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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  

\bi
\item[]
This is the back of {\huge Page \arabic{pagono}}
\item[] of the paper
\ei

\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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Set-valued differentials and a nonsmooth version of Chow's Theorem$\,^{*}$}
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\begin{document}

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\twocolumn[\titlestuff]
\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

\ssection{Introduction}
\label{sec-intro}

\noindent In this note we present a \mbox{simple} \mbox{application}
of the method \mbox{developed} by one of us in recent years to prove
general smooth, nonsmooth, high-order, and \mbox{hybrid} 
\mbox{versions} of the
\mbox{maximum} \mbox{principle} (abbr. MP) for finite-dimensional,
\mbox{deterministic}
\mbox{optimal} control \mbox{problems} without state \mbox{space}
\mbox{constraints}. As \mbox{explained} in \mbox{Sussmann}
\cite{sussshyb,susssmf2000,sussncn2000,susssyd2000}, 
such \mbox{versions} can be \mbox{derived} in a \mbox{unified} way, by using
a \mbox{modified} version of the \mbox{approach} of the classical book
\cite{pontrybook} by Pontryagin {\it et al.}. In the classical approach,
one
\mbox{constructs} ``\mbox{packets} of needle \mbox{variations},'' 
\mbox{linearly} \mbox{approximates} these
\mbox{packets} at the base \mbox{value} of the \mbox{variation} parameter,
and \mbox{propagates} the \mbox{resulting} \mbox{linear} approximations to the
terminal point of the \mbox{trajectory} by means of the \mbox{differentials}
of the \mbox{reference} flow maps.  This technique
must be \mbox{modified} by \mbox{(a)~replacing}
the \mbox{classical} differential by
other \mbox{objects}, called {\em \mbox{generalized}
\mbox{differentials}}
(abbr. GDs), \mbox{(b)~replacing} the time-varying vector fields that
\mbox{occur} in the classical MP by {\em flows}, and (c)~replacing
the \mbox{needle} variations by {\em \mbox{abstract}
variations}.  A \mbox{notion} of GD will yield a \mbox{version} of the MP
provided it \mbox{satisfies}
some \mbox{natural} properties such as the chain rule
and an \mbox{appropriate} ``\mbox{directional} open mapping property.''
(\mbox{Details} are provided in
\cite{sussshyb,susssmf2000,sussncn2000,susssyd2000}.)

Here we will show how to apply the general \mbox{theory} to derive a
``nonsmooth'' version of Chow's theorem, in which some of the vector
fields involved are just \mbox{continuous}, while the others are Lipschitz
continuous but not necessarily of class $C^1$. (We only consider the 
version involving brackets of two system vector fields. Versions involving
higher-order brackets can be proved in a similar way, but require
more regularity of the \mbox{vector} fields.) The result is the one
to be expected, that is, that small-time local \mbox{controllability} from a
point $x_*$ follows if the values at $x_*$ of the system vector
fields, together with the Lie brackets at $x_*$ of the vector fields
that are Lipschitz continuous, \mbox{linearly} \mbox{span} the whole space.  To
render this meaningful, one needs to give a precise definition of the
notion of ``Lie \mbox{bracket}'' of two Lipschitz continuous vector fields
$f$, $g$. This is done in \S\ref{notpremlip}, where we give a definition
according to which $[f,g](x_*)$ is always a nonempty compact \mbox{convex}
set of vectors. Since the values of the Lie \mbox{brackets} are now sets,
rather than single vectors, the notion of ``\mbox{linearly} \mbox{spanning}
the whole space''
needs to be made precise. This is done in the statement of our main 
result, \mbox{Theorem} \ref{mnth}.  

The GD theory used to prove Theorem \ref{mnth} is that of 
``generalized differential quotients'' (abbr. GDQs), so 
we include a section---\S\ref{sec-gdq}---briefly reviewing
the \mbox{definition} and basic properties of this GD theory.


\vbox{
\ssection{Statement of the main theorem}\label{sec-main}

\noindent The result stated below involves 
the notion of ``Lie bracket'' $[f,g]$ of
two locally Lipschitz vector fields $f$, $g$. 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 $f$ and $g$ the bracket $[f,g](x)$ is a {\em nonempty
set} of vectors, rather than a single vector as in the classical
notion of Lie bracket of vector fields of class $C^1$. The precise
definition of the Lie bracket of locally Lipschitz vector fields is
given in \S\ref{notpremlip} below, and is such that the set $[f,g](x)$
is always compact and convex.
}


\begin{theorem}\label{mnth}
Assume that $n\in\zz$, $n\ge 0$, $\Omega$ is an
open subset of $\rr^n$, $f_1,\ldots,f_m,g_1,\ldots,g_r$
are \mbox{vector} fields on $\Omega$, $f_1,\ldots,f_m$
are continuous, and $g_1,\ldots,g_r$ are locally Lipschitz.
Let $\Sigma$ be the driftless control system
$$
\Sigma:\qquad \dot x=\sum_{i=1}^mu_if_i(x)+
\sum_{j=1}^rv_jg_j(x)\,,
$$
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 $\Omega$
such that, for every choice $V=\{v_{k\ell}\}_{1\le k<\ell\le r}$
of members $v_{k\ell}\in[g_k,g_\ell](x_*)$ the set of vectors
\begin{eqnarray*}
&\,&\{f_i(x):i=1,\ldots,m\}\cup
\{g_j(x):j=1,\ldots,r\}\\
&\,&\quad\cup\{v_{k\ell}(x):k=1,\ldots,r-1,\ell=k+1,\ldots,r\}
\end{eqnarray*}
linearly spans $\rr^n$. 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} $A\sqrt{\eps}$ by means
of a piecewise constant bang-bang \mbox{control} 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}

\ssection{Generalized differential quotients (GDQs)}\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,
{\bem the set $GDQ(F;\bar x,\bar y;S)$  does not depend on the choice 
of the charts $\xi$, $\eta$.} Moreover, the following four results
can be proved.

\begin{theorem}\label{isgdt}
GDQ is a generalized differentiation \mbox{theory} in the sense
of \cite{susssmf2000}.\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}, 
$F:\rr^n\tto\rr^m$,  $v\!\in\!\rr^m$,
\mbox{$C$ is a closed} \mbox{convex} cone in $\rr^n$, 
$\Lambda$ \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

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

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

\ssection{Flows of continuous vector fields}\label{notprem}

\noindent We assume throughout this section that 
\begin{itemize}
\item[(A)] {\em $n\in\zz$, $n\ge 0$, and $\Omega$ is an open subset
of $\rr^n$.}
\end{itemize}
If $f:\Omega\mapsto\rr^n$ is a continuous vector field
on $\Omega$, we use $\Phi^f$ to denote the {\em flow} of $f$.
By definition, $\Phi^f$ is the set-valued map from
$\Omega\times\rr$ to $\Omega$ whose value, for a given
$(x,t)\in\Omega\times\rr$, is the set of all
$y\in\Omega$ having  the property that there exists an integral curve 
$\xi:I\to\Omega$ of $f$ which is defined on some subinterval
$I$ of $\rr$ such that $0\in I$ and $t\in I$ and is such 
that $\xi(0)=x$ and $\xi(t)=y$.
If $x_*\in\Omega$, we let $L^f(x_*)$ be the linear map
{}from $\rr^n\times\rr$ to $\rr^n$ given by 
$$
L^f(x_*)(v,r)=v+rf(x_*)\,.
$$

\begin{theorem}\label{diffocont}
Assume (A) holds, and $f:\Omega\mapsto\rr^n$ is a continuous vector field
on $\Omega$. Let $x_*\in\Omega$. Then the set 
$\{L^f(x_*)\}$ is a generalized differential quotient
of $\Phi^f$ at $((x_*,0),x_*)$ in the direction of $\rr^n$.
\end{theorem}
\begin{proof} If $v\in\rr^\nu$, $w\in\rr^\mu$, 
and $v\neq 0$, we let $E_{v,w}^{\nu,\mu}$ be the linear map
{}from $\rr^\nu$ to $\rr^\mu$ given by
$$
E_{v,w}^{\nu,\mu}(u)=\frac{\langle v,u\rangle}{\|v\|^2}w\,\,\,{\rm for}\,\,\,
u\in\rr^\nu\,.
$$
Then 
\begin{equation}
E_{v,w}^{\nu,\mu}(v)=w\,\,\,\quad{\rm and}\,\,\,\quad
\|E_{v,w}^{\nu,\mu}\|=\frac{\|w\|}{\|v\|}\,.
\end{equation}

Fix a positive number $\delta$. Let $\hat\rho$ be such that
$\hat\rho>0$ and 
the compact ball $\hat V=\{x\in\rr^n:\|x-x_*\|\le\hat\rho\}$ is 
contained in $\Omega$. Let $\kappa=\sup\{\|f(x)\|:x\in\hat V\}$.

For every $\rho_1,\rho_2$ such that $0<\rho_1\le\rho_2\le\hat\rho$, let 
\begin{eqnarray*}
&\,&\omega(\rho_1,\rho_2)\equaldef\sup\{\|f(x)-f(x')\|:\\
&\,&\|x-x'\|\le\rho_1\,,\,\|x-x_*\|\le\rho_2\,,\,\|x'-x_*\|\le\rho_2\}\,,
\end{eqnarray*}
so that 
$$
\lim_{\rho_1\downarrow 0}\omega(\rho_1,\rho_2)=0\,\,\,
{\rm if}\,\,\,0<\rho_2\le\hat\rho\,.
$$
Now fix $\bar\rho$ such that 
$$
\bar\rho>0\,,\quad (2+\kappa)\bar\rho\le\hat\rho\,,\quad
{\rm and}\quad\omega((1+\kappa)\bar\rho,\hat\rho)\le\delta\,.
$$
Let $\bar V$ be the closed ball in $\rr^n$ of radius
$\bar\rho$, and center $x_*$, and let
$$
\bar W=\bar V\times[-\bar\rho,\bar\rho]\,.
$$
More generally, if $0<\rho\le\bar\rho$, we let 
$V(\rho)$ be the closed ball in $\rr^n$ of radius $\rho$ and center
$x_*$, and then define $W(\rho)=V(\rho)\times[-\rho,\rho]$.

For $(x,t)\in\bar W$, $y\in\rr^n$, $(x,t)\neq(x_*,0)$, define a linear map 
$M_{x,t,y}:\rr^{n}\times\rr\mapsto\rr^n$ by letting
\begin{equation}\label{defofm}
M_{x,t,y}=L^f(x_*)+E_{(x-x_*,t),y-x-tf(x_*)}^{n+1,n}\,.
\end{equation}
We then let $\Lambda(x,t)$ be the set of all linear maps 
$M_{x,t,y}$, for all $y\in\Phi^{f}(x,t)$.  

We remark that if $t=0$ then $\Phi^f(x,t)=\{x\}$, so 
$\Lambda(x,t)=\{L^f(x_*)\}$. We extend the definition
of $\Lambda$ by defining $\Lambda(x_*,0)=\{L^f(x_*)\}$. 

Now let $(x,t)\in\bar W$, $(x,t)\neq(x_*,0)$.
If a linear map $M$ belongs to $\Lambda(x,t)$, and 
$M=M_{x,t,y}$, $y\in\Phi^{f}(x,t)$,
then 
\begin{eqnarray*}
x_*+M\cdot(x-x_*,t)&=&x_*+L^f(x_*)(x-x_*,t)\\ 
&\,&\quad +\mathcal{E}_{x,t,y}\cdot(x-x_*,t)\\
&=&x_*+x-x_*+tf(x_*)\\
&\,&\quad +y-x-tf(x_*)\\
&=&y\,.
\end{eqnarray*}
where
$$
\mathcal{E}_{x,t,y}\equaldef E_{(x-x_*,t),y-x-tf(x_*)}^{n+1,n}\,.
$$
Therefore 
\begin{equation}\label{was-a}
\Lambda(x,t).(x-x_*,t)\subseteq\Phi^{f}(x,t)\,.
\end{equation}

Now suppose that $0<\rho\le\bar\rho$,  $(x,t)\in W(\rho)$, 
and $M\in\Lambda(x,t)$.
Write $M=M_{x,t,y}$, $y\in\Phi^{f}(x,t)$, and let
$I_t$ denote the interval $[\min(0,t),\max(0,t)]$. 
Then 
\begin{eqnarray*}
\|M_{x,t,y}-L^f(x_*)\|&=&\|E_{(x-x_*,t),y-x-tf(x_*)}^{n+1,n}\|\\
&=&\frac{\|y-x-tf(x_*)x
\|}{\sqrt{\|x-x_*\|^2+t^2}}\,.
\end{eqnarray*}
But
$$
\|y-x-tf(x_*)\|=\Big\|\int_0^t\Big(f(\xi(s))-f(x_*)\Big)\,ds\Big\|\,,
$$
if $\xi:I_t\to\Omega$ is an integral curve of $f$ such that
$\xi(0)=x$ and $\xi(t)=y$. Now, the curve $\xi$ satisfies
$\dot\xi(s)=f(\xi(s))$, and then 
$\|\dot\xi(s)\|\le\kappa$, whenever $\xi(s)\in\hat V$. 
Let $\tilde I$ be the set of all $s\in I_t$ such that
$\xi(\sigma)\in\hat V$ for all \mbox{$\sigma\in I_s$}. Then 
$\tilde I$ is a compact subinterval of $I_t$ such that $0\in I_s$. 
So $\tilde I=I_\tau$ for some $\tau\in I_t$. If $\tau\neq t$ then
$\|\xi(\tau)-x_*\|=\hat\rho$. But 
$\|\xi(s)-x\|\le\kappa|s|$ for all $s\in\tilde I$, and then
the inequality $\|\xi(\tau)-x_*\|\le\kappa|s|+\rho$ holds
for all such $s$. Therefore
$\|\xi(\tau)-x_*\|\le(1+\kappa)\rho\le(1+\kappa)\bar\rho<\hat\rho$
contradicting the fact that $\|\xi(\tau)-x_*\|=\hat\rho$. 
It follows that $\tau=t$ and $\tilde I=I_t$, so that
\begin{equation}\label{asinthe}
\|\xi(s)-x_*\|\le(1+\kappa)\rho\quad{\rm whenever}\quad s\in I_t\,. 
\end{equation}
Then 
$$
\|f(\xi(s))-f(x_*)\|\le\omega((1+\kappa)\rho,\hat\rho)\,\,\,
{\rm if}\,\,\,s\in I_t,.
$$
Therefore
$$
\|y-x-tf(x_*)\|\le|t|\omega((1+\kappa)\rho,\hat\rho)\,.
$$
If we pick $\rho=\bar\rho$, we get
$$
\|y-x-tf(x_*)\|\le|t|\delta\,,
$$
{}from which it follows that 
$$
\frac{\|y-x-tf(x_*)\|}{\sqrt{\|x-x_*\|^2+t^2}}\le\delta\,,
$$
and then $\|M_{x,t,y}-L^f(x_*)\|\le\delta$.

So we have shown that
\begin{equation}\label{was-b}
\|M-L^f(x_*)\|\le\delta\quad{\rm whenever}\quad
M\in\Lambda(x,t)\,,
\end{equation}
as long as $(x,t)\in\bar W$ and $(x,t)\neq(x_*,0)$.

On the other hand, given any $(x,t)\in\bar W$, we can pick
$\rho=\max(\|x-x_*\|,|t|)$, and then get the estimate
\begin{eqnarray}
&\,&\|M-L^f(x_*)\|\le\omega\Big(\max(\|x-x_*\|,|t|)\,,\,
\hat\rho\,\Big)\nonumber\\
&\,&\quad{\rm whenever}\quad (x,t)\in\bar W\,,\,M\in\Lambda(x,t)\,,
\label{was-ba}
\end{eqnarray}

It is clear that equations (\ref{was-a}) and (\ref{was-b}),
which have been proved under the assumption that 
$(x,t)\neq(x_*,0)$, are also valid when 
$(x,t)=(x_*,0)$. So (\ref{was-a}) and (\ref{was-b}) hold
for all $(x,t)\in\bar W$. 

To complete our proof, we have to show that the set-valued
map $\bar W\ni(x,t)\mapsto\Lambda(x,t)\subseteq{\rm Lin}(\rr^{n+1},\rr^n)$
is regular.   To prove this, we first have to prove that the set 
$$
G=\{(x,t,M):(x,t)\in\bar W,M\in\Lambda(x,t)\}
$$
is compact.

For this purpose, we 
let $\{(x^j,t^j,M^j)\}_{j\in\nn}$ be a \mbox{sequence} of members
of $G$, and try to extract a \mbox{subsequence} that converges
to a member of $G$. 

If $(x^j,t^j)=(x_*,0)$ 
for infinitely many indices $j$, then 
$M=L^f(x_*)$ for all such indices, so the \mbox{existence}
of the desired subsequence is obvious. Otherwise, we may assume
that  $(x^j,t^j)\neq(x_*,0)$ for all indices $j$. 
Write $M^j=M_{x^j,t^j,y^j}$, where
$y^j\in\Phi^f(x^j,t^j)$. We may clearly assume that 
the sequence $\{x^j,t^j\}_{j\in\nn}$ itself converges to a limit
$(\bar x,\bar t)$.  Now choose integral curves
$\xi^j$ of $f$, defined on the intervals $I^j=I_{t^j}$,
such that $\xi^j(0)=x^j$ and $\xi^j(t^j)=y^j$. 
As in the proof of (\ref{asinthe}), we conclude that 
$\xi^j(s)\in\hat V$ whenever $s\in I^j$. 
Then $\|\dot\xi^j(s)\|=\|f(\xi^j(s))\|\le\kappa$, so
the sequence $\{\xi^j\}_{j\in\nn}$ is equicontinuous.
Since this sequence is also uniformly bounded, we may assume,
by passing to a subsequence, that the curves
$\xi^j$ converge uniformly to a limit curve $\xi$, defined
on $I_t$. It then follows
by familiar arguments that $\bar y=\xi(t)\in\Phi^f(\bar x,\bar t)$.

Let $\bar M=M_{\bar x,\bar t,\bar y}$ if $(\bar x,\bar t)\neq
(x_*,0)$, $\bar M=L^f(x_*)$ if $(\bar x,\bar t)=(x_*,0)$.
Then $\bar M\in\Lambda(\bar x,\bar t)$. Let us prove that
$M^j\to\bar M$ as $j\to\infty$. Assume first that
$(\bar x,\bar t)\neq(x_*,0)$. 
Since $x^j\to\bar x$, $t^j\to\bar t$, and $y^j\to\bar y$, if follows
{}from (\ref{defofm}) that $M^j\to \bar M$, since the maps
$(v,w)\mapsto E_{v,w}^{\nu,\mu}$ are continuous on
$(\rr^\nu\backslash\{0\})\times\rr^\mu$.
Now assume that \mbox{$(\bar x,\bar t)=(x_*,0)$}. Then 
$x^j\to x_*$ and $t^j\to 0$. So (\ref{was-ba}) implies that
$M^j\to L^f(x_*)=\bar M$. 

We must now show that $\Lambda$ is a limit, in the sense of 
inward graph convergence of set-valued maps from
$\bar W$ to ${\rm Lin}(\rr^{n+1},\rr^n)$, of single-valued continuous
maps. To prove this, we first let $F$ be a continuous
map from $\rr^n$ to $\rr^n$ which coincides with $f$ on $\hat V$ 
and is such that $\|F(x)\|\le\kappa$ for all $x\in\rr^n$.
We then let $F^\zeta$ be the usual regularizations of $F$,
defined---for $\zeta>0$---by letting
\begin{equation}
F^\zeta(x)=\int_{\rr^n}\varphi(h)F(x+\zeta h)\,dh\,,
\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 $\|h\|>1$. .

Then $F^\zeta\in C^\infty$ for all positive $\zeta$,
$\|F^\zeta(x)\|\le\kappa$ for all $x\in\rr^n$ and all positive
$\zeta$, and $F^\zeta\to F$ \mbox{uniformly} on compact sets as
$\zeta\downarrow 0$. The \mbox{flow maps} $\Phi^{F^\zeta}$ are then
\mbox{globally} defined, single-valued, and smooth.  We 
\mbox{construct} maps $\Lambda^\zeta$, defined on $\bar W$, 
exactly as we \mbox{constructed}
$\Lambda$, using $F^\zeta$ instead of $f$. Then the $\Lambda^\zeta$
are single-valued and continuous. Moreover, the moduli of continuity
\begin{eqnarray*}
&\,&\omega^\zeta(\rho_1,\rho_2)\equaldef\sup\{\|F^\zeta(x)-F^\zeta(x')\|:\\
&\,&\|x-x'\|\le\rho_1\,,\,\|x-x_*\|\le\rho_2\,,\,\|x'-x_*\|\le\rho_2\}\,,
\end{eqnarray*}
of the $F^\zeta$ are related to the modulus of continuity
$\omega$ of $f$ by 
\begin{equation}\label{modbound}
\omega^\zeta(\rho_1,\rho_2)\le\omega(\rho_1,\rho_2+\zeta)\,.
\end{equation}
Let us show that $\Lambda^\zeta\togr\Lambda$. Let 
$$
\alpha(\zeta)=\sup\{d(q,G):q\in G^\zeta\}\,,
$$
where 
$$
G^\zeta=\{(x,t,M):(x,t)\in\bar W,M\in\Lambda^\zeta(x,t)\}\,.
$$
We want to prove that $\alpha(\zeta)\to 0$ as $\zeta\downarrow 0$. 
Suppose this is not true. Then there exist
a sequence $\{\zeta_j\}_{j\in\nn}$ of positive 
numbers and an $\bar\alpha$ such that
$\zeta_j\downarrow 0$, $\bar\alpha>0$, and 
$\alpha(\zeta_j)\ge 2\bar\alpha$. Write
$G_j=G^{\zeta_j}$. Pick $q_j\in G_j$ such that 
$d(q_j,G)\ge\bar\alpha$.  Write 
$q_j=(x_j,t_j,M_j)$. We will reach a contradiction 
by finding a subsequence of $\{q_j\}_{j\in\nn}$ 
such that $d(q_j,G)\to 0$. For this purpose, we may assume,
after passing to a subsequence, that $(x_j,t_j)$ converges
to a limit $(x,t)\in\bar W$. 

Let $J$ be the set of all indices $j$ such that 
\mbox{$(x_j,t_j)=(x_*,0)$}. If $J$ is infinite, then 
\mbox{$M_j=L^{F^{\zeta_j}}(x_*)$}
for $j\in J$, so $\lim_{j\to\infty,j\in J}M_j=L^{f}(x_*)$.
Then $\lim_{j\to\infty,j\in J}q_j=(x_*,0,L^{f}(x_*))\in G$
so $\lim_{j\to\infty,j\in J}d(q_j,G)=0$, showing the existence 
of the desired subsequence.

Next, let us assume that $J$ is finite. Then, by passing to a subsequence,
we may assume that $J=\emptyset$, i.e., that 
$(x_j,t_j)\neq(x_*,0)$ for all $j$. Let $y_j=\Phi^{F^{\zeta_j}}(x_j,t_j)$.
Then 
\begin{equation}\label{defofmsi}
M_j=L^{F^{\zeta_j}}(x_*)+
E_{(x_j-x_*,t_j),y_j-x_j-t_jF^{\zeta_j}(x_*)}^{n+1,n}\,.
\end{equation}
For each $j$, find an integral curve $\xi_j:I_{t_j}\mapsto\rr^n$ 
of $F^{\zeta_j}$ 
such that $\xi_j(0)=t_j$ and $\xi_j(t_j)=y_j$. Then 
$\|\dot\xi_j(s)\|\le\kappa$ for all $j$, so 
$\xi_j$ is entirely contained in $\hat V$, and then
the sequence $\{\xi_j\}_{j\in\nn}$ is bounded and 
\mbox{equicontinuous.}
By passing to a subsequence, we may assume that 
$\xi_j\to\xi$ uniformly, for some curve $\xi:I_t\to\rr^n$. 
Since $\xi_j(0)=x_j$ and $\xi_j(t_j)=y_j$, it is clear that 	
$\xi(0)=x$ and \mbox{$\xi(t)=y=\lim_{j\to\infty}y_j$.}
Moreover, $\xi$ is an integral curve of $f$, so 
$y\in\Phi^f(x,t)$. 

Now assume that $(x,t)\neq(x_*,0)$. Then, \mbox{using} the facts that
(a)~the maps $(v,w)\mapsto E_{v,w}^{\nu,\mu}$ are 
\mbox{continuous} on $(\rr^\nu\backslash\{0\})\times\rr^\mu$ and 
\mbox{(b)~$L^{F^{\zeta_j}}(x_*)\to L^f(x_*)$,} we can conclude from 
(\ref{defofmsi}) that $M_j\to M$, where
\mbox{$M=L^{f}(x_*)+E_{(x-x_*,t),y-x-t f(x_*)}^{n+1,n}$.}
But this tells us, \mbox{using} the definition of $\Lambda$, that
$M\in\Lambda(x,t)$. \mbox{Therefore} $q=(x,t,M)\in G$. 
Since $q_j\to q$, we see that \mbox{$d(q_j,G)\to 0$}, as desired.

Finally, we have to consider the case when 
\mbox{$(x,t)=(x_*,0)$.} In that case, we use the estimate
\begin{eqnarray}
&\,&\|M-L^{F^{\zeta}}(x_*)\|
\le\omega^\zeta\big(\max(\|x-x_*\|,|t|),\hat\rho\big)\nonumber\\
&\,&\quad{\rm whenever}\quad (x,t)\in\bar W\,,\,M\in\Lambda^\zeta(x,t)\,,
\label{was-baba}
\end{eqnarray}
which is proved in the same way as (\ref{was-ba}).
In view of (\ref{modbound}), we have 
\begin{equation}\label{was-bababa}
\|M_j-L^{F^{\zeta_j}}(x_*)\|
\le\omega\big(\max(\|x_j-x_*\|,|t_j|),\hat\rho+\beta\big)
\end{equation}
for sufficiently large $j$, if $\beta$ is a fixed \mbox{positive}
\mbox{number.}
Therefore $\|M_j-L^{F^{\zeta_j}}(x_*)\|\to 0$. \mbox{Since}
\mbox{$L^{F^{\zeta_j}}(x_*)\to L^f(x_*)$,} we can conclude that 
\mbox{$M_j\to L^f(x_*)$}. Now, by the definition of $\Lambda$, 
$L^f(x_*))\in\Lambda(x_*,t)$. So, if we let 
$q=(x_*,0, L^f(x_*))$, then $q\in G$. Since $q_j\to q$, we see
that $d(q_j,G)\to 0$ in this case as well, and our proof is complete.
\end{proof}

\ssection{Lie brackets of locally Lipschitz vector fields}\label{notpremlip}

\noindent We assume throughout this section 
that (A) holds. If $f$ is a locally Lipschitz vector field on
$\Omega$, we let $DIFF(f)$ denote the set of points $x\in\Omega$ such
that $f$ is \mbox{differentiable} at $x$.  Then Rademacher's theorem 
\mbox{implies} that $DIFF(f)$ is a subset of full measure of $\Omega$.
If $x\in DIFF(f)$, then $Df(x)$ will denote the differential of
$f$ at $x$.

Now let $f,g$ be two locally Lipschitz vector 
fields on $\rr^n$, and let $x\in\Omega$. We define 
$[f,g](x)$---the {\em 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}\Big(Dg(x_j)\cdot f(x_j)
-Df(x_j)\cdot g(x_j)\Big)\,,
\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}
It is clear that the set $[f,g](x)$ is convex, compact, and nonempty
for every $x\in\Omega$. Moreover, the {\em skew-symmetry
identity}
$$
[f,f](x)=\{0\}
$$
clearly holds for all $x$.

If $f$ is a locally Lipschitz vector field, $x\in\Omega$, and $t\in\rr$,
we use $x\,e^{tf}$ to denote the value at time $t$ of the integral
curve $\xi$ of $f$ such that $\xi(0)=x$. (That is, $x\,e^{tf}$ is the
unique member of $\Phi^f(x,t)$.)

Given two locally Lipschitz vector
fields, $x\in\Omega$, and $t,s\in\rr$, we define
$$
\Psi^{f,g}(x,t,s)=x\,e^{tf}e^{sg}e^{-tf}e^{-sg}\,.
$$
It is clear that if $K$ is an arbitrary compact subset 
of $\Omega$, then there exists a positive time $\tau_K$ such that 
$\Psi^{f,g}(x,t,s)$ is defined whenever $x\in K$,
$|t|\le\tau_K$ and $|s|\le\tau_K$. 

We now derive an integral formula for $\Psi^{f,g}(x,t,s)-x$ when
$f$ and $g$ are smooth.  Formally,
\begin{eqnarray*}
&\,&\!\!\!\!\!
\!\Psi^{f,g}(x,t,s)-x\\
&=&x\,e^{tf}e^{sg}e^{-tf}e^{-sg}-x\\
&=&\int_0^s\Big(x\,e^{tf}e^{\sigma g}ge^{-tf}e^{-\sigma g}
-x\,e^{tf}e^{\sigma g}e^{-tf}ge^{-\sigma g}\Big)\,d\sigma\\
&=&\int_0^s\Big(x\,e^{tf}e^{\sigma g}e^{-tf}\Big(e^{tf}ge^{-tf}-g\Big)
e^{-\sigma g}\Big)\,d\sigma\\
&=&\int_0^s\int_0^t\Big(x\,e^{tf}e^{\sigma g}e^{-tf}\Big(e^{\tau f}[f,g]
e^{-\tau f}\Big)e^{-\sigma g}\Big)\,d\tau\,d\sigma\\
&=&\int_0^s\int_0^t\Big(x\,e^{tf}e^{\sigma g}e^{(\tau-t)f}[f,g]
e^{-\tau f}e^{-\sigma g}\Big)\,d\tau\,d\sigma\,.
\end{eqnarray*}
The above formula can be given a precise interpretation 
and a rigorous proof, by regarding all the expressions 
involved as distributions (cf. for example \cite{kawski}). 

For our purposes, what we need to know about the 
integral formula for $\Psi^{f,g}(x,t,s)-x$ is that the term
$x\,e^{tf}e^{\sigma g}e^{(\tau-t)f}[f,g]e^{-\tau f}e^{-\sigma g}$
is the vector determined as follows: we compute the Lie 
bracket $[f,g]$ of $f$ and $g$ at the point 
\begin{equation}\label{def-theta}
\theta(x,\sigma,t,\tau)\equaldef x\,e^{tf}e^{\sigma g}e^{(\tau-t)f}\,,
\end{equation}
and transport this vector to the point
\begin{eqnarray*}
\Psi^{f,g}(x,t,\sigma)&=&x\,e^{tf}e^{\sigma g}e^{-tf}e^{-\sigma g}\\
&=&\theta(x,\sigma,t,\tau)\,e^{-\tau f}e^{-\sigma g}
\end{eqnarray*}
by means of the differential at 
$\theta(x,\sigma,t,\tau)$ of the \mbox{diffeomorphism}
$y\mapsto y\,e^{-\tau f}e^{-\sigma g}$. 

If $X$ is a smooth vector field, then to transport a tangent vector
$\bar v$ from a point $\bar y$ to the point $\bar y\,e^{\bar tX}$ by 
means of the differential of the map $y\mapsto y\,e^{\bar tX}$, one solves
the variational equation 
$$
\dot v(t)=DX(y\,e^{tX})\cdot v(t)
$$
with initial condition $v(0)=\bar v$. The desired vector is then 
$v(\bar t)$. If $C$ is a constant such that $\|DX(x)\|\le C$ for 
all $x$, then Gronwall's inequality implies the bound
$$
\|v(t)\|\le e^{|t|C}\|\bar v\|\,,
$$
{}from which it follows that $\|v(t)-\bar v\|\le C|t|e^{|t|C}\|\bar v\|$.
Therefore
$$
\|v(r)-\bar v\|\le C|r|e^{|r|C}\|\bar v\|\,.
$$

If we apply this to the integrand 
$x\,e^{tf}e^{\sigma g}e^{(\tau-t)f}[f,g]
e^{-\tau f}e^{-\sigma g}$, we conclude that 
\begin{eqnarray*}
&\,&\!\!\!\!\!\|x\,e^{tf}e^{\sigma g}e^{(\tau-t)f}[f,g]e^{-\tau f}e^{-\sigma g}
-[f,g](\theta(x,\sigma,t,\tau))\|\\
&\,&\le C|s|e^{|s|C}(1+C|t|e^{|t|C})\|[f,g](\theta(x,\sigma,t,\tau))\|,
\end{eqnarray*}
{}from which we get the estimate
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\|\Psi^{f,g}(x,t,s)-x-
\int_0^s\int_0^t\Big([f,g](\theta(x,\sigma,t,\tau))\Big)
\,d\tau\,d\sigma\|\nonumber\\
&\,&\le 2 C^3|s|^2|t|e^{|s|C}(1+C|t|e^{|t|C})\,,\label{inte-est}
\end{eqnarray}
where $C$ is an upper bound for $\|f(y)\|$, $\|Df(y)\|$, 
$\|g(y)\|$, and $\|Dg(y)\|$ for all $y$.

Now suppose that $f$ and $g$ are just Lipschitz. 
Fix a point $x_*\in\Omega$.  Fix a positive number $\hat\rho$ such that
the compact ball 
$$
\hat V=\{x\in\rr^n:\|x-x_*\|\le\bar\rho\}
$$
is contained in $\Omega$. Let $C_1$ be an
upper bound for $\|f(x)\|$ and $\|g(x)\|$ for all $x\in\hat V $,
and let $C_2$ be a \mbox{Lipschitz} constant for $f$ and $g$ 
on $\hat V$.  Then  let $C=\max(C_1,C_2)$. 

For $0<\rho\le\hat\rho$, use
$\gamma(\rho)$ to denote the supremum
of the distances $d\Big(Dg(x)\cdot f(x)-Df(x)\cdot g(x),[f,g](x_*)\Big)$,
for $x\in DIFF(f)\cap DIFF(g)$ such that $\|x-x_*\|\le\rho$. 

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

Now let $f^\zeta$,  $g^\zeta$ be the regularizations of $f$, $g$, 
defined as before for positive $\zeta$. Then, after replacing
$\hat V$ by a smaller ball, and as long as $\zeta$ is sufficiently
small, the constant $C$ is also an
upper bound for $\|f^\zeta(x)\|$ and $\|g^\zeta(x)\|$ for all $x$
and a Lipschitz constant for $f$ and $g$. Then
we can apply (\ref{inte-est}) to the regularized
vector fields, and conclude that
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\Big\|\Psi^{f^\zeta,g^\zeta}(x,t,s)-x\nonumber\\ 
&-&\int_0^s\int_0^t\Big([f^\zeta,g^\zeta](\theta^\zeta(x,\sigma,t,\tau))
\Big)\,d\tau\,d\sigma\Big\|\nonumber\\
&\,&\le 2 C^3|s|^2|t|e^{|s|C}(1+C|t|e^{|t|C})\label{inte-est-reg}
\end{eqnarray}
if $|s|$, $|t|$ and $\zeta$ are small enough,
where $\theta^\zeta$ is defined as in (\ref{def-theta}), using
$f^\zeta$ and $g^\zeta$ instead of $f$ and $g$.


For any given $y$, we would like to estimate the distance
$$
\Delta(y,\zeta)=d([f^\zeta,g^\zeta](y),[f,g](x_*))\,.
$$
We write
\begin{eqnarray*}
&\,&[f^\zeta,g^\zeta](y)=Dg^\zeta(y)\cdot f^\zeta(y)-Df^\zeta(y)\cdot 
g^\zeta(y)\\
&=&\int_{\rr^n}\varphi(h)Dg(y+\zeta h)\cdot f^\zeta(y)\,dh\\
&\,&\,\,\,\,\,\,\,-\int_{\rr^n}\varphi(h)
Df(y+\zeta h)\cdot g^\zeta(y)\,dh\\
&=& I_1(t)+I_2(y)\,,
\end{eqnarray*}
where
\begin{eqnarray*}
I_1(y)&=&\int_{\rr^n}\varphi(h)w(y+\zeta h)\,dh\,,\\
w(z)&=&Dg(z)\cdot f(z)-Df(z)\cdot g(z)\,,\\
I_2(y)&=&\int_{\rr^n}\varphi(h)Dg(y\!+\!\zeta h)\cdot\Big(
f^\zeta(y)\!-\!f(y\!+\!\zeta h)\Big)dh\\
&\,&\!\!\!\!\!\!-\int_{\rr^n}\varphi(h)
Df(y\!+\!\zeta h)\cdot\Big(g^\zeta(y)\!-\!g(y\!+\!\zeta h)\Big)\,dh\,.
\end{eqnarray*}
The integral $I_1(y)$ is an average of vectors of the form
$Dg(z)\cdot f(z)-Df(z)\cdot g(z)$,  for points $z=y+\zeta h$ 
such that $\|z-x_*\|\le\|y-x_*\|+\zeta$. Therefore
$$
d(I_1(y),[f,g](x_*))\le \gamma(\|y-x_*\|+\zeta)\,.
$$
On the other hand, it is clear that
$\|f^\zeta(y)-f(y)\|\le C\zeta$ for every $y$, so 
\begin{eqnarray*}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!
\|f^\zeta(y)\!-\!f(y\!+\!\zeta h)\|
\!\le\!\|f^\zeta(y)\!-\!f(y)\|\!+\!\|f(y)\!-\!f(y\!+\!\zeta h)\|\\
&\,&\,\le  2C\zeta\,,
\end{eqnarray*}
and then  $\|I_2(y)\|\le 4C^2\zeta\,$.
It follows that 
$$
\Delta(y,\zeta)\le\gamma(\|y-x_*\|+\zeta)+4C^2\zeta\,.
$$
Now, if $y=\theta^\zeta(x,\sigma,t,\tau)$, where, as before,
$\sigma\in I_s$ and $\tau\in I_t$, then	
$\|y-x\|\le C(2|t|+|s|)$, so 
$$
\|y-x_*\|\le C(2|t|+|s|)+\|x-x_*\|)\,,
$$
and then the distance from $[f^\zeta,g^\zeta](y)$ to $[f,g](x_*)$ 
is bounded by $\gamma(C(2|t|+|s|)+\|x-x_*\|)+\zeta)+4C^2\zeta$.

If $st>0$, it follows that
\begin{eqnarray*}
&\,&d\Big((ts)^{-1}(\Psi^{f^\zeta,g^\zeta}(x,t,s)-x),
[f,g](x_*)\Big)\\
&\,&\le 2 C^3|s|e^{|s|C}(1+C|t|e^{|t|C})\\
&\,&+\gamma(C(2|t|+|s|)+\|x-x_*\|+\zeta)+4C^2\zeta\,.
\end{eqnarray*}
We can now let $\zeta\downarrow 0$, and conclude that 
\begin{eqnarray}
&\,&d\Big((ts)^{-1}(\Psi^{f,g}(x,t,s)-x),[f,g](x_*)\Big)\nonumber\\
&\,&\le 2 C^3|s|e^{|s|C}(1+C|t|e^{|t|C})+\gamma(C(2|t|+|s|))\nonumber\\
&\,&\le \Gamma(\max(|t|,|s|),x)\,,\label{lnr}
\end{eqnarray}
where 
$$
\Gamma(r,x)=2 C^3re^{rC}(1+Cre^{rC})+\gamma(3rC+\|x-x_*\|)\,.
$$

We now define a map $(x,\eps)\mapsto\Xi^{f,g}(x,\eps)$
by letting
$$
\Xi^{f,g}(x,\eps)=
\left\{\begin{array}{lll}
\Psi^{f,g}(x,\sqrt{\eps},\sqrt{\eps})&{\rm if}&\eps\ge 0\,,\\
\Psi^{f,-g}(x,\sqrt{-\eps},\sqrt{-\eps})&{\rm if}&\eps\le 0\,.
\end{array}\right.
$$
Then $\Xi^{f,g}$ is continuous. (Notice that
when $\eps=0$ the vectors
$\Psi^{f,g}(x,\sqrt{\eps},\sqrt{\eps})$ and
$\Psi^{f,-g}(x,\sqrt{-\eps},\sqrt{-\eps})$ are both equal to $x$.)
Moreover, (\ref{lnr}) implies the inequalities
\begin{eqnarray}
d\Big(\eps^{-1}(\Xi^{f,g}(x,\eps)-x),[f,g](x_*)\Big)\!
&\!\le\!&\!\!\Gamma(\sqrt{\eps},x)\,,\label{lnr-a}\\
d\Biggl(\frac{\Xi^{f,g}(x,\eps)-x}{-\eps},[f,-g](x_*)\Biggr)
\!&\!\le\!&\!\!\Gamma(\sqrt{-\eps},x)\,,\label{lnr-b}
\end{eqnarray}
valid, respectively, when $\eps>0$ and $\eps<0$.
Since $[f,-g](x_*)=-[f,g](x_*)$, (\ref{lnr-b}) is equivalent
to 
\begin{equation}\label{lnr-c}
d\Big(\eps^{-1}(\Xi^{f,g}(x,\eps)-x),[f,g](x_*)\Big)
\le\Gamma(\sqrt{-\eps},x)\,.
\end{equation}
We have therefore proved that the inequality 
\begin{equation}\label{lnr-d}
d\Big(\eps^{-1}(\Xi^{f,g}(x,\eps)-x),[f,g](x_*)\Big)
\le\Gamma(\sqrt{|\eps|},x)
\end{equation}
is valid for all sufficiently small $\eps$, positive or negative.
Equivalently, we have shown that

\vuppp
\begin{eqnarray}
&\,&\inf\Big\{\,\|\Xi^{f,g}(x,\eps)-x-\eps v\|:v\in[f,g](x_*)\,
\Big\}\nonumber\\
&\,&\quad =o(\eps+\|x-x_*\|)\,.\label{equu}
\end{eqnarray}

%\medbreak
\noindent 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  
\mbox{$\mathcal{Q}^W\equaldef \{Q^w: w\in W\}$}. 

Then (\ref{equu}) says---since $\Xi^{f,g}(x_*,0)=x_*$---that 
\begin{eqnarray}
&\,&\inf\Big\{\,\|\Xi^{f,g}(x,\eps)-\Xi^{f,g}(x_*,0)-
Q^v(x-x_*,\eps)\|\nonumber\\
&\,&\quad : v\in[f,g](x_*)\,\Big\}=o(\eps+\|x-x_*\|)\,,\label{equu-aa}
\end{eqnarray}
i.e., that
\begin{eqnarray}
&\,&\inf\Big\{\,\|\Xi^{f,g}(x,\eps)-\Xi^{f,g}(x_*,0)-
M(x-x_*,\eps)\|\nonumber\\
&\,&\quad :M\in\mathcal{Q}^{[f,g](x_*)}\,\Big\}
=o(\eps+\|x-x_*\|)\,.\label{equu-a}
\end{eqnarray}

\noindent The following result then follows from the
\mbox{general} sufficient condition given in \cite{sussncn2000}
for a set to be a GDQ:
\begin{theorem}\label{diffobrac}
Assume (A) holds. Let $f$, $g$ be locally Lipschitz vector fields 
on $\Omega$. Let $x_*\in\Omega$. Then the set $\mathcal{Q}^{[f,g](x_*)}$
is a generalized differential quotient of the map $\Xi^{f,g}$ at 
$(x_*,0)$ in the direction of $\rr^n$.\proofend
\end{theorem}

\ssection{Proof of the main theorem}\label{sec-proof}

\noindent First, we make a slight change of notation, 
and write
$$
\Phi^{f,\eps}(x)=\Phi^{f}(x,\eps)\,,\quad
\Xi^{f,g,\eps}(x)=\Xi^{f,g}(x,\eps)\,.
$$
Let $S=\{(i,j)\in\zz\times\zz:1\le i<j\le r\}$, 
$s=\frac{r(r-1)}{2}$, so ${\rm card}(S)=s$. Fix a bijection
\mbox{$\{1,\ldots,s\}\!\ni\! k\mapsto(i_k,j_k)\!\in\!S$}
{}from $\{1,\ldots,s\}$ onto $S$. Let
\mbox{$X\!=\!\rr^{m+r+s}\!\sim\!\rr^m\!\times\!\rr^r\!\times\!\rr^{s}$}.
Define a map $\Theta$ from a neighborhood of $0$ in $X$ to $\rr^n$, by
letting
\begin{eqnarray*}
&\,&\Theta(\eps_1,\ldots,\eps_m,\alpha_1,\ldots,\alpha_r,
\beta_1,\ldots,\beta_s)=\\
&\,&\Big(\Xi^{g_{i_s},g_{j_s},\beta_s}\circ\ldots\circ\Xi^{g_{i_1},
g_{j_1},\beta_1}\circ\\
&\,&\Phi^{g_r,\alpha_r}\circ\ldots\circ\Phi^{g_1,\alpha_1}
\circ\Phi^{f_m,\eps_m}\circ\ldots\circ\Phi^{f_1,\eps_1}\Big)(x_*)\,.
\end{eqnarray*}
Let $\mathcal{V}$ be the set of all indexed families 
$V=\{v_{k}\}_{k=1,\ldots,s}$ of vectors $v_k\in[g_{i_k},g_{j_k}](x_*)$.
For each member \mbox{$V=\{v_{k}\}_{k=1,\ldots,s}$}
of $\mathcal{V}$, let $\mathcal{L}^V$ 
be the linear map from $X$ to $\rr^n$ given by 
\begin{eqnarray*}
&\,&\mathcal{L}^V(\eps_1,\ldots,\eps_m,\alpha_1,\ldots,\alpha_r,
\beta_1,\ldots,\beta_s)\\
&=&\sum_{i=1}^m\eps_if_i(x_*)+\sum_{j=1}^r\alpha_jg_j(x_*)
+\sum_{k=1}^s\beta_kv_k\,.
\end{eqnarray*}
Then $\Theta(0)=x_*$, and Theorems \ref{diffocont} and \ref{diffobrac}, 
together with
the chain rule for GDQs (cf. \cite{susssmf2000}, \cite{sussncn2000})
imply that the set $\{\mathcal{L}^V:V\in\mathcal{V}\}$ is a 
generalized differential quotient of $\Theta$ at $0$ 
in the direction of $X$. 
The hypothesis of Theorem \ref{mnth} implies that all the maps
$\mathcal{L}^V$ are surjective. The desired conclusion then follows
{}from the open mapping property for GDQs (Theorem \ref{hasomp}).\proofend

\vspace{-3mm}
\begin{thebibliography}{99}

\vupp
\bibitem{kawski}{Kawski,~M., and H.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{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}

