilon)$ is an $x$-dependent
 neighborhood of zero. Of course,
the map  $(x,\epsilon)\to
x\Xi^{h_1,\dots,h_s}(\epsilon)$  is continuous.

The following result puts in evidence  the strict  relation occurring
 between the flows
 $(x,\epsilon)\to
x\Xi^{h_1,\dots,h_s}(\epsilon)$
 and the (possibly set-valued) Lie brackets introduced above.

\vskip 0.7 truecm
\begin{theorem}\label{nonsmooth-appr-th}
Let $q\in \zz$, $q>0$,let $h_1,\dots,h_{q+1}$
be vector fields belonging to
 $C_{loc}^{q-1,1}(\Omega)$, and let $x_*\in\Omega$. Let $s\in\zz$,
$0<s\leq q+1$. Then, for every $(i_1,\dots,i_s) \in \{1,\dots,q+1\}^s$,
 the set
${\cal Q}^{[..[h_{i_1},h_{i_2}],..],h_{i_{s}}](x_{*})}$
is a Generalized Differential
Quotient of the map  $(x,\epsilon)\to
x\Xi^{h_1,\dots,h_s}(\epsilon)$ at $((x_{*},0),x_{*})$ in the direction of
$\rr^n$.
\end{theorem}

\vskip 0.7 truecm
\begin{remark}
Notice that ${\cal Q}^{[..[h_{i_1},h_{i_2}],..],h_{i_{s}}]}$ is a
singleton as soon as $s<q+1$. Since in this case the map
$\Xi^{h_{i_1},\dots,h_{i_s}}$ is univalued, it follows that --see
Remark \ref{univalued}-- it
 is differentiable at $(x_*,0)$ and its differential coincides with the
map
 $$
\rr^n\times\rr\ni(v,r)\mapsto
v+r[..[h_{i_1},h_{i_2}],..],h_{i_{s}}](x_*).
$$
\end{remark}

\section{Proof of Theorem \ref{nonsmooth-appr-th}}\label{Proof of Theorem}
 Let us observe that it is sufficient to prove Theorem
\ref{nonsmooth-appr-th}
 {\it just the case when $s=q+1$ and $(i_1,\dots,i_{q+1})=
 (1,\dots,q+1)$.}
 Moreover,  in view of the local character of the thesis,  {we are allowed
to prove the  theorem under the following, simplifying,
extra assumption}:

\vskip 0.4truecm

{\bf Hypothesis XA.}
{\it The manifold
 $M$ coincides with ${\rr}^n$, and the vector fields
 $h_1,\dots,h_{q+1}$  are of class $C_c^{q-1,1}$. This means that
they  have compact supports,   they are of class $C^{q-1}$, and
 there exists a  real number  $L>0$ such that
their derivatives   up
 to the $(q-1)$-th order are bounded by
$L>0$, and the $(q-1)$-th order derivatives are globally Lipschitz
continuous
with $L$ as a Lipschitz constant.}

\vskip 0.8truecm

\begin{remark} The fact that this hypothesis can be assumed {\it without
loss of generality}
follows from the following, standard cut-off function  arguments.
 To begin with, consider the local expression of the vector fields $h_i$
in a neighborhood of $x_*$ in a system of coordinates whose range is
an open subset $U\subset\rr^n$. Let $y_*$ be  the image of $x_*$ in
this system of coordinates and let $\eta>0$ be such that the closed
ball $B[y_*,\eta]$ is contained in $U$. Let $\xi$ be a $C^{\infty}$
map on $U$ such that $\xi=1$ on $B[y_*,{\eta\over 2}]$ and $\xi(y)=0$
for any $y\in U\backslash B[y_*,\eta]$. Let us consider
 the vector fields
$$
\tilde h_i=\xi h_i\qquad i=1,\dots,q+1
$$
(where, with  abuse of notation,  we have written $h_i$ to denote what
in fact is the local expression of $h_i$) and let us extend each of
them to all of $\rr^n$ by setting $\tilde h_i=0$ outside $\Omega$.
Then the maps $\tilde h_1, \dots,\tilde h_{q+1}$ verify {\bf(XA)}.
Moreover, for any $i=1, \dots, q+1$, the flow of $h_i$ coincides with
the flow of $\tilde h_i$ in a neighborhood of $(y_*,0)$.  So, in view
of the local character of the definition of Generalized Differential
Quotient,
 it is straightforward to check that proving
Theorem \ref{nonsmooth-appr-th} is equivalent to proving
the same theorem with the $\tilde h_i$'s in place of the
 $h_i's$.
Hence we can definitely assume hypothesis {\bf(XA)}.
\end{remark}

The proof of
 Theorem \ref{nonsmooth-appr-th} will  consist of three essential steps.
We shall begin by proving the result in the smooth case. Secondly,
we shall consider regularizations  and establish some estimates of the
distances between Lie brackets
made of  the given vector fields and those made of the regularized vector
fields.
 Finally, we shall exploit these estimates and a general approximation
property of GDQ's to prove the thesis of Theorem
\ref{nonsmooth-appr-th}.
 \ref{nonsmooth}.





\subsection{Step 1: smooth vector fields}

\begin{theorem}\label {approximation-th}
For every $q\in\nn$ there exists a function
$\phi=\phi(K,t_1,\dots,t_{q+1})$
defined on
$[0,+\infty)\times \rr^{q+1}$ such that the following hold:
\begin{itemize}
\item[1.]  for every
$(t_1,\dots,t_{q+1})\in \rr^{q+1}$
$\phi(\cdot,t_1,\dots,t_{q+1})$ is non decreasing, and
$$
\lim_{(t_1,\dots,t_{q+1})\to
0}{{\phi(K,t_1,\dots,t_{q+1})}\over{|(t_1,..., t_{q+1})|^{q+1}}} =
0\qquad \forall K\in [0,+\infty);
$$
\item[2.] If $h_1,\dots,h_{q+1}$ are vector fields of class $C^q$ and $K$
is
a
bound for the vector fields $h_1,\dots,h_{q+1}$ together with their
derivatives up to the $q$-th  order, then for every $x\in {\rr}^n$ one has
\end{itemize}
\begin{eqnarray*}
\Big\| x\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1})- x
 -
t_1\cdot...\cdot
t_{q+1}[...[[h_1,h_2],h_3],...,h_{q+1}](x)\Big\|\\
\leq
\phi(K,t_1,\dots,t_{q+1})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\qquad\,\,\,\,&.
\end{eqnarray*}
\end{theorem}
\vs {\bf HECTOR: this was the point where I had found a problem, but
now it seems to me that it can be easily proved by Taylor's expansion}
\vs
\subsection{Step 2: regularizations}
\label{nonsmooth}


Let $h_1,\dots,h_{q+1}$ be vector fields verifying hypothesis {(\bf
XA)}, and
 let us consider the regularizations
$h_i^\zeta$ introduced in Definition \ref{regularization}.

Then, for every  $i=1,\dots,q$, and  for all positive $\zeta$,
 $h_i^\zeta$ is of class  $C^\infty$
  Moreover, by the elementary properties of
the convolution product and by hypothesis {(\bf XA)},
  all  vector fields $h_i^{\zeta}$
and their derivatives {\it up to the $(q)$-th order} are bounded by the
constant
$L>0$.
 Furthermore, we have Proposition \ref{derivative convercence} below,
 where  $\phi$  denotes any
derivative of order less than or equal to $q-1$ of a  vector field
$h_i$ and  $\phi^\zeta$ stand  for  the corresponding derivative of
the
 $\zeta$-regularization $h_i^\zeta$. (By elementary properties of convolutions
 $\phi^\zeta$ coincides with the
 $\zeta$-regularization of
$\phi$.)

 \begin{itproposition}\label{derivative convercence}
 If {\bf (XA)} is assumed,  then  ($\phi$ is continuous and)
there exists a constant $C$
 such that
\begin{equation}\label{derivative convergence}
\|\phi^\zeta(x) - \phi(x)\| \leq C\zeta
\end{equation}
for any $x\in\rr^n$ and $\zeta>0$.
\end{itproposition}
\vs


\vskip 1truecm

As a basic step to prove  Theorem \ref{nonsmooth-appr-th}, the next
proposition provides an estimate of
 the distance
$$
d\Big([\dots[h_1^\zeta,h_2^\zeta],\dots],h^\zeta_{q+1}]_{ord}(x)\,\,,\,\,
[\dots[h_1,h_2],\dots],h_{q+1}](y)\Big).
$$



\begin{itproposition}\label{distanza}
Let us assume hypothesis {\bf(XA)}. Then there is a constant $M>0$
such that for all $x,y\in \rr^n$ such that $\|x-y\|\leq 1$  and any
positive $\zeta$ one has
\begin{eqnarray}\label{dist-reg-bracket}
d\Big([[\dots[h_1^\zeta,h_2^\zeta],\dots],h_{q+1}^\zeta]_{ord}(x)\,\,,\,\,
[[\dots[h_1,h_2],\dots,]h_{q+1}](y)\Big)\\
\leq \gamma_y (\|x-y| + \zeta) + M(\zeta + \zeta^q)\!\!\!
\end{eqnarray}
\end{itproposition}
\vs

In order to prove this proposition we need to introduce the functions
$A^{q-1}$ below.

Let $i$ be a positive integer, and let $P$ a multi-linear map from
$(\rr^n)^i$ into $\rr^n$. Let us  denote the image of an element
$(v_1,\dots,v_i)\in (\rr^n)^i$ by
$$
P\Big<v_1,\dots,v_i\Big>.
$$
 Let  $X$ be a vector field differentiable up to the $i$-th order
at a point $x\in\rr^n$, let $D^iX(x)$ designate the $i-th$ order
 differential of $X$ at $x$.
In particular,  $D^iX(x)$ is a multi-linear map on $\rr^i$ whose
entries are suitable  $i$-th order partial derivatives of $X$ at $x$.

\vskip 0.8truecm

 By
 Rademacher's theorem, the vector
fields $h_1,\dots,h_{q+1}$ have derivatives up to the $q$-th order on
the complement of a null set $E$.

  For any $x\in \rr^n\backslash E$ let us set
\begin{eqnarray*}
&\,&A^{q-1}(h_1,\dots,h_{q+1})(x) = \\
&\,& [\dots[[h_1,h_2],...,]h_{q+1}] \\
&-& (-1)^qD^{q}h_1(x)\Big<h_2(x),h_3(x),\dots,h_{q+1}(x) \Big>\\
&-& (-1)^{q-1}D^{q}h_2(x)\Big<h_1(x),h_3(x),\dots,h_{q+1}(x) \Big>
\end{eqnarray*}

It is just a matter of elementary computation to prove the following
lemma.
\begin{lemma}\label{functional dep}Let us assume hypothesis {\bf (XA)}.
For any  $x\in \rr^n\backslash E$, $$A^{q-1}(h_1,\dots,h_{q+1})(x)$$
is a polynomial of the variables
$$
h_1(x),..,h_{q+1}(x),D^1h_1(x),..,D^{1}h_{q+1}(x),..,D^{q-1}h_1(x),..,
D^{q-1}h_{q+1}(x).
$$
\end{lemma}

In particular, since the vector fields $h_i$ are of class $C^{q-1}$ ,
we obtain: \begin{corollary} The map
$$
x\to A^{q-1}(h_1,\dots,h_{q+1})(x)
$$
 has a unique continuous extension
to all of $\rr^n$, which will be still denoted by
$A^{q-1}(h_1,\dots,h_{q+1})(\cdot)$ \vskip 0.5truecm
\end{corollary}


Next convergence result  follows trivially
 by   Lemma \ref{functional dep}  and  Proposition
\ref{derivative convercence}:

\begin{lemma}\label{a-conv}Let us assume hypothesis {\bf (XA)}. Then,
 there exists a
constant $C'$ such that
 $$
\| A^{q-1}(h_1^\zeta,\dots,h^\zeta_{q+1})(x) -
A^{q-1}(h_1,\dots,h_{q+1})(x)\| \leq C' \zeta
$$
for any $x\in\rr^n$ and $\zeta>0$.
 \end{lemma}

\vskip 1 truecm

 For any $y\in\rr^n$ let us set

$$
\gamma_y(\rho)=\sup_x d\Big([\dots[h_1,h_2],\dots],h_{q+1}](y)\,\,
,\,\, [\dots[h_1,h_2],\dots],h_{q+1}]_{ord}(x)\Big)
$$
where the supremum is taken over the set  $(\rr^n\backslash E)\cap
B[y,\rho]$.

It  is easy to see that
$$
\lim_{\rho\downarrow 0}\gamma_y(\rho)=0\,.
$$

 {\it Proof of Proposition \ref{distanza}.} Let $E\subset\R^n$ be a subset of measure zero such
 that the vector fields $h_1,...,h_{q+1}$ are differentiable up to the
 $q$-th order.
For any $x\in\rr^n$ and $\zeta>0$, let us define
    the zero-measure subset

$$
E_{x,\zeta}= \{z\in\rr^n\,\,:\,\,\, x+\zeta z\in E\}.
$$

Then
\begin{eqnarray*}
&\,&[[\dots[h_1^\zeta,h_2^\zeta],\dots,]h_{q+1}^\zeta](x) =
A^q(h^\zeta_1,\dots,h^\zeta_{q+1})(x) + \\
&\,&\int_{\rr^n\backslash E_{x,\zeta}}\Big(\varphi(z)(-1)^qD^qh_1(x+\zeta
z)
\Big<(h_2^\zeta(x),h_3\zeta(x),
\dots,h_{q+1}^\zeta(x)\Big>\Big) dz +\\
&\,& \int_{\rr^n\backslash E_{x,\zeta}
}\Big(\varphi(z)(-1)^{q-1}D^qh_2(x+\zeta z)
\Big<h_1^\zeta(x),h_3^\zeta(x),
\dots,h_{q+1}^\zeta(x)\Big>\Big) dz\\
&\,&=
(I)_\zeta(x) + (II)_\zeta(x) + (III)_\zeta(x)\, ,
\end{eqnarray*}
 where
$$
(I)_\zeta = \int_{\rr^n\backslash E_{x,\zeta}}
\varphi(z)[[\dots[h_1,h_2],\dots,]h_{q+1}]_{ord}(x+\zeta z)dz \, ,
$$
$$
(II)_\zeta(x) = \int_{\rr^n\backslash E_{x,\zeta}}
\varphi(z)\Big(A^q(h^\zeta_1,\dots,h^\zeta_{q+1})(x) -
A^q(h_1,\dots,h_{q+1})(x+\zeta z)\Big) dz\, ,
$$
and
\begin{eqnarray*}
(III)_\zeta(x)=
\int_{\rr^n\backslash E_{x,\zeta}}\Big(\varphi(z)(-1)^qD^qh_1(x+\zeta z)
\Big<(h_2^\zeta(x),h_3^\zeta(x),\dots,h_{q+1}^\zeta(x))-\\
(h_2(x+\zeta z),h_3(x+\zeta z),\dots,h_{q+1}(x+\zeta z))\Big> \Big) dz +\\
\int_{\rr^n\backslash E_{x,\zeta}}\Big(\varphi(z)(-1)^{q-1}D^qh_2(x+\zeta
z)
\Big<(h_1^\zeta(x),h_3^\zeta(x),\dots,h_{q+1}^\zeta(x))- \\
(h_1(x+\zeta z),h_3(x+\zeta z),\dots,h_{q+1}(x+\zeta z))\Big> \Big) dz\, .
\end{eqnarray*}

Hence
\begin{eqnarray*}
d\Big([[\dots[h_1^\zeta,h_2^\zeta],\dots],h_{q+1}^\zeta](x),
[[\dots[h_1,h_2],\dots],h_{q+1}](y)\Big)\\
\leq
d\Big((I)_\zeta\,,\,[[\dots[h_1,h_2],\dots],h_{q+1}](y)\Big) +
\|(II)_\zeta\| + \|(III)_\zeta\|
\end{eqnarray*}


Now $(I)_\zeta$ is a convex combination of vectors
$[[\dots[h_1,h_2],\dots,]h_{q+1}](z)$ with $z\in B[x,\zeta]\backslash E$,
 therefore
$$
d\Big((I)_\zeta\, , \,[[\dots[h_1,h_2],\dots],h_{q+1}](y)\Big)
 \leq \gamma_y(\zeta+\|x-y\|).
$$
Moreover,  by hypothesis {\bf (XA)},
the functions $A^q(h_1,\dots,h_{q+1})(\cdot)$,
$h_1(\cdot),$ $\dots$, $h_{q+1}(\cdot)$ turn out to be
 Lipschitz continuous, say with a Lipschitz
 constant $G$. Finally, since the $(q-1)$-th order derivatives of the
fields
$h_i$ are Lipschitz continuous (with constant $L$), it follows that
there exists a constant $G'$ such that
$$
\|D^qh_i(z)\Big<w_1,\dots, w_{q}\Big>\| \leq  G' \|w_1,\dots,w_q\|^q
$$
for all $z\in\rr^n\backslash E$ and all  $(w_1,\dots,w_q)\in \rr^q$.
Hence, in view of Lemma \ref{a-conv} and of (\ref{derivative
convergence}), one has
$$
\|(II)_\zeta\| \leq \Big(G + C'\Big)\zeta
$$
and
$$
\|(III)_\zeta\|\leq G" \zeta^q,
$$
for a suitable $G''$, Hence, (\ref{dist-reg-bracket}) is proved, with
$M=\sup\{G", G'+ C\}$. $\diamond$ \vskip 1truecm

\subsection{Proving Theorem \ref{nonsmooth-appr-th}}
{\it Proof of Theorem \ref{nonsmooth-appr-th}.}
Notice that if the vector fields $h_i$
verify {\bf(XA)}, then the same holds true for their regularizations
$h_i^\zeta$ with the same constant $L$.
 Hence we can apply Theorem \ref{approximation-th} to the set of
vector fields $\{h_1^\zeta, \dots, h_{q+1}^\zeta\}$, for all $\zeta>0$.

Therefore, if $t_1\cdot\dots\cdot t_{q+1} >0$, we have
\begin{eqnarray}\label{quasi}
d\Big((t_1\cdot\dots\cdot t_{q+1})^{-1}\cdot \Big(x\psi^{h_1^\zeta,
\dots, h_{q+1}^\zeta}(t_1,\dots,t_{q+1}) - x\Big)\,\,,\,\,
[[\dots [h_1,h_2].\dots, h_{q+1}](x_*)\Big) \nonumber \\
 \leq d\Big((t_1\cdot\dots\cdot t_{q+1})^{-1}\cdot \Big(x\psi^{h_1^\zeta,
\dots, h_{q+1}^\zeta}(t_1,\dots,t_{q+1}) - x\Big)\,\,,\,\,
[[\dots [h_1^\zeta,h_2^\zeta].\dots, h_{q+1}^\zeta]\Big)  + \nonumber \\
d\Big([[\dots[h_1^\zeta,h_2^\zeta],\dots],h_{q+1}^\zeta](x)\,\,,\,\,
[[\dots[h_1,h_2],\dots,]h_{q+1}](x_*)\Big)  \nonumber\\
\leq
 (t_1\cdot\dots\cdot t_{q+1})^{-1}\phi(L,t_1,\dots,t_{q+1})\,\,\,\,
+ \gamma_{x_*} (\|x-x_*| + \zeta) + M(\zeta + \zeta^q),
\end{eqnarray}
so, letting $\zeta$ go to zero, we obtain
 \begin{eqnarray}
d\Big((t_1\cdot\dots\cdot t_{q+1})^{-1}\cdot \Big(x\psi^{h_1,
\dots, h_{q+1}}(t_1,\dots,t_{q+1}) - x\Big)\,\,,\,\,
[[\dots [h_1,h_2].\dots, h_{q+1}](x_*)\Big) \nonumber \\
\leq \gamma_{x_*} (\|x-x_*\|)
 +
 (t_1\cdot\dots\cdot t_{q+1})^{-1}\phi(L,t_1,\dots,t_{q+1}).
\,\,\,\,\label{quasiquasi}
\end{eqnarray}
In turn, (\ref{quasiquasi}) implies
\begin{eqnarray}
&\,&\inf\Big\{\,\|x\Xi^{h_1,\dots,h_{q+1}}(\eps)-x-\eps v\|:v\in
[[\dots [h_1,h_2].\dots, h_{q+1}](x_*)\,
\Big\}\nonumber\\
&\,&\quad =o(\eps+\|x-x_*\|)\,.\label{quasiquasiquasi}
\end{eqnarray}
i.e.,
 \begin{eqnarray}
&\,&\inf\Big\{\,\|x\Xi^{h_1,\dots,h_{q+1}}(\eps)-
\Xi^{h_1,\dots,h_{q+1}}(x_*,0)-
M(x-x_*,\eps)\|\nonumber\\
&\,&\quad :M\in\mathcal{Q}^{[[\dots [h_1,h_2].\dots,
h_{q+1}](x_*)}\,\Big\}
=o(\eps+\|x-x_*\|)\,.\label{quasissimo}
\end{eqnarray}
Hence the thesis follows from
the
\mbox{general} sufficient condition given in \cite{sussncn2000}
for a set to be a GDQ.

%%%%%%%%%%%%%%%%%%%%%%%%%HECTOR%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1truecm {\bf HECTOR

Perhaps it is better to write down the sufficient condition of
 \cite{sussncn2000}
 we have  mentioned in the last line of this  proof}
\vskip 1.5truecm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%HECTOR%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{
 Proof of Theorem \ref{LCST}}\label{Proof-of-main}

{\it Proof of Theorem \ref{LCST}.}
If  $q>2$,  for any integer $p$ such that $2\leq p\leq q-1$ let us choose
a subsets
$V'_p\subset V_p$
such that the following hold:
\begin{itemize}
{\it \item[1. ] Each element of $V'_p$   is of the form
$$
[..[g_{i_1},[g_{i_2},[......],g_{i_p}]
$$
for some $(i_1,\dots,i_p)\in \{1,\dots r\}^p$.

\item[2.] The elements of $V'_p$    are linearly independent.

\item[3.] Every element of $V_p$  is a linear combination of elements
of $V'_p$ .}
\end{itemize}
Moreover, if $q>1$  let us choose  a subset  $W'_q =
\{Z'_1,\dots,Z'_{s'}\}\subset W_q$
such that the following hold:
\begin{itemize}
{\it
\item[1$_q$. ]  Each element of $Z'_q$   is of the form
$$
[..[g_{i_1},[g_{i_2},[......],g_{i_q}]
$$
for some $(i_1,\dots,i_q)\in \{1,\dots r\}^q$.

\item[2$_q$. ]  The elements of  $W'_q$   are linearly independent.
This means that there exists a selection   $(\zeta_1,\dots,\zeta_s)\in
Z_1\times \dots\times Z_s$ such that if
$a_1\zeta_1+\dots+a_s\zeta_s=0$ then all coefficients $a_i$ have to be
zero.

\item[3$_q$. ] Every element $Z_i$  of $W_q$ is a linear combination
of elements of  $W'_q$. This means that if $z\in Z_i$ then there exist
real coefficients $a_1,\dots,a_{s'}$ and a selection
$(\zeta'_1,\dots,\zeta'_{s'})\in Z'_1\times\dots\times Z'_{s'}$ such
that
$$
z = a_1{\zeta'}_1+\dots+a_{s'}{\zeta'}_{s'}
$$.
}\end{itemize}

In view of Proposition \ref{smooth-q-lie} and
\ref{nonsmooth-(q+1)-lie}  such subsets $V'_p$ and $W'_q$ do exist,
though the choice is not  unique.
\vskip 8truemm

It is trivial to verify that proving Theorem \ref{LCST} is equivalent to
proving the
result below.

\begin{theorem}\label{LCST'}
Under the same assumptions on the vector fields $f_1,\dots,f_m$,
$g_1,\dots,g_r$ as in  Theorem \ref{LCST}, let $x_*$ be a point of $M$
such that, {\em for all  choices}
$$
(z'_1,...,z'_{s'})\in
Z'_1(x_*)\times\dots\times Z'_{s'}(x_*),
$$
 the set of vectors
$$
\{f_1(x_*),..,f_m(x_*)\}\cup
\{g_1(x_*),..,g_r(x_*)\}
\cup V'_1(x_*)\cup..\cup V'_{q-1}(x_*)\cup \{z_1,..,z_{s'}\}
$$
linearly spans $T_{x_*}M$.

Then $\Sigma$ is locally controllable from
$x_*$ in small time. More precisely, there exists a
\mbox{positive} constant
$A$ having the property that for every \mbox{sufficiently} small
$\eps$ it is possible to reach every point within a distance
$\le\eps$ from $x_*$ in time not \mbox{exceeding} $A{\eps}^{1\over q}$
by means
of a piecewise constant bang-bang
 \mbox{controls} such that at each
time $t$ only one of the
 numbers
$$u_1(t),\ldots,u_m(t), v_1(t),\ldots,
v_r(t)
$$
 is nonzero.

\end{theorem}


\begin{proof}
%For any integer $p$ such that $2\leq p\leq q-1]$ let $n(p)$ be the
cardinality of $V'_p$ and let $n(q)$
%be the cardinality of $W'_q$.
%Let us choose an order in any of the sets
${V'}_1,\dots,{V'}_{q-1},{W'}_q$. For
%every $p\in[2,q-1]$  let us denote the elements
%of ${V'}_p$ by $X_{p,i}$, $i=1,\dots, n(p)$, and let us denote the
elements of
%${W'}_q$ by $X_{q,,j}$. Observe that
% $X_{p,i}$ is a vector field if $p<q$, while $X_{q,,j}$ is a set-valued
vector
%field.
Let $\omega(\in \Z)$ the cardinality of the set
$$
{\cal W} = {V'}_2\cup\dots\cup{V'}_{q-1}\cup{W'}_q
$$
Let us choose an order on $\cal W$ in such a way   we can write
$$
{\cal W} = \{Y_1, \dots,Y_{\omega}\}
$$
and
$$
{W'}_q =  \{Z'_1,\dots,Z'_{s'}\} = \{Y_{{\omega-s'}+1},\dots,Y_{\omega}\},
$$

Hence  there is a one-to-one function which maps any index $j =
1,...,\omega$
 into
$$
(p(j),g_{i_1}^j,\dots,g_{i_{p(j)}}^j),
$$
where  $2\leq p(j)\leq q$ and
$(g_{i_1}^j,\dots,g_{i_{p(j)}}^j)\in \{g_1,\dots,g_r\}^{p(j)}$, and such
that
$$
Y_j = [..[g_{i_1}^j,\dots],g_{i_{p(j)}^j}]
$$
For every integer $j$ such that $1\leq j\leq \omega$ let us also consider
the map
$$
F_j:(x,\epsilon)\to
xF_j(\epsilon) = x\Xi^{g_{i_1}^j,......,g_{i_p(j)}^j}(\epsilon)
$$
Furthermore, let us define the  map $\cal F$ from
$\rr^m\times\rr^r\times\rr^{\omega}$ into $\rr^n$
 by setting
$$\begin{array}{c}
{\cal F}
(\alpha_1,\dots,\alpha_m,\beta_1,\dots,\beta_r,\gamma_1,\dots,\gamma_{\omega})
= \\
\,\\
x_{*}e^{\alpha_1f_1}\dots e^{\alpha_1f_m}e^{\beta_1g_1}\dots
e^{\beta_rg_r}F_1(\gamma_1)\dots F_{\omega}(\gamma_{\omega})
\end{array}
$$
Finally, for any choice
$$
(z'_1,...,z'_{s'})\in
 Y_{{\omega-s'}+1}\times\dots\times Y_{\omega} ,
$$
let $L^{(z'_1,...,z'_{s'})}$ be the linear functional  from
$\rr^m\times\rr^r\times\rr^{\omega}$ into $\rr^n$
defined by
$$
\begin{array}{c}
L^{(z'_1,...,z'_{s'})}(\alpha,\beta,\gamma) = \\
\sum_{i=1}^m \alpha_i f_i(x_{*}) + \sum_{j=1}^r \beta_j g_j(x_{*}) +
\sum_{l=1}^{\omega-s'} \gamma_l Y_l(x_{*}) + \sum_{p=1}^{s'} \gamma_{p
+\omega-s'} z'_{p}
\end{array}
$$

Now, by Theorem \ref{diffocont} the singletons ${\cal Q}^{\{f_i(x)\}}$,
$i=1,\dots,m$, and ${\cal Q}^{\{g_j(x)\}}$,
$j=1,\dots,r$, are Generalized Differential Quotients (at $((x,0),0)$) of
the maps
$$
(x,\epsilon)\mapsto xe^{\epsilon
f_i}\quad\,\,\,\,\,\,\hbox{and}\,\,\,\,\,\quad
(x,\epsilon)\mapsto xe^{\epsilon g_j},
$$
respectively. Moreover,  Theorem  \ref{nonsmooth-appr-th} implies that
for any  $j=1,...,\omega$  ${\cal Q}^{F_s(x)}$ is a Generalized
Differential Quotient (at $((x,0),0)$) of the map
$$
(x,\epsilon)\mapsto xF_j(\epsilon)
$$

Therefore the chain rule for GDQ's (see \cite{...})implies that the family
of linear functionals
$$\Big\{L^{(z'_1,...,z'_{s'})},\,\, (z'_1,...,z'_{s'})\in
 Y_{{\omega-s'}+1}\times\dots\times Y_{\omega}\Big\}
$$
is a GDQ of $\cal F$ at $((0,0,0),x_{*})$ in the direction of $\rr^n$.
Hence the desired conclusion follows from the open mapping result stated
in Theorem \ref{hasomp}.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}

\vupp
\bibitem{kawski}{Kawski,~M., andH.J.~Sussmann,
``Noncommutative power series and formal Lie-algebraic
techniques in nonlinear control theory.''  In
\textit{Operators, \mbox{Systems} and Linear Algebra: Three Decades
of Algebraic Systems Theory, Kaiserslautern, 1997},
U.~Helmke, D.~Praetzel-Wolters, E.~Zerz Eds.;
B.~G.~Teubner Stuttgart, 1997, pp~111-129.}

\vupp
\bibitem{pontrybook}{Pontryagin,~L.S., V.G.~Boltyanskii,
\mbox{R.~V.} \mbox{Gamkrelidze}
and E.F.Mischenko, \textit{The \mbox{Mathematical} Theory of Optimal
Processes.} Wiley, New York, 1962.}

\vupp

\bibitem{ramsus1}{Rampazzo,~F., Sussmann,~H.J., ``Set-valued differentials
and a nonsmooth version of Chow's theorem.'' In \textit{Proc. 40th IEEE
Conference on Decision and Control; Orlando, Florida, Dec. 2001}}.

UPDATE!

\vupp
\bibitem{sussshyb}{Sussmann,~H.J., ``A maximum principle
for \mbox{hybrid} optimal control problems.'' In \textit{Proc. 38th IEEE
Conf. Decision and Control, Phoenix, AZ, Dec. 1999}.  IEEE
publications, New York, 1999, \mbox{pp.~425--430}.}

\vupp
\bibitem{susssmf2000}{Sussmann,~H.J., ``R\'esultats r\'ecents
sur les courbes optimales.''  In \textit{15$\,^e$ Journ\'ee Annuelle
de la Soci\'et\'e Math\'emathique de France (SMF)}, Publications de
la SMF, Paris, 2000, pp.~1-52.}

\vupp
\bibitem{sussncn2000}{Sussmann,~H.J., ``New theories of set-valued
\mbox{differentials}
and new versions of the maximum \mbox{principle} of optimal control
theory.'' In \textit{Nonlinear Control in the year 2000},
A.~Isidori, F.~Lamnabhi-Lagarrigue and  W.~Respondek Eds.,
Springer-Verlag, London, 2000, pp.~487-526.}

\vupp
\bibitem{susssyd2000}{Sussmann,~H.J.,
``Set-valued differentials and the hybrid maximum principle.''
In \textit{Proc. 39th IEEE Conf. Decision
and Control, Sydney, \mbox{Australia}, \mbox{December} 12-15, 2000}, IEEE
\mbox{publications}, \mbox{New York}, 2000.}
\end{thebibliography}

\end{document}







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%sul teorema 8.2%%%%%%%%%%%%%%%%%%%
\begin{proof}
The proof follows directly from point 2. in   Proposition
\ref{approximation-prop}
 below. Indeed, if $\|y_q(\sigma,\tau)\|$, $k_q$,and $\tilde \phi$ are
as in the statement of Proposition \ref{approximation-prop}, then
 in view of the
boundedness hypotheses there exists a constant $L$ such that
\begin{eqnarray*}
\Big\|y_q(\sigma,\tau)[...[[h_1,h_2],h_3]...],h_{q+1}]e^{\tau
k_q}e^{\sigma h_{q+1}}
e^{-(t_1\cdot\dots\cdot t_q)k_q} -\\
x[...[[h_1,h_2],h_3]...],h_{q+1}]\Big\| \leq L \|t_1\cdot\dots\cdot
t_q|,
\end{eqnarray*}
which yields the thesis, with
$$
\phi (K,t_1,\dots,t_{q+1})=\tilde\phi(K,t_1,\dots,t_{q+1}) +
L\|t_1\cdot\dots\cdot t_q\|^2\|t_{q+1}\|
$$
\end{proof}




\begin{itproposition}\label{approximation-prop}
For every $q\in\nn$ there exists a function
$\tilde\phi=\tilde\phi(K,t_1,\dots,t_{q+1})$ defined on
$[0,+\infty)\times \rr^{q+1}$ such that the following hold:
\begin{itemize}
\item[1.]  for every $(t_1,\dots,t_{q+1})\in \rr^{q+1}$
$\tilde\phi(\cdot,t_1,\dots,t_{q+1})$ is non decreasing, and
$$
\lim_{(t_1,\dots,t_{q+1})\to
0}{{\tilde\phi(K,t_1,\dots,t_{q+1})}\over{|(t_1,..., t_{q+1})|^{q+1}}}
=0;
$$
\item[2.] If the vector fields $h_1,\dots,h_{q+1}$ are of class  $C^q$
and $K$ is a bound for the vector fields $h_1,\dots,h_{q+1}$ together
with their derivatives up to the $q$-th  order, then for every $x\in
\rr^n$ one has
\end{itemize}
$$
\begin{array}{l}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Big| x\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) - x
 - \\
\int^{(t_1\cdot...\cdot t_{q})}_0 \int_0^{t_{q+1}} \Big(
y_q(\sigma,\tau)[...[[h_1,h_2],h_3],...],h_{q+1}] \, e^{\tau k_q}
e^{\sigma h_{q+1}} e^{-(t_1\cdot...\cdot
t_{q})k_q}\Big)\,d\tau\,d\sigma\,\Big|
\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
 \leq  \tilde\phi(K,t_1,\dots,t_{q+1})
\end{array}
$$
with
$$k_q= [...[[h_1,h_2],h_3],...],h_{q}]$$
and
$$
y_q(\sigma,\tau)=xe^{-\sigma h_{q+1}} \,e^{(t_1\cdot...\cdot
t_{q}-\tau)k_q}.
$$
\begin{itemize}
\item[3.] If the vector fields $h_1,\dots,h_{q+1}$ are as in point 2.
\, and $w$ is any vector field,
 then for every $x\in
\rr^n$ one has
\end{itemize}
$$
\begin{array}{l}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Big| xw\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) - xw
 - \\
\int^{(t_1\cdot...\cdot t_{q})}_0 \int_0^{t_{q+1}} \Big(
w_q(\sigma,\tau)[...[[h_1,h_2],h_3],...],h_{q+1}] \, e^{\tau k_q}
e^{\sigma h_{q+1}} e^{-(t_1\cdot...\cdot
t_{q})k_q}\Big)\,d\tau\,d\sigma\,\Big|
\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\
 \leq  \tilde\phi(K,t_1,\dots,t_{q+1})
\end{array}
$$
where
$$
 w_q(\sigma,\tau)
=xwe^{-\sigma h_{q+1}} \,e^{(t_1\cdot...\cdot t_{q}-\tau)k_q}
$$
\end{itproposition}
\begin{proof}
Let us prove the theorem by induction on $q$. The case when $q=1$ has
been already proved, with $\tilde\phi = 0$, for it
 fits the hypotheses of Lemma \ref{lemmaintform} as soon as one sets
$f=-h_2$, $g=h_1$, and $(t,s)=(t_1,t_2)$.

Let us assume that the thesis holds true for a  $q\geq 1$ and let us
show that it holds true for $q+1$ as well.

%Since the derivatives up to the $q+1$-th order of the vector fields
%$h_1,\dots,h_{q+2}$
%are assumed to be  bounded by a constant $K$,
%it is trivial to check that there exists a constant $K'$ such that for
%any bracket  $h$ of degree {\it less than $q+2$} the flow
%$$
%(x,s)\to xe^{th}
%$$ is defined for all $(x,s)\in\rr^n\times\rr$
%and verifies
%\begin{equation}\label{bound}
%\|xe^{sh} -x\|\leq K'|s|.
%\end{equation}


Let us set
$$
k_{q+1} = [...[[h_1,h_2],h_3],...],h_{q+1}],
$$
and let  $\Pi$ be  either the identity map on $\rr^n$ (so that $x\Pi =
x$) or the vector field $w$ of point 3. (so that $x\Pi = xw$).
 This position allows us to  avoid the repetition of similar
passages in the proofs of  2.  and 3. One has
$$
 x\Pi\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) -
x\Pi\,e^{(t_1\cdot\dots\cdot t_{q+1})h_{q+1}} = A_1 +A_2
$$
where
$$
 \begin{array}{l}
A_1 =  \Big(x\Pi\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) - x\Pi -
\\
\int^{(t_1\cdot...\cdot t_{q})}_0 \int_0^{t_{q+1}} \Big(
y_q(\sigma,\tau)\Pi[...[[h_1,h_2],h_3],...],h_{q+1}] \, e^{\tau k_q}
e^{\sigma h_{q+1}} e^{-(t_1\cdot...\cdot
t_{q})k_q}\Big)\,d\tau\,d\sigma\,
\Big)\\
A_2= \Big(\int^{(t_1\cdot...\cdot t_{q})}_0 \int_0^{t_{q+1}} \Big(
y_q(\sigma,\tau)\Pi[...[[h_1,h_2],h_3],...],h_{q+1}] \, e^{\tau k_q}
e^{\sigma h_{q+1}}
e^{-(t_1\cdot...\cdot t_{q})k_q}\Big)\,d\tau\,d\sigma\, \\
+x\Pi - x\Pi\,e^{(t_1\cdot\dots\cdot t_{q+1})h_{q+1}}\Big)\,\,=
 \end{array}
$$
Now, by the inductive hypothesis one has
$$
|A_1| \leq  \tilde\phi(K,t_1,\dots,t_{q+1}),
$$
while
$$
A_2 = \int^{t_{q+1}}_0 \int_0^{(t_1\cdot...\cdot t_{q})}
F\Big(\tau,\sigma,t_1\cdot...\cdot t_{q}\Big)d\tau d\sigma,
$$
where
$$
 F\Big(\tau,\sigma,r\Big) =
 \Big(x\Pi e^{-\sigma h_{q+1}}e^{(r-\tau)k_q}k_{q+1}
 e^{\tau k_q}
e^{\sigma h_{q+1}} e^{-rk_q}-xe^{r\sigma k_{q+1}}k_{q+1} - \Big)
$$
Observe that there exists a positive increasing  function $L:\rr\to
\rr$
 such that the  Lipschitz constants of $h_{q+1}, k_q, k_q+1$ are bounded
by $L(K)$. Hence it is easy to check that
 the integral
$$
\int^{t_{q+1}}_0 \int_0^{(t_1\cdot...\cdot t_{q})}
F\Big(\tau,\sigma,t_1\cdot...\cdot t_{q}\Big)d\tau d\sigma
$$
is bounded by a map $I(K,t_1,\dots,t_{q+1})$ such that
$$
\lim_{(t_1,\dots,t_{q+1})\to
0}{{I(K,t_1,\dots,t_{q+1})}\over{|(t_1,..., t_{q+1})|^{q+1}}} =0.
$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%$$
%F_1 = \Big(xe^{-\sigma h_{q+1}}e^{(t_1\cdot...\cdot t_{q})k_q}
%e^{\sigma h_{q+1}}
%e^{-(t_1\cdot...\cdot t_{q})k_q}k_{q+1}-xe^{(t_1\cdot...\cdot
%t_{q})\sigma k_{q+1}}\Big)
%$$
%and
%$$
%F_2 =
%xe^{-\sigma h_{q+1}}e^{(t_1\cdot...\cdot t_{q}-\tau)k_q}}\Big(k_{q+1}
% e^{\tau k_q}
%e^{\sigma h_{q+1}}
%e^{-(t_1\cdot...\cdot t_{q})k_q}- e^{\tau k_q}
%e^{\sigma h_{q+1}}
%e^{-(t_1\cdot...\cdot t_{q})k_q}k_{q+1}\Big)
%$$



%\tilde\phi(K,t_1,\dots,t_{q+1})\,\,+ \\
%\int^{t_{q+1}}_0
%\int_0^{(t_1\cdot...\cdot t_{q})}
%\Big(xe^{-\sigma h_{q+1}}e^{(t_1\cdot...\cdot t_{q}-\tau)k_q}k_{q+1}
% e^{\tau k_q}
%e^{\sigma h_{q+1}}
%e^{-(t_1\cdot...\cdot t_{q})k_q}-xe^{(t_1\cdot...\cdot t_{q}\sigma k_{q+1}}\Big)\,d\tau\,d\sigma\,\,\,\,\,\,\,\,
%=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
%\\
%\int^{t_{q+1}}_0
%\int_0^{(t_1\cdot...\cdot t_{q})}
%\Big(xe^{-\sigma h_{q+1}}e^{(t_1\cdot...\cdot t_{q})k_q}
%e^{\sigma h_{q+1}}
%e^{-(t_1\cdot...\cdot t_{q})k_q}k_{q+1}-xe^{(t_1\cdot...\cdot t_{q})\sigma k_{q+1}}\Big)\,d\tau\,d\sigma  \,\,\,
%\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\
%\int^{t_{q+1}}_0
%\int_0^{(t_1\cdot...\cdot t_{q})}
%xe^{-\sigma h_{q+1}}e^{(t_1\cdot...\cdot t_{q}-\tau)k_q}}\Big(k_{q+1}
% e^{\tau k_q}
%e^{\sigma h_{q+1}}
%e^{-(t_1\cdot...\cdot t_{q})k_q}- e^{\tau k_q}
%e^{\sigma h_{q+1}}
%e^{-(t_1\cdot...\cdot t_{q})k_q}k_{q+1}\Big)\,d\tau\,d\sigma\\
% + \tilde\phi(K,t_1,\dots,t_{q+1}).
%\end{eqnarray*}
%In view of
%Lemma \ref{lemmaintform} $F_1$ is clearly infinitesimal as $(t_1,..., t_{q})$ goes to zero, uniformly
% with respect to $x$. In order to check that $F_2$ tends to zero as well,
let us set
%$$
%y = xe^{-\sigma h_{q+1}}e^{(t_1\cdot...\cdot t_{q}-\tau)k_q}}
%$$
%so that
%$$
%F_2 =
%\Big(yk_{q+1}e^{\tau k_q}e^\sigma h_{q+1}e^{-(t_1\cdot\dots\cdott_q)k_q} - yk_{q+1}\Big) -\\
%\Big( ye^{\tau k_q}e^\sigma h_{q+1}e^{-(t_1\cdot\dots\cdott_q)k_q}k_{q+1} -yk_{q+1}\Big),
%$$
%It is easy to check that the  two terms tend to zero,
%in that  $k_{q+1}$ is Lipschitz continuous, and both $h_{q+1}$ and
%$k_q$ have uniformly bounded derivatives.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As a consequence,
\begin{equation}\label{formula 1}
\| x\Pi\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) -
x\Pi\,e^{(t_1\cdot\dots\cdot t_{q+1})k_{q+1}}\| \leq \hat\phi(K,t
_1,\dots,t_{q+1})
\end{equation}
where,
 $\hat \phi$ is defined by
$$
 \hat\phi(K,t
_1,\dots,t_{q+1}) = I(K,t _1,\dots,t_{q+1})  + \tilde\phi(K,t
_1,\dots,t_{q+1}).
$$
so that it verifies
$$
\lim_{(t_1,\dots,t_{q+1})\to
0}{{\hat\phi(K,t_1,\dots,t_{q+1})}\over{|(t_1,..., t_{q+1})|^{q+1}}}
=0.
$$
Let us set
 $$
\psi_{q+1} = \psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}),
$$
which gives
$$
\begin{array}{c}
 x\Pi\psi^{h_1,\dots,h_{q+2}}(t_1,\dots,t_{q+2}) - x\Pi =
x\Pi e^{-t_{q+2}h_{q+2}}\psi_{q+1}e^{t_{q+2}h_{q+2}}(\psi_{q+1})^{-1}
- \Pi x =\\ x\Pi e^{-t_{q+2}h_{q+2}}e^{(t_1\cdot\dots\cdot
t_{q+1})k_{q+1}}e^{t_{q+2}h_{q+2}}(\psi_{q+1})^{-1} - x\Pi
 +  (I)
\end{array}
$$
with
$$
(I)= x\Pi e^{-t_{q+2}h_{q+2}}(\psi_{q+1}- e^{(t_1\cdot\dots\cdot
t_{q+1})k_{q+1}})e^{t_{q+2}h_{q+2}}(\psi_{q+1})^{-1}.
$$
>From  the case when $q=1$ we then get
\beq
\begin{array}{c}
\Big|  x\Pi\psi^{h_1,\dots,h_{q+2}}(t_1,\dots,t_{q+2}) - x\Pi\\
=\int_0^{(t_1\cdot\dots\cdot t_{q+1})}\int_0^{t_{q+2}}\Big(x\Pi
e^{-\sigma h_{q+2}}e^{(t_1\cdot\dots\cdot t_{q+1}-\tau)k_{q+1}}
[\dots[h_1,h_2],\dots],h_{q+2}]\\
e^{\tau k_{q+1}}e^{\sigma h_{q+2}}e^{-(t_1\cdot\dots\cdot
t_{q+1})k_{q+1}}\Big) d\sigma\,d\tau  \,\Big| +(I)
\end{array}
\eeq In order to conclude the proof we only need to prove that there
is a function $\phi(K,t_1,\dots,t_{q+2})$ satisfying point  1. in
 the statement of the theorem and such that
$$
(I)\leq \phi(K,t_1,\dots,t_{q+2})
$$
Now
\begin{eqnarray}
(I)= \int_0^{t_{q+2}}\Big(-x\Pi e^{-\sigma h_{q+2}} h_{q+2}(\psi_{q+1}
-e^{(t_1\cdot\dots\cdot t_{q+1})k_{q+1}})e^{\sigma h_{q+2}}\Big)
(\psi_{q+1})^{-1}
d\sigma\label{stimaI} \\
+\int_0^{t_{q+2}}\Big(x\Pi e^{-\sigma h_{q+2}} (\psi_{q+1}
-e^{(t_1\cdot\dots\cdot t_{q+1})k_{q+1}}) h_{q+2}e^{\sigma
h_{q+2}}\Big) (\psi_{q+1})^{-1} d\sigma\nonumber
\end{eqnarray}
Now observe that  (\ref{formula 1}) maintain validity   if  $x$ is
replaced by
 any $y\in\rr^n$ and consider the special case when $w=k_{q+2}$. It
follows  that
$$
\| y h_{q+2}(\psi_{q+1} -e^{(t_1\cdot\dots\cdot
t_{q+1})k_{q+1}})\|\leq C \hat\phi(K,t_1\dots,t_{q+1})
$$
and
$$
\| y (\psi_{q+1} -e^{(t_1\cdot,\dots\cdot t_{q+1})k_{q+1}})\|\leq
\hat\phi(K,t_1,\dots,t_{q+1})
$$
for any $y\in\rr^n$.

These two estimates and (\ref{stimaI}) allow us to conclude that there
exists a constant $N$ such that
\begin{equation}\label{stimaI'}
(I)\leq N |t_{q+2}|\hat\phi(K,t_1,\dots,t_{q+1})
\end{equation}
which is the desired estimate, with
$$
\phi(K,t_1,\dots,t_{q+2}) = N |t_{q+2}|\hat\phi(K,t_1,\dots,t_{q+1}).
$$
The proposition is thus proved.
\end{proof}


