% Cher Hector,
% 
% L'article ci-dessous, que tu connais peut-etre, a ete
% ecrit avec Jean-Michel Coron (paru en 1994). C'est le
% dernier que je t'envoie, sauf si tu as besoin d'un autre
% papier.
%
% Amities, 
% 
% Lionel
\documentclass[12pt]{article}
\title{A relation between continuous time-varying\\
and discontinuous feedback stabilization}
%
\author{Jean-Michel CORON$^*$, Lionel ROSIER$^{**}$}
\date{}
%
\usepackage{amsthm,amsmath,amssymb,a4wide,psfig}
\textheight250mm
\textwidth170mm
\baselineskip=15pt
\vsize=26 true cm
\hsize=18 true cm
\topmargin=-2cm
\oddsidemargin=-0.5cm
\evensidemargin=0.5cm
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%$
% Quelques definitions abregees utiles
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%$
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\bkR{\R}
\def\bkZ{\Z}
\def\va{\hfill\break}
%
\begin{document}
\maketitle
\vspace*{-1cm}
\begin{center}
$^*$Laboratoire d'Analyse Num\'erique, Universit\'e Paris-Sud  \\
B\^atiment 425, 91405 Orsay, France \\
$^{**}$CMLA, ENS Cachan, \\
61, av. du pr\'esident Wilson,  \\
94235 Cachan, France \\
\end{center}

\begin{abstract}
We prove that a control system which can be
asymptotically stabilized {\sl in the Filippov sense\/} by means of a
discontinuous feedback law can be asymptotically stabilized by
means of a continuous periodic time-varying feedback law. If,
moreover, the control system is affine then it can be
asymptotically stabilized by means of a continuous feedback law.
\va\va
{\bf\sc Key Words:} {\sl Asymptotic stabilization, discontinuous feedback law,
time-varying feedback law} 
\va
{\bf\sc AMS subject classification: 93D15, 93C10}
\end{abstract}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
\section{Introduction}
\va\indent
Let $ \Sigma $ be the control system
$$ \Sigma \ :\ \dot  x=f(x,u) \eqno (1.1) $$
where $ u\in \bkR^ m $ is the control, $ x\in \bkR^ n $ the state, and $ f $ a
map from $ \bkR^ n\times \bkR^ m $
into $ \bkR^ n $ such that
$$ f\in C^0(\bkR^ n\times \bkR^ m;\bkR^ n). \eqno (1.2) $$
Throughout this paper we assume
$$ f(0,0)=0. \eqno (1.3) $$
It has been proved by R.W. Brockett in $\lbrack$B$\rbrack$ that the local
controllability of $ \Sigma $ does not imply that $ \Sigma $ can be
asymptotically
stabilized by means of a continuous feedback law. To get around
this problem two main strategies have been proposed \par
\begin{itemize}
\item[(i)] Asymptotic stabilization by means of a discontinuous
feedback law -see e.g. the pioneer work by H. Sussmann $\lbrack$Su$\rbrack$ as
well as $\lbrack$FM$\rbrack$, $\lbrack$M$\rbrack$, $\lbrack$AB$\rbrack$, and
$\lbrack$CS$\rbrack$, and
the references therein-,
\item[(ii)] Asymptotic stabilization by means of a continuous
periodic time-varying feedback law -see e.g. the pioneer works by
E. Sontag and H. Sussmann $\lbrack$SS$\rbrack$ and by C. Samson
$\lbrack$Sa$\rbrack$ as well as
$\lbrack$C1$\rbrack$, $\lbrack$C2$\rbrack$, and the references therein-. 
\end{itemize}
\par
\smallskip
The main goal of this paper is to show a relation between
these two strategies. Before stating this relation we need to
specify the meaning of asymptotic stability for a system $ \dot  x=X(x) $
where $ X $ is a discontinuous vector field. Many definitions are
possible but, following H. Hermes $\lbrack$He$\rbrack$, it seems natural to
adopt \par
\smallskip
\noindent{\bf Definition 1.1.} {\sl Let\/} $ X\in L^{\infty}_{ {\rm
loc}}(\bkR^ n;\bkR^ n). $ {\sl Then\/} 0 {\sl is a locally
asymptotically stable point of\/} $ \dot  x=X(x) $ {\sl if\/} \par
\begin{itemize}
\item[(i)] {\sl for any\/} $ \epsilon \geq 0 $ {\sl there exists\/}
$ \delta > 0 $ {\sl such that any solution,\/}
\underbar{in the Filippov sense}, 
{\sl of\/} $ \dot  x=X(x) $ {\sl which is
defined at\/} $ t=0 $ {\sl and is
maximal, is defined on\/} $ \lbrack 0,+\infty ) $ {\sl if\/} $ \vert x(0)\vert
< \delta $ {\sl and moreover\/}
$$ \vert x(0)\vert < \delta \Longrightarrow  \vert x(t)\vert <
\epsilon ,\ \ \forall t\in \lbrack 0,+\infty ), \eqno (1.4) $$
\item[(ii)] {\sl there exists\/} $ \eta $ {\sl in\/} (0,+$\infty \rbrack$
{\sl such that any maximal solution,\/} \underbar{in the Filippov sense},
 {\sl of\/} $ \dot  x=X(x) $ {\sl which is defined at\/} $ t=0
$ {\sl is defined on\/}
$\lbrack$0,+$\infty$) {\sl if\/} $ \vert x(0)\vert < \eta $ {\sl and
moreover\/}
$$ \vert x(0)\vert < \eta \Longrightarrow  (x(t)\rightarrow 0\ {\rm as} \
t\rightarrow \infty ). \eqno (1.5) $$
{\sl Furthermore, if in\/} (ii) {\sl one can take\/} $ \eta =+\infty , $ {\sl
then \/} 0 {\sl is a globally
asymptotically stable point of\/} $ \dot  x=X(x). $ 
\end{itemize}
\par
\smallskip
Let us recall that a solution in the Filippov
sense of $ \dot  x=X(x) $ on an interval $ I $ is (see $\lbrack$F$\rbrack$) a
locally
absolutely continuous map from $ I $ into $ \bkR^ n $ such that
$$ \dot  x (t)\in F(x(t))\ {\rm for\ almost\ all\ }
 t\ {\rm in\ } I \eqno (1.6) $$
with
$$ F(x)\ := \bigcap^{ }_{ \epsilon > 0} \bigcap^{ }_{\vert N\vert =0}
{\overline{\rm conv}} X((x+\epsilon B)\backslash N),
\eqno (1.7) $$
where $ B $ is the unit ball of $ \bkR^ n, $ and, for a set $ A $,
$\vert A\vert$ is the Lebesgue measure of $A$ and
${\overline{\rm conv}} A $
is the smaller closed convex set containing $ A. $ A maximal solution
(in the Filippov sense) of $ \dot  x=X(x) $ is a solution $ x $ on some
interval $ I $ such that there exists no solution defined on an
interval which contains strictly $ I $ and which is equal to $ x $ on $ I. $
\par
\smallskip
Note that, if $ X $ is continuous,
$$ F(y)=\{ X(y)\} \eqno (1.8) $$
and therefore in this case our definition of asymptotic stability
coincide with the usual one. \par
\smallskip
We now define "asymptotically stabilizable by means of a
discontinuous feedback law (resp. periodic time-varying
continuous feedback law)". \par
\smallskip
\noindent{\bf Definition 1.2} {\sl System\/} $ \Sigma $ {\sl is locally (resp.
globally) asymptotically
stabilizable by means of a discontinuous feedback law if there
exists\/} $ u\in L^{\infty}_{ {\rm loc}}(\bkR^ n;\bkR^ m) $ {\sl such that\/}
$$ {\rm Essential\ Sup\ }\{\vert u(x)\vert ;\ \vert x\vert < \epsilon\}
\rightarrow 0\ {\rm as} \ \epsilon \rightarrow 0, \eqno (1.9) $$
$$ \left\{ 
\begin{array}{l}
0\ {\rm is\ a\ locally\ (resp.\ globally)\ asymptotically} \\ 
{\rm stable\ point\ of\ }\dot  x=f(x,u(x)). 
\end{array}
\right.
\eqno (1.10) $$
\par
\noindent{\bf Definition 1.3.} {\sl System \/}$ \Sigma $ {\sl is locally
(resp. globally)
asymptotically stabilizable by means of a continuous time-varying
feedback law of period\/} $ T $ {\sl if there exists\/} $ u\in C^0(\bkR^
n\times \bkR ;\bkR^ m) $ {\sl such that\/}
\setcounter{equation}{10}
\begin{eqnarray}
u(x,t+T)&=&u(x,t)  \ \ \forall (x,t)\in \bkR^ n\times \bkR ,  \\ 
u(0,t)&=&0  \ \ \forall t\in \bkR , \nonumber 
\end{eqnarray}
%$$
%\begin{array}{rl}
%u(x,t+T)&=u(x,t)  \ \ \forall (x,t)\in \bkR^ n\times \bkR ,
%\eqno(1.11) \\
%u(0,t)&=0  \ \ \forall t\in \bkR ,
%\begin{array}
%$$
$$ \left\{ 
\begin{array}{l}
0\ {\rm is\ a\ locally\ (resp.\ globally)\ asymptotically} \\ 
{\rm stable\ point\ of\ }\dot  x=f(x,u(x,t)).
\end{array}
\right.
\eqno (1.12) $$
The reason for considering in Definition 1.1 solutions in the
Filippov sense is, as explained in $\lbrack$He$\rbrack$, the following one :
the
feedback law $ u(x(t)) $ is determined after making a measurement of
the state $ x(t) $ at time $ t; $ of course this measurement gives only
an approximation of $ x(t):$ there is an "error" $ e(t) $ between $ x(t) $
and its measurement. A direct consequence of $\lbrack$He;Lemma 3$\rbrack$ is
\par
\smallskip
\noindent{\bf Proposition 1.4.} {\sl Assume that\/} $ f $ {\sl is locally
Lipschitzian with
respect to\/} $ x. $ {\sl Let\/} $ x\ :\ \lbrack 0,T\rbrack \rightarrow \bkR^
n $ be a Filippov solution of $ \dot  x=f(x,u(x)) $
where $ u\in L^{\infty}_{ {\rm loc}}(\bkR^ n;\bkR^ m). ${\sl\ Let\/} $
\epsilon $ {\sl be a positive real number. Then there
exist\/} $ e\in L^{\infty} ((0,T);\bkR^ m) $ {\sl and an absolutely continuous
function \/}$ y:\lbrack 0,T\rbrack \rightarrow \bkR^ n $
{\sl such that\/}
$$ \vert e(t)\vert \leq \epsilon \ {\rm for\ all\ } t\ {\rm in\ } (0,T),
\eqno (1.13) $$
$$ \dot  y(t)=f(y(t),u(y(t)+e(t)))\ {\rm for\ almost\ all\ } t\ {\rm in }
\ (0,T), \eqno (1.14) $$
$$ y(0)=x(0), \eqno (1.15) $$
and
$$ \vert y(t)-x(t)\vert \leq \epsilon \ {\rm for\ all\ } t\ {\rm in\ }\ 
\lbrack 0,T\rbrack . \eqno (1.16) $$
\par
\noindent\underbar{Proof}.  Let $ \delta > 0. $ By $\lbrack$He;Lemma
3$\rbrack$ there exists $ \omega \in L^{\infty} ((0,T);\bkR^ m), $ and
an absolutely continuous function $ z\ :\ \lbrack 0,T\rbrack \rightarrow
\bkR^ n $ such that
$$ \dot  z(t)=f(z(t)+\omega (t),\ u(z(t)+\omega (t)))\ {\rm for\ almost\ all\
} t\ {\rm in}\  (0,T), \eqno (1.17) $$
$$ z(0)=x(0), \eqno (1.18) $$
and
$$ \vert z(t)-x(t)\vert \leq \delta ,\ {\rm and} \ \vert \omega (t)\vert \leq
\delta \ {\rm for\ all\ } t\ {\rm in }\  \lbrack 0,T\rbrack . \eqno (1.19) $$
Let $ y $ be the maximal solution of
$$ \dot  y=f(y,u(z(t)+\omega (t))),\ y(0)=x(0). \eqno (1.20) $$
By standard estimates on ordinary differential equations (see e.g.
$\lbrack$Ha;II Thm 3.2$\rbrack$) we have, if $ \delta $ is small enough
$$ y\ {\rm is\ defined\ on\ } \lbrack 0,T\rbrack , \eqno (1.21) $$
(1.16), (1.13), and (1.14) with e.g. $ e(t)=z(t)+\omega (t)-y(t). $ \par
\smallskip
Proposition 1.4 justifies our definition of asymptotically
stabilization by means of a discontinuous feedback law. \par
\smallskip
Our main result is \par
\smallskip
\noindent{\bf Theorem 1.5.} {\sl Assume that\/} $ \dot  x=f(x,u) $ {\sl can be
locally (resp. globally)
asymptotically stabilized by means of a discontinuous feedback law.
Then, for any\/} $ T> 0, $ $\dot x =f(x,u) $ {\sl can be locally (resp.
globally) stabilized
by means of a continuous time-varying feedback law of period\/} $ T; $ {\sl if,
moreover,\/} $ \dot  x=f(x,u) $ {\sl is an affine system (i.e.\/} $
f(x,u)=f_0(x)+ \sum^ m_{i=1}u_if_i(x)), $
{\sl then\/} $ \dot  x=f(x,u) $ {\sl can be locally (resp. globally)
asymptotically stabilized
by means of a continuous feedback law \/}(\underbar{independent of} $ t $ : $
u=u(x)). $ \par
\smallskip
\noindent\underbar{Remark 1.6}. There are completely controllable affine
systems which are
globally asymptotically stabilized by means of a continuous periodic
time varying feedback law which cannot be locally asymptotically
stabilized by means of a continuous feedback law (e.g. $ \dot  x_1=u_1, $ $
\dot  x_2=u_2, $
$ \dot  x_3=x_1u_2-x_2u_1; $ see $\lbrack$Sa$\rbrack$ or
$\lbrack$C1$\rbrack$). By Theorem 1.5 these systems cannot be
locally asymptotically stabilized by means of a discontinuous feedback
law. Note that this is not in contradiction with $\lbrack$Su$\rbrack$ (or
$\lbrack$CS$\rbrack$ and $\lbrack$M$\rbrack$
for the above example) : indeed our definition of asymptotic stability
is more restrictive than the one used in $\lbrack$Su$\rbrack$,
$\lbrack$CS$\rbrack$ and $\lbrack$M$\rbrack$ (see e.g.
the definition of "steers $ M $ to $ p $" in $\lbrack$Su,p.42$\rbrack$; in
particular we do not
have any "exit rule" $ E $ on the singular set of $ u $ in our definition of
asymptotic stability). \par
In Section 2 we will give the proof of Theorem 1.5. In Section 3 we will
show that, in opposition to smooth stabilization, $  \dot  x=f(x,u) $
may be asymptotically stabilized by means of a discontinuous feedback
law even if $ \dot  x=f(x,y),\ \dot y =v$ cannot be asymptotically
stabilized by means of a discontinuous feedback law. 
\section{Proof of Theorem 1.5}
\va\indent
For simplicity we prove only the global statement (the proof of
the local statement is similar). Our first lemma is \par
\smallskip
\noindent{\bf Lemma 2.1.} {\sl Assume that\/}  $ \dot  x=f(x,u) $ {\sl can be
globally asymptotically
stabilized by means of a discontinuous feedback law. Then there exists \/}$
V\in C^1(\bkR^ n;\lbrack 0,+\infty )) $
{\sl such that\/}
$$ V(x) \rightarrow  +\infty \ {\rm as\ }\vert x\vert \rightarrow +\infty ,
\eqno (2.1) $$
$$ \forall x\not= 0\ \ \exists u\in \bkR^ m\ {\rm s.t.} \ f(x,u)\cdot \nabla
V(x)< 0, \eqno (2.2) $$
{\sl and, for all $ \epsilon > 0, $ there exists $ \delta > 0 $ such that }
$$ \vert x\vert < \delta  \Longrightarrow  (\exists u\in \bkR^ m\ {\rm
with} \ \vert u\vert < \epsilon \ {\rm and} \ f(x,u)\cdot \nabla
V(x)< 0). \eqno (2.3) $$
Let us remark that Theorem 1.5 in the affine case follows from Lemma 2.1
and a theorem due to A. Artstein $\lbrack$A;Thm 5.2$\rbrack$. \par
Let us also remark that the converse of Lemma 2.1 is true. Indeed if
the conclusion of Lemma 2.1 holds, then there exist $V\in {\rm C}^1
(\bkR^ n;[0,+\infty ))$ satisfying (2.1),
$\delta\in {\rm C}^0(\bkR^ n \setminus\{ 0\} ;( 0,+\infty ))$, and
$u\in L_{\rm loc}^{\infty}(\bkR^ n ;\bkR^ m )$ satisfying (1.9) and such
that, for all $x$ in $\bkR^ n\setminus \{ 0\}$, $f(x,u(x))\cdot\nabla V(x)
\le -\delta (x)$ from which it follows, as in the regular case, that $0$
is a globally asymptotically stable point of $\dot x =f(x,u(x))$.
\va\va
\underbar{Proof of Lemma 2.1}. Let $ u\in L^{\infty}_{ {\rm
loc}}(\bkR^ n;\bkR^ m) $ satisfying (1.9) be such that 0
is a globally asymptotically stable point of $ \dot  x=f(x,u(x)). $
Multiplying,
if necessary, $ f(x,u(x)) $ by $ \rho (x) $ where $ \rho $ is
a suitable function in
$ C^0(\bkR^ n;\lbrack 0,+\infty )) $ which is positive on $ \bkR^ n\backslash
\lbrace 0\rbrace $, we may assume, without loss of generality, that
$$ \vert f(x,u(x+e))\vert \leq {\rm Min} (1,\vert x\vert /2)\ \ \forall
(x,e)\in \bkR^ n\times \bkR^ m\ {\rm with} \ \vert e\vert \leq 1. \eqno (2.4)
$$
Using $\lbrack$R;Thm 2$\rbrack$, we get \par
\smallskip
\noindent{\bf Lemma 2.2.}{\sl\ There exist\/} $ V^\ast$ {\sl in}
 ${\rm C}^0(\bkR^
n;\lbrack 0,+\infty )) ${\sl\ and\/} $ \eta$ in
 $C^0(\bkR^ n;\lbrack 0,+\infty
)) $ {\sl such that\/}
\setcounter{equation}{4}
\begin{eqnarray}
V^\ast (x) \rightarrow +\infty \ &{\rm as}& \ \vert x\vert
\rightarrow +\infty , \\ 
V^\ast (x) =0 &\Longleftrightarrow&  x=0,\\
\eta (x) =0 &\Longleftrightarrow&  x=0, 
\end{eqnarray}
{\sl and\/}
$$ V^\ast (x(1))\leq V^\ast (x(0))-\eta (x(0)) \eqno (2.8) $$
{\sl for any Filippov solution of \/}$ \dot  x=f(x,u(x)) $ {\sl defined on\/}
$\lbrack$0,1$\rbrack$. 
\par
\smallskip
Let, for a measurable set $ A, $ $ \vert A\vert $ be its Lebesgue measure. Let $
B(x,r) $
be the open ball of radius $ r $ centered at $ x. $ For $ \delta $ in 
$C^0(\bkR^ n;\lbrack 0,1\rbrack ) $ such that
$$ \delta (x)=0 \Longleftrightarrow  x=0 \eqno (2.9) $$
let $ X_\delta \ :\ \bkR^ n\rightarrow \bkR^ n $ be defined by
$$ X_\delta (x)=\frac{1}{\vert B(x,\delta (x))\vert}  \int^{ }_{ B(x,\delta
(x))}f(x,u(y))\, dy\ \ \  {\rm if} \ x\not= 0, \eqno (2.10) $$
$$ X_\delta (0)=0. \eqno (2.11) $$
Note that by (2.4), (2.9), (2.10) and (2.11)
$$ X_\delta \in {\rm C}^0 (\bkR^ n;\{ x\in\bkR^ n ;\vert x\vert
\le 1\} ). \eqno (2.12) $$
Our next lemma is \par
\smallskip
\noindent{\bf Lemma 2.3. }{\sl There exists\/} $ \delta $ {\sl in\/} $
C^0(\bkR^ n;[0,1]) $ {\sl satisfying \/}(2.9) {\sl such that}
$$ 0\  \hbox{\sl  is a globally asymptotically stable point
of } \dot x =X_\delta (x). \eqno (2.13) $$
Lemma 2.1 is a corollary of Lemma 2.3. Indeed, 
by an inverse of Lyapunov's second theorem due to J. Kurzweil
$\lbrack$Ku,Thm 7$\rbrack$
there exists $ V\in C^{\infty} (\bkR^ n;[0,+\infty) ) $ satisfying (2.1) such
that
$$ X_\delta (x)\cdot \nabla V(x)< 0\ \ \ \forall x\in \bkR^ n\backslash\{
0\} ; \eqno (2.14) $$
it remains only to notice that (2.2) and (2.3) follow from (1.9), (2.9)
and (2.10). \par
\medskip
We now prove Lemma 2.3. For $ i\in \bkZ $ let
\setcounter{equation}{14}
\begin{eqnarray}
D_i &=& \left\{ x\in \bkR^ n;\ \ i\leq\vert x\vert \leq i+1
\right\} \ {\rm if\ } i> 0,  \\ 
D_i &=& \left\{ x\in \bkR^ n;\ \
2^{i-1}\leq\vert x\vert \leq 2^i \right\} \ {\rm if\ } i\leq 0, 
\end{eqnarray}
By (2.4) and (2.10), for any $ \delta $ in $ C^0(\bkR^ n;[0,1]), $ any $ i
$ in $ \bkZ $, and any
maximal solution of $ \dot  x=X_\delta (x), $
$$ x(0)\in D_i \Longrightarrow  (x(t)\in D_{i-1} \cup D_i\cup D_{i+1}\ \ \
\forall t\in[ 0,1]). \eqno (2.17) $$
Note also that
$$ X_\delta =f_\delta +g_\delta \eqno (2.18) $$
with
$$ f_\delta (x)=\frac{1}{\vert B(x,\delta (x))\vert}  \int^{ }_{ B(x,\delta
(x))}f(y,u(y))\, dy \eqno (2.19) $$
and
$$ g_\delta (x)=\frac{1}{\vert B(x,\delta (x))\vert}  \int^{ }_{ B(x,\delta
(x))}\big( f(x,u(y))-f(y,u(y))\big) dy. \eqno (2.20) $$
Arguing by contradiction, we get, using (2.7), (2.8), (2.17), (2.18),
(2.19), (2.20) and a compactness theorem due to A. Filippov
$\lbrack$F,Thm 3$\rbrack$
that for any $ i $ in $ \bkZ $ there exists $ \delta_ i> 0 $ such that, if
$ \delta (x)\leq \delta_ i\ \ \forall x\in D_{i-1}\cup D_i\cup D_{i+1} $
and if $x(\cdot )$ is any maximal solution of $\dot x =X_\delta (x)$ then
$$ (x(0)\in D_i)\Rightarrow 
(V^\ast (x(1))\leq V^\ast (x(0))-\frac{1}{2} \eta (x(0))\, ). \eqno (2.21) $$
 Therefore (recall $\vert X_\delta \vert\le 1$) any $ \delta $ in $ C^0(\bkR^
n;[0,1]) $
satisfying (2.9) and
$$ \delta (x)\le {\rm Min} (\delta_{ i-1},\delta_ i,\delta_{ i+1})\ \ \
\forall x\in D_i,\ \forall i\in \bkZ \eqno (2.22) $$
has the properties required by Lemma 2.3. \par
\smallskip\noindent\underbar{Remark 2.4}. We need Lemma 2.2 only to prove
Lemma 2.3. One can obtain also Lemma 2.3 by using Theorem A.2 of the
appendix instead of Lemma 2.2. \par\smallskip
In the nonaffine case Theorem 1.5 is a consequence of
Lemma 2.1 and of \par
\smallskip
\noindent{\bf Proposition 2.5.} {\sl Assume that the conclusion of Lemma\/}
2.1 {\sl holds.
Then, for any positive\/} $ T, $ $ \dot  x=f(x,u) $ {\sl can be globally
asymptotically
stabilized by means of a continuous\/} $ T- ${\sl periodic time-varying
feedback law.\/} \par
\smallskip
\noindent\underbar{Remark 2.6}. a) Proposition 2.5 has been previously proved
by E.
Sontag and H. Sussmann in $\lbrack$SS;(3.5)$\rbrack$ when $ n=1 $ (the
periodicity of
the stabilizing time-varying feedback law is not stated in
$\lbrack$SS$\rbrack$
but can be easily obtained from the proof of $\lbrack$SS;(3.5)$\rbrack$). b)
It
follows from $\lbrack$A;Thm 4.3$\rbrack$ and Proposition 2.5 that, if $
\dot  x=f(x,u) $
can be stabilized by means of a relaxed feedback law which is
continuous at 0 (and vanishes at zero), it can be, for any positive
$ T, $ globally asymptotically stabilized by means of a continuous $ T-
$periodic time-varying feedback law. \par
\smallskip
\noindent\underbar{Proof of Proposition 2.5}. Clearly by changing the scale of
time
if necessary we may assume
$$ T=2. \eqno (2.23) $$
Our first lemma follows from $\lbrack$A$\rbrack$. \par
\smallskip
\noindent{\bf Lemma 2.7.} {\sl Under the assumptions of Proposition\/}
2.5{\sl\ there exists\/}
$ v\ :\ \bkR^ n\times \bkR \rightarrow \bkR^ m $ {\sl such that\/}
$$ v\in C^{\infty} ((\bkR^ n\backslash\{ 0\} )\times \bkR ;\bkR^ m)\cap
C^0 (\bkR^ n\times \bkR ;\bkR^ m) $$
$$ v(x,s+1)=v(x,s)\ \ \ \forall (x,s)\in \bkR^ n\times \bkR , \eqno (2.24) $$
$$ v(0,s)=0\ \ \ \forall s\in \bkR , \eqno (2.25) $$
and
$$ \nabla V(x)\cdot \int^ 1_0f(x,v(x,s))ds< 0\ \ \forall x\in \bkR^
n\backslash\{ 0\} . \eqno (2.26) $$
\par
\noindent\underbar{Proof of Lemma 2.7}. With the notations of
$\lbrack$A;p.1167$\rbrack$ (with $ W=\bkR^ n\backslash\{ 0\} ) $
we define $ \tilde  v\ :\ (\bkR^ n\backslash\{ 0\} )\times \lbrack
0,1)\rightarrow \bkR^ m $ in the following way : let $ x $ be a
point belonging to the simplex $ \left[z_{i_1},...,z_{i_k} \right] $ with $
i_1< \cdot \cdot \cdot < i_k; $ in
particular
$$ x= \sum^ k_{j=1}t_{i_j}z_{i_j} \eqno (2.27) $$
with $ t_{i_j}\in[ 0,1] $ $ \forall j\in[ 1,k] $ and
$$ \sum^ k_{j=1}t_{i_j}=1; \eqno (2.28) $$
For $ t\in \left[ \sum^{%1
 \ell -1}_{j=1}t_{i_j}\, ,\, \sum^{ \ell}_{ j=1}t_{i_j} \right) $ (with the
convention $ \sum^ 0_{j=1}t_{i_j}=0) $ and $ \ell \in[ 1,k] $
let
$$ \tilde  v(x,t)=u_{i_{\ell}} . \eqno (2.29) $$
Using (2.3) we may require that if $ z_{i_q}\rightarrow 0 $ as $ q\rightarrow
\infty $ then $ u_{i_q}\rightarrow 0 $ as $ q\rightarrow \infty . $
Therefore
$$ {\rm Sup} \left\{ \left\vert\tilde  v(x,t) \right\vert ;\ t\in[ 0,1)
\right\} \rightarrow 0\ {\rm as} \ x\rightarrow 0. \eqno (2.30) $$
By construction
$$ \nabla V(x)\cdot \int^ 1_0f(x,\tilde  v(x,t))dt<0\ \ \forall x\in \bkR^
n\backslash\{ 0\} \eqno (2.31) $$
It remains only to extend $ \tilde  v $ to all $ \bkR^ n\times \bkR $ by
requiring (2.24) and
(2.25) and then to smooth it to get $ v. $ \par
We now go back to the proof of Proposition 2.5. Let $ v $ be
as in Lemma 2.7 and let for $ \varepsilon >0 $
$$ X_\varepsilon (x,t)=f(x,v(x,t/\varepsilon )). $$
Let $ K $ and $ K^\prime $ be two compact sets of $ \bkR^ n\backslash\{ 0\} $
such that
%$$ K\subset \doubleup{ \circ \cr K \cr} ^\prime . \eqno (2.31) $$
$ K\subset \hbox{ Interior}(\, K'\, ) $. 
Let
$$ 0<\delta = {\rm Min} \left\{ -\nabla V(x)\cdot 
\int^ 1_0f(x,v(x,t))dt;\ x\in K^\prime
\right\} . \eqno (2.32) $$
Then we have \par
\smallskip
\noindent{\bf Lemma 2.8. }{\sl There exists\/} $ \varepsilon_ 0>0 $ {\sl such
that for all\/} $ \varepsilon $ {\sl in\/} $ (0,\varepsilon_ 0\rbrack $
\setcounter{equation}{32}
\begin{eqnarray}
(\dot  x=X_\varepsilon (x,t),\ x(0)\in K) &\Rightarrow&
(x(t)\in K^\prime ,\ \forall t\in[ 0,\varepsilon] ),\qquad\\
(\dot x=X_\varepsilon (x,t),\ x(0)\in K) 
&\Rightarrow& (V(x(\varepsilon ))\leq
V(x(0))-(\delta /2)\varepsilon ). 
\end{eqnarray}
\par
\noindent\underbar{Proof of Lemma 2.8}. Clearly there exists $ \varepsilon_
1>0 $ such that, for
$ \varepsilon \in (0,\varepsilon_ 1\rbrack $, (2.33) holds. We now assume that
$ \varepsilon \in (0,\varepsilon_ 1\rbrack $ and let $ x $ be
a solution of $ \dot  x=X_\varepsilon (x,t) $ with $ x(0)\in K. $ We have
$$ V(x(\varepsilon ))-V(x(0))= \int^ \varepsilon_ 0\nabla V(x(t))\cdot
f(x(t),v(x(t),t/\varepsilon ))dt, \eqno (2.35) $$
$$ \vert \nabla V(x(t))-\nabla V(x(0))\vert \leq C(\varepsilon ), \eqno (2.36)
$$
$$ \vert f(x(t),v(x(t),t/\varepsilon ))-f(x(0),v(x(0),t/\varepsilon ))\vert
\leq C(\varepsilon ), \eqno (2.37) $$
where $ C(\varepsilon ) $ is a constant independent of $ x(0) $ in $ K $ and $
t $ in $ [0,\varepsilon) $
and satisfies
$$ \lim_{\varepsilon \rightarrow 0} C(\varepsilon )=0. \eqno
(2.38) $$
>From (2.32), (2.35), (2.36), (2.37), and (2.38) we get (2.34) if
$ \varepsilon \in (0,\varepsilon_ 0 ] $ with $ \varepsilon_ 0\in
\left(0,\varepsilon_ 1 \right] $ small enough. \par
Let $ r=(r_i;\ i\in \bkZ ) $ be a sequence of positive real numbers
such that
$$ r_i<r_{i+1} \ \ \ \forall i\in \bkZ , \eqno (2.39) $$
$$ \lim_{i\rightarrow +\infty}  r_i=+\infty , \eqno (2.40) $$
$$ \lim_{i\rightarrow -\infty}  r_i=0. \eqno (2.41) $$
Replacing $ V $ by $ V-V(0) $ we may assume
$$ V(0)=0. \eqno (2.42) $$
For two real numbers $ a $ and $ b $ let
$$ D(a,b)= \left\{ x\in \bkR^ n;\ a\leq V(x)\leq b \right\} . \eqno (2.43) $$
Let $ \mu =(\mu_ i;i\in \bkZ ) $ be the sequence of positive real numbers
defined
by
$$ 0<\mu_ i= {\rm Min} \left\{ -\nabla V(x)\cdot \int^ 1_0f(x,v(x,t))dt;\ \
x\in D(r_i,r_{i+1}) \right\} . \eqno (2.44) $$
Let $ a=(a_i;i\in \bkZ ) $ be a sequence of positive real numbers such that
$$ r_i+2a_i<r_{i+1}-2a_{i+1}\ \ \forall i\in \bkZ \eqno (2.45) $$
and, for a sequence, let $ A\in C^0((0,+\infty );[0,1]) $ be defined by
$$ A(s)= \left\vert s-r_i \right\vert /a_i\ {\rm if} \ s\in
\left[r_i-a_i,r_i+a_i \right] $$
$$ A(s)=1\ {\rm if} \ s\in \left[r_i+a_i,r_{i+1}-a_{i+1} \right]. \eqno (2.46)
$$
For a sequence of positive real numbers $ \varepsilon =(\varepsilon_ i;i\in
\bkZ ) $ we define a
continuous time-varying feedback law $ w $ on $ \bkR^ n\times[ 0,1] $ by
$$ w(x,t)=v(x,t/\varepsilon_ i)A(V(x))\ {\rm if} \ x\in D(r_i,r_{i+1}) \eqno
(2.47) $$
$$ w(0,t)=0. \eqno (2.48) $$
If $ \bar  \varepsilon =(\bar \varepsilon_ i;i\in \bkZ ) $ is another sequence of
positive real
numbers we will say that $ \varepsilon \leq\bar  \varepsilon $ if
$$ \varepsilon_ i\leq\bar  \varepsilon_ i,\ \ \forall i\in \bkZ . \eqno (2.49)
$$
Then, using Lemma 2.8, one easily gets \par
\smallskip
\noindent{\bf Lemma 2.9.} {\sl Let\/} $ a=(a_i;i\in \bkZ ) $ {\sl be a
sequence of positive real numbers
such that\/} (2.45) {\sl holds. Then there exists a sequence\/} $ \bar
\varepsilon $ {\sl of
positive real numbers such that if\/} $ \varepsilon \leq\bar  \varepsilon $
{\sl then any maximal solution
of\/} $ \dot  x=f(x,w(x,t)) $ {\sl defined at\/} 0 {\sl is defined on\/} $
[0,1] $ {\sl and satisfies
for all\/} $ i $ {\sl in\/} $ \bkZ $ {\sl and for all} $t$ {\sl in} $[0,1]$
\setcounter{equation}{49}
\begin{eqnarray}
V(x(0))\in \left[r_{i-1},r_i \right] &\Longrightarrow&  V(x(t))\leq
V(x(0))+2 {\rm Max} (a_{i-1},a_i) , \\
V(x(0))\leq r_i-2a_i &\Longrightarrow&  \nonumber\\ 
V(x(1))&\leq& {\rm Max}
(r_{i-1}+2a_{i-1},V(x(0))-(\mu_ i/2)). 
\end{eqnarray}
Let now $ r^\prime =(r^{\prime}_ i;i\in \bkZ ) $ be the sequence of positive
real
numbers defined by
$$ r^{\prime}_ i=\frac{r_i+r_{i+1}}{2}. \eqno (2.52) $$
For two sequences $ a^\prime ,\varepsilon ^\prime $ of positive real numbers
we define as
before but, with $ r^{\prime}_ i $ instead of $ r_i, $ a time-varying feedback
law
$ w^\prime \in C^0(\bkR^ n\times[ 0,1];\bkR^ m). $ Finally let $ \bar  u\ :\
\bkR^ n\times[ 0,2]\rightarrow \bkR^ m $ be defined by
\setcounter{equation}{52}
\begin{eqnarray}
\bar  u(x,t) &=& w(x,t)\ {\rm if} \ t\in[ 0,1], \\
\bar u(x,t) &=& w^\prime (x,t-1)\ {\rm if} \ t\in (1,2\rbrack .
\end{eqnarray}
Note that
$$ \bar  u\in{\rm C}^0(\bkR^ n\times ([0,2]\backslash\{ 1\}) ;\bkR^ m), \eqno
(2.55) $$
$$ {\rm Sup} \left\{ \left\vert\bar  u(x,t) \right\vert ;t\in[ 0,2] \right\}
\rightarrow 0\ {\rm as} \ x\rightarrow 0. \eqno (2.56) $$
One easily checks using Lemma 2.9 that if $ a $ and $ a^\prime $ are small
enough (i.e. $ a_i+a^{\prime}_ i\leq \eta_ i $ $ \forall i\in \bkZ $ for a
suitable choice of $ \eta_ i>0) $ then,
if $ \varepsilon $ and $ \varepsilon ^\prime $ are small enough, any maximal
solution of $ \dot  x=f(x,\bar  u(x,t)) $
defined at $ t=\tau \in\{ 0,1\} $ is defined on $ [\tau ,2] $ and satisfies
$$ V(x(t))\leq 2V(x(\tau ))\ \ \forall t\in[ \tau ,2], \eqno (2.57) $$
$$ V(x(2))\leq V(x(0))-\bar\theta (x(0))\ {\rm if} \ \tau =0, \eqno (2.58) $$
where $ \bar  \theta \in C^0(\bkR^ n;\lbrack 0,+\infty )) $ satisfies
$$ \bar  \theta (x)=0 \Longleftrightarrow  x=0. \eqno (2.59) $$
Let $ \eta \in C^0(\bkR^ n;[0,1/2]) $ with $ \eta (x)=0 \Longleftrightarrow
x=0 $. We define $ u\ :\ \bkR^ n\times[ 0,2]\rightarrow \bkR^ m $
by $ u(0,t)=0 $ for all $ t $ in $ [0,2] $, and for $ x\not= 0 $ \par
\noindent (i) if $ t\leq 1-\eta (x) $ or $ t\in[ 1,2-\eta (x)] $
$$ u(x,t)=\bar  u(x,t), \eqno (2.60) $$
(ii) if $ t\in[ 1-\eta (x),1] $
$$ u(x,t)=\frac{1-t}{\eta (x)} w(x,t)+\frac{\eta (x)+t-1}{\eta (x)} w^\prime
(x,0), \eqno (2.61) $$
(iii) if $ t\in[ 2-\eta (x),2] $
$$ u(x,t)=\frac{2-t}{\eta (x)}w^\prime (x,t)+\frac{t-2+\eta (x)}{\eta
(x)}w(x,0). \eqno (2.62) $$
Note that
$$ u\in C^0(\bkR^ n\times[ 0,2];\bkR^ m), \eqno (2.63) $$
$$ u(x,0)=u(x,2),\ \ \forall x\in \bkR^ n. \eqno (2.64) $$
We extend $ u $ to all $ \bkR^ n\times \bkR $ by requiring
$$ u(x,t+2)=u(x,t)\ \ \forall (x,t)\in \bkR^ n\times \bkR . \eqno (2.65) $$
By (2.64) and (2.65)
$$ u\in C^0(\bkR^ n\times \bkR ;\bkR^ m). \eqno (2.66) $$
Let $ D_k= \left\{ x\in \bkR^ n;3^{k-1}\leq V(x)\leq 3^k \right\} $ for $
k\in \bkZ . $ From the definition of $ u $ we
get that there exists a sequence $ \varepsilon =(\varepsilon_ k;k\in \bkZ ) $
of positive real numbers such that, if
$$ \eta (x)\leq \varepsilon_ k\ \ \forall x\in D_k\ \ \forall k\in \bkZ ,
\eqno (2.67) $$
then any maximal solution of $ \dot  x=f(x,u(x,t)) $ defined at $ t=0 $ which
satisfies $ V(x(0))\in \left[3^{k-1},3^{k+1} \right] $ is defined on $ [0,2] $
and satisfies
\setcounter{equation}{67}
\begin{eqnarray}
V(x(t)) &\leq& 3V(x(0))\ \ \ \forall t\in[ 0,2], \\
V(x(2)) &<& V(x(0)). 
\end{eqnarray}
Indeed this can be easily seen by noticing that, given $ k $ in $ \bkZ , $
there exists $ \delta =\delta (k)>0 $ such that for every solution $ x\ :\
[0,2]\rightarrow \cup_{l\le k+2} D_l $
of $ \dot  x=f(x,\bar  u(x,t)) $ and every solution $ \tilde  x\ :\
[1,2]\rightarrow \cup_{l\le k+2} D_l$ of the same
equation then
$$ \left\vert x(1)-\tilde  x(1) \right\vert <\delta \Longrightarrow  V(\tilde
 x(2))<V(x(0))-\frac{1}{2}\theta (x(0)). \eqno (2.70) $$
We now choose $ \eta \in C^0(\bkR^ n;[0,1/2]) $ satisfying (2.67) for all $ k $
in $ \bkZ . $
Then 0 is globally asymptotically stable point of $ \dot  x=f(x,u(x,t)). $
%\bye
%\input definit.tex
\def\va{\hfill\break}
\def\sou#1{$\underline{\smash{\hbox{#1}}}$}
\def\R{{\rm I\!R}}%
\def\Rn{{\R ^n}}
\def\Rnn{{\Rn\backslash \{ 0\} }}
\def\ce#1{{\mathaccent"17 #1}}
\def\N{{\rm I\!N}}%
\def\.#1{{\mathaccent"5F #1}}
\def\x{{\. x}}
\def\nf{\infty}
\def\pa{{\partial ^{\alpha}}}
\def\wa{\vskip 2 true cm }
\def\wb{\vskip 1 true cm }
\def\wc{\vskip 0.5 true cm }
\section{Stabilization by Means of Discontinuous Feedback
Laws and Integrator}
\va\indent
Let us recall first a result proved independently in [BI] and [T]
(see also [CP] for related results): if the smooth (e.g. of class $C^1$)
system $\x =f(x,u)$ can be locally asymptotically stabilized by means
of a smooth (e.g. of class $C^1$) feedback law then $\x =f(x,y),\
\. y =v$ can be locally asymptotically stabilized by means of a smooth
(e.g. of class $C^0$) feedback law.\par
In this section we give an example showing that, even if $f$ is analytic,
this no longer is true if we replace smooth feedback laws by discontinuous
feedback laws. Our example will be with $n=2,\, m=1$. Note that by a
theorem due to W.P. Dayawansa and C.F. Martin ([DM]) there is no such
example with $n=1,\, m=1$. Our example is given in the following
\par\smallskip
\noindent{\bf Proposition 3.1}. {\sl The system $\.{x_1}=u(\in\R  ),\
\.{x_2}={x_1}^2-2x_1 u^2$
 can be locally asymptotically stabilized by means
of a discontinuous feedback law. But the system $\.{x_1}=y,\
\.{x_2}={x_1}^2-2x_1y^2,\ \. y =v$ cannot be locally asymptotically
stabilized by means of a discontinuous feedback law. In fact it cannot also
be locally asymptotically stabilized by means of a time-varying feedback
law.}\va\va
\sou{Remark 3.2}. If $f$ is only continuous there are systems with $n=m=1$
such that $\x =f(x,u)$ can be locally asymptotically stabilized by means
of a continuous feedback law and such that $\x =f(x,y),\ \. y =v$ cannot
be asymptotically stabilized by means of a feedback law (time-varying
 or discontinuous). This is the case for example if 
 $f(x,u)=x^{\frac{1}{3}}-u$:
 $0$ is a globally asymptotically stable point of $\x =f(x,u(x))$ with
$u(x)=2x^{\frac{1}{3}}$, but, if 
$V(x,y)=x-y^2,\ \x =x^{\frac{1}{3}}-y,\ \. y =v$
then $\. V =x^{\frac{1}{3}}-y-2yv\ge (y^2)^{\frac{1}{3}}-y-2yv\ge 0$ if
$V\ge 0, \vert v\vert\le 1$, and $\vert y\vert$ small; this shows that
$\x =f(x,y),\ \. y=v$ cannot be locally asymptotically stabilized by
means of a feedback law (time-varying or discontinuous).\va\va
\sou{Proof of proposition 3.1} . We first prove that
$\.{x_1} =u,\ \.{x_2} ={x_1}^2-2x_1u^2$ can be locally asymptotically
stabilized by means of a discontinuous feedback law. Let $u:\R ^2\to \R$
be defined by
\setcounter{equation}{0}
\begin{eqnarray}
u(x_1,x_2)&=&(x_2-2x_1)^{\frac{1}{3}}\hbox{ if }
\vert x_2-x_1\vert\ge ({x_2}^2+{x_1}^2)^{\frac{33}{32}}, \\
u(x_1,x_2)&=&\vert x_2\vert ^{\frac{1}{3}}\hbox{ if } \vert x_2-x_1\vert\le
 ({x_2}^2+{x_1}^2)^{\frac{33}{32}}.
\end{eqnarray} 
Note that $(1.9)$ holds. Let $V\in C^1(\R ^2;\R )$ be defined by
$$V(x_1,x_2)=x_1^2-x_1x_2+{x_2}^2.\eqno(3.3)$$
Note that
$$V(x_1,x_2)>V(0)\hbox{ if } (x_1,x_2)\ne (0,0).\eqno(3.4)$$
Moreover straightforward computations show that, with $X(x)=$\va
$(u(x),{x_1}^2
-2x_1u^2)\,$  $ (=f(x,u(x))\, )$,
$$X(x)\cdot \nabla V(x)\le -({x_1}^2+{x_2}^2)^2
\hbox{ if } \vert x\vert \hbox{ is small enough } \eqno(3.5)$$
which shows that $0$ is a locally asymptotically stable point of
$\x =f(x,u(x))$.\par
We now prove that $\.{x_1} =y,\ \.{x_2} ={x_1}^2-2x_1y^2,\ \.{y} =v$ cannot
be locally asymptotically stabilized by means of (discontinuous or
time-varying) feedback law. Let $H\in  C^1(\R  ^3,\R )$ be defined by
$$ H(x_1,x_2,y)=x_2+y{x_1}^2.\eqno(3.6)$$
Then
$$\.{H}={x_1}^2(1+v).\eqno(3.7)$$
Therefore $\.{H}\ge 0$ if $v\ge -1$ which gives the desired statement
(note that $(0,0,0)$ is in the closure of $\{ (x,y);H(x,y)>0\}$).
\va
\va
\medskip\noindent
\centerline{\bf APPENDIX}\va
\centerline{{\bf Robustness of asymptotical stability for
discrete-time systems}}
\par
\smallskip
It is sometimes useful to study the discrete-time system associated
 with an ordinary differential equation
$$\dot x =f(x) \eqno(A.1)$$
in the following way: if $f$ is locally Lipschitzian, we define $F(x)$ to
be the point reached from $x$ at time 1, that is $F(x)=x(1)$ where $x(\cdot )$
is the solution of $\dot x =f(x)$ such that $x(0)=x$. The discrete-time
system we consider is then
$$x_{n+1}=F(x_n). \eqno(A.2)$$
When (A.1) is an asymptotically stable system, then (A.2) is
also an asymptotically stable system.
If $f$ is now only assumed to be continuous, $F$ must be regarded as a
set-valued map, since there exist in general many points reached
from $x$ in time 1. Our aim is to prove, under some general
hypotheses -which are satisfied when $F$ is constructed from $f$ as
above- that asymptotic stability for systems of
the type
$$x_{n+1}\in F(x_n). \eqno(A.3)$$
is a robust property, i.e: if we perturb slightly 
 such systems the asymptotic stability
property  is not lost, (even if $F$ is not the time one map of
an ordinary differential equation.) \par
>From now on we consider an open set $G\subset \R ^n$, containing the origin,
a continuous map $\omega :G\to \R ^+ $ , such that $\omega (x)=0\iff x=0$
and $\omega (x)\to +\nf$ as $x\to \partial G$, and a set-valued map
$F:\, x\in G\mapsto F(x)\subset G$ such that
\begin{itemize}
\item[(i)] $\forall x\in G,\, F(x)$ is a nonempty compact set;
\item[(ii)] $F$ is upper semicontinuous (usc): given $x_0\in G$ and
an open set $\Omega \supset F(x_0)$, we may find $\delta >0$ such that
$\vert x-x_0\vert < \delta \Rightarrow F(x)\subset \Omega$.\va\va
It may be noted that (i) together with (ii) are equivalent to
\item[(iii)] For every nonempty compact subset $K$, the set
$\big\{ (x,y)\in G^2; \, x\in K,\, y\in F(x)\big\}$ is a nonempty
compact set.
\end{itemize}
\par
For $A\subset G$, $F(A)$ denotes the set $\cup_{x\in A} F(x)$ and we
define, for $x\in G$, the sequence
$\big( F^n(x)\big) _{n\ge 0}$ by the natural
following induction:
$F^0(x)=\{ x\} ,\, F^1(x)=F(x)$, and $F^n(x)=F(F^{n-1}(x))
=\cup_{y\in F^{n-1}(x)}F(y)$ for $n\ge 1$. The following proposition is
an immediate translation of a result obtained by Kurzweil in [Ku].\par
\smallskip
\noindent {\bf Proposition A.1}. 
{\sl Let $F:G\to {\cal P} (G)$ be a set-valued map
satisfying (i), (ii) and such that $F(0)=\{ 0\}$. Then the following two
properties are equivalent:\va\va
${\cal P}_1$: $0$ is an asymptotically stable (on $G$) point for the system
(A.3), that is 
\begin{itemize}
\item[a)] $\forall \epsilon ,\ \exists\delta$ such that
$\omega (x) <\delta\Rightarrow
\forall n,  \ \ \omega (F^n (x))\subset [0,\epsilon ];$
\item[b)] all trajectories of (A.3)
 tend to $0$ as $n\to +\nf$.
\end{itemize} 
${\cal P}_2$: There exist an increasing function $B:(0,+\nf )\to (0,+\nf )$,
with $\lim_{\beta\to 0}B(\beta )=0$ and a function $T:(0,+\nf )^2\to (0,+\nf )$
 such that $\forall \epsilon >0
,\, \forall \beta >0 $, for any trajectory $(x_n)_{n\ge 0}$ of (A.3), with
$\omega (x_0)\le \beta$, we have
\begin{itemize}
\item[c)] $\omega (x_n)\le B(\beta )\ \forall n\ge 0$.
\item[d)]
$\omega (x_n)\le \epsilon\ \forall n\ge T(\beta ,\epsilon )$.
\end{itemize}
}
\va
We now state the main result of this appendix, which can be used 
instead of Lemma 2.2 to prove Lemma 2.3.
\par\smallskip\noindent
{\bf Theorem A.2}. {\sl Assume that ${\cal P }_1$  holds, then there exists
$\delta$ {\sl in}
${\rm C}^0(G;[0,+\nf ))$ such that  \par
\begin{itemize}
\item[i)] $\delta >0$ on $G\setminus \{ 0\}$;
\item[ii)] $\delta (0)=0$;
\item[iii)] $F(x)+B(0,\delta (x))\subset G, \ \ \forall x\in G
\setminus\{ 0\}$;
\item[iv)] $0$ is an asymptotically stable
point  (on $G$)  for the system
$$x_{n+1}\in F(x_n) + B(0,\delta (x_n)).$$ 
\end{itemize}
}\par
Theorem A.2 is an immediate consequence of 
\def\d{\delta}
\def\O{\Omega}
\def\o{\omega}
\par\smallskip\noindent
{\bf Lemma A.3}. {\sl Let $\beta >0$ and $N\in \N$ be such that
$\omega (x)\le\beta \Rightarrow \omega (F^n (x))\subset
[0,\frac{\beta}{4}]\ \forall n\ge N$. 
Then there exists $\delta >0$
such that, for any sequence $\hat x_0,\hat x_1,\ldots ,\hat x_N$
in $G$ satisfying: $\o (\hat {x_0})\le\beta$ and $\hat x_{n+1}\in
F(\hat x_n)+B(0,\d )$ for $0\le n\le N-1$, we have
$$
\left\{ 
\begin{array}{ll}
\o (\hat x_N) & < \frac{\beta}{2} \\
\o (\hat x_n) & < 2B(\beta )\ \ \hbox{for  } n=0,1,\ldots ,N.
\end{array}   
\right.
\eqno{(A.4)}$$
($B(\cdot )$ is the function given by ${\cal P}_2$.)  }
\par\smallskip\noindent
\sou{Proof of Lemma A.3}.
Since the set-valued map $F$ is usc, for any compact subset $K$ of $G$ and any
neighborhood $V$ of $F(K)$, there exists a neighborhood $U$ of
$K$ such that $F(U)\subset V$. $x_0$ being any point 
with $\o (x_0) \le \beta$, we may apply
inductively this fact to the sequence of compact sets $F^{N-1} (x_0),
F^{N-2} (x_0),\ldots ,F(x_0),$ $\{ x_0\}$
for proving the existence of $\d _{x_0}$ such that,
if $\tilde x_0\in B(x_0,\delta _{x_0})$ and 
   $\tilde x_{n+1}\in F(\tilde x_n)+B(0,\d _{x_0})$ for 
all $n$ in $[0,N-1]$, then
$$ d(\tilde x_n, F^n(x_0)) < \min (B(\beta ),
\frac{\beta}{4})\quad 0\le n\le N.\eqno{(A.5)}$$
Using the compacity of $\{ x; \o (x)\le \beta\}$ the lemma follows.
\va\va
\centerline{{\bf REFERENCES}}
\va
$\lbrack$AB$\rbrack$ A. ANDREINI, A. BACCIOTTI, Stabilization by
piecewise constant feedback, Proc. of the first European Control Conference,
July 2-5 1991, vol. 1 p. 474-479. \va
$\lbrack$A$\rbrack$ Z. ARTSTEIN, Stabilization with relaxed controls,
{\sl Nonlinear
Anal., T.M.A. 7\/} (1983) 1163-1173. \va
$\lbrack$BI$\rbrack$ C.I. BYRNES and A. ISIDORI, New results and
counterexamples in nonlinear feedback stabilization, Systems and
Control Letters, 12 (1989) 437-442. \va
$\lbrack$B$\rbrack$ R.W. BROCKETT, Asymptotic stability and feedback
stabilization, in {\sl Differential Geometric Control Theory,\/} (R.W.
Brockett, R.S. Millman and H.J. Sussmann Eds.,) Birkha\"user, 1983. \va
$\lbrack$CS$\rbrack$ C. CANUDAS DE WIT and O.J. SORDALEN, Exponential
stabilization of mobile robots with nonholonomic constraints,
Proc. of the 30th Conference on Decision and Control, Brighton,
England, December 1991, W.2-12-2:10, p. 692-697. \va
$\lbrack$C1$\rbrack$ J.-M. CORON, Global asymptotic stabilization for
controllable systems without drift, {\sl Math. Control Signals
Systems,\/} 5 (1992) 295-312. \va
$\lbrack$C2$\rbrack$ J.-M. CORON, Links between local controllability
and local
continuous stabilization, Preprint Universit\'e Paris-Sud, October
1991 and NOLCOS Bordeaux, June 1992. \va
$\lbrack$CP$\rbrack$ J.-M. CORON and L. PRALY, Adding an
integrator for the stabilization problem, Systems and Control Letters,
17 (1989) 89-104.\va
$\lbrack$DM$\rbrack$ W.P. DAYAWANSA and C.F. MARTIN, Asymptotic
stabilization of two dimensional real analytic systems, Systems and
Control Letters, 12 (1989) 205-211. \va
$\lbrack$F$\rbrack$ A. FILIPPOV, Differential equations with
discontinuous
right-hand side, {\sl Amer. Math. Soc. Translations 42,\/} (1964)
199-231. \va
$\lbrack$FM$\rbrack$ M. FLIESS and F. MESSAGER, Vers une
stabilisation non
lin\'eaire discontinue, Proc. 9th International Conf. {\sl Analysis and
Optimization of Systems,\/} 
Lecture Note Control Inform., Springer, Berlin 1990.\va
$\lbrack$Ha$\rbrack$ P. HARTMAN, {\sl Ordinary Differential
Equations,\/} Wiley,
New York, 1964.\va
$\lbrack$He$\rbrack$ H. HERMES, Discontinuous vector fields and feedback
control,
in {\sl Differential Equations and Dynamical Systems,\/} (J.K. Hale and
J.P. La Salle, Eds.), Academic Press, New-York and London, 1967. \va
$\lbrack$Ku$\rbrack$ J. KURZWEIL, On the inversion of Ljapunov's
second theorem on
stability of motion, {\sl Amer. Math. Soc. Translation 24\/} (1956)
19-77. \va
$\lbrack$M$\rbrack$ F. MESSAGER, Two nonlinear examples of
discontinuous
stabilization, in Colloque International sur l'Analyse des
Syst\`emes Dynamiques Contr\^ol\'es, vol. 1, Lyon, France, 1990. \va
$\lbrack$R$\rbrack$  L. ROSIER, Inverse of Lyapunov's second theorem
for
measurable functions, preprint,
 Universit\'e Paris-Sud, 1991, and
NOLCOS Bordeaux, 1992. \va
$\lbrack$Sa$\rbrack$ C. SAMSON, Velocity and torque feedback control
of a wheeled
mobile robot, stability analysis, Preprint INRIA,
Sophia-Antipolis, 1990.\va
$\lbrack$Su$\rbrack$ H.J. SUSSMANN, Subanalytic sets and feedback
controls, {\sl J.
Diff. Equ. 31,\/} (1979) 31-52.\va
$\lbrack$SS$\rbrack$ E.D. SONTAG and H.J. SUSSMANN, Remarks on
continuous
feedback, in {\sl Proc. of the 19th IEEE Conference on Decision and
Control,\/} Albuquerque, NM, 1980, p. 916-921.\va
$\lbrack$T$\rbrack$ J. TSINIAS, Sufficient Lyapunovlike
conditions for stabilization, Math. Control Signal Systems,
2 (1989) 343-357.\va
\smallskip
\end{document}

