%From mathgsto@adele.rutgers.edu Fri Nov 29 17:11:00 2002
%From: Math Guest O <mathgsto@adele.rutgers.edu>
%To: "Hector J. Sussmann" <sussmann@adele.rutgers.edu>
%Subject: file you own from now on

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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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%                  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 atnd 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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\vspace{3cm}

{\Large  

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This is the back of {\huge Page \arabic{pagono}}
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$\rr^n
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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\\
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Piscataway, NJ 08854-8019, USA\\
{\small\tt sussmann@hamilton.rutgers.edu}
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H\'ector J. Sussmann\\
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Department of Mathematics; Rutgers, the State University of New Jersey\\
Hill Center---Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ
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E-mail: sussmann@hamilton.rutgers.edu\quad\quad
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{\bf
Set-valued differentials and a nonsmooth version of Chow's 
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\begin{document}

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\footnotetext[1]{The second author was supported in part by
NSF Grant DMS98-03411-00798.  He is grateful to the
Weizmann Institute Institute of Science, in Rehovot, Israel, where 
most of the work leading to this paper was
done,  while he was a Weston Visiting Professor during the Fall
of 2000.}

\vskip4.5cm

\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 word  {\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 Lipschitz continuous at $x$} 
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 
neighbourhoods $U$ 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. (In that case,
$\psi\circ F\circ\varphi^{{-1}}$ is Lipschitz continuous near $\varphi(x)$ 
for {\em every} pair $(\varphi,\psi)$.)  We call $F$ {\it 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.\proofend
\end{remark}

The following version of Rademacher's Theorem on 
manifolds is an obvious consequence of the standard
version on Euclidean spaces. (Recall that the concept of a ``null subset''
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 
$M\backslash E$.\proofend
\end{theorem}
    
\subsection{Notation on manifolds}
In  \cite {kawski}  the classical 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 a set of formal rules which highly simplify some 
computations involving vector fields,
brackets, flows and their derivatives, an so on. Since we shall use this 
notation in the next sections, 
for the sake of selfconsistency,  let
 us recall some basic facts from \cite{kawski}.
 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 1truecm
{\bf HECTOR  here I have inserted  pages 2-3 of your 
paper wth 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}}
We wish to extend to the case of locally Lipschitz continuous vector fields
the notion of Lie bracket of two vector fields of class $C^1$. Since the 
resulting ``Lie bracket'' $[f,g](x)$ of two vector fields
$f$, $g$, at a point $x$, will not necessarily  agree with the ordinary 
Lie bracket $[f,g](x)$ whenever the latter is defined (i.e., 
whenever $f$ and $g$ are classically differentiable at $x$),
we will use $[f,g]_{ord}(x)$ to denote the ordinary Lie bracket 
of $f$ and $g$ at $x$. That is, $[f,g]_{ord}(x)$ is defined 
to be the differential operator 
${\cal E}\ni\varphi\mapsto (fg\varphi-gf\varphi)(x)\in\rr$, 
acting on test functions $\varphi\in{\cal E}$. (Under the hypothesis
that $f$ and $g$ are classically differentiable at $x$, the functions
$g\varphi$ and $f\varphi$ are differentiable at $x$, so the directional
derivatives $f(g\varphi)(x)$ and $g(f\varphi)(x)$ exist.) It follows
that 
$$
x[f,g]_{ord} =  x(fg-gf)\,, 
$$
in the sense that, if we write
$x\sigma$, $xV$, rather than $\sigma(x)$, $V(x)$, 
for the value  at $x$ of a (possibly 
partially defined) function $\sigma$ or 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. \proofend
\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 $x\in
M$.  Hence in this case we will use of the notation $[f,g]$ for the
ordinary bracket, instead of writing $[f,g]_{ord}$.\proofend
\end{remark}


\subsection{Multivalued iterated brackets}\label{sectliemon}

If  $F$ is a family of $C^{\infty}$ vector fields,
let us say that a vector field $h\in F$ is  {\it a Lie 
bracket of degree one} and that
any  standard Lie bracket is {\it a Lie bracket of degree 2}. Moreover
we say that the Lie bracket of a Lie bracket of degree $i$
and a Lie bracket of degree $j$ is {\it a Lie bracket of degree $i+j$}.
So, 
 it remains inductively defined what a {\it  Lie bracket of degree r } is, 
for any
natural number $r$.




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.
 
{\bf Multivalued Lie bracket of higher degree.}
If $p$ is an integer such that $1\leq p \leq q$ and $h_1,\dots,h_p$ are 
$C^{q-1}$ vector fields, the notion of bracket
of degree $p$ remains unchanged.  On the other hand, if  $p=q+1$,  the 
notion of  a $(q+1)$-th degree bracket
 is now natural: indeed  
all  brackets having a degree less than $q+1$ turn out to be locally 
Lipschitz
continuous vector fields, so Definition \ref{DEFlie} applies.

 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 fiels $h_i$ can be weakened.
For instance, in order to define $[[h_1,h_2],h_3]$ 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 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 $q>2$ (or, more generally, when the $g_i$'s 
are twice differentiable), 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-simmetry, many of these 
 Lie brackets are identically equal to zero)
\end{itproposition}
As an example, it is straightforward to verify that
$$
[[h_1,h_2],[h_3,h_4]]= -[[[h_3,h_4],h_1],h_2] + [[[h_3,h_4],h_2],h_1]
$$ 
 By arguing as in the case of the Jacobi's identity one easily obtains 
the following {\it nonsmooth} extension of the foregoing 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-simmetry, many of these 
 Lie brackets are identically equal to singleton $\{0\}$).
\end{itproposition}
Notice that the nonsmooth version of the foregoing example reads as 
follows:
$$
[[h_1,h_2],[h_3,h_4]](x) \subseteq -[[[h_3,h_4],h_1],h_2](x) +
 [[[h_3,h_4],h_2],h_1](x).
$$                  

\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
$[f,g]=0$ 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 
\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 the following result for smooth vector fields.

\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 $\min(s ,0)\leq \sigma \leq\max(s ,0)$ the identity
\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
\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 $x[f,g]+o(1)$ 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 example, that $[f,g]$ vanishes if the flows
of $f$ and $g$ commute. On the other hand, (\ref{integralformula}) is
interesting in itself, because that it gives an {\em exact integral
formula} rather that an asymptotic result such as (\ref{esti}).
This will enable us to use (\ref{integralformula}) to do things that
cannot be done with (\ref{esti}), such as, for example, writing 
(\ref{integralformula}) for vector fields $f_j$, $g_j$ instead of
$f$, $g$, and taking the limit as $j\to\infty$.\proofend
\end{remark}
 \vskip 0.8 truecm

\begin{proof} (Already sketched in \cite{ramsus1}.)
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

\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 neighbourhood 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 neighbourhood 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 neighbourhood $Q_1$  of
$\phi(\bar x)$ and $K=0$ on $\phi(U)\backslash Q_2$, where
 $Q_2\subset\phi(U)$ is a compact neighbourhood 
 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 and 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 neighbourhood 
$Q_0\subset Q_1$ 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  $y\,e^{tf}e^{sg}e^{-tf}e^{-sg}-x$ 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  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  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}
 
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 multivalued $q-th$ 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  multivalued. 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.
Moreover the set-valued Lie brackets defined in the above sections allow 
one to build 
 Generalized Differential Quotients of suitable multi-flows.
 



%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 the 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 
$C^{p-q,1}_{loc}$,
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 $x\in M$, let $V_{p}(x)\subset T_xM$ denote the set of the 
evaluations
        Date of this version: August 30, 2001
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\vspace{3cm}

{\Large  

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This is the back of {\huge Page \arabic{pagono}}
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$\rr^n
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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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H\'ector J. Sussmann\\
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Set-valued differentials and a nonsmooth version of Chow's 
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of the elements of $V_{p}$ at $x$, and let $Z_1(x),\dots,Z_s(x)$  be the 
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 specifed which Riemannian structure we are 
referring
 to. However the statement is meaningful beacause 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 the needed 
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 differentiate these 
composed maps; iii)
 finally, one applies an open mapping theorem.
Though this scheme is standard, its pursue requires severals 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}

\noindent 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 $Y$. \proofend
\end{definition}
It is easy to see that if $F:X\mapsto Y$ is an ordinary 
(that is, single-valued and everywhere defined) map, then
$F$ belongs to $\regu{X}{Y}$ if and only if $F$ is continuous.

It is not hard to prove the following.
\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 $\regu{X}{Z}$.\proofend
\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 $x\in U\cap S$.\proofend
\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
HECTOR

EXACT REFERENCE OF 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
$\{DF(x)\}\in GDQ(F;x,F(x);\rr^m)$.\proofend
\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 $GDQ(F;x,F(x);\rr^m)$.\proofend
\end{theorem}

\begin{theorem}\label{hasomp}
GDQ has the strong directional open mapping property with linear 
rate (SDOMPLR).\proofend
\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 
$(x,r)\!\in\!Z_y\,$.}\proofend
\een


\vskip 1truecm

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

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

The occurence of the continuous vector fields $f_1,\dots,f_m$ in Theorem 
\ref{LCST} lead us to build 
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 neighbourhood $K$ 
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.
 

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.tex}.

\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 shall now address an issue connected eith the more involved part of of 
Theorem\ref{LCST}'s proof,
 namely the one concerning the vector fields  $g_i$. 

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
${\mathcal{Q}}^L$ as a Generalized Differential Quotient at
$((x_*,0),x_*)$. Notice that $L$ 
 is a single-valued map 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 $s\le q$. 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 neighbourhood
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_{s-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
 neighbourhood 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 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 tha 
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}
 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 neighbourhood 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 neighbourhood of $(y_*,0)$.  So, 
in 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  consists 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}.will be proved in subsection
 \ref{nonsmooth}.




 
\subsection{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
$$
\lim_{(t_1,\dots,t_{q+1})\to
0}{{\phi(K,t_1,\dots,t_{q+1})}\over{|(t_1,..., t_{q+1})|^{q+1}}} = 0;
$$
\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}

\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 constan $L$ 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 use $\Lambda$ to denote either the identity map on $\rr^n$ (so 
that $x\Lambda = x$) or the vector field $w$ (so that 
$x\Lambda = xk$).
 We make this position in order to avoid the repetition of similar 
passages to prove points 2.  and 3.  in the statement. 
By the induction hypothesis one
has
$$
\begin{array}{l}
 x\Lambda\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) - 
x\Lambda\,e^{(t_1\cdot\dots\cdot
t_{q+1})h_{q+1}} = 
 \\
 \Big(x\Lambda\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) - x\Lambda\Big)+
\Big(x\Lambda - x\Lambda\,e^{(t_1\cdot\dots\cdot
t_{q+1})h_{q+1}}\Big)\,\,= \\
\int^{t_{q+1}}_0
\int_0^{(t_1\cdot...\cdot t_{q})} F\Big(\tau,\sigma,t_1\cdot...\cdot 
t_{q}\Big)d\tau d\sigma + 
\tilde\phi(K,t_1,\dots,t_{q+1})
\end{array}
$$ 
where
$$
 F\Big(\tau,\sigma,r\Big) =
 \Big(x\Lambda e^{-\sigma h_{q+1}}e^{(r-\tau)k_q}k_{q+1}  
 e^{\tau k_q} 
e^{\sigma h_{q+1}}
e^{-rk_q}-xe^{r\sigma k_{q+1}}k_{q+1}\Big)
$$
Observe that there exists a positive increasing  function $L:\rr\to \rr$
 such that the  Lipschtz 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, uniformely
% 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 Lipscitz continuous, and both $h_{q+1}$ and
%$k_q$ have uniformly bounded derivatives.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As a consequence, 
\begin{equation}\label{formula 1}
\| x\Lambda\psi^{h_1,\dots,h_{q+1}}(t_1,\dots,t_{q+1}) -
x\Lambda\,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
$$
\begin{array}{c}
 x\Lambda\psi^{h_1,\dots,h_{q+2}}(t_1,\dots,t_{q+2}) - x\Lambda = 
x\Lambda e^{-t_{q+2}h_{q+2}}\psi_{q+1}e^{t_{q+2}h_{q+2}}(\psi_{q+1})^{-1} 
-
\Lambda x 
=\\ x\Lambda e^{-t_{q+2}h_{q+2}}e^{(t_1\cdot\dots\cdot 
t_{q+1})k_{q+1}}e^{t_{q+2}h_{q+2}}(\psi_{q+1})^{-1} - x\Lambda
 +  (I)
\end{array}
$$
with
$$
(I)=
x\Lambda e^{-t_{q+2}h_{q+2}}(\psi_{q+1}-
e^{(t_1\cdot\dots\cdot
t_{q+1})k_{q+1}})e^{t_{q+2}h_{q+2}}(\psi_{q+1})^{-1}.
$$
>From  the case when $q=1$ we then get
\beq
\begin{array}{c}
\Big|  x\Lambda\psi^{h_1,\dots,h_{q+2}}(t_1,\dots,t_{q+2}) - x\Lambda\\
=\int_0^{(t_1\cdot\dots\cdot t_{q+1})}\int_0^{t_{q+2}}\Big(x\Lambda 
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\Lambda 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\Lambda 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}

\subsection{Regularizations}
\label{nonsmooth}


Let $h_1,\dots,h_{q+1}$ be vector fields verfying 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, denoting by $\phi$  any 
derivative of order less than or equal to $q-1$ of any of the vector 
fields
$h_i$ and writing  $\phi^\zeta$ for  the corresponding derivative of the
 $\zeta$-regularization $h_i^\zeta$ ---which coincides with the
 $\zeta$-regularitaztion of
$\phi$---
 the following elementary fact holds true:
 \begin{itproposition}\label{derivative convercence}
 If {\bf (XA)} is assumed,   ($\phi$ 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}


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 having derivatives 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 polinomial 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 $C^{q-1}$ , 
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


Next convergence result  follows trivially
 by   Lemma \ref{functional dep}  and  Prtoposition 
\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
 $\gamma_y(\rho)$  denote the supremum
of the distances 
$$
d\Big([\dots[h_1,h_2],\dots],h_{q+1}](y)\,\, ,\,\,
x[\dots[h_1,h_2],\dots],h_{q+1}]_{stand}\Big) 
$$
when $x$ ranges over  $(\rr^n\backslash E)\cap B[y,\rho]$.

It  is easy to see that 
$$
\lim_{\rho\downarrow 0}\gamma_y(\rho)=0\,. 
$$
\vskip 1truecm

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



\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](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}

 \begin{proof}
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
$$
(I)_\zeta = \int_{\rr^n\backslash E_{x,\zeta}}
\varphi(z)[[\dots[h_1,h_2],\dots,]h_{q+1}]_{stand}(x+\zeta z)dz \, , 
$$
$$
(II)_\zeta(x) = \int_{\rr^n\backslash E_{x,\zeta}}
\varphi(z)\Big(A^q(h^\zeta_1,\dots,h^\zeta_{q+1})(x) -
A^q(h_1,\dots,h_{q+1})(x+\zeta z)\Big) dz\, ,
$$
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 $M=\sup\{G", G'+ C\}$.
\end{proof}
\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}
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 \gamma_{x_*} (\|x-x_*| + \zeta) + M(\zeta + \zeta^q) + 
 (t_1\cdot\dots\cdot t_{q+1})^{-1}\phi(L,t_1,\dots,t_{q+1}),\,\,\,\,
\label{quasi}
\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.
\proofend
%%%%%%%%%%%%%%%%%%%%%%%%%HECTOR%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1truecm {\bf HECTOR

Perhaps it is better to write down the sufficient condition of
 \cite{sussncn2000}
 we have  mentioned at the end of the previous 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   $(z_1,\dots,z_s)\in Z_1\times \dots\times Z_s$ such that if
$a_1z_1+\dots+a_sz_s=0$   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 $(z'_1,\dots,z'_{s'})\in 
Z'_1\times\dots\times Z'_{s'}$ such that 
$$
z = a_1{z'}_1+\dots+a_{s'}{z'}_{s'}
$$.
}\end{itemize}
Of course there is not a unique such choice of subsets $V'_p$ and  $W'_q$.
\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},
tet $x_*$ 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
$$
j\in \{j\in \Z,\,1\leq j\leq \omega\}
$$ 
 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
$$
\begin{array}{c}
L^{(z'_1,...,z'_{s'})}(\alpha,\beta,\gamma) = \\
\sum_{i=1}^m \alpha_i f_i(x_{*}) + \sum_{j=1}^r \beta_j g_j(x_{*}) + 
\sum_{l=1}^{\omega-s'} \gamma_l Y_l(x_{*})
+ \sum_{p=1}^{s'} \gamma_p z'_{s'}
\end{array}
$$

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, by Theorem  \ref{nonsmooth-appr-th} implies that 
if $j$ is an integer such that
$1\leq j \leq s'$ (resp. $s'+1\leq j\leq \omega$), then  
${\cal Q}^{F_s(x)}$ is a GDQ (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}


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

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UPDATE!

\vupp
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\end{thebibliography}

\end{document}





