\documentclass[12pt,leqno]{article} 
\title{Smooth Lyapunov Functions for \\
Discontinuous Stable Systems} 
\author{Lionel Rosier}
\date{}
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\newtheorem{thm}{Theorem}
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\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\begin{document} 
\maketitle
\vspace*{-1cm}
\begin{center}
Laboratoire d'Analyse Num\'erique et EDP,\\
Universit\'e Paris 11, B\^at. 425,\\
91405 Orsay Cedex, France.\\ 
{\footnotesize
e-mail: Lionel.Rosier@math.u-psud.fr}\\
\end{center}

\begin{abstract}
%\baselineskip0.9cm
It has been proved in \cite{BR} that a differential system
$dx/dt =f(t,x)$
with a discontinuous right-hand side
admits some {\it continuous} weak Lyapunov function
if and only if it is {\it robustly} stable.
This paper focuses on the smoothness of such a Lyapunov function.
An example of an (asymptotically) stable
system for which there does not
exist any (even weak) Lyapunov functions of class $C^1$
is given. In the more general context
of differential inclusions, the existence of
a weak Lyapunov function of class $C^1$ (or $C^\infty$) 
is shown to be equivalent to the robust stability 
of some perturbed system obtained in introducing measurement error
with respect to $x$ {\it and } $t$.
This condition is proved to be satisfied by 
most of the robustly stable systems 
encountered in the literature. Analogous results are given for the 
Lagrange stability.
As an application to the study of the links between
internal and external stability for control systems,
an extension of a result by Bacciotti and Beccari is obtained by means of a 
{\em smooth} Lyapunov function associated with a robustly Lagrange 
stable system.
\va\va
{\bf Mathematics Subject Classifications (1991):} 34A60, 34D20,
93D05, 93D25.
\va\va
{\bf Key Words:} Converse first Lyapunov theorem, 
smooth Lyapunov function, Lagrange stability, bounded input bounded state
stability, differential inclusion, Filippov's solution.
\end{abstract}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section{Introduction}
\va
Let us consider the ODE
\begin{equation}
\label{eq:1}   
\dot{x} = f(t,x), 
\end{equation}
where $t\in \R  ^+$ is time, $x\in \R ^n$ is the state variable,
$\dot{x}=dx/dt$ and $f$ is
some field defined on $\R ^+ \times \R ^n$. Let 
$Q$ be an open subset of $\R ^n$ and let  
$V=V(t,x)$ be a function
defined on $\R ^+ \times Q$.
\va\va
{\bf Definition 1.}
$V$ is said to be a {\em weak Lyapunov function}  for (\ref{eq:1})
on $\R ^+ \times Q$ if 
\begin{equation*}
%\label{eq:2}
t_1\le t_2 \Rightarrow V\big( t_1,x(t_1)\big) \ge V\big( t_2,x(t_2)\big)
\end{equation*}
for each pair of nonnegative numbers $t_1,t_2$ and each
solution $x(t)\, : [t_1,t_2]\rightarrow Q$ of (\ref{eq:1}).
If, moreover, $Q=\{ x\, :\, \Vert x\Vert <h\}$ for some $h>0$ (resp.
$Q=\R ^n\setminus\{ 0\}$) and if there exist two increasing continuous
maps $a:\R ^+\to \R ^+$ and $b:\R ^+\to \R ^+$ such that \va
(I) $a(0)=0$, $a(r)>0$ for $r>0$,\va
(II) $b(0)=0$ (resp. $a(r)\to +\infty$ as $r\to +\infty$),\va
(III) $a(\Vert x\Vert ) \le V(t,x) \le b(\Vert x \Vert )$ for
$t\ge 0$, $x\in Q$, \va
then $V$ is said to be a weak Lyapunov function {\em in the small}
(resp. {\em in the large}).
\va

A well-known version of the first Lyapunov theorem asserts that if there
exists a weak Lyapunov function in the small for (\ref{eq:1}),
then the origin is a (uniformly) {\em stable} equilibrium 
position for (\ref{eq:1}).
The question of whether such a theorem is invertible, with some
(hopefully) smooth Lyapunov function, constitutes an important issue in 
control theory. (Indeed, many applications of stabilization theory are
based on the so-called Jurdjevic-Quinn method for the design of the
feedback, which, in turn, rests on the knowledge of a {\it smooth}
weak Lyapunov function for the unforced associated system.)  
The first positive answers to this question were given independently by
Krasovski \cite{K}, Kurzweil \cite{K1} and Yoshizawa \cite {Y1}.
In these papers, rather strong assumptions are made as far as the
regularity of the field $f$ is concerned.  (For instance, $f$ is 
assumed to be $C^1$ in
\cite{K1}, or continuous and locally Lipschitz continuous
w.r.t. $x$ in \cite{Y1}.)
These assumptions guarantee the uniqueness of the solution of (\ref{eq:1})
issuing from $x_0$ at $t=t_0$ for any pair $(t_0,x_0)$, and also the
smoothness of the Lyapunov function which is constructed. It turns out that
the situation is more involved when $f$ is only assumed to be continuous.
Indeed Kurzweil and Vrko\v c  \cite{KV} have shown that
a stable system (\ref{eq:1}) with a continuous r.h.s. may fail to
have a continuous weak Lyapunov function $V=V(t,x)$. In fact the first
Lyapunov theorem may be inverted provided that the stability concept is
slightly strengthened. Let us denote by $x(t;t_0,x_0)$ any solution of
(\ref{eq:1}) such that $x(t_0)=x_0$. 
\va\va
{\bf Definition 2.}
Assume that $x=0$ is an equilibrium position for (\ref{eq:1}). We say that 
the origin is {\em robustly stable} for (\ref{eq:1}) 
if there exist a sequence
$( G_{-i}) _{i=0,1,2,\ldots }$ of open sets in $\R ^+\times \R ^n$ and two
sequences $(a_{-i}) _{i=0,1,2,\ldots }$,
$(b_{-i}) _{i=0,1,2,\ldots }$ of real numbers such that:\va
(i) $0<b_{-i-1}<a_{-i}\le b_{-i}$ for each $i=0,1,2,\ldots $ and
$b_{-i}\to 0$ as $i\to +\infty$; \\
(ii) $\R^{+}\times \{ x:\, \Vert x\Vert <a_{-i}\} \subset
G_{-i} \subset \R^{+}\times \{ x:\, \Vert x\Vert <b_{-i}\}$ for each
$i=0,1,2,\ldots $;  \\
(iii) for each $i=0,1,2,\ldots $, each initial pair $(t_0,x_0)\in G_{-i}$ and
each solution $x(.;t_0,x_0)$, one has 
$\big( t,x(t;t_0,x_0)\big) \in G_{-i}$ for
each $t\ge t_0$.
\va

Obviously the robust stability implies the usual stability and it may
be proved that these two concepts agree when the continuous field $f$ is
{\em also} assumed to be locally Lipschitz continuous w.r.t. $x$. The main
result in \cite{KV} asserts that for
any continuous field $f=f(t,x)$ such that $f(t,0)=0$ for any $t$,
the existence of a  continuous weak
Lyapunov function in the small $V=V(t,x)$ 
for (\ref{eq:1}) is
{\em equivalent} to the
robust stability of the origin.
This result is extended in \cite{BR} to discontinuous systems,  i.e. 
$f\in {\cal L}^{\infty}_{\rm loc}(\R ^+\times \R ^n,\R ^n)$ and
solutions are taken in Filippov's sense \cite{F}. The
aim of this paper is to investigate the existence of a {\it smooth} 
Lyapunov function for a {\it discontinuous} robustly stable system. 

In the (somewhat different) context of asymptotic stability, significant
progress have recently been made by Clarke {\em et al.} in the construction
of {\em smooth} Lyapunov functions for discontinuous (autonomous) systems. 
Indeed, it follows from \cite[Thm. 1.2]{CLS} that a globally 
asymptotically stable system $\dot{x}=f(x)$ (with 
$f\in {\cal L}^{\infty}_{\rm loc}(\R ^n,\R ^n)$ 
and solutions of (\ref{eq:1}) taken in Filippov's sense) admits
a (strong) Lyapunov function of class $C^{\infty}$. 
Notice that such a result cannot be generalized to time-dependent
systems, as it is shown by the following result (to be proved in this paper).
\begin{prop}
\label{prop1}
There exists a bounded Borel map $\lambda \! :\R \to \R$ such that the origin
is globally asymptotically stable {\rm (}hence, robustly stable{\rm )} 
for the system
\begin{equation}
\dot{x}=\lambda (t) x
\label{eq:10}
\end{equation}
and such that there does not exist any (even weak) Lyapunov function
in the large {\rm (}or in the small{\rm )} \sou{of class $C^1$}.
\end{prop}

Hence, the local Lipschitz continuity of a (strong) Lyapunov function for 
a discontinuous (time-dependent) asymptotically stable 
system (\ref{eq:1}), asserted by \cite[Thm. 2]{R}, cannot be improved
in general. Notice that the counterexample given in Proposition
\ref{prop1} proves to be rather pathological,
since the restriction of the function
$\lambda$ to the complement of any zero measure set is {\em nowhere}
continuous. If we intend to 
construct a {\em smooth} Lyapunov function, some
additional assumption has to be made as far as the regularity of $f$
w.r.t. time is concerned.
\va\va
{\bf Definition 3.}
Let $f\in {\cal L}_{\rm loc}^{\infty}(\R^+\times \R ^n,\R ^m)$, 
$m,n\ge 1$. The function $f$ is said to 
be {\em essentially continuous w.r.t. time} ({\bf e.c.t.} in short) 
if there exists
a set ${\cal N}\subset \R^+\times \R ^n$ of measure zero such that
for each
pair $(t_0,x_0)\in\R^+\times \R ^n$ and for each $\varepsilon >0$, there exists 
$\delta >0$  such that for each pair $(t,x)\in\R^+\times \R ^n$
\begin{equation}
\label{eq:13}
\left.
\begin{array}{l}
\vert t-t_0\vert +\Vert x-x_0\Vert <\delta \\
(t,x)\not\in {\cal N} \\
(t_0,x)\not \in {\cal N}
\end{array}
\right\} \im \Vert f(t,x)-f(t_0,x)\Vert < \varepsilon .
\end{equation}
\va\va
Roughly speaking, the function $f$ is {\bf e.c.t.}
if the restriction of $f$ to the
complement of some set of measure zero is continuous w.r.t. time, in a
(locally) uniform way w.r.t. $x$.
Such a property is fulfilled when, for instance, $f$ takes the form
\begin{equation*}
f(t,x):=h(u(t),v(x)),
\end{equation*}
where $u\! :\R^+ \to\R ^p$ is a (locally) Riemann integrable map,
$v\in {\cal L}_{\rm loc}^{\infty}(\R^n,\R^q)$ and
$h\in C^0(\R^{p+q},\R^m )$, $m,n,p,q\ge 1$.
Indeed, a bounded function defined on a segment of
$\R$ is Riemann integrable iff the set of points at which the function fails
to be continuous is of measure zero \cite{Rd1}. Notice that 
every field $f$ of the form
\begin{equation*}
f(t,x):= f_0(x) + \sum_{i=1}^p u_i(t)\, f_i(x),
\end{equation*}
where  for each
$i\ge 0$ $f_i\in {\cal L}_{\rm loc}^{\infty}(\R^n,\R^n)$ and
for each $i\ge 1$
$u_i$ is a piecewise continuous (or piecewise monotonous) function, is
concerned.\va


The following result, which asserts the existence of a smooth weak Lyapunov
function for a discontinuous (but {\bf e.c.t.}) 
robustly stable system, is one of the main results in this paper.
\begin{thm}
\label{thm1}
Let $f\in{\cal L}^{\infty}_{\rm loc}(\R ^+\times \R ^n,\R ^n)$ be 
{\bf e.c.t.}
If the origin is an equilibrium position which is 
robustly stable for {\rm (\ref{eq:1}) (}with Filippov's solutions{\rm )},
then there exists a weak Lyapunov function in the small
for {\rm (\ref{eq:1})} which is \sou{of class $C^\infty$}.
\end{thm}
We recall that a Filippov solution
$x(t)$ of (\ref{eq:1}) on some interval $I\subset \R ^+$ is a map
$x\! :I\to \R^n$ which is absolutely continuous on any $[a,b]\subset I$
and such that 
\begin{equation}
\hbox{for a.e. } t\in I, \quad \dot x(t)\in K_x f(t,x(t)),
\label{eq:3000}
\end{equation} 
where
\begin{equation}
K_x  f(t,x):=\bigcap_{\delta >0}\, \bigcap_{\mu (N)=0} \overline{\hbox{co}}
\{ f(t,B(x,\delta)\setminus N)\}. 
\label{eq:4}
\end{equation}
In (\ref{eq:4}), $B(x,\delta )$, $\overline{\hbox{co}}$ and $\mu$
denote, respectively, the closed ball of center $x$ and radius $\delta$, 
the convex
closure and the usual Lebesgue measure. 
Notice that Filippov's system (\ref{eq:3000}) is a
special instance of a differential inclusion   
\begin{equation}
\dot{x}\in F(t,x) 
\label{eq:5}
\end{equation}
in which the multivalued map $F$ assumes compact convex values and
is upper semicontinuous w.r.t. $x$. Actually Theorem 1 will follow
from a more general result 
which applies to differential inclusions (Theorem 4, see below). 
Moreover a smoothing procedure for multivalued maps 
(inspired from the one in \cite{CLS}) appears as
a key step in the proofs of Theorems 1, 2 and 4, and we emphasize
that this procedure cannot be achieved within the framework of 
ODE. Consequently, the exposition in the following 
sections will be developed within the framework of differential 
inclusion theory. 

The same ideas may be applied to the problem of boundedness of 
solutions, the so-called Lagrange stability.
Recall that system
(\ref{eq:1}) is said to be {\em Lagrange stable} if for each $R>0$ there exists
$S>0$  such that for each pair $(t_0,x_0)$  with $t_0\ge 0$, each solution
$x(.;t_0,x_0)$ defined on some interval $[t_0,t_1]$ one has 
\begin{equation}
\Vert x_0\Vert < R \im \forall t\in [t_0,t_1] \quad 
\Vert x(t;t_0,x_0)\Vert <S.
\label{eq:7}
\end{equation}
As for the stability, this concept has to be
slightly strengthened in order to be characterized  by means of some
continuous weak Lyapunov function.\va\va
{\bf Definition 4.}
Let $f(t,x)$ (resp. $F(t,x)$) be defined on $\R ^+ \times \R ^n$. We say that
(\ref{eq:1}) (resp. (\ref{eq:5})) is {\em robustly Lagrange stable} 
if there exist a sequence
$\{ G_i \} _{i=0,1,2,\ldots }$ of open sets in $\R ^+\times \R ^n$ and two
sequences $\{ a_{i}\} _{i=0,1,2,\ldots }$, 
$\{b_{i}\} _{i=0,1,2,\ldots }$ of real numbers such that:\va
(i') $0<a_{i}\le b_{i}<a_{i+1}$ for each $i=0,1,2,\ldots $ and
$a_{i}\to +\infty$ as $i\to +\infty$; \\
(ii') $\R^{+}\times \{ x:\, \Vert x\Vert <a_{i}\} \subset
G_{i} \subset \R^{+}\times \{ x:\, \Vert x\Vert <b_{i}\}$ for each
$i=0,1,2,\ldots $;  \\
(iii') for each $i=0,1,2,\ldots $, each initial pair $(t_0,x_0)\in G_{i}$ and
each solution $x(.;t_0,x_0)$ one has $\big( t,x(t;t_0,x_0)\big) \in G_{i}$ for
each $t\ge t_0$.
\va\va
Obviously a result analogous to Theorem 1 still holds for the robust Lagrange 
stability, but only the version for differential inclusions 
(Theorem 3, see below) will be reported here. 

The paper is organized as follows: The main results (existence of smooth 
Lyapunov functions for robustly stable or Lagrange stable 
differential inclusions) are presented in Section 2. The proofs are given
in Sections 3 and 4. Section 5 contains an application to control
theory, namely an extension of a result by Bacciotti and Beccari 
which asserts that an affine Lagrange stable system is UBIBS 
stabilizable. 

\setcounter{equation}{0}
\section{The Main Results}
\va
In what follows, $\langle \cdot ,\cdot \rangle$ 
denotes the usual scalar product in
$\R ^n$, $\Vert \cdot \Vert$ the corresponding Euclidean norm, 
$\hbox{dist}(x,S)$ 
the Euclidean distance from a point $x$ to a set $S$
and $h(S_1,S_2)$ the Hausdorff distance between two closed sets 
$S_1$ and $S_2$, defined by 
\begin{equation*} 
h(S_1,S_2):=\max \left( \sup _{x_1\in S_1} \hbox{dist}(x_1,S_2),
\sup _{x_2\in S_2} \hbox{dist}(x_2,S_1)\right) .
\end{equation*}
We shall adopt the notation $B$ (resp. $N$) for a closed ball 
in $\R _x^n$ (resp. a zero measure set in $\R _t^+$), and the notation 
$\cal B$ (resp. $\cal N$) for a closed ball (resp., a zero measure
set) in the extended space $\R _{(t,x)} ^{n+1}$.
\va

Let $F(t,x)$ be a multivalued map. For the sake of brevity we shall
denote by ({\bf H})
the following hypothesis on $F$: There exists a 
zero measure set $N_0\subset \R ^+$ such that \va
({\bf H1}) 
For each $(t,x)\in (\R ^+\setminus N_0)\times \R ^n$,
$F(t,x)$ is a nonempty convex compact subset of $\R ^n$;\\
({\bf H2}) For each $R>0$, there exists $M>0$ such that 
\begin{equation*}
(t\in [0,R]\setminus N_0  \hbox{ and } \Vert x\Vert \le R )\ \im \ 
F(t,x)\subset  B(0,M); 
%\label{eq:6}
\end{equation*}
({\bf H3}) For every  $t\in \R ^+\setminus N_0$, $F(t,x)$, as a
multivalued map of $x$, is upper semicontinuous;\\
({\bf H4}) For every $x\in \R ^n$, $F(t,x)$, as a multivalued map of $t$, is
measurable.\va

By a solution $x(t)$ of (\ref{eq:5})
on some interval $I\subset \R ^+$ we mean a map
$x\! :I\to \R^n$ which is absolutely continuous on any $[a,b]\subset I$
and such that $\dot x(t)\in F(t,x(t))$ for a.e. $t\in I$. It is well known
(see \cite{AC}, \cite{D})
that any Cauchy problem
\begin{equation*}
\left\{
\begin{array}{rl}
\dot{x}
&\in F(t,x) \\
x(t_0)&=x_0,
\end{array}
\right.
\end{equation*}
admits a (local) solution provided that ({\bf H}) holds true, and
that ({\bf H}) is fulfilled by Filippov's multivalued map
$F(t,x):=K_x\, f(t,x)$ for any 
$f\in {\cal L}^{\infty}_{\rm loc}(\R ^+\times \R ^n,\R ^n)$.
\va 

Proposition \ref{prop1} is obtained as a direct consequence of the following 
result, which claims that the existence of a smooth Lyapunov 
function in the large is actually equivalent to 
the robust Lagrange stability of the perturbed
system we get when introducing  measurement error w.r.t.
the {\em full} state $(t,x)$. (Compare with \cite{LS}, in which
 the existence
of a {\em smooth} uniform Control Lyapunov Function is shown to be equivalent 
 to the existence of a stabilizing feedback law which is robust
 w.r.t. {\em measurement error} and external disturbances.)
\begin{thm}
\label{thm2}
Let $F(t,x)$ be a multivalued map such that {\rm  ({\bf H})} holds
and let $N\supset N_0$ be a zero measure set in $\R ^+$.
Consider the following assertions:\\
{\bf (A1)}: There exists a weak Lyapunov function in the large 
$V$ \sou{of class $C^1$} for (\ref{eq:5})
such that  
\begin{equation}
\label{eq:11}
\forall t\in \R^+\setminus N , \quad \forall x\in \R^n\setminus\{ 0\} ,
\quad \forall v\in F(t,x)\quad\quad 
\frac{\partial V}{\partial t}(t,x) +\langle \nabla _x V(t,x),v\rangle
 \le 0; 
\end{equation}
{\bf (A2)}: There exists a continuous map 
$\delta\! :\R ^+\times\R ^n \to\R ^+$ 
such that
$\delta (t,x)=0\Leftrightarrow x=0$ and the system
\begin{equation}
\label{eq:12}
\dot{x}\in G(t,x):=
\left\{ 
\begin{array}{ll}
\overline{\hbox{co}}\, \big\{ 
F\big( \B \big( (t,x),\delta (t,x)\big)\cap (\R^+\setminus N)\times \R ^n
\big)\big\} 
&\hbox{ if } x\ne 0; \\
F(t,0) &\hbox{ if } x=0,         
\end{array}
\right.
\end{equation}
is also robustly Lagrange stable.
Then (A1) $\Leftrightarrow$ (A2). Moreover, the Lyapunov function 
$V$ in (A1) may as well be taken of class $C^{\infty}$.
\end{thm}
\noindent
{\sc Remark 1.} Consider the following multivalued map (defined on
$\R ^+\times \R$)
\begin{equation*}
F(t,x):=\left\{
\begin{array}{ll}
[-x,1],&\hbox{ if }x=t\in \Q ^+;\\
\{ -x\} , &\hbox{ otherwise.}
\end{array}
\right.
\end{equation*}
Obviously ({\bf H}) is satisfied (with $N_0=\emptyset$),
the origin is globally asymptotically stable
for (\ref{eq:5}) and (\ref{eq:11})
holds true with $N:=\Q ^+$ and $V(t,x):=x^2$. We claim that there does not
exist any (smooth) weak Lyapunov
function $V$ such that (\ref{eq:11}) holds true for
{\em every} $t\ge 0$ (that is with $N=\emptyset$): Indeed, $x(t)=t$ is a
trajectory of (\ref{eq:12}) for {\em any} function $\delta$ as in (A2) if
$N=\emptyset$.

Since condition (A2) is not always easy to be checked in practice,
we introduce a strengthened version of ({\bf H}) which ensures the existence
of smooth Lyapunov functions for robustly stable (or Lagrange stable)
multivalued systems:\\
\noindent
Let $({\bf H'})$ denote the following assumption for the multivalued
map $F(t,x)$:
There exists a zero measure set $N_0\subset \R ^+$ such that ({\bf H1}),
({\bf H2}) hold true together with\va
$({\bf H3'})$: The multivalued map $F_{
\vert _{(\R ^+\setminus N_0)\times \R^n}}$
is {\em upper semicontinuous},
i.e. for each pair $(t_0,x_0)\in (\R ^+\setminus N_0)\times \R^n$,
for each $\varepsilon >0$, there exists 
$\delta >0$ such that for any  $(t,x)\in (\R ^+\setminus N_0)\times \R ^n$,
\begin{equation*}
\vert t-t_0\vert +\Vert x-x_0\Vert <\delta \im F(t,x)\subset 
F(t_0,x_0) + B (0,\varepsilon ).
\end{equation*}    
It is easy to see that $({\bf H'})\im ({\bf H})$. Moreover, it 
follows from the next 
proposition that ({\bf H'}) holds true when $F(t,x)=K_x\, f(t,x)$ for a wide
class of discontinuous fields $f$.
\begin{prop}
\label{prop2}
Let $f\in {\cal L}^{\infty}_{\rm loc}(\R ^+\times \R ^n,\R ^n)$. If $f$ is
{\bf e.c.t.}, then $({\bf H'})$ is fulfilled by
$F(t,x):=K_x\, f(t,x)$.
\end{prop}
We are now in position to state the main result of the paper.
\begin{thm}
\label{thm3}
Let $F(t,x)$ be a multivalued map such that $({\bf H'})$ holds. If the
system (\ref{eq:5}) is robustly Lagrange stable, then  
there exists a weak Lyapunov
function in the large $V\! :\R ^+\times \R ^n\setminus \{ 0 \} \to \R ^+$
which is \sou{of class $C^{\infty}$} and such that
for almost every $t\ge 0$ and every $x\ne 0$,
\begin{equation*}
%\label{eq:200}
\forall v\in F(t,x)\quad
\frac{\partial V}{\partial t}(t,x)+\langle \nabla _x V(t,x),v\rangle  \le 0.
\end{equation*}
\end{thm}

Of course, a result analogous to Theorem \ref{thm3}
also holds true for a {\em robustly} stable differential inclusion,
provided that $({\bf H'})$ is fulfilled.
\begin{thm}
\label{thm4}
Let $F(t,x)$ be a multivalued map such that $({\bf H'})$ holds.  If the 
origin is robustly stable for (\ref{eq:5}), then there exists a weak Lyapunov
function in the small $V\! :\R ^+\times \{ x:\ \Vert x\Vert <h \}\to \R ^+$
which is \sou{of class $C^{\infty}$} and such that
for almost every $t\ge 0$ and every $x$ with $\Vert x\Vert <h$,
\begin{equation*}
%\label{eq:200}
\forall v\in F(t,x)\quad
\frac{\partial V}{\partial t}(t,x)+\langle \nabla _x V(t,x),v\rangle \le 0.
\end{equation*}
\end{thm}
Theorem 1 is therefore a direct consequence of Theorem 4 and Proposition 2.\va
{\sc Remark 2.}
The converse of the second Lyapunov theorem is not investigated in this
paper, but we believe that, for {\em time-dependent}
asymptotically stable
differential inclusions, $(H')$ is a relevant assumption to ensure
the existence of a {\em smooth} (strong) Lyapunov function.
\setcounter{equation}{0}
\section{Proofs of Theorem \ref{thm2} 
and Proposition \ref{prop1}}
\va
Before going to the proof of Theorem \ref{thm2}, let us remark that for
any function $\delta (t,x)$ as in (A2), 
\begin{equation*}
\forall t\not\in N,\ \forall x\in \R ^n\quad\ 
F(t,x)\subset G(t,x).
\end{equation*}
Therefore any solution of
\begin{equation*}
%\label{eq:23}
\left\{
\begin{array}{rl}
\dot{x}
&\in F(t,x) \\
x(t_0)&=x_0,
\end{array}
\right.
\end{equation*}
is also a solution of
\begin{equation}
\label{eq:24}
\left\{
\begin{array}{rl}
\dot{x}
&\in G(t,x) \\
x(t_0)&=x_0,
\end{array}
\right.
\end{equation}
and we do not need to check whether assumption ({\bf H})
is fulfilled by $G$ in order to claim the
(local) existence of solutions for the Cauchy problem (\ref{eq:24}).\va\va
{\sc Proof of Theorem \ref{thm2}.}
Let us first deal with (A1)$\im$(A2).
Assume that (A1) holds. Let $a(\cdot )$, $b(\cdot )$ be as in Definition 1.
We first use a trick by Yoshizawa \cite{Y2}, which
allows us to strengthen the inequality (\ref{eq:11}). We substitute 
\begin{equation*}
\tilde{V} (t,x):=\frac{1+t}{1+2t}V(t,x)
\end{equation*}
for $V(t,x)$, and we observe that 
\begin{equation*}
\frac{1}{2} a(\Vert x\Vert )\le \tilde{V} (t,x)\le b(\Vert x\Vert )
\end{equation*}
but this time
\begin{equation}
\label{eq:30}
\begin{array}{rl}
\displaystyle\frac{\partial\tilde{V}}{\partial t}(t,x) 
+ \langle {\nabla _x\tilde{V}}(t,x),v\rangle
&= -\frac{1}{(1+2t)^2}V(t,x) +\frac{1+t}{1+2t}
\big( \displaystyle\frac{\partial V}{\partial t}(t,x) 
+ \langle {\nabla _x V}(t,x),v\rangle \big)\\
&\le -\frac{1}{(1+2t)^2} a(\Vert x\Vert )
\end{array}
\end{equation}
for every $t\in \R ^+ \setminus N$, every $x\in \R ^n\setminus \{ 0\}$
 and every
$v\in F(t,x)$. To prove (A2), it is clearly sufficient (thanks to 
the first Lyapunov theorem \cite[Thm. 2.1]{BR} and to a standard compactness
argument) to check that for
any $(t_0,x_0)\in \R ^+\times (\R ^n\setminus \{ 0\}) $, there exists
$\delta >0$ s.t. for each pair 
$(t,x)\in \B ((t_0,x_0),\delta )\cap \R ^+\times (\R ^n\setminus \{ 0\} )$ 
\begin{equation*}
\forall v\in  F\bigg(\B ((t,x),\delta)\cap
(\R ^+ \setminus N)\times  \R ^n\bigg)
\quad \frac{\partial\tilde{V}}{\partial t}(t,x) + 
\langle {\nabla _x\tilde{V}}(t,x), v\rangle
\le 0.
\end{equation*}
By (H2) there exists $M>0$ such that 
\begin{equation}
\label{eq:31}
\Vert (t-t_0,x-x_0)\Vert\le 1 \ \hbox{ and }\  t\not\in N_0 \im
\forall \, v\in F(t,x)\quad \Vert v\Vert \le M.
\end{equation}
\def\dvt{\frac{\partial \tilde{V}}{\partial t}}
\def\dvx{\nabla _x \tilde{V}}
Since $\tilde{V}$ is $C^1$ on $\R ^+\times (\R ^n\setminus \{ 0\} )$
there exists $\delta\in (0, \min (1/2,
\Vert x_0\Vert /2))$
such that
for all pairs $ (t,x),(\bar{t}, \bar{x})$ in $\B ((t_0,x_0), 2\delta )
\cap \R ^+\times \R ^n$
\begin{equation}
\label{eq:32}
\left\vert \dvt (t,x) -\dvt (\bar{t},\bar{x})\right\vert +
\left\Vert \dvx (t,x) -\dvx (\bar{t},\bar{x})\right\Vert \cdot M
\le\frac{1}{2} \frac{a(\Vert x_0\Vert )}{(1+2t_0)^2}
\end{equation}
and
\begin{equation}
\label{eq:33}
\frac{a(\Vert \bar{x}\Vert )}{(1+2\bar t)^2} \ge 
\frac{1}{2} \frac{a(\Vert x_0\Vert )}{(1+2t_0)^2} .
\end{equation}
Then let 
$(t,x)\in \B ((t_0,x_0),\delta )\cap
\R ^+\times (\R ^n\setminus \{ 0\})$
and
$v\in F\big( \B ((t,x),\delta)\cap
(\R ^+ \setminus N)\times  \R ^n\big)$.
This means that
$v\in F(\bar t, \bar x)$ for some pair
$(\bar t,\bar x)\in \B ((t,x),\delta)\cap
(\R ^+ \setminus N)\times  \R ^n$.
Hence
$\Vert (\bar t -t_0,\bar x -x_0)\Vert \le 2\delta 
<\min (1,\Vert x_0\Vert )$, $\bar t\not\in N$
and $\bar x\ne 0$.
We infer from (\ref{eq:30}), (\ref{eq:31}), (\ref{eq:32}) and 
(\ref{eq:33}) that
\begin{equation*}
\begin{array}{l}
\dvt (t,x) +\langle \dvx (t,x),v\rangle
\\ 
\quad \le \left( \dvt (\bar{t},\bar{x}) +
\langle \dvx (\bar{t},\bar{x}),v\rangle \right) \\ 
\qquad + 
\left\vert \dvt (t,x)-\dvt (\bar{t},\bar{x})\right\vert +
\left\Vert \dvx (t,x)-\dvx (\bar{t},\bar{x})\right\Vert \cdot M \\
\quad \le -\displaystyle\frac{a(\Vert \bar{x}\Vert )}{(1+2\bar{t})^2}  +
\displaystyle\frac{1}{2} \frac{a(\Vert x_0\Vert )}{(1+2t_0)^2}\le 0.
\end{array}
\end{equation*}
\qed
We now turn to the \va
{\sc Proof of (A2)$\im$(A1).}
We assume that (A2) holds. Without loss of generality, we may also
assume that the function $\delta$ is Lipschitz continuous with a Lipschitz 
constant equal to $1$, that is for all pairs $(t_1,x_1)$ and $(t_2,x_2)$
in $\R ^+\times \R ^n$
\begin{equation*}
\label{eq:35}
\vert \delta (t_1,x_1)-\delta (t_2,x_2)\vert \le 
\Vert (t_1-t_2,x_1-x_2)\Vert .
\end{equation*}
Notice that for any $t\ge 0$ and any $x\in \R ^n$
\begin{equation*}
\vert \delta (t,x)\vert =\vert \delta (t,x) -\delta (t,0)\vert \le 
\Vert x\Vert .
\end{equation*}
In order to get a Lyapunov function which is smooth w.r.t. $(t,x)$ 
up to $t=0$, we are led to work on the open set
\begin{equation*}
\U :=(-1,+\infty )\times (\R ^n \setminus \{ 0\} ).
\end{equation*}
Set $\delta (t,x)=\delta (0,x)$ for
$t\in (-1,0 )$, $x\in \R ^n$  
and
\begin{equation} 
\label{eq:36}
W(t,x) := \left\{ (s,y)\in \U ;\
\Vert(s-t,y-x)\Vert <\min \left( 
\frac{t+1}{2} , 
\frac{\delta (t,x)}{3} \right) \right\}
\end{equation}
for $(t,x)\in \U$.
\va
\sou{\sc Claim 1.}  
{\em The family 
\begin{equation*}
\big( W(t,x) \big) _{(t,x)\in \big( (-1,+\infty )\setminus N\big)\times 
\big(\R ^n \setminus \{ 0\} \big) }
\end{equation*}
constitutes an open covering of $\U$.}\va
Indeed, for each pair $(t,x)$ in $N\times (\R ^n\setminus \{ 0\})$, 
there exists $\tilde{t} \in (-1,+\infty )\setminus N$ such that 
\begin{equation*}
\vert t-\tilde{t}\vert <
\min (\frac{t+1}{4}, \frac{\delta (t,x)}{6} ) ,\quad
\tilde{t}+1 >\frac{t+1}{2}
\hbox{ and } 
\delta (\tilde{t},x) > \frac{\delta (t,x)}{2} .
\end{equation*}
($(-1,+\infty )\setminus N$ is dense in $(-1, +\infty)$ since $\mu (N)=0$.)
It follows that $(t,x)\in W_{(\tilde{t},x)}$.\qed
Set 
\begin{equation*}
{\cal N}=N\times \R ^n.
\end{equation*}
 ($\cal N$ is a zero measure set in $\R ^{n+1}$.)
\def\cN{{\cal N}}
Let $(\psi _i)_{i\ge 1}$  denote some $C^{\infty}$ partition of unity
such that the carriers  of the $\psi _i$'s form a refinement  of
$\big( W(t,x) \big) _{(t,x)\in \U \setminus {\cal N}}$.
Setting, for every $i\ge 1$, 
\bee 
%\label{eq:100}
S_i:=\overline{ \{ (t,x)\in \R ^{n+1}; \ \psi _i(t,x)\ne 0\}} ,
\end{equation*}
this means that (i) each $\psi _i$ is a nonnegative function of
class $C^{\infty}$ on $\R ^{n+1}$;
(ii) for each $i\ge 1$, there exists $(t_i,x_i)\in \U \setminus \cal{N}$
such that $S_i \subset W(t_i,x_i)$;
(iii) for each pair $(t,x)\in \U$,  
$\sum_{i=1}^{\infty} \psi _i (t,x) =1$ and 
there exist a positive number $\rho >0$ and a 
{\em finite} subset $I$ of $\N ^*$
such that
\begin{equation*}
{\cal B}\big( (t,x),\rho \big)\cap  S_i\ne \emptyset\  \im \ i\in I.
\end{equation*}
By (H2) there exists a continuous function $M\! :\R ^+\to \R ^+$ such that
\begin{equation*}
(t\in \R ^+\setminus N_0,\ \Vert x\Vert \le 1)\im 
F(t,x)\subset B\big( 0,M(t)\big) .
\end{equation*}
We regularize the multivalued map  $F$ as follows.
Set for $t\ge 0$ and $x\in \R ^n$
\be
\label{eq:37}
F_L(t,x):=
\left\{
\begin{array}{l}
\sum_{i\ge 1} \psi _i(t,x)\, 
\overline{\hbox{co}}\ \left\{ F\left( \B ((t_i,x_i),
\frac{\delta (t_i,x_i)}{3}\big) \cap \big( \R ^+ \setminus 
N\big) \times \R ^n \right)\right\} \hbox{ if } x\ne 0, \\
B\big( 0,M(t)\big) \hbox{ if } x=0,
\end{array}
\right.
\end{equation}
and $F_L(t,x)=F_L(0,x)$ for $-1<t<0$ and $x\in \R ^n$.
It is easy to see that (1) $F_L(t,x)$ is a (nonempty) convex compact
set in $\R ^n$ for any $(t,x)\in (-1,+\infty )\times \R ^n$,
(2) the multivalued map 
$F_L$ is locally Lipschitz continuous for the Hausdorff metric on $\U$
(since on compact subsets of $\R ^+\times (\R ^n \setminus \{ 0\} )$
the sum in (\ref{eq:37}) is finite) and (3) the assumption ({\bf H}) is also
satisfied by the multivalued map $F_L\at{\R ^+\times \R ^n}$.
(Thanks to (\ref{eq:36}), we see that
$\psi _i(t,x)\ne 0\im 
\Vert (t-t_i,x-x_i)\Vert <
\frac{\delta (t_i,x_i)}{3} \le \frac{\Vert x_i\Vert }{3}$,
hence $\Vert x_i\Vert < \frac{3}{2}\Vert x\Vert$ and
$\vert t-t_i\vert <\frac{1}{2}\Vert x\Vert $.
This is useful when
proving that for each $t\in \R ^+\setminus N_0$ the multivalued map
$x\mapsto F_L(t,x)$ be upper semicontinuous at $x=0$.)
\va
\sou{\sc Claim 2.}
\be
\label{eq:38}
\forall t\in \R ^+\setminus N,\ \forall x\in \R ^n \ \ 
F(t,x)\subset F_L(t,x).
\end{equation}
The result is obvious for $x=0$, so let $x\ne 0$ and $t\in\R ^+\setminus N$.
Let $i\ge 1$ be such that
$\psi _i(t,x)>0$.
It follows from (\ref{eq:36}) that
$\Vert (t-t_i,x-x_i)\Vert <\frac {1}{3}\delta (t_i,x_i)$
hence
\begin{equation*}
F(t,x)\subset
F\bigg( \B ((t_i,x_i),
\frac{\delta (t_i,x_i)}{3}\big) \cap \big( \R ^+\setminus
N\big) \times \R ^n \bigg) .
\end{equation*}
and then
$F(t,x)\subset F_L(t,x)$. \qed
\sou{\sc Claim 3.} 
\begin{equation*}
%\label{eq:39}
\forall (t,x)\in \R ^+\times (\R ^n\setminus \{ 0\})
\quad F_L(t,x)\subset G(t,x).
\end{equation*}
To prove this claim, let $x\ne 0$, $t\in \R ^+$, and $i\ge 1$ be
such that $\psi _i(t,x)>0$. Since $\delta$ is a Lipschitz continuous 
function with
a Lipschitz constant equal to 1, we may write
\begin{equation*}
\delta (t_i,x_i) -\delta (t,x) \le \Vert (t-t_i,x-x_i)\Vert 
< \frac{1}{3}\delta (t_i,x_i),
\end{equation*}
hence 
\begin{equation*}
\B\big( (t_i,x_i),\frac{1}{3}  \delta (t_i,x_i) \big) \subset 
\B\big( (t,x), \frac{2}{3} \delta (t_i,x_i) \big) \subset
\B\big( (t,x), \delta (t,x) \big) 
\end{equation*}
from which we infer that
$F_L(t,x)\subset G(t,x)$. \qed
As a consequence, the system 
\begin{equation}
\label{eq:40}
\dot{x}\in F_L(t,x)\qquad (t\ge 0,x\in \R ^n)
\end{equation}
\def\V{\tilde{V}}
is also (robustly) Lagrange stable. By \cite[Thm. 2.2]{BR},
there exist continuous functions
$\V \! :\R ^+\times (\R ^n \setminus \{ 0\} )
\to \R ^+$, $a,b\!  :\R ^+\to \R ^+$ such that
$\V$ is locally Lipschitz continuous,
$a$ and $b$ are increasing,
$a(0)=0$, $a(r)>0$ for $r>0$, $a(r)\to \infty$ as $r\to \infty$,
$a(\Vert x\Vert )\le \V (t,x) \le b(\Vert x\Vert )$
for each pair $(t,x)\in \R ^+\times (\R ^n \setminus \{ 0\} )$,
and for any nonvanishing
solution $x(t)$ of (\ref{eq:40}) on some interval $[t_1,t_2]\subset \R ^+$
we have $ \V (t_2,x(t_2))\le \V (t_1,x(t_1))$.
(In other words $\tilde V$ is a locally Lipschitz continuous weak Lyapunov
function in the large for (\ref{eq:40}).)
We now turn to the
\va
\sou{\sc Claim 4.}
\be
\label{eq:41}
\quad \hbox{\em  For a.e. } (t_0,x_0)
\in \R ^+\times (\R ^n \setminus \{ 0\} ),\  
\forall v\in F_L(t_0,x_0)
\quad \dvt (t_0,x_0) +\langle \dvx (t_0,x_0),v\rangle  \le 0.
\end{equation}
Recall that it follows from Rademacher's theorem (see \cite{EG})
that $\V$ is differentiable at $(t_0,x_0)$ for almost every
$(t_0,x_0)$.
Therefore, it is sufficient to prove that for {\em every}
pair $(t_0,x_0) \in \R ^+\times (\R ^n \setminus \{ 0\} )$,
\begin{equation*}
\forall v\in F_L(t_0,x_0)
\quad
\limsup _{h\to 0^+}
\frac{\tilde{V}(t_0+h,x_0+hv)-\tilde{V}(t_0,x_0)}{h} \le 0.
\end{equation*}
Notice first that for each $v\in F_L(t_0,x_0)$
there exists a solution $x(t)$ of (\ref{eq:40}) such that 
$x(t_0)=x_0$ {\em and} $\dot{x}(t_0)=v$. Indeed, we may infer from 
the local Lipschitz continuity of $F_L$ that the map 
\be
\label{eq:42}
g(s,y):=\pi _{F_L(s,y)}(v)
\end{equation}
(where $\pi _{F_L (s,y)}$ denotes the projection on the convex compact set
$F_L(s,y)$) is {\em continuous}, hence, by Peano's theorem, there exists a
solution $x(t)$ on some interval
$[{t_0},{t_0}+\varepsilon ]$ of the (classical) Cauchy problem
\begin{equation*}
\left\{ 
\begin{array}{rl}
\dot{x}&=g(t,x)\  (\in F_L(t,x))\\
x(t_0)&=x_0.
\end{array}
\right.
\end{equation*}
Obviously ${\dot x}(t_0)=g(t_0,x_0)=v$.
Now there exists $L>0$ such that for any $(t_1,x_1)$, $(t_2,x_2)$ 
in some neighborhood of
$(t_0,x_0)$
\begin{equation*}
\label{eq:43}
\vert \V (t_1,x_1)-\V (t_2,x_2)\vert \le L\, \Vert (t_1-t_2,x_1-x_2 )\Vert. 
\end{equation*}
It follows that for
$h>0$ small enough 
\begin{equation*}
\begin{array}{rl}
\frac{\V (t_0+h,x_0+hv)-\V (t_0,x_0)}{h}&=
\frac{\V (t_0+h,x_0+hv)-\V (t_0+h,x(t_0+h))}{h} +
\frac{\V (t_0+h,x(t_0+h))-\V (t_0,x_0)}{h}  \\
&\le  L\left\Vert 
\frac{x(t_0+h)-x_0}{h}-v
\right\Vert ,
\end{array}
\end{equation*}
hence
\begin{equation*}
\limsup _{h\to 0^+}\frac{\V (t_0+h,x_0+hv)-\V (t_0,x_0)}{h} \le 0.
\end{equation*}
\qed
Set for $t\ge 0$ and $x\in \R ^n \setminus \{ 0\}$
\begin{equation*}
V_L(t,x):= \frac{1+t}{1+2t} \V (t,x),
\end{equation*}
\begin{equation*}
W_L(t,x):=\frac{1}{(1+2t) ^2}a(\Vert x\Vert ).
\end{equation*}
We infer from (\ref{eq:41}) that for a.e. 
$(t_0,x_0)\in \R ^+\times (\R ^n \setminus \{ 0\} )$
\begin{equation}
\label{eq:440}
\forall  v\in F_L (t_0,x_0)
\quad\quad
\frac{\partial V_L}{\partial  t}(t_0,x_0) +
\langle \nabla _x V_L (t_0,x_0),v\rangle 
\le - W_L(t_0,x_0).
\end{equation}
Before applying a smoothing procedure, we are led to extend
$V_L$ and $W_L$ on a neighborhood $\Omega$ of
$\R ^+\times (\R ^n \setminus \{ 0\} )$ in such a way that (\ref{eq:440})
remains valid for a.e. $(t_0,x_0)\in \Omega$. 
Since $V_L(0,\cdot )$ is locally  Lipschitz continuous 
on $\R ^n\setminus \{ 0\}$ and $F_L$ is locally bounded on $\U$,
there exists a continuous map $l\! :\R ^n\setminus \{ 0\}\to (0,+\infty )$
such that
\begin{equation*}
\begin{array}{rl}
\hbox{ for a.e. } x\in \R ^n\setminus \{ 0\}  
\quad
&\Vert \nabla _x V_L(0,x)\Vert \le l(x) \ \hbox{ and } \\
\forall (t,x)\in (-\frac{1}{2},0)\times  (\R ^n\setminus \{ 0\} ),\
\forall v\in F_L(t,x)\quad &\Vert v\Vert \le l(x).
\end{array}
\end{equation*}
Let $d:\R ^n\setminus \{ 0\}\to (0,+\infty )$ denote some function of class 
$C^{\infty}$ such that
\begin{equation*}
\forall x\in \R ^n\setminus \{ 0\}\quad d(x)>l(x)^2+a(\Vert x\Vert ).
\end{equation*}
For $(t,x)\in (-1,0)\times (\R ^n\setminus \{ 0\} )$ we now set
\begin{equation*}
\begin{array}{rl}
V_L(t,x)&=V_L(0,x)-td(x), \\
W_L(t,x)&=W_L(\vert t\vert,x).
\end{array}
\end{equation*}
$V_L$ is obviously locally Lipschitz continuous on $\U$. Set
\begin{equation*}
\begin{array}{rl}
{\cal O}:=&\{ \, (t,x)\in (-\frac{1}{2},0)\times  (\R ^n\setminus \{ 0\} ) ;\
V_L(t,x)>\, \frac{1}{2}\, a(\Vert x\Vert)\ \hbox{ and }\\
&-d(x)+l(x)\big( l(x) +\vert t\vert\cdot 
\Vert \nabla _x d(x)\Vert \big)
<- W_L(t,x) \, \} ,
\end{array}
\end{equation*}
and $\Omega := \R ^+\times (\R ^n\setminus \{ 0\}) \cup {\cal O}$. 
It is easy to
see that $\Omega$ is an open neighborhood of 
$\R ^+\times (\R ^n \setminus \{ 0\} )$.
On the other hand, 
\begin{equation*}
\forall (t,x)\in \Omega , \qquad \frac{1}{2} a(\Vert x\Vert )\le 
V_L(t,x) \le b(\Vert x\Vert ),\qquad  W_L(t,x)>0.
\end{equation*}
\sou{\sc Claim 5.} {\em  For a.e. $(t,x)\in \Omega$, 
\begin{equation*}
\label{eq:44}
\forall v\in F_L(t,x),\qquad
\displaystyle\frac{\partial V_L}{\partial  t}(t,x) +
\langle \nabla _x V_L (t,x),v\rangle
\le - W_L(t,x).
\end{equation*}}
Since the result is already proved for a.e.
$(t,x)\in \R ^+\times (\R ^n\setminus \{ 0\})$ (see (\ref{eq:440})),
it is sufficient to observe that for a.e. $(t,x)\in {\cal O}$ and
every $v\in F_L (t,x)$
\begin{equation*}
\begin{array}{rl}
\frac{\partial V_L}{\partial  t}(t,x) +
\langle \nabla _x V_L (t,x),v\rangle
&= -d(x) + \langle \nabla _x V_L (0,x) -t\,
 \nabla _x d(x),v\rangle \\
&\le   -d(x) + l(x) (l(x)+\vert t\vert \cdot \Vert \nabla _x d(x)\Vert ) \\  
&\le -W_L(t,x).
\end{array}
\end{equation*}
\qed
Regularization of $V_L$ on compact subsets of $\Omega$ is performed 
by means of convolution with mollifiers:
\begin{lem}
\label{lem1}
Let $\phi,\psi :\Omega \to (0,+\infty )$ be continuous functions such that 
$\phi$ is locally Lipschitz continuous 
w.r.t. $(t,x)$ on $\Omega $ and
\be
\label{eq:45}
\hbox{ for a.e. } (t_0,x_0)\in \Omega ,\ \forall  v\in F_L (t_0,x_0)
\quad\quad
\frac{\partial \phi}{\partial  t}(t_0,x_0) +
\langle \nabla _x \phi (t_0,x_0),v\rangle
\le - \psi (t_0,x_0).
\end{equation}
Let $S\subset \Omega$ be any compact set and let $\ep > 0$. Then there exists
a function $\tilde{\phi}$ of class $C^{\infty}$ with 
compact support in $\Omega$ such that 
\be
\label{eq:450}
\Vert \phi -\tilde{\phi}\Vert _{L^{\infty}(S)} <\ep 
\hbox{ and }
\end{equation}
\be
\label{eq:46}
\forall (t_0,x_0)\in S,\ \forall v\in F_L(t_0,x_0)\quad
\frac{\partial \tilde{\phi}}{\partial t}(t_0,x_0) +
\langle \nabla _x  \tilde{\phi}(t_0,x_0),v\rangle
  \le -\frac{1}{2} \psi (t_0,x_0).
\end{equation}
\end{lem}
\va
{\sc Proof of Lemma \ref{lem1}:}
Let $\rho\in C^{\infty}(\R ^{n+1}, \R )$ be a nonnegative function such that
$\hbox{supp} (\rho )\subset \B (0,1)$ 
and $\int_{\R ^{n+1}} \rho (t,x)dtdx=1$.
Set for $\delta >0$,  $\rho _{\delta }(t,x)=\frac{1}{\delta ^{n+1}}\rho
\left( \frac{t}{\delta},\frac{x}{\delta}\right)$
and
\begin{equation*}
\begin{array}{rl}
\phi _{\delta}(t,x):&=\phi \ast \rho _{\delta}(t,x)  \\
&=\int_{\R^{n+1}}\phi (t-s,x-y)\rho _{\delta}(s,y)\, ds\, dy \\
&=\int_{\Vert (\bar{s},\bar{y})\Vert \le 1} 
\phi (t-\delta \bar{s},x-\delta\bar{y})
\rho (\bar{s},\bar{y})d\bar{s}\, d\bar{y},\\
\psi _{\delta }(t,x):&=\psi\ast\rho _{\delta}(t,x).
\end{array}
\end{equation*}
\def\f{\phi _{\delta}}
\def\p{\psi _{\delta}}
Fix some open neighborhood $\Omega '$ of $S$ such that
$\Omega '\subset\subset \Omega$, i.e. $\overline{\Omega '}$ is a compact 
subset of $\Omega$.
Then for $\delta >0$ small enough the functions
$\f$ and $\p$ are defined and of class $C^{\infty}$  on $\Omega '$,
and $\f (t,x)\to \phi (t,x)$ (resp. $\p (t,x)\to \psi (t,x)$) 
{\em uniformly} on
$S$ as $\delta \to 0$.
Let $\theta$ be a function of class $C^{\infty}$ with compact support in
$\Omega '$ and such that $\theta (t,x)=1$ for every $(t,x)$ in a
neighborhood of $S$.
Setting
$\tilde{\phi}:=\theta \f$  we see that the function
$\tilde{\phi}$ is of class $C^{\infty }$ on $\R ^{n+1}$ 
with compact support in $\Omega '$ and  that
(\ref{eq:450}) holds true provided that 
$\delta$ is small enough. To complete 
the proof of Lemma \ref{lem1}, we now show that there exists
$\delta _0>0$  such that for any $\delta \in (0,\delta _0 )$
\begin{equation*}
%\label{eq:47}
\forall (t_0,x_0)\in S,\ \forall v\in F_L(t_0,x_0)\quad
\frac{\partial \f}{\partial t}(t_0,x_0) +
\langle \nabla _x  \f (t_0,x_0),v\rangle \le -\frac{1}{2} \psi (t_0,x_0).
\end{equation*}
Let $\delta _1>0$ be such that  $S+\B (0,\delta _1)\subset
\Omega$
and $L>0$ be such that for any pairs $(t_1,x_1),(t_2,x_2)$ in 
$S+\B (0,\delta _1)$
\be
\label{eq:48}
h\big( F_L(t_1,x_1),F_L(t_2,x_2)\big) + 
\vert \phi (t_1,x_1)-\phi (t_2,x_2)\vert  \le L\, \Vert (t_1-t_2,x_1-x_2)\Vert .
\end{equation}
It follows that for a.e. $(t,x)\in S+\B (0,\delta _1)$,
$\phi$ is differentiable at $(t,x)$ and
\be
\label{eq:480}
\Vert \big(\frac{\partial\phi}{\partial t},\nabla _x\phi\big) 
(t,x)\Vert \le L. 
\end{equation}
Let $\delta \in (0,\delta _1)$, $(t_0,x_0)\in S$ and $v\in F_L (t_0,x_0)$.
Applying Lebesgue's theorem we infer 
from (\ref{eq:48}) that
\be
\label{eq:481}
\begin{array}{rl}
\frac{\partial \f}{\partial t}(t_0,x_0) &+
\langle \nabla _x  \f (t_0,x_0),v\rangle \\
&=\lim_{\eta \to 0}\int_{\Vert (\bar{s},\bar{y})\Vert \le 1} 
\frac{\phi (t_0-\delta\bar{s}+\eta ,x_0-\delta \bar{y}+\eta v)-
\phi (t_0-\delta \bar{s},x_0-\delta \bar{y})}{\eta}\rho (\bar{s},\bar{y})
d\bar{s}\, d\bar{y}\\
&=\int_{\Vert (\bar{s},\bar{y})\Vert \le 1}
\left(
\frac{\partial \phi}{\partial t}(t_0-\delta\bar{s},x_0-\delta \bar{y}) +
\langle \nabla _x \phi (t_0-\delta\bar{s},x_0-\delta \bar{y}),v\rangle 
\right)
\rho (\bar{s},\bar{y})d\bar{s}\, d\bar{y}.
\end{array}
\end{equation}
$g$ also denoting the map defined by (\ref{eq:42}), we may write
(thanks to (\ref{eq:45}), (\ref{eq:480}) and (\ref{eq:481})) 
\be
\label{eq:490}
\begin{array}{rl}
\frac{\partial \f}{\partial t}(t_0,x_0) &+
\langle \nabla _x  \f (t_0,x_0),v\rangle \\
&=\int_{\Vert (\bar{s},\bar{y})\Vert \le 1}
\left(
\frac{\partial \phi}{\partial t} +
\langle \nabla _x \phi ,g\rangle \right)
(t_0-\delta\bar{s},x_0-\delta \bar{y})
\rho (\bar{s},\bar{y})d\bar{s}\, d\bar{y}\\
 &\ +
\int_{\Vert (\bar{s},\bar{y})\Vert \le 1}
\langle \nabla _x \phi
(t_0-\delta\bar{s},x_0-\delta \bar{y})
, v-g(t_0-\delta\bar{s},x_0-\delta \bar{y})\rangle
\rho (\bar{s},\bar{y})d\bar{s}\, d\bar{y},\\
&\le -\psi _{\delta}(t_0,x_0) +
L\, \int_{\Vert (\bar{s},\bar{y})\Vert \le 1}
\Vert v-g(t_0-\delta\bar{s},x_0-\delta \bar{y})\Vert
\rho (\bar{s},\bar{y})d\bar{s}\, d\bar{y}.
\end{array}
\end{equation}
Now observe that for $\Vert (\bar{s},\bar{y})\Vert \le 1$,
\begin{equation*}
\begin{array}{rl}
\Vert v-g(t_0-\delta \bar{s},x_0-\delta \bar{y}) \Vert
&= \hbox{dist}\big( v,F_L (t_0-\delta \bar{s},x_0-\delta \bar{y})\big) \\
&\le h\big( F_L(t_0,x_0),F_L(t_0-\delta \bar{s},x_0-\delta \bar{y})\big) \\
&\le L\delta ,
\end{array}
\end{equation*}
owing to (\ref{eq:48}).
Pick any $\delta _0\in (0,\delta _1)$ such that
\be
\label{eq:50}
L^2\delta _0 <\frac{1}{4} \min_{(t,x)\in S} \psi (t,x)
\end{equation}
and such that for any
$0<\delta <\delta _0$
\be
\label{eq:51}
\max_{(t,x)\in S}\vert \psi (t,x)-\p (t,x)\vert
<\frac{1}{4} \min_{(t,x)\in S} \psi (t,x) .
\end{equation}
Then we infer from (\ref{eq:490}), (\ref{eq:50}) and (\ref{eq:51}) that 
for $0<\delta <\delta _0$
\begin{equation*}
\frac{\partial \f}{\partial t}(t_0,x_0) +
\langle \nabla _x  \f (t_0,x_0) ,v\rangle
\le -\p (t_0,x_0)  +\frac{1}{4} \psi (t_0,x_0)
\le -\frac{1}{2}\psi (t_0,x_0) \cdot
\end{equation*}
The proof of Lemma 1 is complete.
\qed
For $i\ge 1$, set
\begin{equation*}
J_i=\left\{ j\ge 1,\ S_i\cap S_j\ne \emptyset\right\} .
\end{equation*}
(Recall that $S_i$ is the support of $\psi _i$ and that
$S_i\subset \U$.)
It follows from the compactness of $S_i$ and the local finiteness of
the covering of $\U$ by the $S_j,\ j\ge 1$, that $J_i$ is {\em finite}.
Applying Lemma \ref{lem1}, we see that for each $i\ge 1$ there exists a
function $V_i$ of class $C^{\infty }$ with compact support in
$\Omega $ and such that, if we set
\begin{equation*}
S_i ^+:=S_i\cap \big( \R ^+\times \R ^n\big) , 
\end{equation*}
\begin{equation*}
%\label{eq:52}
\ep _i=
\left\{
\begin{array}{ll}
\frac{1}{2}\min \big( \min _{(t,x)\in S_i^+}  V_L(t,x),
\min _{(t,x)\in S_i ^+}  W_L(t,x) \big)\  (>0)
&\hbox{ if } S_i^+\ne\emptyset ,\\
1 &\hbox{ otherwise,}
\end{array}
\right.
\end{equation*}
\begin{equation*} 
%\label{eq:53}
q_i=
\left\{
\begin{array}{ll}
\max_{(t,x)\in S_i^+,\, v\in F_L(t,x)}
\left\vert
\frac{\partial \psi _i}{\partial t}(t,x) + 
\langle \nabla _x \psi _i(t,x),v\rangle
\right\vert 
&\hbox{ if } S_i^+\ne\emptyset ,\\
0 &\hbox{ otherwise,}
\end{array}
\right.
\end{equation*}
we have
\be
\label{eq:54}
\forall (t,x)\in S_i^+\quad \vert V_i(t,x)-V_L(t,x)\vert
< \frac{\min_{j\in J_i} \ep _j}{2^i(1+q_i)} 
\end{equation}
and
\be
\label{eq:540}
\forall (t,x)\in S_i^+,\ \forall v\in F_L(t,x) \quad
\frac{\partial V_i}{\partial t}(t,x)\, +\, 
\langle\nabla _x V_i (t,x), v \rangle  
\le -\frac{1}{2} W_L(t,x).
\end{equation}
(Notice that if $S_i^+=\emptyset$, (\ref{eq:54}) and (\ref{eq:540}) are
fulfilled by {\em any} function $V_i$ of class $C^{\infty}$ with compact
support in $\Omega$.) 
Set
\begin{equation}
\label{eq:3333}
V=\sum_{i=1}^{+\infty}\psi _i V_i.
\end{equation}
It is clear that $V$ is of class $C^{\infty}$ on $\Omega$.
Moreover  for each $(t,x)\in \R ^+\times (\R ^n\setminus \{ 0\} )$ 
\begin{equation*}
\vert V(t,x)  -V_L(t,x)\vert \le
\sum_{i=1}^{+\infty}\psi _i(t,x)
\vert V_i(t,x)-V_L(t,x)\vert
\le \sum_{i=1}^{+\infty}\frac{1}{2^{i+1}} V_L(t,x) =
\frac{V_L(t,x)}{2} ,
\end{equation*}
hence
\begin{equation*}
\frac{1}{4} a(\Vert x\Vert )\le V(t,x)\le
\frac{3}{2} b(\Vert x\Vert ) .
\end{equation*}
Let $(t,x)\in \R ^+\times (\R^n\setminus \{ 0\})$ and  $v\in F_L (t,x)$, we
aim to prove that $\frac{\partial V}{\partial t}(t,x) 
+\langle \nabla _x V(t,x), v\rangle  \le 0$. 
By (\ref{eq:3333})
\begin{equation*}
\begin{array}{rl}
\frac{\partial V}{\partial t} +\langle \nabla _x V, v\rangle
&=
\sum _i \left(
\frac{\partial \psi _i}{\partial t} + 
\langle \nabla _x \psi _i, v\rangle \right) V_i
+
\sum _i \psi _i \left(
\frac{\partial V _i}{\partial t} + \langle\nabla  _x V_i , v\rangle\right) \\
&= 
\sum _i \left(
\frac{\partial \psi _i}{\partial t} + \langle\nabla _x \psi _i, v\rangle
 \right) 
\left( V_i -V_L\right)
+
\sum _i \psi _i \left(
\frac{\partial V _i}{\partial t} + \langle\nabla  _x V_i , v\rangle\right) \\
&\le 
\sum_{ \{ i;\ (t,x)\in S_i\} } q_i 
\frac{\min_{j\in J_i}\ep _j}{2^i(1+q_i)} +
\sum_i \psi _i\cdot \big( -\frac{1}{2}W_L(t,x) \big) .
\end{array}
\end{equation*}
Let $i_0\ge 1$ be such that $(t,x)\in S_{i_0}$ and 
\begin{equation*}
\ep _{i_0}=\min _{ \{ i;\ (t,x)\in S_i\} }\ep _i \cdot 
\end{equation*}
If $i\ge 1$ is such that $(t,x)\in S_i$, then $i_0\in J_i$ and 
$\ep _{i_0}\ge \min _{j\in J_i} \ep _j$. It follows that
\begin{equation*} 
\sum_{ \{ i;\ (t,x)\in S_i\} } q_i 
\frac{\min_{j\in J_i}\ep _j}{2^i(1+q_i)} 
\le 
\sum_{ \{ i;\ (t,x)\in S_i\} }
\frac{\ep _{i_0}}{2^i} \le \ep _{i_0}.
\end{equation*}
On the other hand
\begin{equation*} 
\ep _{i_0} = 
\sum_{ \{ i;\ (t,x)\in S_i\} } \psi _i(t,x)\ep _{i_0}
\le \sum_{i} \psi _i(t,x)\ep _{i},
\end{equation*}
hence
\be
\label{eq:541}
\frac{\partial V}{\partial t} + \langle\nabla _x V, v\rangle \le 
\sum _i \psi _i\left( \ep _i -\frac{1}{2} W_L(t,x) \right)
\le 0.
\end{equation}
Now (\ref{eq:11}) follows from (\ref{eq:38}) and (\ref{eq:541}).
The proof of Theorem  \ref{thm2} is complete.\qed

The end of this section is devoted to the \va
{\sc Proof of Proposition \ref{prop1}:} 
Let $B\subset \R$ be a Borel set such that 
\begin{equation*}
0<\mu (B\cap I) < \mu (I)
\end{equation*}
for every nondegenerate finite interval $I\subset \R$. 
(Proving the existence of
$B$ is a classical exercise in measure theory, see \cite[ex. 8 p. 59]{Rd2}.)
Set
\begin{equation*}
F=\{ 0 \} \cup \big( B\cap [0,\frac{1}{2} ]\big) \cup \big( \frac{1}{2} +
[0,\frac{1}{2} ]\setminus B \big) .
\end{equation*}
Obviously $\mu (F)=\frac{1}{2}$ and also 
\begin{equation*}
0<\mu \big( F\cap (a,b) \big) <b-a
\end{equation*}
for any $0\le a<b\le 1$. Finally, set $E=\bigcup_{n\in \Z}(F+n)$ and 
\begin{equation*}
\lambda (t)= \left\{
\begin{array}{ll}
1&\hbox{ if }t\in E, \\
-2&\hbox{ otherwise. } \\
\end{array}
\right.
\end{equation*}
$\lambda$ is a bounded 1-periodic Borel function. Let $\Lambda (t)=
\int_0^t\lambda (\tau )\, d\tau $.
Since $\Lambda$ is absolutely continuous on any $[a,b]\subset \R$,  
there exists a zero measure Borel set $N\subset \R$ 
such that for any $t\not\in N$, $\Lambda$ is differentiable at time $t$ and
$\Lambda '(t)=\lambda (t)$.
\def\l{\lambda}
\def\L{\Lambda}
Obviously, for any $(t_0,x_0)\in \R ^2$, the (unique) solution in
Carath\'eodory's sense of the Cauchy problem  
\be
\label{eq:60}
\left\{ 
\begin{array}{rl}
\dot{x} &=\l (t)x \\
x(t_0)&=x_0 
\end{array}
\right.
\end{equation}
is given by $x(t)={\rm e}^{\L (t)-\L (t_0)}x_0$. If $s$ is a real number  
let $[s]$ denote its integral part. Since $\l$ is periodic, we get for 
any $t\ge t_0$
\begin{equation*}
\begin{array}{rl}
\L (t)-\L (t_0) 
&=\int_{t_0}^{t_0+[t-t_0]}\lambda (\tau )d\tau  +
\int_{t_0+[t-t_0]}^t \lambda (\tau )d\tau\\
&\le [t-t_0]\int_0^1\lambda (\tau )d\tau  +
 \int_{t_0+[t-t_0]}^t d\tau \\
&\le [t-t_0]\big( \mu (F)-2\mu ([0,1]\setminus F)\big) +1, 
\end{array}
\end{equation*}
hence 
\begin{equation*} 
\L (t) -\L (t_0) \le -\frac{1}{2}[t-t_0] +1,
\end{equation*}
and 
\begin{equation}
\label{eq:61}
\vert x(t)\vert \le 
\big( e \vert x_0\vert \big)\,  e^{  -\frac{1}{2}[t-t_0] }.
\end{equation}

We infer from (\ref{eq:61}) that the origin is globally asymptotically stable
(hence robustly stable and Lagrange stable)
for system (\ref{eq:10}).
To complete the proof of Proposition \ref{prop1}
we argue by contradiction: We assume that
there exists a weak Lyapunov function
in the large $V=V(t,x)$, which is of class $C^1$. We shall infer from this
assumption that the system $\dot{x}=x$ is (robustly) Lagrange stable,
a property which is obviously false. Let $x(t)$ denote the solution of
(\ref{eq:60}). If $t_0\not\in N$ and
$x_0\ne 0$, then $x(\cdot )$ is differentiable at $t_0$ and 
$\dot x(t_0)=\l (t_0)x_0$. Hence
\begin{equation*}
\frac{\partial V}{\partial t} (t_0,x_0) +
\l (t_0)\,  x_0\,  \frac{\partial V}{\partial x} (t_0,x_0)
=\frac{ d[V(t,x(t))] }{dt} _{\vert _{t=t_0}}\le 0
\end{equation*}
(since
$V(t,x(t))$ is a nonincreasing function), i.e. (A1) holds true
(here $N_0=\emptyset$ and $F(t,x)=\{ \l (t) x\}$).
It follows from Theorem \ref{thm2} that there exists
a nonnegative continuous
function $\delta (t,x)$ such that $\delta (t,x)=0 \iff x=0$, and such that
the system $\dot x\in G(t,x)$ (where $G(t,x)$ is defined in (\ref{eq:12}))
is also (robustly) Lagrange stable.
We now define a (Borel) function $s=s(t,x)$ as follows: If $x=0$ and $t\ge 0$,
we set $s(t,0)=0$. If $n\le t< n+1$, $2^k\le \vert x\vert <2^{k+1}$
with $n\in\N$, $k\in \Z$
we first pick $p\in \N ^*$ such that
\begin{equation*} 
\frac{1}{p} < \min \{\delta (\bar{t},\bar{x});\ n\le \bar{t}\le n+1 
\hbox{ and } 
2^k\le \vert \bar{x}\vert \le 2^{k+1} \} . 
\end{equation*}
Next for each $i\in \{ 1,\ldots ,p\}$ we choose
$t_i\in E\cap (n+ \frac{i-1}{p},n+\frac{i}{p})\setminus N$.
(This is possible, since
$\mu \left( E\cap (n+\frac{i-1}{p}, n+\frac{i}{p}) \right) >0$
whereas $\mu  (N)=0$.) Then we set $s(t,x)=t_i\, (\in E\setminus N)$ for
$t\in [n+\frac{i-1}{p},n+\frac{i}{p})$, $1\le i\le p$ and
$2^k\le \vert x\vert <2^{k+1}$. Observe that
$\vert s(t,x)-t\vert <\frac{1}{p}<\delta (t,x)$ and that $\l (s(t,x))=1$.
Obviously, $0\in G(t,0)$ and for $x\ne 0$
\begin{equation*}
x=\l (s(t,x))x\in 
F\bigg(\B \big( (t,x),\delta (t,x)\big) \cap (\R ^+\setminus N)
\times \R \bigg) \subset G(t,x).
\end{equation*}
%Finally $0\in G(t,0)=F(t,0}=\{ 0\}$.
It follows that the system
$\dot x=x$, whose trajectories are also solutions of $\dot x\in G(t,x)$,
should be Lagrange stable, a property which is clearly false. A slight
modification of previous reasoning shows that there does not exist 
any weak Lyapunov function of class $C^1$ {\em in the small}. The 
proof of Proposition \ref{prop1} is complete.\qeed
\setcounter{equation}{0}
\section{Proofs of Proposition \ref{prop2}, 
Theorem \ref{thm3} and Theorem \ref{thm4}}
\va
{\sc Proof of Proposition \ref{prop2}:} Since 
$f\in {\cal L}_{\rm loc}^{\infty}(\R^+\times \R ^n,\R ^n)$, we may assume
without loss of generality (by extending the zero measure set ${\cal N}$
in Definition 3) that 
\be
\label{eq:62}
\forall R>0,\ \exists M>0, \quad
(0\le t\le R,\ \Vert x\Vert \le R \ \hbox{ and }\  (t,x)\not\in {\cal N}
\im \Vert f(t,x)\Vert \le M ).
\end{equation}
Set 
\begin{equation*}
N=\{ t\ge 0;\ \int_{\R ^n}{\chi _{\cal N}}(t,x)dx > 0\}.
\end{equation*}
($\chi _{\cal N}$ denotes the characteristic function of the set $\cal N$.)
$N$ is a zero measure set in $\R ^+$, and it is clear that
\begin{equation*}
0\le t\le R,\ \Vert x\Vert < R\  \hbox{ and } \ t\not\in N
\im K_x f(t,x)\subset B(0,M).
\end{equation*}
Now let $t_0\not\in N$, $x_0\in \R ^n$ and $\varepsilon >0$. We aim to show
that there exists $\delta '>0$ s.t.
\be
\label{eq:65}
\big( \vert t-t_0\vert +\Vert x-x_0\Vert < \delta '\hbox{ and }
t\in \R ^+\setminus N  \big) \  \im \ 
K_x f(t,x)\subset K_x f(t_0,x_0) +  B(0,\varepsilon ).
\end{equation}
Let $\delta >0$ be such that (\ref{eq:13}) holds true with
$\frac{\ep}{2}$ instead of $\ep$. Pick any 
$(t,x)\in (\R ^+\setminus N)\times \R ^n$ such that  
$\vert t-t_0\vert +\Vert x-x_0\Vert  < \delta / 2$.
Since $t\not\in N$ and
$t_0\not\in N$, for almost every $y\in B(x, \delta /2 )$, we have
$(t,y),(t_0,y)\not\in {\cal N}$ 
hence (by (\ref{eq:13})) $\Vert f(t,y)-f(t_0,y)\Vert < 
\varepsilon /2$.
We infer from
\cite[Thm. 1]{PS} 
that
\be
\label{eq:66}
K_x f(t,x)\subset K_x f(t_0,x) +K_x\big( f(t,x)-f(t_0,x)\big)
\subset K_x f(t_0,x) + B(0,\frac{\ep}{2} ).
\end{equation}
Since the multivalued map $x\mapsto K_x f(t_0,x)$ is upper semicontinuous,
there exists $0<\delta ' < \delta / 2$ such that 
\be
\label{eq:67}
\Vert x-x_0\Vert <\delta ' \im K_x f(t_0,x)\subset K_x f(t_0,x_0) +
B(0,\frac{\ep}{2} )
\end{equation}
Then (\ref{eq:65}) follows from (\ref{eq:66}) and (\ref{eq:67}).
\qed
We now turn to the \va
{\sc Proof of Theorem \ref{thm3}:} By Theorem \ref{thm2} it is sufficient
to prove that (A2) holds, that is there exists a nonnegative continuous
function $\delta (t,x)$ such that $\delta (t,x)=0\iff x=0$ and
the system (\ref{eq:12})
is robustly Lagrange stable. Since it proves to be difficult to 
(directly) construct a sequence $\{ G_i \}_{i\ge 1}$ of open sets 
(as in Definition 4) for the flow of (\ref{eq:12}),
we are led  to define together with $\delta (t,x)$
a (locally Lipschitz) continuous weak Lyapunov function in the large
$V=V(t,x)$
for (\ref{eq:12}), which ensures the robust Lagrange stability of
(\ref{eq:12}).
The construction of $V$ being
essentially the same as that given in the proof of 
\cite[Thm. 2.2]{BR} we limit ourselves
to giving the main steps,
the reader being referred to \cite{BR} for the details
of the proof.


Let us  first remark  that for any function $\delta (t,x)$
as in (A2), the multivalued map $G$ defined by (\ref{eq:12})
(with $N=N_0$) satisfies 
(${\rm H}'$), as well. Indeed (H1) and (H2) are obvious and
(${\rm H3}'$) follows easily from the upper semicontinuity of
the multivalued maps $F_{\vert _{(\R ^+\setminus  N_0)\times\R ^n}}$ and
$(t,x)\mapsto \B ((t,x),\delta(t,x)) 
\cap (\R ^+\setminus N_0)\times \R^n$.
 
The following compactness lemma, which may be seen as a 
synthesis of  \cite[Lemma  4.1]{BR} and  
\cite[Lemma 2.3]{CLS}, 
is the first step in the proof of Theorem \ref{thm3}.\va

\begin{lem}
\label{lem2}
Assume that (${\rm H}'$) holds true for the multivalued map
$F(t,x)$. Let $R>0$, $T>0$ be given.
Let $(t_1^j)_{j=0,1,2,\ldots }$,
$(t_2^j)_{j=0,1,2,\ldots }$ and
$(\delta ^j)_{j=0,1,2,\ldots }$ be sequences of
numbers such that $[t_1^j,t_2^j]\subset [0,T]$, $\delta ^j\to 0^+$ 
as $j\to +\infty$ and
let $(y^j)_{j=0,1,2,\ldots }$ be a sequence of
absolutely continuous functions 
\begin{equation*}
y^j:[t_1^j,t_2^j]\to B(0,R).
\end{equation*}
Assume that 
\begin{equation*}
\label{eq:68}
\lim_{j\to +\infty}\int_{t_1^j}^{t_2^j}
\hbox{dist}
(\dot y^j(\tau ),\overline{co}
\big\{  F\big( {\cal B}\big( (\tau ,y^j(\tau )),\delta ^j\big) \cap 
(\R ^+\setminus N_0)\times \R ^n\big) \big\} ) d\tau   =0.
\end{equation*}
Then there exist $t_1,t_2\in [0,T]$, a function
$z:[t_1,t_2]\to B(0,R)$
and a sequence $(j_l)$ such that $z(\cdot )$ is a solution
of (\ref{eq:5})  on $[t_1,t_2]$, $\lim_{l\to\infty}j_l=+\infty$, 
$t_1^{j_l}\to t_1$, $t_2^{j_l}\to t_2$, and
\begin{equation*}
\lim_{l\to\infty} y^{j_l}(t_1^{j_l})=z(t_1),\quad\quad
\lim_{l\to\infty} y^{j_l}(t_2^{j_l})=z(t_2).
\end{equation*}
\end{lem}
\va
{\sc Proof of Lemma \ref{lem2}:}
Without loss of generality, we may assume that $\delta ^j <1$  for $j\ge 0$.
Let $M>0$ be such that
\begin{equation*}
\label{eq:680}
t\in [0,T+1]\setminus N_0, \quad \Vert y\Vert \le R+1\ \im \
F(t,y)\subset B(0,M).
\end{equation*}

For $j\ge 0$ and $t\in [0,T]$, we set 
\begin{equation*}
\tilde{y}^j(t)= 
\left\{
\begin{array}{ll}
y^j(t_1^j)&\hbox{ if }     0\le t\le t_1^j, \\
y^j(t)    &\hbox{ if } t_1^j\le t\le t_2^j, \\
y^j(t_2^j)&\hbox{ if } t_2^j\le t\le T,
\end{array}
\right.
\end{equation*}
and
we let $p^j(t)$ denote the projection of $\dot{\tilde y ^j}(t)$
on $\overline{\hbox{co}} \big\{ F\big( {\cal B}((t,\tilde{y} ^j(t )),
\delta ^j)\cap  (\R ^+\setminus N)\times \R ^n) \big\}$.
Then the (almost everywhere defined) map $p^j$ is measurable  (thanks
to  \cite[Cor. 8.2.13]{AF}) and it satisfies
$\Vert p^j(t)\Vert \le M$  and 
\begin{equation*}
%\label{eq:681}
\hbox{dist}
(\dot{\tilde y^j}(t ),\overline{ \hbox{co} }\big\{  
F\big( {\cal B}((t ,\tilde{y}^j(t )),
\delta ^j) \cap
(\R ^+\setminus N)\times \R ^n\big) \big\} ) 
=\Vert \dot{\tilde y^j}(t)-p^j(t)\Vert
\end{equation*}
for each $j\ge 0$ and every $t\in [0,T]$ such that 
the derivative $\dot{\tilde y ^j} (t)$ exists.

Arguing as in the proof of \cite[Lemma 4.1]{BR}, we
see that there exist two numbers $t_1,t_2\in [0,T]$, an increasing
sequence
$j_l\to \infty$ and an absolutely continuous map $z(\cdot ):[0,T]\to B(0,R)$
such that
$\tilde{y}^{j_l}(t)\to z(t)$ uniformly on $[0,T]$,
$t_1^{j_l}\to t_1$,
$t_2^{j_l}\to t_2$,  $\tilde{y}^{j_l}(t_1^{j_l})\to z(t_1)$,
$\tilde{y}^{j_l}(t_2^{j_l})\to z(t_2)$ and $(p^{j_l})$ converges to $\dot{z}$
weakly in $L^2([0,T],\R ^n)$  as $l\to +\infty$.
\def\cf{\rightharpoonup}

It remains to prove that $\dot z (t)\in F(t,z(t))$ for a.e. $t\in (t_1,t_2)$.
We assume $t_1<t_2$, otherwise there is nothing to prove. Let us consider
the functional 
\begin{equation*}
J(w)=\int_{t_1}^{t_2} \hbox{dist}\big( w(t),F(t,z(t))\big) dt
\end{equation*}
for $w\in L^2([t_1,t_2],\R ^n)$.
The same argumentation as in the proof of \cite[Lemma 4.1]{BR} shows that the
functional $J$ is well-defined, convex 
and continuous (for the strong topology), hence
lower semicontinuous w.r.t. the weak topology 
of $L^2([t_1,t_2],\R ^n)$. Since 
$p^{j_l}\rightharpoonup \dot{z}$ in $L^2([t_1,t_2],\R ^n)$, we get
\begin{equation*}
J(\dot z)\le \liminf_{l\to\infty}J(p^{j_l}) .
\end{equation*}
To prove that $J(\dot z)=0$ (which means
that $\dot z(t)\in F(t,z(t))$ for a.e. $t\in (t_1,t_2)$),
we are led to show that $J(p^{j_l})\to 0$ as $l\to\infty$.
Since $\hbox{dist}(p^{j_l}(t), F(t,z(t)))\le 2M$ a.e., it is sufficient
(by Lebesgue's theorem) to prove that for a.e. $t\in (t_1,t_2)$:
\begin{equation*}
\hbox{dist}(p^{j_l}(t),F(t,z(t)))\to 0 \ \hbox{ as } l\to\infty .
\end{equation*}
Let $t_0\in (t_1,t_2)\setminus N_0$ be such that 
$\dot{\tilde y^{j_l}}(t_0)$ exists for all $l\ge 0$ 
and let $\varepsilon >0$. 
By (${\rm H3}'$), there exists $\delta >0$  such that
\be
\label{eq:691}
\Vert (t-t_0,x-z(t_0))\Vert\le \delta \hbox{ and }
t\not\in N_0\im F(t,x)\subset F(t_0,z(t_0)) +B(0,\ep ).
\end{equation}
Let $L$ be a positive integer such that
\be
\label{eq:692}
l\ge L \im \delta ^{j_l}< \frac{\delta}{2}\  \hbox{ and }\  
\Vert \tilde{y}^{j_l}(t_0)-z(t_0)\Vert < \frac{\delta}{2}
\end{equation}
It follows from (\ref{eq:692}) that for any $l\ge L$
\begin{equation*}
\B \big( (t_0,\tilde{y}^{j_l}(t_0)),\delta  ^{j_l}\big)\subset
\B \big( (t_0,z(t_0)),\delta \big) ,
\end{equation*}
hence, by (\ref{eq:691}),
\begin{equation}
\label{eq:69}
\overline{\hbox{co}}\big\{
F(\B ((t_0,\tilde{y}^{j_l}(t_0)),\delta ^{j_l})
\cap (\R ^+\setminus N_0)\times \R ^n\big)\big\}
\subset F(t_0,z(t_0))+B(0,\ep ).
\end{equation}
We infer from the definition of $p^{j_l}(t_0)$ 
and (\ref{eq:69}) that for $l\ge L$,
\begin{equation*}
dist( p^{j_l}(t_0),F(t_0,z(t_0)))\le\ep .
\end{equation*}
The proof of Lemma \ref{lem2} is complete. \qed

Since system  (\ref{eq:5}) is robustly Lagrange stable, there exist positive
numbers $a_0,a_1,\ldots $, $b_0,b_1,\ldots $ and open sets $G_0,G_1,\ldots $
in $\R ^+\times \R ^n$ such that conditions (i'), (ii') and (iii') in
Definition 4  are fulfilled for the flow of (\ref{eq:5}).

The following two results are easy consequences of Lemma \ref{lem2}.
The first one (Proposition \ref{prop3}) is, word by word, 
\cite[Proposition 4.1]{BR}, and the proof of the second one (Proposition
\ref{prop4})
does not differ from the proof of \cite[Proposition 4.2]{BR}.

\begin{prop}
\label{prop3}
Let $i$ and $k$ be fixed nonnegative integers. Let 
$K\subset ([k,k+1]\times \R ^n)\cap G_{i+1}$ be a (nonempty) compact set.
Finally, let
\begin{equation*}
\begin{array}{rl}
C(K)=&\{ (t_2,x_2)\in [k,k+1]\times \R ^n: \hbox{ there exist }
(t_1,x_1)\in K \hbox{ and a solution } \\
&\ x(\cdot ):[t_1,t_2]\to \R ^n 
\hbox{ of (\ref{eq:5}) s.t. } x(t_1)=x_1,x(t_2)=x_2\ \} .
\end{array}
\end{equation*}
Then $C(K)$ is a compact subset of $G_{i+1}$. 
\end{prop}
\begin{prop}
\label{prop4}
Let $i,k,K,C(K)$ be as in Proposition \ref{prop3}, and let 
\begin{equation*}
\alpha =\hbox{dist}
\big( C(K),\R ^+ \times \R ^n\setminus G_{i+1} \big) .
\end{equation*}
($\alpha >0$ by Proposition \ref{prop3}.) 
Then there exists a number $\psi >0$ which enjoys the following property:
For every $y(\cdot ):[t_1,t_2]\subset [k,k+1] \to \R ^n$ such that \va
(1) $y(\cdot )$ is absolutely continuous,\va
(2) $\big( t_1,y(t_1)\big) \in K$,\va
(3) $\int_{t_1}^{t_2}\hbox{dist}( \dot{y}(\tau ),
\overline{\hbox{co}}\big\{ F\big( \B ((\tau ,y(\tau )),\psi )\cap
(\R ^+ \setminus N)\times \R ^n \big) \big\}) d\tau  <\psi $,\va
we have
\begin{equation*}
\min_{t\in[t_1,t_2]} \hbox{dist}\big(\, (t,y(t)) ,
\R ^+ \times \R^n\setminus G_{i+1} \big) \ge \frac{\alpha}{2} .
\end{equation*}
\end{prop}

Let $i\ge 0$ be fixed.
We define two sequences $(\alpha _i^k)_{k\ge 0}$,
$(\psi _i^k)_{k\ge 0}$ of positive numbers and a sequence
$(K_i^k)_{k\ge 0}$
of compact subsets of $\R ^+\times \R ^n$ by induction on $k\ge 0$.
For $k=0$, set $K_i^0=[0,1]\times \R ^n \cap\overline{G_i}$,
$\alpha _i^0=\alpha$,
$\psi _i^0=\psi$, where $\alpha$ and $\psi$  are given 
in Proposition \ref{prop4} (applied with $K=K_i^0$,
$k=0$).

Then assume that 
$K_i^l$, $\alpha _i^l$, $\psi _i^l$  have been defined
for $0\le l<k$. Set
\begin{equation*}
K_i^k=\big( [k,k+1]\times\R^n\cap \overline{G_i}\big) \cup 
\{ (t,x); \ t=k \hbox{ and } \hbox{dist}\big(  (t,x),
\R ^+ \times \R^n\setminus G_{i+1}\big)  \ge 
\frac{\alpha _i^{k-1}}{2} \} ,
\end{equation*}
$\alpha _i^k=\alpha$ and
$\psi _i^k=\psi$, where $\alpha$ and $\psi$ 
are given in Proposition \ref{prop4} (applied with $K=K_i^k$).\va

We now take a function $\psi\! :\R ^+\times \R ^n\to (0,+\infty )$
in such a way that\va
(1) $\psi (\cdot ,\cdot )$ is of class $C^1$;\va
(2) $\psi (t,x)> 1/{\psi _i^k}$ for $t\in [k,k+1]$ and
$(t,x)\in  G_{i+1}\setminus G_i$ ($i\ge 0$);\va
(3) $\psi (t,x)>1$ for $t\in \R ^+$ and $x\in \R ^n$.\va

We also pick a function $\delta \! :\R ^+\times \R ^n\to \R ^+$ such that\va
(1) $\delta (\cdot ,\cdot )$ is continuous;\va
(2) $\delta (t,x)=0  \iff  x=0$;\va
(3) $\delta (t,x) < \psi _i^k$  for
$t\in [k,k+1]$ and
$(t,x)\in  G_{i+1}\setminus G_i$ ($i\ge 0$).\va
>From now on $G$ denotes the multivalued map (defined by
(\ref{eq:12})) associated with this function $\delta$ (with $N=N_0$).
We finally define, for each pair
$(t_0,x_0)\in \R ^+\times\R ^n$, the class of
functions 
\begin{equation*}
\begin{array}{rl}
{\cal F}(t_0,x_0)=
&\bigg\{ y(\cdot )\! :[t_0,+\infty )\to \R ^n
\hbox{ such that } y(\cdot ) \hbox{ is absolutely continuous on every}\\
 &\hbox{segment }[t_0,b],\ y(t_0)=x_0, \hbox{ and }
\int_{t_0}^{+\infty}\psi (t,y(t))\, 
\hbox{dist}\big( \dot y (t),G(t,y(t))\big) dt <1\bigg\} \cdot
\end{array}
\end{equation*}

The proof of the following result is the same as the one of 
\cite[Proposition 4.3]{BR}.  

\begin{prop}
\label{prop5}
Let $i\ge 1$ be a fixed integer, and let $(t_0,x_0)\in G_i$. For each
$y\in {\cal F}(t_0,x_0)$ and each $t\ge t_0$ we have $(t,y(t))\in G_{i+1}$.
\end{prop}

We are now ready to define a (locally Lipschitz) continuous weak Lyapunov
function in the large for $G$. For $(t_0,x_0)\in \R ^+\times\R ^n$ we set
\begin{equation*}
V(t_0,x_0):=\sup_{y\in {\cal F}(t_0,x_0)}
\left\{ \left(
1-\int_{t_0}^{+\infty}\psi (t,y(t))\,
\hbox{dist} \big( \dot y (t),G(t,y(t))\big) dt\right)
\cdot\sup_{t\ge t_0}  \Vert y(t)\Vert \right\} \cdot 
\end{equation*}
The same reasoning as in the proofs of \cite[Propositions 4.4,
4.5 and 4.6]{BR}
shows that $V$ is a (locally Lipschitz) continuous weak Lyapunov
function in the large for system (\ref{eq:12}), and therefore we infer
from the first Lyapunov theorem \cite[Thm. 2.1]{BR} that
(\ref{eq:12}) is robustly Lagrange stable. Next it follows from
Theorem \ref{thm2} that (A1) holds true for $F$.
The proof of Theorem \ref{thm3} is complete.\qed
{\sc Proof of Theorem \ref{thm4}:} The proof is
almost identical to the proof
of Thm \ref{thm3}, the main difference being that the function $V$ is
now required to be smooth up to $x=0$.
Let $V$ be a continuous  weak Lyapunov function such that
(I), (II) and (III) in Definition 1 hold true
(with $Q=\{ x:\ \Vert x\Vert <h \}$) 
for some continuous functions $a,b:[0,h) \to \R ^+$,
$V_{\vert _{\R ^+\times \{x:\  0< \Vert x\Vert <h \} }}$ is of class
$C^{\infty}$ and (\ref{eq:11}) holds for 
$(t,x)\in (\R ^+\setminus N)\times  \{ x:\ 0<\Vert x\Vert <h\}$.
By \cite[Thm. 6]{K2} there exists a function $\nu \in C^{\infty}(\R )$
such that $\nu (r)=0$ for $r\le 0$, $\nu (r)>0$ and $\nu '(r)>0$ for $r>0$, 
$\nu (r)\to +\infty$ as $r\to +\infty$ and 
\begin{equation*}
\nu \circ V\in C^{\infty}(\R ^+\times \{x:\ \Vert x\Vert <h\}), 
\end{equation*}
\begin{equation*}
\partial ^{\alpha}(\nu \circ V)(t,0)=0\quad \forall \alpha\in \N ^{n+1}, \ 
\forall t\ge 0.
\end{equation*}
Notice that for every $(t,x)\in \R^+\times \{ x:\ \Vert x\Vert <h\} $
\be
\label{eq:55}
\nu (a(\Vert x \Vert )) \le \nu (V(t,x)) \le \nu (b(\Vert x\Vert ))
\end{equation}
and for every $(t,x)\in (\R^+ \setminus N)\times \{ x:\ \Vert x\Vert <h\} $
\begin{equation}
\label{eq:56}
\forall v\in F(t,x)\quad
\frac{\partial (\nu \circ V)}{\partial t}(t,x)+
\langle \nabla _x (\nu\circ V)(t,x), v\rangle \le 0.
\end{equation}
We infer from  (\ref{eq:55}) and (\ref{eq:56}) that
the assertions in Theorem \ref{thm4} hold true when $\nu\circ V$,
$\nu\circ a$ and $\nu\circ b$ are substituted for $V$, $a$ and $b$,
respectively.\qeed 
\setcounter{equation}{0}
\section{Some Application to Control Systems} 
Let us now consider the following control system:
\begin{equation}
\label{eq:15}
\dot{x}=f(t,x,u),
\end{equation}
where $t\ge 0$ is time, $x\in \R^n$ is the state variable and $u\in \R ^m$
is the control variable. We aim to investigate the links between the (robust)
Lagrange stability of the unforced system
\begin{equation}
\label{eq:16}
\dot{x}=f(t,x,0)
\end{equation}
and the {\em uniform bounded input bounded state {\rm (}UBIBS{\rm )}
 stability} of
(\ref{eq:15}), defined as follows:
\begin{equation*}
%\label{eq:17}
\forall R>0,\quad \exists M>0 \ \hbox{ s.t. }\ 
\left. 
\begin{array}{l}
t_0\ge 0\\
\Vert x_0\Vert <R\\
\Vert u\Vert _{L^{\infty}(t_0,+\infty )} <R 
\end{array}
\right\}
\quad \im\quad \forall t\ge t_0 \quad \Vert x(t;t_0,x_0,u) \Vert < M
\end{equation*}
for any solution $x(.;t_0,x_0,u)$ of (\ref{eq:15}) such that $x(t_0)=x_0$.
Obviously the UBIBS stability of (\ref{eq:15}) implies the Lagrange stability
of (\ref{eq:16}), but the converse is false \cite{B}. We shall say that 
(\ref{eq:15}) is UBIBS {\em stabilizable} if there exists some feedback law
$u=k(t,x)+v$ such that the system $\dot{x}=f(t,x,k(t,x)+v)$
is UBIBS stable (w.r.t. the new control variable $v$). 
In practice, we look for maps
$k$ with the same regularity as $f$.  It is well known that in general 
the (even robust) Lagrange stability of (\ref{eq:16}) does not imply the UBIBS
stabilizability of (\ref{eq:15}) \cite{B}. However,
when dealing with (discontinuous) \sou{affine} time-independent systems
\begin{equation*}
%\label{eq:20}
\dot{x}=f_0(x)+\sum_{i=1}^m u_i\, f_i(x),
\end{equation*}
it has been
proved by Bacciotti and Beccari \cite{BB} that the existence
of a \sou{time-independent} weak Lyapunov function
$V=V(x)$ \sou{of class $C^1$} such that 
$\langle \nabla V(x),f_0(x)\langle \le 0$ for each
$x\in\R ^n$ guarantees the UBIBS stabilizability of (\ref{eq:15}).
Notice that the existence of a {\em time-independent
continuous} weak Lyapunov function $V=V(x)$
(in the large) is equivalent \cite[Thm 9]{AS}
to the so-called {\em absolute boundedness} (or {\em absolute}
Lagrange stability),
a concept which is \sou{stronger}
than the (robust) Lagrange stability, since it accounts not only
for the behaviour of trajectories, but also for the behaviour of their
prolongations. On the other hand, even for a smooth field $f_0$, the
existence of a (time-independent) \sou{continuous} weak Lyapunov function
does not imply the existence of a (time-independent) weak Lyapunov function
\sou{of class $C^1$} \cite{BR2}.

A time-dependent version of the 
Bacciotti-Beccari result in which we merely assume
the (robust) Lagrange stability of the unforced system is as follows.
\begin{thm}
\label{thm5}
Consider a system 
\begin{equation}
\label{eq:21}
\dot{x}=f_0(t,x)+\sum_{i=1}^m u_i\, f_i(t,x)
\end{equation}
where for each $i\in\{0,\ldots ,m\}$ the field
$f_i\in {\cal L}_{loc}^{\infty}(\R ^+\times \R ^n,\R^n)$ and it is {\bf
e.c.t.}. Assume that the unforced system
$\dot{x}=f_0(t,x)$
is (robustly) Lagrange stable. Then the system  (\ref{eq:21}) is
UBIBS stabilizable by means of a feedback law $u=k(t,x)+v$ such that
$k\in {\cal L}_{\rm loc}^{\infty}(\R ^+\times \R ^n,\R^m)$ and $k$ is
{\bf e.c.t.}, as well. Moreover, if for each $i\ge 0$ $f_i$ is of class
$C^r$ for some $r\in \N \cup \{\infty \}$, then $k$ is also of class $C^r$.
\end{thm}
Notice that a preliminary version of
this result was announced in \cite{BR}.
\va
\proof
It follows from Theorem 3 
(applied with  $F(t,x)=K_x \, f_0(t,x)$)
that there exist a  set
$N\subset \R ^+$ of measure zero and three functions 
$V=V(t,x)$, $a=a(r)$ and $b=b(r)$
such that (I), (II), (III) and (A1) hold true.
Without  loss  of generality,   we may assume that  $V$ and
$a_{\vert _{(0,+\infty )}}$ are of class $C^{\infty}$
and also (thanks to Yoshizawa's trick, see (\ref{eq:30})) that
\begin{equation*}
%\label{700}
\begin{array}{rl}
\forall t\in \R^+\setminus N,\ \  &\forall x\in \R^n\setminus \{ 0\},\
\forall v\in K_x \, f_0(t,x), \\
&\frac{\partial V}{\partial t}(t,x) + 
\langle\nabla _x V(t,x), v\rangle \le
-\frac{1}{(1+2t)^2}a(\Vert x\Vert ).
\end{array}
\end{equation*}

Let $\psi\in C^{\infty }(\R )$ be such that
$\psi (r)=0$ for $r\le \frac{1}{2}$, $\psi (r)=1$ for $r\ge 1$. Define
the components $k_i\, (1\le i\le m)$ of $k=k(t,x)$ as
follows: $k_i(t,0)=0$ for $t\ge 0$ and
\begin{equation*}
\begin{array}{l}
k_i(t,x)\\
\ \  =-\Vert x\Vert \psi (\Vert x\Vert )\, \hbox{sgn} 
\big( \langle \nabla _x V(t,x),f_i(t,x)\rangle \big)\psi \left(
\vert \langle\nabla _x V(t,x),f_i(t,x)\rangle \vert\cdot \Vert x \Vert\cdot
\frac{(1+2t)^2}{a(\Vert x\Vert )}
\right)
\end{array}
\end{equation*}
for $t\ge 0$ and $x\in \R ^n \setminus \{ 0\}$.

Notice  that if $f_i$ is continuous,
($\langle \nabla _x V(t_0,x_0),f_i(t_0,x_0)\rangle =0$ or
$x_0=0$) $\im$ $k_i\equiv 0$ in a neighborhood of $(t_0,x_0)$. It follows
that $k$ is of class $C^r$, $r\in \N \cup \{\infty \}$, 
if the $f_i$'s are of class $C^r$.
When the $f_i$'s are merely assumed to be
measurable, locally (essentially)
bounded and {\bf e.c.t.}, it is clear that
$k$ is a measurable locally bounded function. 
We now prove that \sou{$k$ is {\bf e.c.t.}}. Let
${\cal N}\subset \R ^+\times \R ^n$ be a zero measure (Borel) set such that 
(\ref{eq:62}) is satisfied by  $f_i$ (for $0\le i\le m$) and such that 
for any $(t_0,x_0)\in \R ^+\times \R ^n$, for any $\varepsilon >0$
there exists $\delta >0$ s.t.
\begin{equation}
\label{eq:70}
\left.
\begin{array}{l}
\vert t-t_0\vert +\Vert x-x_0\Vert <\delta \\
(t,x)\not\in {\cal N} \\
(t_0,x)\not \in {\cal N}
\end{array}
\right\} \im 
\forall i\in \{0,\ldots ,m\} ,\quad \Vert f_i(t,x)-f_i(t_0,x)\Vert < 
\varepsilon .
\end{equation}
Fix $i\in\{ 1,\ldots ,m\}$, 
a pair $(t_0,x_0)\in \R^+\times \R ^n$ and $\varepsilon >0$. 
We are done if the following claim is proved. 
\va 
\sou{\sc Claim 6.} 
{\em There exists $\tilde\delta >0$ s.t. 
\begin{equation*}
\left.
\begin{array}{l}
\vert t-t_0\vert +\Vert x-x_0\Vert <\tilde\delta \\
(t,x)\not\in {\cal N} \\
(t_0,x)\not \in {\cal N}
\end{array}
\right\} \im \vert k_i(t,x)-k_i(t_0,x)\vert < \varepsilon .
\end{equation*}}
If $x_0=0$, then 
$\big( \Vert x\Vert <\frac{1}{2},\ t\ge 0\big) \im k_i(t,x)=0$ and
$\tilde\delta =\frac{1}{2}$ is convenient.
Assume now $x_0\ne 0$. Set 
$Q:=[0,t_0+1]\times B(x_0, \Vert x_0 \Vert /2)$. There exists
$M>0$ s.t. 
$\forall (t,x)\in Q\setminus {\cal N},\ \Vert f_i(t,x)\Vert\le M$.
By an uniform continuity argument we infer from (\ref{eq:70}) that there
exists $\tilde{\delta} > 0$ such that for each pair $(t,x)\in Q$,
\begin{equation}
\label{eq:71}
\left.
\begin{array}{l}
\vert t-t_0\vert +\Vert x-x_0\Vert <\tilde{\delta} \\
(t,x)\not\in {\cal N} \\
(t_0,x)\not \in {\cal N}
\end{array}
\right\} \im 
\end{equation}
\begin{equation}
\label{eq:72}
\left\vert \langle \nabla _x V(t,x), f_i(t,x)\rangle - 
\langle \nabla _x V(t_0,x), f_i(t_0,x)\rangle
\right\vert < \frac{1}{20} \frac{1}{(1+2t_0)^2} 
\frac{a(\Vert x\Vert )}{\Vert x\Vert },
\end{equation}
\be
\label{eq:73}
\begin{array}{rl}
\Vert x\Vert \psi (\Vert x\Vert )
&\left\vert \psi \left( \vert \langle\nabla _x V(t,x),f_i(t,x)\rangle
\vert\,
\Vert x \Vert \frac{(1+2t)^2}{a(\Vert x\Vert )} \right) \right. \\
&-
\left. 
\psi \left( \vert \langle\nabla _x V(t_0,x),f_i(t_0,x)\rangle \vert \,
\Vert x \Vert \frac{(1+2t_0)^2}{a(\Vert x\Vert )} \right) \right\vert 
<\varepsilon 
\end{array}
\end{equation}
and also 
\begin{equation}
\label{eq:74}
\frac{1}{(1+2t_0)^2} <
\frac{2}{(1+2t)^2} \cdot  
\end{equation}
Now let $(t,x)\in Q$ be such that the l.h.s. of (\ref{eq:71}) holds. If 
\be
\label{eq:75}
\vert \langle \nabla _x V(t_0,x), f_i(t_0,x)\rangle \vert < \frac{1}{10}
\frac{a(\Vert x\Vert )}{(1+2t_0)^2\Vert x\Vert}, 
\end{equation}
then, by (\ref{eq:72}) and (\ref{eq:74}),
\begin{equation*}
\vert \langle \nabla _x V(t,x),f_i(t,x)\rangle \vert < \frac{2}{10}
\frac{a(\Vert x\Vert )}{(1+2t_0)^2\Vert x\Vert}< 
\frac{4}{10}\frac{a(\Vert x\Vert )}{(1+2t)^2\Vert x\Vert} ,
\end{equation*}
hence $k_i(t,x)=k_i(t_0,x)=0$ and we are done. If, instead,  
(\ref{eq:75}) does not hold, we infer from (\ref{eq:72}) that
\begin{equation*}
\hbox{sgn} ( \langle \nabla_x V(t,x), f_i(t,x)\rangle )
=\hbox{sgn}( \langle \nabla_x V(t_0,x), f_i(t_0,x)\rangle ) .
\end{equation*}
Hence, thanks to (\ref{eq:73}),
\begin{equation*}
\vert k_i(t,x)-k_i(t_0,x)\vert<\varepsilon .
\end{equation*}
The proof of Claim 6 is complete. \qed
We now proceed to the UBIBS stability. 
\va
\sou{\sc Claim 7.} {\em  The UBIBS stability holds true
if for every $R>1$, every $x\in \R ^n$ and every $v\in\R ^m$
\begin{equation}
\label{eq:76}
\begin{array}{rl}
\big( \sum_{i=1}^m \vert v_i\vert <R<\Vert x\Vert \big) 
&\im \frac{\partial V}{\partial t}(t,x) +
\langle \nabla _x V(t,x),w\rangle \le 0
\quad \hbox{ for every } t\in \R ^+\setminus N \\
\hbox { and every } &w\in 
K_x\left( f_0(t,x)+\sum _{i=1}^m(k_i(t,x)+v_i) f_i(t,x) \right) .
\end{array}
\end{equation}}
(Notice that in the expression $K_x(\ldots )$ the $v_i$'s are reviewed as 
parameters.) Indeed if (\ref{eq:76}) holds true, if a control map 
$v(t)$ and
an initial state $x_0$ are such that $\sum_{i=1}^m \vert v_i(t)\vert <R$
for a.e. $t\ge t_0$ and $\Vert x_0\Vert <R$, then for every solution $x(t)$
in Filippov's sense of
\begin{equation*}
\left\{
\begin{array}{rl}
\dot{x}&=f_0(t,x)+\sum_{i=1}^m 
\big( k_i(t,x)+v_i(t)\big) f_i(t,x) \\
x(t_0)&=x_0
\end{array}
\right.
\end{equation*}
we have $\Vert x(t)\Vert\le a^{-1}(b(R))$ for every $t\ge t_0$: Otherwise
there
would exist times $t_1<t_2$ such that $\Vert x(t_1)\Vert =R$,
$\Vert x(t)\Vert >R$ for $t_1<t\le  t_2$ and $a(\Vert x(t_2)\Vert )>b(R)$.
Then, thanks to (\ref{eq:76}), 
$\dot{V}(t)\le 0$ for a.e. $t\in (t_1,t_2)$, hence
$a(\Vert x(t_2)\Vert )\le V(t_2,x(t_2))\le V(t_1,x(t_1))\le b(R)$,
a contradiction.\qed
Now it follows
from \cite[Thm. 1]{PS} that
%\begin{equation*}
%\quad\quad K_x\left( f_0(t,x)
%+\sum_{i=1}^m(k_i(t,x)+v_i) f_i(t,x)\right) \subset
%K_x\ f_0(t,x)+\sum_{i=1}^m\, K_x\bigg( (k_i(t,x)+v_i) f_i(t,x)\bigg)
%\end{equation*}
any $w\in K_x\left( f_0(t,x)+\sum _{i=1}^m(k_i(t,x)+v_i) f_i(t,x) \right)$
may be decomposed as $w=w_0+\sum_{i=1}^m w_i$ with
$w_0\in K_x \, f_0(t,x)$ and $w_i\in K_x \big( (k_i(t,x)+v_i) f_i(t,x)\big)$
for $1\le i\le m$. Fix $R>1$.
Since for each pair
$(t,x)\in (\R ^+\setminus N)\times (\R ^n\setminus \{ 0\} )$
and each $w_0\in \,  K_x\, f_0(t,x)$, we have
\begin{equation*}
\frac{\partial V}{\partial t}(t,x)
+\langle \nabla _x V(t,x), w_0\rangle
\  \le \ -\frac{a(\Vert x\Vert )}{(1+2t)^2} ,
\end{equation*}
to get (\ref{eq:76}), it is sufficient to prove that for each $t\ge 0$, each
$x\in\R ^n\setminus B(0,R)$ and each $v\in \R ^m$ s.t.
$\sum_{i=1}^m \vert v_i\vert <R$ we have
\begin{equation}
\label{eq:77}\ 
\forall i=1,\ldots ,m,\ 
\forall w_i\in K_x\big( (k_i(t,x)+v_i)f_i(t,x)\big) \
 \ \langle \nabla _x V(t,x), w_i\rangle 
\le \frac{a(\Vert x\Vert )}{(1+2t)^2}\frac{\vert v_i\vert}{\Vert x\Vert}.
\end{equation}
Fix $i\in\{ 1,\ldots ,m\}$ and $w_i\in K_x \big( (k_i(t,x)+v_i)f_i(t,x)\big)$.
By \cite[Thm. 1]{PS} and Carath\'eodory's theorem, we may write
$w_i=\sum_{j=1}^{n+1}\alpha ^j\omega ^j$,
where $\alpha ^j\ge 0$ for each $j\ge 1$, 
$\sum_{j=1}^{n+1}\alpha ^j =1$ and
for $j\ge 1$
\begin{equation*}
\omega ^j=\lim_{k\to\infty}(k_i(t,x_k^j)+v_i)f_i(t,x_k^j)
\end{equation*}
for some sequence $x_k^j\to x$ (as $k\to +\infty$),
and we may assume that for every $k\ge 1$,
$\Vert x_k^j\Vert >R>1$, as well.
Therefore $\psi (\Vert  x_k^j\Vert )=1$ for each $k\ge 1$.
We get
\begin{equation*}
\label{eq:78}
\langle \nabla _x V(t,x), w_i\rangle =\lim _{k\to\infty} \sum_{j=1}^{n+1} 
\alpha ^j \langle \nabla _x V(t,x_k^j), f_i(t,x_k^j)\rangle 
(k_i(t,x_k^j)+v_i).
\end{equation*}
\sou{\sc Claim 8.} {\em  For any $k$
\be
\label{eq:79}
\langle \nabla _x V(t,x_k^j), f_i(t,x_k^j)\rangle 
\big( k_i(t,x_k^j)+v_i\big)
\le \frac{a(\Vert x_k^j\Vert )}{(1+2t)^2}\frac{\vert v_i\vert}
{\Vert x_k^j\Vert}.
\end{equation}}
Notice first that 
\be
\label{eq:80}
\begin{array}{l}
\langle \nabla _x V(t,x_k^j), f_i(t,x_k^j)\rangle \, k_i(t,x_k^j) \\
\quad =\ -\Vert x_k^j \Vert \,
\vert \langle\nabla _xV(t,x_k^j), f_i(t,x_k^j )\rangle \vert 
\psi \left( \vert \langle \nabla _xV(t,x_k^j), f_i(t,x_k^j )\rangle \vert\,
\Vert x_k^j\Vert \frac{(1+2t)^2}{a (\Vert x_k^j  \Vert )} \right) \\
\quad \le \ 0.
\end{array}
\end{equation}
If 
\begin{equation*}
\vert \langle \nabla _x V(t,x_k^j), f_i(t,x_k^j)\rangle  \vert
\le \frac{a(\Vert x_k^j\Vert )}{(1+2t)^2 \Vert x_k^j\Vert}
\end{equation*}
then (\ref{eq:79}) follows from (\ref{eq:80}). 
Otherwise
\begin{equation*}
\vert \langle \nabla _x V(t,x_k^j), f_i(t,x_k^j)\rangle \vert
> \frac{a(\Vert x_k^j\Vert )}{(1+2t)^2 \Vert x_k^j\Vert}
\end{equation*}
and we get
\begin{equation*}
\begin{array}{rl}
\langle \nabla _x V(t,x_k^j), f_i(t,x_k^j)\rangle
\big( k_i(t,x_k^j)+v_i\big)
&\ =\ \vert \langle \nabla _x V(t,x_k^j), f_i(t,x_k^j)\rangle \vert
\big( -\Vert x_k^j\Vert \pm v_i\big) \\
&\ \le \ 0 \quad \hbox{(since }\ \vert v_i\vert 
<R< \Vert x_k^j\Vert \hbox{ )} \\
&\ \le \ \frac{a(\Vert x_k^j\Vert )}{(1+2t)^2}
\frac{\vert v_i\vert}{\Vert x_k^j\Vert} , 
\end{array}
\end{equation*}
as required.\qed
We now infer from (\ref{eq:79}) that
\begin{equation*}
\langle \nabla _xV(t,x), w_i\rangle \le\lim_{k\to\infty}
\sum_{j=1}^{n+1}\alpha^j 
\frac{a(\Vert x_k^j\Vert )}{(1+2t)^2}
\frac{\vert v_i\vert}{\Vert x_k^j\Vert} =
\frac{a(\Vert x\Vert )}{(1+2t)^2}
\frac{\vert v_i\vert}{\Vert x\Vert},
\end{equation*}
i.e. (\ref{eq:77}) holds true.
This completes the proof of Theorem \ref{thm5}.\qed
{\bf Acknowledgement.} The author would like to thank Professor A. Bacciotti
for useful comments.
\begin{thebibliography}{99}
\bibitem{AC} AUBIN  J.P.   and  CELLINA A., {\em   Differential 
Inclusions}, Springer-Verlag, New York, 1984.    
\bibitem{AF} AUBIN J.P. and FRANKOWSKA H., {\em Set-Valued Analysis},
Birkh\"{a}user, Basel, 1990.
\bibitem{AS}  AUSLANDER J.  and SEIBERT P.,  Prolongations and 
Stability  in  Dynamical  Systems,   {\em Ann.  Inst.  Fourier 
(Grenoble)} {\bf 14} (1964), 237-268.  
\bibitem{B} BACCIOTTI A., External Stabilizability of Nonlinear
Systems with some applications, {\em Internat. J. Robust
Nonlinear Control} {\bf 8} (1998), 1-10.
\bibitem{BB}   BACCIOTTI  A.   and  BECCARI  G.,  External
Stabilizability  by Discontinuous Feedback, {\em  Proc. 2nd
 Portuguese Conference on Automatic Control}, 1996.
\bibitem{BR} BACCIOTTI A. and ROSIER L., Liapunov and Lagrange 
Stability: Inverse Theorems for Discontinuous Systems, 
{\em Math. Control Signals Systems} {\bf 11} (1998), 101-128.
\bibitem{BR2} BACCIOTTI A. and ROSIER L., On the Converse of 
First Liapunov Theorem: The Regularity Issue, {\em submitted}.
\bibitem{CLS} CLARKE F.H., LEDYAEV Yu. S. and STERN R.J., Asymptotic
Stability and Smooth Lyapunov Functions, {\em J. Differential Equations}
{\bf 149}  (1998), 69-114.
\bibitem{D} DEIMLING K., {\em Multivalued Differential Equations}, 
de Gruyter, 1992.
\bibitem{EG} EVANS L.C. and GARIEPY R.F., {\em Measure theory and
fine properties of functions}, Stud. adv. math., CRC Press,
London, 1992.
\bibitem{F}  FILIPPOV  A.F.,   Differential  Equations  with 
Discontinuous    Right-hand  Side, {\em Mat. Sb.} {\bf 5} (1960),
 99-127 (in Russian). English translation in 
{\em Amer.  Math. Soc. Transl.} {\bf 42} (1964),  199-231.
\bibitem{K}  KRASOVSKI N.N.,  The Converse of the
Theorem of K.P. Persidskij on Uniform Stability,
{\em Prikl. Mat. Mekh.} {\bf 19} (1955), 273-278 
(in Russian)  
\bibitem{K1}  KURZWEIL J., On the Invertibility of the  First 
Theorem of Lyapunov Concerning the Stability of Motion,  
{\em Czech. Math.  J.} 
{\bf 80} (1955), 382-398 (in Russian)  
\bibitem{K2} KURZWEIL J., On the Inversion of Liapunov's Second 
Theorem  on  Stability  of  Motion, {\em  Trans. Amer. 
Math. Soc.} {\bf 24}  (1963), 19-77.
\bibitem{KV}  KURZWEIL J. and VRKO\v C I.,  The Converse 
Theorems of Lyapunov and Persidskij Concerning the Stability of Motion,  
{\em Czech. Math.  J.} {\bf 82} (1957),
254-272 (in Russian)  
\bibitem{LS} LEDYAEV Yu.S. and SONTAG E.D., A Lyapunov Characterization
of Robust Stabilization, to appear in {\em Nonlinear Analysis, TMA}.  
\bibitem{PS} PADEN B.E. and SASTRY S.S., A Calculus for
Computing Filippov's Differential Inclusions with Applications to the 
Variable Structure Control of Robot Manipulators, {\em IEEE Trans.
Circuits Systems} {\bf 34} (1987), 73-81.
\bibitem{R} ROSIER L.,  Inverse of Lyapunov's Second Theorem
for Measurable Functions, in: M. Fliess (ed.) {\em Proc. of IFAC-NOLCOS 92}, 
pp. 655-660.  
\bibitem{Rd1} RUDIN W., {\em Principles of Mathematical Analysis},
McGraw-Hill, New York, 1987.
\bibitem{Rd2} RUDIN W., {\em Real and Complex Analysis},   
McGraw-Hill, New York, 1974.
\bibitem{S}  SONTAG E.D., Smooth  Stabilization   Implies   Coprime
Factorization,  {\em IEEE Trans. Autom. Control} {\bf 34} 
(1989), 435-443. 
\bibitem{VL}  VARAIYA P.P. and LIU R., Bounded-input
Bounded-output  Stability of Nonlinear Time-varying  Differential 
Systems,  {\em SIAM J. Control} {\bf 4} (1966),  698-704.
\bibitem{Y1}   YOSHIZAWA T.,  On the Stability of
Solutions of a System of Differential Equations, {\em Mem. 
Coll. Sci. Univ. Kyoto Ser. A} {\bf 29} (1955), 27-33.
\bibitem{Y2}    YOSHIZAWA  T.,  Liapunov's  Functions  and
Boundedness  of Solutions,  {\em Funkc. Ekvac.} {\bf 2}  (1959), 
95-142.   
     
\end{thebibliography}                                             

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