%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
%            This is a LateX file, containing  the paper 
%
%                      THE EULER-LAGRANGE EQUATION 
%                  UNDER WEAK REGULARITY CONDITIONS
%
%                       by H. J. Sussmann
%
%           This version has been submitted for presentation at the 
%             38th IEEE CDC, to be held in Phoenix, Arizona, Dec. 1999,
%               and will appear in the conference proceedings.
%  
%               Date of this version: February 3, 1999
%
%                  The paper is 6 printed pages long.
%
%       You should run Latex twice on this file. After the second run,
%      there will still be several ``underfull  hbox'' messages.  Just
%             ignore them and print the .dvi file.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\beq}{\begin{equation}}
\newcommand{\vup}{\vspace{-1mm}}
\newcommand{\vupp}{\vspace{-2mm}}
\newcommand{\vuppp}{\vspace{-3mm}}
\newcommand{\vupppp}{\vspace{-4mm}}
\newcommand{\vuppppp}{\vspace{-5mm}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqn}{\begin{eqnarray}}
\newcommand{\eeqn}{\end{eqnarray}}
\newcommand{\beqno}{\begin{eqnarray*}}
\newcommand{\eeqno}{\end{eqnarray*}}
\newcommand{\bequot}{\begin{quotation}}
\newcommand{\equot}{\end{quotation}}
\newcommand{\rf}[1]{[\ref{#1}]} %bibliographical references
\newcommand{\text}[1]{\hbox{\rm \ #1\ \/}}
%\newcommand{\beql}[1]{\begin{equation}\label{#1}}
\newcommand{\bt}{\begin{theorem}}
\newcommand{\bl}{\begin{lemma}}
\newcommand{\br}{\begin{remark}}
\newcommand{\bc}{\begin{corollary}}
\newcommand{\bp}{\begin{proposition}}
\newcommand{\bd}{\begin{definition}}
\newcommand{\bfact}{\begin{fact}}
\newcommand{\bex}{\begin{example}}
\newcommand{\ber}[1]{\begin{exercise}\label{#1}}
\newcommand{\bem}[1]{\begin{example}\label{#1}}  %Yes, 2 different ones...
\newcommand{\ec}{\end{corollary}}
\newcommand{\el}{\end{lemma}}
\newcommand{\er}{\end{remark}}
\newcommand{\et}{\end{theorem}}
\newcommand{\ed}{\end{definition}}
\newcommand{\ednoen}{\end{definition}}
\newcommand{\eprop}{\end{proposition}}
\newcommand{\epr}{\end{proof}}
\newcommand{\bpr}{\begin{proof}}
\def\tto{\mbox{$\,\,\rightarrow\kern-13pt\rightarrow\,\,$}}
\def\transv{{\cap\kern-5.2pt\lower
    -2.4pt\hbox{$|$}\kern-5.6pt\lower -8pt\hbox{$-$}}}
\def\baba{|\kern-1.9pt|} 
\def\stransv{{\cap\kern-5.7pt\lower
    -2.4pt\hbox{$\baba$}\kern-6.3pt\lower -8pt\hbox{$-$}}}
\def\supportnote{Research supported in part by %
NSF Grant DMS95-00798 and AFOSR Grant 0923.}
\def\bdel{\bar\delta}
\def\ffi{\varphi}
\def\beps{\bar\varepsilon}
\def\co{{\rm co}}
\def\cco{\overline{{\rm co}}}
\def\togr{{\buildrel \rm gr \over \to}}
\def\bfact{\begin{quotation}}
\def\efact{\end{quotation}}
\def\cvfadomi{TVVF_{AD}(\Om,I)}
\def\cvfomi{TVVF(\Om,I)}
\def\cvf{TVVF}
\def\omi{\Omega\!\times\!I}
\def\msr{{\rm meas}}
\def\bloi{{\cal BL}(\Omega,I)}
\def\mbloiy{{\cal M}_e(\bloi,Y)}
\def\ndc{{\bf W}}
\def\eps{\varepsilon}
\def\bbxr{{\cal B}(X\times\rr)}
\newcommand{\mybox}{\hfill \mbox{$\diamondsuit$}} %put at right margin (white square)
\def\dom{{\rm Dom}}
\newcommand{\pubnote}{\vskip-5.0cm
\framebox{Proc.\ {\it IEEE Conf. Decision and Control, Tucson, Dec.\ 1992}} 
\vskip4.5cm}
\font\myverysmall=cmr5
\def\midbreak{\vskip 0.01true cm}
\def\steer#1#2{$#1$\steero$#2$}
\def\steert#1#2#3{\hbox{$#1$\hbox{\lower0.4ex\hbox{\steero}}\kern-0.59cm\raise0.53ex\hbox{$#2$}}\kern0.35cm$#3$}
\def\Om{\Omega}
\def\compo{\hbox{$\,${\myverysmall o}$\,$}}
\def\plplus{(+\kern-0.02in\compo\kern -0.02in\oplus)}
\def\om{\omega}
\def\vphi{\varphi}
\def\mtraj{{\rm MTraj\,}}
\def\indi{\raise 1.5pt\hbox{$\chi$}}
\def\lowsub#1{\lower 2pt\hbox{${_{#1}}$}}
\def\pfo#1{{\cal P}_{#1}^{\rightarrow}}
\def\pba#1{{\cal P}_{#1}^{\leftarrow}}  
\def\aabx{{\cal A}(a,b,X)}
\def\icone{IC\kern0.1pt$^1$}
\def\iconeit{IC\kern0.5pt$^1$}
\def\mdno{\medbreak\noindent}
\def\inn{\int_{0}^{\tau}}
\def\bel{\beta}
\def\al{\alpha}
\def\ctraj#1{{\rm CTraj}(#1)}
\def\carc#1{{\rm CArc}(#1)}
\def\traj{{\rm Traj}}
\def\arc#1{{\rm Arc}(#1)}
\def\restr{\lceil}
\def\ep{\varepsilon}
\def\dki{{\partial\,\,\,\over\partial\kappa^i}}
\def\dkia{\partial_i^\kappa}
\def\dkonea{\partial_1^\kappa}
\def\dkna{\partial_n^\kappa}
\def\pdto{\,\rightarrow\kern -12pt ^{pd}\,}
\def\tvto{\,\rightarrow\kern -12pt ^{tv}\,}
\def\opdto{\,\longrightarrow\kern -18pt ^{opd}\,}
\def\pdtvto{\,\hbox to 1.1true cm{\rightarrowfill}\kern -20pt ^{pdtv}\,\,\,}
\def\tpdtvto{\,\hbox to 1.1true cm{\rightarrowfill}\kern -23pt ^{tpdtv}\,\,\,}
\def\adjo#1{{\rm Adj}(#1)}
\def\adjonul#1{{\rm Adj}^{\nu,L}(#1)}
\def\cch{\overline{co}}
\def\rrinf{\bar{\rr}}
\def\nn{{\rm I}\kern-.2em{\rm N}}
\def\nninf{\bar{\nn}}
\def\rrsmall{{\mysmall I}\kern-.2em{\mysmall R}}
\def\compo{\hbox{$\,${\myverysmall o}$\,$}}
\newcommand{\Psiu}{\underline{\Psi}}
\newcommand{\Psiud}{\underline{\dot\Psi}}
\newcommand{\Deltau}{\underline{\Delta}}
\newcommand{\psiu}{\underline{\psi}}
\newcommand{\Phiu}{\underline{\Phi}}
\newcommand{\thetau}{\underline{\theta}}
\newcommand{\sigmau}{\underline{\sigma}}
\newcommand{\varphiu}{\underline{\varphi}}
\newcommand{\nuu}{\underline{\nu}}
\newcommand{\hu}{\underline{h}}
\newcommand{\Deltaud}{\underline{\dot\Delta}}
\newcommand{\psiud}{\underline{\dot\psi}}
\newcommand{\Phiud}{\underline{\dot\Phi}}
\newcommand{\thetaud}{\underline{\dot\theta}}
\newcommand{\sigmaud}{\underline{\dot\sigma}}
\newcommand{\varphiud}{\underline{\dot\varphi}}
\newcommand{\nuud}{\underline{\dot\nu}}
\newcommand{\hud}{\underline{\dot h}}
\newtheorem{defi}{Definition}%[section]
\newtheorem{definition}{Definition}
\newtheorem{exam}{Example}%[section]
\newtheorem{propos}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{theorem}{Theorem}
\newcommand{\pa}{\partial}
\newcommand{\raw}{\rightarrow}
\newcommand{\vare}{\varepsilon}
\newcommand{\nis}{\cal N}
\def\beginproof{\medbreak\noindent{\sl PROOF.\enspace}}
\def\equaldef{{\buildrel \rm def \over =}}
\def\equivdef{{\buildrel \rm def \over \equiv}}
\def\proofend{\mybox}
\def\bi{\begin{itemize}}
\def\ei{\end{itemize}}
\def\br{\begin{remark}}
\def\er{\end{remark}}
\def\bt{\begin{theorem}}
\def\et{\end{theorem}}
\documentclass{article}
\usepackage{amsfonts}
\def\zz{\mathbb{Z}}
\def\qq{\mathbb{Q}}
\def\cc{\mathbb{C}}
\def\rr{\mathbb{R}}
\def\nn{\mathbb{N}}
\def\kk{\mathbb{K}}
\def\bmath{\mathbb{B}}
%\renewcommand{\baselinestretch}{0.95} %to fine-tune - use .95 if you need more
\renewcommand{\baselinestretch}{0.85} %to fine-tune - use .95 if you need more
\newcommand{\auone}{
\vspace{-15pt}
\begin{center}
H\'ector J. Sussmann\\
%\footnote[1]{\supportnote}\\
%\footnote{\supportnote}\\
Department of Mathematics; Rutgers, the State University of New Jersey\\
Hill Center---Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ
08854-8019, USA\\ 
\vskip5pt
E-mail: sussmann@hamilton.rutgers.edu\quad\quad
Web site: http://www.math.rutgers.edu/$\widetilde{\,\,}$sussmann\\

\end{center}
\vskip5pt
} 

\newcommand{\authors}{
\auone
}

\renewcommand{\title}{
\vspace{-15 pt}

The Euler-Lagrange equation under weak regularity conditions$\,^{1}$}

\def\spaceabovetitle{\hspace{1cm}}
\def\spacebelowtitle{\vspace{2pt}}
\def\spaceafterauthors{}

\def\titlestuff{
{\small \null\vspace{-5mm}This paper has been submitted for presentation to
the 38th IEEE Conference on Decision and Control, to be held in
Phoenix, Arizona, in December 1999,  and  for publication in the conference 
proceedings. The date of this version is February 3, 1999.}\\
\vspace{-5.6mm}
\null\vspace{4mm}
\vspace{2mm}
\begin{center}
\spaceabovetitle
\begin{center}
{\Large \bf \title}\\
\end{center}
\spacebelowtitle
\authors
\spaceafterauthors
\end{center}}
\tolerance=20 %allow VERY LITTLE tolerance in extending after line.
\overfullrule=0pt %no dark lines if overfull

\newcommand{\ssection}[1]{\vspace{0.25cm}\setcounter{subsection}{0}
  \protect\addtocounter{section}{1}%
  \centerline{{\bf \thesection.~#1}}\vspace{0.05cm}}
\newcommand{\subssection}[1]{\vspace{0.16cm}
  \protect\addtocounter{subsection}{1}
  \noindent{{\bf\thesubsection.~#1}}\vspace{0.1cm}} 
\topmargin -2.3cm
\textheight 24.4cm
\oddsidemargin -0.3in
\evensidemargin -0.3in
\textwidth 178mm

\sloppy
\flushbottom
\parindent 1em
\leftmargini 2em
\leftmarginv .5em
\leftmarginvi .5em

\marginparwidth 48pt 
\marginparsep 10pt 
\columnsep 10mm

%\pagestyle{empty}
%\renewcommand{\thefootnote}{\fnsymbol{footnote}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Macros needed for this sample paper
\def\vbar{\mathchoice{\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height4.1ptdepth-.35ptwidth.6pt\kern-.6pt}
   {\vrule height3.1ptdepth-.25ptwidth.5pt\kern-.5pt}}
\def\fudge{\mathchoice{}{}{\mkern.5mu}{\mkern.8mu}}
\def\bbc#1#2{{\rm \mkern#2mu\vbar\mkern-#2mu#1}}
\def\bbb#1{{\rm I\mkern-3.5mu #1}}
\def\bba#1#2{{\rm #1\mkern-#2mu\fudge #1}}
\def\bb#1{{\count4=`#1 \advance\count4by-64 \ifcase\count4\or\bba A{11.5}\or
   \bbb B\or\bbc C{5}\or\bbb D\or\bbb E\or\bbb F \or\bbc G{5}\or\bbb H\or
   \bbb I\or\bbc J{3}\or\bbb K\or\bbb L \or\bbb M\or\bbb N\or\bbc O{5} \or
   \bbb P\or\bbc Q{5}\or\bbb R\or\bbc S{4.2}\or\bba T{10.5}\or\bbc U{5}\or
%   \bbb P\or\bbc Q{5}\or\bbb R\or\bba S{8}\or\bba T{10.5}\or\bbc U{5}\or
   \bba V{12}\or\bba W{16.5}\or\bba X{11}\or\bba Y{11.7}\or\bba Z{7.5}\fi}}
%
\def\R{{\bb R}}
\def\C{{\bb C}}

\newcommand{\new}{\newcommand}
\new{\rnew}{\renewcommand}
\new\Empty{/\mkern-9.5mu O}
\newcommand{\Rn}[1]{\R^{#1}}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\renewcommand{\c}{\gamma}
\newcommand{\Ca}{\Gamma}
\renewcommand{\d}{\delta}
\newcommand{\D}{\Delta}
\newcommand{\e}{\varepsilon}
\newcommand{\G}{\bigtriangledown}
%\newcommand{\k}{\kappa}
\renewcommand{\l}{\lambda}
\renewcommand{\L}{\Lambda}
\newcommand{\p}{\varphi}
\newcommand{\s}{\sigma}
\renewcommand{\t}{\theta}
\newcommand{\w}{\omega}
\newcommand{\tx}{\tilde x}
\new{\ty}{\tilde y}
\new\ttt{\tilde t}
\new{\te}{\tilde e}
\new{\bx}{\bar{x}}
\new{\by}{\bar{y}}
\new{\bz}{\bar{z}}
\new{\dx}{\dot{x}}
\new{\dy}{\dot{y}}
\new{\dz}{\dot{z}}
\new\sat{\rm sat}
\new\sign{\rm sign}
\newcommand{\dd}[2]{\frac{d#1}{d#2}}
\newcommand{\pdo}[1]{\frac{\partial}{\partial#1}}
\renewcommand{\vec}[2]{(#1_1,\cdots,#1_{#2})}
\newcommand{\vecc}[3]{(#1_{#2},\cdots,#1_{#3})}
\newcommand{\vecf}[3]{(#1_1#3,\cdots,#1_{#2}#3)}
\newcommand{\row}[2]{#1_1,\cdots,#1_{#2}}
\newcommand{\roww}[3]{#1_{#2},\cdots,#1_{#3}}
\newcommand{\newt}{\newtheorem}
\newcommand{\rarrow}{\rightarrow}
\newtheorem{Corollary}[lemma]{Corollary}
%\newcommand{\beq}{\vspace{-0.18cm}\begin{equation}\vspace{-0.12cm}}
%\newcommand{\eeq}{\end{equation}}
%\newcommand{\beqn}{\begin{eqnarray}}
%\newcommand{\eeqn}{\end{eqnarray}}
\newcommand\yangspace{\vspace{-2.45mm}}
\newcommand\yangspacebb{\vspace{-2.25mm}}
\newcommand\yangspacebbc{\vspace{-2.10mm}}
\newcommand\yangspaceb{\vspace{-1.4mm}}
\newcommand\yangspacec{\vspace{-1.2mm}}
\newcommand\yangspaced{\vspace{-.85mm}}
\newcommand{\brem}{\yangspacebb\begin{remark}}
\newcommand{\bedef}{\yangspacebb\begin{defi}}
%\def\bd{\bedef}
%\def\ed{\edef}
\newcommand{\begexam}{\yangspacebb\begin{exam}}
\newcommand{\termiexam}{\end{exam}\yangspacebb}
\newcommand{\beprop}{\yangspacebb\begin{propos}}
\newcommand{\termiprop}{\end{propos}\yangspacebb}
\newcommand{\termidef}{\end{defi}\yangspacebb}
\newcommand{\erem}{\end{remark}\yangspacebb}
\newcommand{\shsection}{\ssection}
\newcommand{\proof}{\noindent {\sc Proof.}\ }
\newcommand{\niindent}{\vspace{1mm}\noindent}
%\new{\eproof}{\hfill $\Box$ \medbreak}
\newcommand{\dg}{\dagger}
\newcommand{\dfn}{\stackrel{\rm def}{=}}

\def\pmb#1{\setbox0=\hbox{$#1$}%
  \kern-.025em\copy0\kern-\wd0
  \kern.05em\copy0\kern-\wd0
  \kern-.025em\raise.0433em\box0 }

\new{\bfs}{{\pmb \s}}
\new{\bfl}{{\pmb l}}
\new{\num}{\mu}
\newcommand{\evle}{\le_{\rm ev}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%    PAPER STARTS HERE    %%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\setcounter{section}{0}

\twocolumn[\titlestuff]
\vskip4.5cm

\ssection{Introduction}

\noindent This note is a continuation of our
recent\footnotetext[1]{\supportnote} work
(cf. Sussmann
\cite{sussloja,susspavel,sussrecent,susskobe,sussmulti,sussbro,susssanto}),
on a general nonsmooth version of the finite-dimensional Pontryagin
Maximum Principle (cf. Pontryagin {\em et al.}
\cite{pontrybook}, Berkovitz \cite{berkovitz}). Our 
purpose here is to announce a further extension of this
result for the special case of finite-dimensional 
classical calculus of variations problems in $\rr^n$, 
with constraints $\dot\xi(t)\in U$, where $U$ is 
a given subset of $\rr^n$. For simplicity, we will only consider problems 
with fixed endpoints, omitting the discussion of more general endpoint
constraints and the corresponding transversality conditions.

Naturally, our result is a generalization of the classical Euler-Lagrange
equations with the Weierstrass side conditions, stated in 
the Hamiltonian language of optimal control theory.  As in our earlier work
on the Maximum Principle, the approach used here is classical, in the sense 
that it involves ``needle variations'' and uses a fixed-point argument. 
The main new feature is that we use a one-sided analogue of the
``semidifferentials'' that we introduced in the earlier papers. This 
concept---called ``subsemidifferential'' here---is particularly 
suitable for calculus of variations problems, because
the differentiability conditions that are truly needed for the Lagrangian 
only involve upper bounds of the form 
$L(\xi_*(t)+h,\dot\xi_*(t),t)\le L(\xi_*(t),\dot\xi_*(t),t)+
\varphi(t,h)+o(\|h\|)$
as $h\to 0$,  where $\varphi$ is a Lipschitz function of $h$, and $\xi_*$
is the reference trajectory. 

The precise definition of the subsemidifferentials is given in \S 2. Our
version of the Euler-Lagrange condition is stated and proved in \S 3. The
Lagrangian $L(q,u,t)$ is required to be continuous in $q$ for each fixed
$(u,t)$ and measurable in $t$ for each fixed $(q,u)$. In addition, the
following subsemidifferentiability requirement is made along the reference
trajectory: the function $q\to L(q,\dot\xi_*(t),t)$ should admit, for
almost every $t$, a subsemidifferential $W(t)\subseteq\rr^n$ at
$q=\xi_*(t)$, with some uniformity with respect to $t$, as specified in
condition (A8) below.  This includes, in particular, the case when
$L(q,\dot\xi_*(t),t)$ is Lipschitz with respect to $q$ for each $t$, with
an integrable Lipschitz constant $k(t)$, but is much more general.

\ssection{Subsemidifferentials} 
\noindent If $Q$ is an open subset of 
$\rr^n$, $f:Q\to\rr$ is a function, and $\bar q\in Q$, a 
{\sl subsemidifferential} of $f$ at $\bar q$ is a nonempty compact
subset $W$ of $\rr^n$ such that 
\vupp
$$
\lim_{\delta\downarrow 0}\frac{1}{\delta}\sup\Big\{f(q)-f(\bar q)-F_{W,\bar q}(q)
:\|q-\bar q\|\le\delta\Big\}=0\,,
$$
where
\vupppp
\beq
F_{W,\bar q}\equaldef\max\Big\{w.(q-\bar q):w\in W\Big\}\,.
\eeq
The following result shows that subsemidifferentials lead to a necessary
condition of the usual kind for a local minimum. It is easy to see that a
similar result is not true for a local maximum, even if $f$ is continuous, 
so subsemidifferentials are a truly ``one-sided'' concept of generalized
differential, suitable for minimization problems. (For example, if
$f(x)=-|x|$ for $x\in\rr$, then $\{r\}$ is a subsemidifferential of $f$ at
$0$ for any $r\in[-1,1]$, but this subsemidifferential does not contain $0$
unless $r=0$, even though $f$ has a maximum at $0$.)
\bt
If $Q$ is an open subset of $\rr^n$, $f:Q\to\rr$ is lower semicontinuous,
$\bar q\in Q$, and  $W$ is a subsemidifferential of $f$ at $\bar q$, then
a necessary condition for $f$ to have a local minimum at 
$\bar q$ is that $0$ belong to the convex hull of $W$. 
\et

\noindent{\sl PROOF.} We can clearly assume, without loss of generality,
that $W$ is convex, $\bar q=0$ and $f(0)=0$. Assume that  $f$ has a local
minimum at $0$. 

We prove our result by contradiction. Assume that $0\notin W$, and
define $\rho=\min\{\|w\|:w\in W\}$, so $\rho>0$. 

Fix a number $\alpha$ such that $0<\alpha<\rho$. 
Let $\delta$ be a positive number such that 
\vup
\bi
\item[(1)] $q\in Q$ and $f(q)\ge 0$ for all $q$ such that 
$\|q\|\le\delta$, 
\vupp
\item[(2)] 
$\sup\Big\{f(q)-\max\{w.q:w\in W\}:\|q\|\le\delta\Big\}\le\alpha\delta$. 
\ei
For $w\in W$, let  $q_w\!=\!-\!\delta\frac{w}{\|w\|}$,
so \mbox{$W\!\ni\! w\!\to\! q_w\!\in\!\rr^n$} is a continuous map such that 
$\|q_w\|\!=\!\delta$ for all \mbox{$w\!\in\! W$}. For each $w\in W$, let $S(w)$
denote the set of all $v\in 
W$ such that $f(q_w)-v\cdot q_w\le\alpha\delta$.  Then $S(w)$ is nonempty, 
compact, and convex. Let \mbox{$U\!=\!\{(w,v):w\!\in\! W,v\!\in\!S(w)\}$}, so 
$U$ is the graph of the set-valued map $S$. Then $U\subseteq W\times W$, 
so $U$ is a bounded subset of $\rr^n\times\rr^n$. If 
$\{(w_j,v_j)\}_{j=1}^\infty$ is a sequence of points 
of $U$ that converges to a limit $(w,v)$, then $w\in W$, because
$W$ is compact. Also, 
$f(q_{w_j})-v_j\cdot q_{w_j}\le\alpha\delta$ for all $j$, and then 
$f(q_{w})-v\cdot q_{w}\le\alpha\delta$, because $w\to q_w$ is continuous
and $f$ is lower semicontinuous. So $v\in S(w)$, and then 
$(w,v)\in U$. So $U$ is compact. Therefore $S$ is an upper semicontinuous
set-valued map from the compact convex subset $W$ of $\rr^n$ with values
nonempty  
compact convex subsets of $W$. By the Kakutani Fixed Point Theorem, 
$S$ has a fixed point. So there exists a $w\in W$ such that
$w\in S(w)$.   This implies that 
$f(q_{w})-w\cdot q_{w}\le\alpha\delta$.
In view of the definition of $q_w$, the scalar product 
$w\cdot q_{w}$ is equal to $-\delta\|w\|$. So 
$f(q_{w})\le(\alpha-\|w\|)\delta\le(\alpha-\rho)\delta<0$.
But $f(q_{w})\ge 0$, because $\|q_w\|=\delta$. So we have reached a
contradiction, and our proof is complete. \proofend

\def\aqul{{\cal A}(Q,U,L,a,b)}
\def\aqulep{{\cal A}_{\bar q,\hat q}(Q,U,L,a,b)}
\def\aquld{{\cal A}^{\delta}(Q,U,L,a,b)}
\def\aquldep{{\cal A}_{\bar q,\hat q}^{\delta}(Q,U,L,a,b)}


\ssection{The Euler-Lagrange equation}  Let us assume that 
\bi\bi
\vupp
\item[(A1)] {\sl $Q$ is an open subset of $\rr^n$.}
\vup
\item[(A2)] {\sl $U$ is a subset of $\rr^n$.}
\vup
\item[(A3)] {\sl $L$ is a function from
$Q\times U\times[a,b]$ to $\rr$.}
\ei\ei
We define $\aqul$---the set of ``admissible arcs''---to be the set of 
all absolutely continuous maps $\xi:[a,b]\to\rr^n$ such that 
\bi
\vup
\item[(1)] {\sl $\xi(t)\in Q$ for all $t\in[a,b]$,}
\vup
\item[(2)] {\sl $(t)\in U$ for almost all $t\in[a,b]$,}
\vup
\item[(3)] {\sl the function $t\!\!\to\!\!L(\xi(t),\dot\xi(t),t)$ is 
integrable on $[a,b]$.}
\ei
\def\nbhq{{\cal N}_{\bar q,\hat q}}
For fixed $\bar q,\hat q\in Q$, we let $\aqulep$ denote the set of all 
$\xi\in{\cal A}$ such that $\xi(a)=\bar q$ and $\xi(b)=\hat q$. 

We then assume 
\bi\bi
\vupp
\item[(A4)] {\sl ${\cal N}$ is a subset of $\aqul$,}
\ei\ei
\vupppp
and write $\nbhq\equaldef{\cal N}\cap\aqulep$.

We then consider a standard fixed-endpoints calculus of variations
problem of the form
\vup
\begin{equation}\label{standard}
\fbox{{$\left.\begin{array}{l}
{\rm minimize\,\,} I(\xi)=\int_a^bL(\xi(t),\dot\xi(t),t)dt\,,\cr
{\rm subject \,\,\,to\,\,\,\,\,}\xi\in\nbhq\,,\end{array}\right.$}}
\end{equation} 
or, equivalently, of the form 
\vup
\begin{equation}\label{standardu}
\fbox{{$\left.\begin{array}{l}
{\rm minimize\,\,}\hat I(\xi,\eta)=\int_a^bL(\xi(t),\eta(t),t)dt\,,\cr
{\rm subject \,\,\,to\,\,\,\,\,}\xi\in\nbhq\,\,\,\,{\rm and}\,\,\cr
\dot\xi(t)=\eta(t)\,\,{\rm for\,\,\,
almost\,\,\,all\,\,\,}t\in[a,b]\,.\end{array}\right.$}}
\end{equation}
We define a function $H:Q\times U\times\rr^n\times[a,b]\to\rr$---the
``normal Hamiltonian''---by letting
\vupp
\beq\label{defnorham}
H(q,u,p,t)=p.u-L(q,u,t)\,.
\eeq

\vupp
\noindent Suppose that $\xi_*$ is a solution of (\ref{standard}),
so $\xi_*\in\nbhq$, and the pair $(\xi_*,\eta_*)$ is a solution of 
(\ref{standardu}), if we let $\eta_*(t)=\dot\xi_*(t)$.  
Let us compare $I(\xi_*)$ with the cost $I(\xi)$ of another trajectory 
$\xi\in\aqul$. We find

\vuppppp
$$
\left.\begin{array}{l}
I(\xi)-I(\xi_*)=\int_a^b\psi(t).\Big(\dot\xi(t)-\eta_*(t)\Big)\,dt\\
\!-\!\int_a^b\!\!\Big(H(\xi(t),\dot\xi(t),\psi(t),t)\!-\!
H(\xi_*(t),\eta_*(t),\psi(t),t)\Big)dt,\end{array}\right.
$$

\vuppp
\noindent where $[a,b]\ni t\to\psi(t)\in\rr^n$ is, for the time being, an
arbitrary  absolutely continuous function of $t$, and we have expressed $L$
terms of $H$ using the identity $L(q,u,t)=p\cdot u-H(q,u,p,t)$. Integration
by  parts---which 
is perfectly justified because $\psi$, $\xi$,  and 
$\xi_*$ are absolutely continuous---then yields

\vuppppp
$$
\left.\begin{array}{l}
I(\xi)-I(\xi_*)=\\
\!-\int_a^b \dot\psi(t).\Big(\xi(t)-\xi_*(t)\Big)\,dt
+\psi(b)\cdot(\xi(b)-\xi_*(b))\\
\!-\!\int_a^b\!\!\Big(H(\xi(t),\dot\xi(t),\psi(t),t)\!-\!
H(\xi_*(t),\eta_*(t),\psi(t),t)\Big)dt,\end{array}\right.
$$

\vuppp
\noindent where we have used the fact that $\xi(a)-\xi_*(a)=0$. 

\def\tube{{\cal T}(\xi_*,\delta)}

For $\delta>0$, use $\tube$ to denote the ``$\delta$-tube about the
reference trajectory $\xi_*$,'' i.e. the set 
\vup
\beq 
\tube\equaldef\{(q,t):a\le t\le b,\|q-\xi_*(t)||\le\delta\}\,.
\eeq 
\vup
Since $\xi_*\in\aqul$, it is clear that there exists a $\bar\delta_1$ such
that 

\medbreak
\noindent $\,\,$(C1) {\sl $\bar\delta_1>0$ and ${\cal
T}(\xi_*,\bar\delta_1)$ is a  subset of $Q\times[a,b]$.}

We then use $\aquld$, $\aquldep$, to denote, respectively,
the set of all $\xi\in\aqul$ such that the graph of $\xi$
is contained in $\tube$, i.e.  such that
$\|\xi(t)-\xi_*(t)\|\le\delta$ for all
$t\in[a,b]$ and the set $\aqulep\cal \aquld$. We define 
\beqn
{\cal N}(\delta)&=&{\cal N}\cap\aquld\,,\nonumber\\
\nbhq&\equaldef&{\cal N}\cap\aquldep\,.\nonumber
\eeqn
We then assume:
\bi\bi
\item[(A5)] {\sl $L(q,u,t)$ is continuous as a function of $q$ for each 
fixed $u\in U$, $t\in[a,b]$.}
\vup
\item[(A6)] {\sl the function $t\to L(q,u,t)$ is Lebesgue measurable
for each fixed $q\in Q$, $u\in U$.}
\vup
\item[(A7)] {\sl the function $t\mapsto L(q,\dot\xi_*(t),t)$ is Lebesgue
measurable for each fixed $q\in Q$.}
\vup
\item[(A8)] {\sl For some $\bar\delta_2\in\,]\,0,\bar\delta_1]$, there exist}
\vup
\item[(A8.1)] {\sl a measurable set-valued map 
$W:[a,b]\to 2^{\rr^n}$ with nonempty compact convex values,}
\vup
\item[(A8.2)] {\sl  measurable functions 
$k_\delta:[a,b]\to[0,+\infty]$, for $0<\delta\le\bar\delta_2$,} 
\vup
\item[(A8.3)] {\sl  an integrable function $k:[a,b]\to[0,+\infty]$,}
\item[]{\sl such that} 
\vup
\item[(A8.a)] {\sl the inequality} 
\ei\ei
\vuppp
\beqn
L(q,\eta_*(t),t)&\le &L(\xi_*(t),\eta_*(t),t)\nonumber\\
&\,\,&\,+F_{W(t),\xi_*(t)}(q)+\delta k_\delta(t)\label{bonobo}
\eeqn
\bi\bi
\vuppp
\item[]{\sl holds whenever $(q,t)\in\tube$,  where}
\ei\ei
\vup
$$
F_{W,x}(q)=\max\{w.(q\!-\!x):w\!\in\! W(t)\}\,.
$$
\bi\bi
\vuppp
\item[(A8.b)]
$\sup\Big\{\|w\|:w\!\!\in\!\!W(t)\Big\}\!\!\le\!\!k(t)$ {\sl whenever 
\mbox{$t\!\in\![a,b]$},}
\vup
\item[(A8.c)] $\lim_{\delta\downarrow 0}\int_a^bk_\delta(t)dt=0$.
\ei\ei

\vupp\noindent 
Assumption (A8) implies that there exist an integrable function
$\hat k:[a,b]\to\rr\cup\{+\infty\}$ and a
$\bar\delta_3\in\,]0,\,\bar\delta_2]$ such that the {\sl
Carath\'eodory-type upper bound}
\vupp
\beq\label{cara}
L(q,\eta_*(t),t)\le\hat k(t)\,\,\,{\rm for}\,\,\,(q,t)\in{\cal
T}(\xi_*,\bar\delta_3)
\eeq

\vuppp\noindent holds. (Indeed, it suffices to  choose  $\hat k(t)$ equal to 
$|L(\xi_*(t),\eta_*(t),t)|+\bar\delta_3(k(t)+k_{\bar\delta_3}(t))$,
where $\bar\delta_3$ is such that $\int_a^bk_{\bar\delta_3}(t)dt<\infty$.)

We then define $W_\delta(q,t)$, for $\delta\in\,]\,0,\bar\delta_3]$ 
and $(q,t)$ in $\tube$, to be the set of those $w\in W(t)$ such that

\vuppppp
$$
L(q,\eta_*(t),t)\!\le\!L(\xi_*(t),\eta_*(t),t)\!+\!
w.(q-\xi_*(t))\!+\!\delta k_\delta(t)\,.
$$

\vupp\noindent Clearly, $W_\delta(q,t)$ is compact, convex and nonempty for
every $q,t\in\tube$, as long as $\delta\le\bar\delta_2$.  

If $\xi\in\aqul$, we define $W_\delta^\xi(t)\equaldef W_\delta(\xi(t),t)$.  
Then $[a,b]\ni t\to W_\delta^\xi\subseteq\rr^n$ is a 
measurable set-valued function with nonempty compact convex values,
provided that $\delta\le\bar\delta_2$, and $\xi\in\aquld$.
(The fact that values are compact, convex and nonempty is trivial. 
To prove measurability of $W_\delta^\xi$, 
we have to show that for each compact subset 
$K$ of $\rr^n$, if we define

\vupp
$$
E_K=\{t\in[a,b]:W_{\delta}^\xi(t)\cap K\neq\emptyset\}\,,
$$

\vup\noindent
then $E_K$ is Lebesgue measurable. Let $\qq^n$ be the set of all
points of $\rr^n$ with rational coordinates. Define

\vupppp
\beqn
\rho(t)&=&L(\xi(t),\eta_*(t),t)-L(\xi_*(t),\eta_*(t),t)\,,\nonumber\\
\sigma_q(t)&=&{\rm dist}(q,W(t))\,\,\,{\rm for}\,\,\,q\in\rr^n\,.\nonumber
\eeqn

\vupp\noindent
Then $k_\delta$, $\rho$ and the $\sigma_q$ are measurable functions.   By
Lusin's  theorem,  there are pairwise disjoint subsets $J_i$,
$i=0,1,2,\ldots$,   of $[a,b]$  such that 
\bi
\item[(i)] for $i>0$, the set $J_i$ is compact and the restrictions 
$k_\delta\restr J_i$, $\rho\restr J_i$ , $\sigma_q\restr J_i$ of 
$k_\delta$, $\rho$, and the $\sigma_q$ for 
$q\in\qq^n$, to $J_i$ are continuous; 
\vup
\item[(ii)] $J_0$ has measure zero;
\vup
\item[(iii)] $[a,b]=\cup_{i=0}^\infty J_i$.  
\ei
To prove that $E_K$ is measurable, it suffices to show that $E_K\cap J_i$ is 
measurable for $i>0$. Let $i>0$. We will show that $E_K\cap J_i$ is closed. Let
$\{t_j\}$ be a sequence of members of $E_K\cap J_i$ that converges to a limit
$t\in J_i$.   Then 
$W_{\delta}^\xi(t_j)\cap K\neq\emptyset$, so we can pick
$w_j\in W_\delta^\xi(t_j)\cap K$.  Since $K$ is compact we may assume,
after passing to a subsequence, that $w=\lim_{j\to\infty}w_j$ exists. 
Clearly, $w$ belongs to $K$.  To prove that $w$ also belongs to
$W_{\delta}^\xi(t)$, we must show that 
$\rho(t)\le w\cdot(\xi(t)-\xi_*(t))+\delta k_\delta(t)$ 
and $w\in W(t)$. The first assertion follows from the fact that 
$\rho(t_j)\le w\cdot(\xi(t_j)-\xi_*(t_j))+\delta k_\delta(t_j)$---since
$w_j\in W_{\delta}(\xi(t_j),t_j)$---and $\rho$, $\xi$, $\xi_*$ and $k_\delta$ 
are continuous on $J_i$. To prove that $w\in W(t)$, it suffices to show 
that ${\rm dist}(w,W(t))\!=\!0$, since $W(t)$ is closed. Assume that
${\rm dist}(w,W(t))\!=\!\alpha\!>\!0$.  Pick $q\!\in\!\qq^n$ such that 
$\|w\!-\!q\|\!<\!\frac{\alpha}{3}$. Then
$\sigma_q(t)\!>\!\frac{2\alpha}{3}$. On the other hand, 
$\sigma_q(t_j)\!\le\!\|q\!-\!w_j\|$, since $w_j\!\in\! W(t_j)$. 
Since $\sigma_q$ is continuous on $J_i$, and $w_j\!\to\! w$, we have
$\sigma_q(t)\!\le\!\|q\!-\!w\|\!<\!\frac{\alpha}{3}$.  
This contradiction proves that $w\in W(t)$, as desired.)

We let $\Gamma(W)$ denote the set of all measurable selections of
$W$, so the members of $\Gamma(W)$ are the measurable maps
$\omega:[a,b]\to\rr^n$ such that $\omega(t)$ belongs to $W(t)$ for almost all 
$t\in[a,b]$. Then $\Gamma(W)$ is a nonempty, convex, weakly compact subset
of $L^1([a,b],\rr^n)$.  Similarly, we use $\Gamma(W_\delta^\xi)$ to
denote the set of all measurable selections of 
$W_\delta^\xi$, so $\Gamma(W_\delta^\xi)$ 
is also a nonempty, convex, weakly compact subset
of $L^1([a,b],\rr^n)$. 

We now define

\vuppppp
\beqn
R_{\xi,\omega}(t)&=&L(\xi(t),\eta_*(t),t)-L(\xi_*(t),\eta_*(t),t)\nonumber\\
\,&\,\,&\,\,-\omega(t)\cdot(\xi(t)-\xi_*(t))\,.\nonumber
\eeqn

\vuppp\noindent
Then the estimate

\vupp
\beq\label{estirem}
R_{\xi,\omega}(t)\le\delta k_\delta(t)
\eeq

\vupp\noindent
holds for $\xi\!\in\!\aqul$, $t\!\in\![a,b]$,
$\omega\!\in\!\Gamma(W_\delta^\xi)$. 

We then  have, writing $\xi,\xi_*,\dot\xi,\eta_*,\omega$,
$R_{\xi,\omega}$, $S_\xi$, rather  than 
$\xi(t)$, $\xi_*(t)$, $\dot\xi(t)$, $\eta_*(t)$, $\omega(t)$, 
$R_{\xi,\omega}(t)$, $S_\xi(t)$, 
$$
\left.\begin{array}{l}
H(\xi,\dot\xi,\psi,t)-H(\xi_*,\eta_*,\psi,t)\\
\,\,=(H(\xi,\dot\xi,\psi,t)-H(\xi_*,\dot\xi,\psi,t))\\
\,\,\,\,\,\,\,+(H(\xi_*,\dot\xi,\psi,t)-H(\xi_*,\dot\xi_*,\psi,t))\\
\,\,=(H(\xi_*,\dot\xi,\psi,t)-H(\xi_*,\eta_*,\psi,t))\nonumber\\
\,\,\,\,\,\,\,-(L(\xi,\dot\xi,t)-L(\xi_*,\dot\xi,t))\\
\,\,=(H(\xi_*,\dot\xi,\psi,t)-H(\xi_*,\eta_*,\psi,t))\\
\,\,\,\,\,\,\,-\omega.(\xi-\xi_*)-R_{\xi,\omega}-S_\xi\,,\end{array}\right.
$$

\vupppp\noindent where 

\vuppp
\beqn
S_\xi(t)&\equaldef &
\Big(L(\xi(t),\dot\xi(t),t)-L(\xi_*(t),\dot\xi(t),t)\Big)\nonumber\\
&\,\,\,&\,-\Big(L(\xi(t),\eta_*(t),t)-L(\xi_*(t),\eta_*(t),t)\Big)\,.\nonumber
\eeqn 


\vupppp\noindent Now let 

\vuppp
$$
\psi_\omega(t)=-\int_t^b\omega(s)ds\,.
$$

\vup\noindent Then, if $\xi\in\aqul$, 
\beq\label{estihamo}
\left.\begin{array}{l}
I(\xi)-I(\xi_*)={\cal R}_{\xi,\omega}\\
\,\,+\!\int_a^b\!\Big(H(\xi_*,\eta_*,\psi_\omega,t)\!-\!H(\xi_*,\eta,\psi_\omega,t)
\Big)dt\,,\end{array}\right.
\eeq
where we have used the fact that $\psi_\omega(b)=0$, 
written $\eta(t)=\dot\xi(t)$, and defined the 
remainder term ${\cal R}_{\xi,\omega}$ to be 
\beq\label{flarem}
{\cal{}R}_{\xi,\omega}\equaldef\int_a^bR_{\xi,\omega}(t)dt+
\int_a^bS_\xi(t)dt\,.
\eeq

Now $t\to\eta(t)$ is essentially an arbitrary function, except only for
the fact that it has to satisfy the integral constraint $\int_a^b\eta(t)dt=0$.
Ignoring this fact temporarily---and ignoring also the presence of the
remainder term ${\cal R}_{\xi,\omega}$, and the fact that
$\psi_\omega$ depends on $\omega$---the above
formal calculation has achieved the ``decoupling'' of the individual
minimization problems at various times $t$.  If we let
$h(t,u)=H(\xi_*(t),u,\psi(t),t)$, 
then (\ref{estihamo}) essentially implies that 

\vuppp
$$
\int_a^bh(t,\eta(t))dt\le \int_a^bh(t,\eta_*(t))dt
$$

\vupp\noindent
for all functions $\eta(\cdot)$, from which it should follow that
$h(t,\eta_*(t))=\max\{h(t,u):u\in\rr^n\}$ for a.~e. $t$,
which is essentially the Hamiltonian maximization condition. 

We now make the above argument rigorous, for which we must take
into account the constraint $\int_a^b\eta(t)dt=0$ and deal with the
remainder term ${\cal R}_{\xi,\omega}$ and the dependence of $\psi$ on 
$\omega$. For this purpose, we will make combined needle 
variations, as we now explain.

We take pairs $({\bf u},{\bf t})$ of 
$m$-tuples ${\bf u}=(u_1,\ldots,u_m)$, ${\bf
t}=(t_1,\ldots,t_m)$ such that the $u_i$ belong to $U$ and the $t_i$ are
points in $[a,b\,[\,$.  We let $v_i=u_i-\eta_*(t_i)$, and use 
${\cal V}$ to denote the convex hull of the vectors $v_i$.  We call the
pair $({\bf u},{\bf t})$ {\sl normal} if $0$ is an interior point of ${\cal
V}$, and we call $({\bf u},{\bf t})$ {\sl abnormal} if it is not normal.
We consider the normal and abnormal cases separately.

Suppose first that $({\bf u},{\bf t})$ is abnormal.  This implies that
there exists a nonzero vector $\bar p\in\rr^n$ such that 
$\bar p\cdot(u_i-\eta_*(t_i))\le 0$ for $i=1,\ldots,m$. Define 
a function $\psi:[a,b]\to\rr^n$ by letting  $\psi(t)\equiv\bar p$. 
Let $\psi_0=0$. Then $\psi$ trivially satisfies the differential inclusion
\beq\label{inclu}
\dot\psi(t)\in\psi_0 W(t)\,\,\,{\rm for\,\,\,almost\,\,\,all}\,\,\,t\in[a,b]\,.
\eeq
Moreover, if we define $H^{s}(q,u,p,t)=p.u-sL(q,u,t)$, we see that

\vspace{0.7mm}
$H^{\psi_0}(\xi_*(t_i),u_i,\psi(t_i),t_i)
\!\le\! H^{\psi_0}(\xi_*(t_i),\eta_*(t_i),\psi(t_i),t_i)$
\vspace{-4mm}
\beq\label{maxi}
{\rm for}\,\,\,i=1,\ldots,m\,.\phantom{aaaaaaaaaaaaaaaaaaaaa}	
\eeq

We now assume that $({\bf u},{\bf t})$ is normal, and seek to prove the
existence of a nontrivial pair $(\psi,\psi_0)$ such that (\ref{inclu}) and
(\ref{maxi}) hold.  In this case, we set $\psi_0=1$, so $H^{\psi_0}$ is the
normal Hamiltonian $H$ defined earlier in (\ref{defnorham}).

We define


\vuppppp
$$
\Sigma^m\equaldef\Big\{(r_1,\ldots,r_m)\in[0,1]^m:r_1+\ldots+r_m=1\Big\}\,,
$$

\vupp\noindent
and use $\Sigma_0^m$ to denote the subset of $\Sigma^m$ 
consisting of all those ${\bf r}\in\Sigma^m$ 
such that ${\bf r}\cdot{\bf v}=0$, where, for 
${\bf r}=(r_1,\ldots,r_m)$, we write 
${\bf r}\cdot{\bf v}\equaldef\sum_{i=1}^mr_iv_i$.
Then $\Sigma_0^m$ is a---possibly empty---compact convex subset 
of $\Sigma^m$. 

We now let

\vuppppp
\beqn
z_i^\omega&=&H(\xi_*(t_i),\eta_*(t_i),\psi_\omega(t_i),t_i)\nonumber\\
&\,\,&\,-H(\xi_*(t_i),u_i,\psi_\omega(t_i),t_i)\,,\nonumber\\
{\bf r}.{\bf z}^\omega&=&\sum_{i=1}^mr_iz_i^\omega\,,\nonumber
\eeqn

\vuppp\noindent
and distinguish two cases, namely,
\bi
\item[(A)] {\sl There exists $\omega\in\Gamma(W)$ such that 
${\bf r}.{\bf z}^\omega\ge 0$ for all ${\bf r}\in\Sigma_0^m$.} 
\vup
\item[(B)] {\sl For every $\omega\in\Gamma(W)$ 
there exists an ${\bf r}\in\Sigma_0^m$ such  that} ${\bf r}.{\bf z}^\omega<0$.
\ei
Suppose (A) holds, and fix an $\omega$ that satisfies the conditions of 
(A). Let $C_0$ denote the set of vectors
$\{(v_i,z_i^\omega):i=1,\ldots,m\}$, so $C_0$ is a finite subset
of $\rr^{n+1}$. Let $C$ be the convex hull of $C_0$, so 
$C$ is a compact convex subset of $\rr^{n+1}$. 
Regarding $\rr^{n+1}$ as the product $\rr^n\times\rr$, let 
$D=\{0\}\times\,]\!-\!\infty,0\,[\,$.  Then $C\cap D=\emptyset$. 
So there exists a nonzero vector $(\bar p,p_0)\in\rr^{n+1}$ 
such that $\bar p\cdot v+p_0z\ge 0$ for all $(v,z)\in C$, and
$\bar p\cdot v+p_0z\le 0$ for all $(v,z)\in D$. Since $(0,-1)\in D$, 
$p_0$ must be nonnegative.  If $p_0=0$, it would follow that
$\bar p\neq 0$, and $\bar p\cdot v_i\ge 0$ for $i=1,\ldots,m$, 
implying that $({\bf u},{\bf t})$ is abnormal, which is a contradiction. 
So $p_0\neq 0$, and then $p_0>0$, so we can clearly assume
that $p_0=1$.  We then have 

\vupp
\beq
\bar p\cdot(u_i-\eta_*(t_i))=\bar p\cdot v_i\ge-z_i\,,\nonumber
\eeq

\vuppp\noindent and


\vuppppp
\beqn
z_i^\omega&=&\psi_\omega(t_i)\cdot(\eta_*(t_i)-u_i)\nonumber\\
&\,\,&\,-(L(\xi_*(t_i),\eta_*(t_i),t_i)-L(\xi_*(t_i),u_i,t_i))\,,\nonumber
\eeqn

\vuppp\noindent so 

\vuppppp
\beq
\left.\begin{array}{l}\Big(\psi_\omega(t_i)-\bar p\Big)\cdot(u_i-\eta_*(t_i))\\
\,\,\le L(\xi_*(t_i),u_i,t_i)-L(\xi_*(t_i),\eta_*(t_i),t_i)\,,
\end{array}\right.\nonumber
\eeq

\vupp\noindent
and then, if we let $\psi(t)=\psi_\omega(t)-\bar p$, we find

\vuppp
$$
\psi(t_i)\cdot(u_i\!-\!\eta_*(t_i))
\le L(\xi_*(t_i),u_i,t_i)\!-\!L(\xi_*(t_i),\eta_*(t_i),t_i)\,,
$$

\vupp\noindent that is, 

\vuppppp
\beq
\left.\begin{array}{l}
\psi(t_i)\cdot u_i-L(\xi_*(t_i),u_i,t_i)\\
\,\,\le\psi(t_i)\cdot\eta_*(t_i)-L(\xi_*(t_i),\eta_*(t_i),t_i)
\,,\end{array}\right.\nonumber
\eeq

\vupp\noindent so 
$H(\xi_*(t_i),u_i,\psi(t_i),t_i)\le 
H(\xi_*(t_i),\eta_*(t_i),\psi(t_i),t_i)$. 

Since $\psi_0=1$, the map $\psi$ satisfies (\ref{maxi}). Moreover, since
$\dot\psi(t)=\omega(t)$ a.~e., it is clear that $\psi$ satisfies
(\ref{inclu}) as well. 

So in case (A) we have proved the existence of a nontrivial 
$(\psi,\psi_0)$ for which $\psi_0\ge 0$ and (\ref{inclu}) and (\ref{maxi})
hold.     

We must now consider the remaining possibility, namely, case (B).  
In this case, we will construct a ``variation'' of $\xi_*$ 
that will lead to a trajectory $\xi$ for which $I(\xi)<I(\xi_*)$,
thereby contradicting the assumption that $\xi_*$ solves our minimization
problem. 

Assumptions (A5), (A6), and (A7), together with the Scorza-Dragoni Theorem,
imply that there exist pairwise disjoint subsets $J_\nu({\bf u})$ of
$[a,b]$, for $\nu=0,1,2,\ldots\,$, such that
\bi
\item[(i)] for $\nu>0$, the set $J_\nu({\bf u})$ is compact and the function 
\ei

\vuppppp\vuppp
$$
{\cal T}(\xi_*,\bar\delta_3)\cap(Q\times J_\nu({\bf u}))\ni(q,t)
\to\Lambda(q,t)\in\rr^{1+m+n}
$$

\vupppp\vup
\bi
\item[] is continuous, where we use $\Lambda(q,t)$ to denote the 
vector
\ei

$(L(q,\eta_*(t),t),L(q,u_1,t),\ldots,L(q,u_m,t),\eta_*(t))$;

\bi
\item[(ii)] $J_0({\bf u})$ has measure zero;  
\vup
\item[(iii)] $[a,b]=\cup_{\nu=0}^\infty J_\nu({\bf u})$. 
\ei

\vupp\noindent
For $\nu=1,2,\ldots\,$,  we let $\tilde J_\nu({\bf u})=\cup_{k=1}^\nu J_k({\bf
u})$, and observe that $\Lambda$ is continuous on 
${\cal T}(\xi_*,\bar\delta_3)\cap(Q\times\tilde J_\nu({\bf u}))$, because
the $J_k({\bf u})$, for $k=1,\ldots,\nu$, are compact and pairwise
disjoint. We let $G_{\nu}({\bf u})$ denote the
set of all $t\in[a,b]$ such that $t$ is a point of density of 
$\tilde J_\nu({\bf u})$ and a Lebesgue point of $\eta_*$.  

\brem
{\rm Recall that a  {\sl Lebesgue point} of a scalar or vector-valued function 
$\mu$ on an interval $[a,b]$ is a point $t$ such that $a<t<b$ and 
$\lim_{h\downarrow 0}h^{-1}\int_{t-h}^{t+h}|\mu(s)-\mu(t)|ds=0$. 

A {\sl point of density} of a measurable subset $E$ of $[a,b]$ is a 
point $t\in E$ which is a Lebesgue point of the indicator function
$\chi_{_E}$.  Equivalently, $t$ is a point of density of $E$ 
iff $t\in E$, $a<t<b$, and 
$\lim_{h\downarrow 0}h^{-1}{\rm meas}(E\cap[t-h,t+h])0$. 
It is well known that if $\mu$ is integrable then almost every point 
of $[a,b]$ is a Lebesgue point of $\mu$.  This implies, in particular, 
that if $E$ is measurable then almost all points of $E$ are points of 
density.} \proofend
\erem 

We let $G({\bf u})=\cup_{\nu=1}^\infty G_{\nu}({\bf u})$, and assume that 
\bi\bi
\item[(C2)] {\sl $t_1,\ldots,t_m$ belong to $G({\bf u})$}. 
\ei\ei
It then follows that there exists a $\nu$ such that 
\bi\bi
\item[(C3)] {\sl $t_1,\ldots,t_m$ belong to $G_{\nu}({\bf u})$}. 
\ei\ei
From now on, we fix a $\nu$ for which (C3) holds, and 
a positive $\bar\eps_1$ such that $t_i+\bar\eps_1<b$ whenever
$i=1,\ldots,m$, and
$[t_i,t_i+\bar\eps_1]\cap[t_j,t_j+\bar\eps_1]=\emptyset$ 
whenever $t_i\neq t_j$. 
   

For $\eps\in\,]\,0,\bar\eps_1]$, and ${\bf r}=(r_1,\ldots,r_m)\in\Sigma^m$, 
we define disjoint measurable subsets $I_1({\bf r},\eps)$,
$\ldots$, $I_m({\bf r},\eps)$ of the interval $[a,b]$
by letting $I_i({\bf r},\eps)$ be the intersection of the interval
$[t_i+(r_1+\ldots+r_{i-1})\eps,t_i+(r_1+\ldots+r_{i})\eps\,[\,$ with the set 
$G_{\nu}({\bf u})$. We emphasize that
these sets are disjoint even if the $t_i$ are not distinct. 

We let 
\beqn
\theta_i(r,\eps)&=&r_i-\frac{{\rm meas}(I_i({\bf r},\eps))}{\eps}\,,\nonumber\\
\theta(\eps)&=&\max\Big\{\theta_i({\bf r},\eps):{\bf r}\in\Sigma^m,
i=1,\ldots,m\Big\}\,,\nonumber\\
\tilde\theta(\eps)&=&\frac{1}{\eps}\sum_i\int_{t_i}^{t_i+\eps}
\|\eta_*(t)-\eta_*(t_i))\|dt\,,\nonumber\\
\kappa_1&=&\max\{\|v_i||:i=1,\ldots,m\}\,,\nonumber\\
\Theta(\eps)&=&\theta(\eps)+\kappa_1\tilde\theta(\eps)\,.\nonumber
\eeqn
and observe that $\Theta(\eps)\to 0$ as $\eps\downarrow 0$. 

We then define, for every pair $({\bf r},\eps)\in\Sigma^m\times\,]\,0,
\bar\eps_1]$,
a function $\eta^{{\bf r},\eps}:[a,b]\mapsto U$, by letting
$$
\eta^{{\bf r},\eps}(t)=\left\{\begin{array}{lcl}
\eta_*(t)&{\rm if}&t\notin\cup_{i=1}^mI_i({\bf r}.\eps),\\
u_i&{\rm if}&t\in I_i({\bf r},\eps),\,\,i=1,\ldots,m\,.\end{array}\right.
$$  
It is clear that each $\eta^{{\bf r},\eps}$ is integrable. Define 
$$
\xi^{{\bf r},\eps}(t)=\bar q+\int_a^t\eta^{{\bf r},\eps}(s)ds\,,
$$
so the $\xi^{{\bf r},\eps}$ are absolutely continuous. Clearly, 
\beqn
\int_a^b\eta^{{\bf r},\eps}(t)dt
&=&\int_a^b\eta_*(t)dt\!+\!\sum_i\int_{I_i({\bf r},\eps)}(u_i\!-\!\eta_*(t))dt
\nonumber\\
&=&\hat q-\bar q+\eps
\sum_ir_i(1-\theta_i({\bf r},\eps))\cdot v_i\nonumber\\
&\,\,&\,\,\,\,-\sum_i\int_{I_i({\bf r},\eps)}(\eta_*(t)-\eta_*(t_i))dt\nonumber\\
&=&\hat q-\bar q+\eps{\bf r}\cdot{\bf v}+\zeta^{{\bf r},\eps}\,,\nonumber
\eeqn
where 
\beq
\|\zeta^{{\bf r},\eps}\|\le\eps\Theta(\eps)\,.
\eeq
Therefore 
$\xi^{{\bf r},\eps}(b)=\hat q+\eps{\bf r}\cdot{\bf v}+\zeta^{{\bf r},\eps}$. 
So, if we choose ${\bf r}\in\Sigma_0^m$, then the map 
$\eps\to\xi^{{\bf r},\eps}$ is a ``variation of $\xi_*$ with fixed
endpoints to first order.'' We will see below how to make a small correction
to obtain an ${\bf r}(\eps)$ such that 
$\xi^{{\bf r}(\eps),\eps}(b)$ is exactly equal to $\hat q$.

If $t\in[a,b]$, and we write $\rho_i=r_1+\ldots+r_i$, then
the integral
$\int_a^t\|\eta^{{\bf r},\eps}(s)-\eta_*(s)\|\,ds$ is equal to 
$\sum_i\int_{I_i({\bf r},\eps)\cap[a,t]}\|u_i-\eta_*(s)\|\,ds$, 
which is bounded above by 
$\sum_i\int_{t_i+\rho_{i-1}\eps}^{t_i+\rho_i\eps}\|u_i-\eta_*(s)\|\,ds$. 
Since $\|u_i-\eta_*(s)\|$ is bounded by 
$\|v_i\|+\|\eta_*(t_i)-\eta_*(s)\|$, we get
\beq
\int_a^t\|\eta^{{\bf r},\eps}(s)-\eta_*(s)\|\,ds
\le\eps(\kappa_1+\tilde\theta(\eps))\,.
\eeq
We now choose $\bar\eps_2\in\,]\,0,\bar\eps_1]$ such that 
$\tilde\theta(\eps)\le 1$ for all $\eps\in\,]\,0,\bar\eps_2]$, and 
$(\kappa_1+1)\bar\eps_2\le\bar\delta_3$.

It then follows that the arcs $\xi^{{\bf r},\eps}$ satisfy 
\beq\label{fiber}
||\xi^{{\bf r},\eps}(t)-\xi_*(t)||\le(\kappa_1+1)\eps\,\,\,{\rm for\,\,\,all}\,\,\,t\in[a,b]\,,
\eeq
whenever $\eps\in\,]\,0,\bar\eps_2]$.   From now on, we choose 
\mbox{$\delta=\delta(\eps)=(\kappa_1+1)\eps$}, so that
\mbox{$\xi^{{\bf r},\eps}\in\aquld$}---and, in particular, 
\mbox{$\xi^{{\bf r},\eps}$ belongs to 
${\cal A}^{\bar\delta_3}(Q,U,L,a,b)$}---whenever 
${\bf r}\in\Sigma^m$ and $0<\eps\le\bar\eps_2$. 

We now estimate the difference $I(\xi^{{\bf r},\eps})-I(\xi_*)$ using 
(\ref{estihamo}), with $\xi^{{\bf r},\eps}$ in the role of $\xi$. We fix an
$\omega\in\Gamma(W)$. Write 
$\alpha(u,t)=H(\xi_*(t),u,\psi_{\omega}(t),t)$, and then define
$\beta_i(t)=\alpha(u_i,t)-\alpha(\eta_*(t),t)$. Then 
\beqn
&\,\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int_a^b\Big(\alpha(\eta^{{\bf r},\eps}(t),t)-
\alpha(\eta_*(t),t)\Big)\,dt\nonumber\\
&\,\,&\,\,=\sum_i\int_{I_i({\bf r},\eps)}\Big(
\alpha(u_i,t)-\alpha(\eta_*(t),t)\Big)\,dt\nonumber\\
&\,\,&\,\,=\,-\eps\sum_ir_i(1-\theta_i({\bf r},\eps))z_i^\omega+{\cal E}_1(\eps)\nonumber\\
&\,\,&\,\,=\,-\eps{\bf r}\cdot{\bf z}^\omega+{\cal E}_2(\eps)\nonumber
\eeqn
where the error terms ${\cal E}_{\ell}(\eps)$ are given by
\beqn
{\cal E}_2(\eps)&=&{\cal E}_1(\eps)+
\eps\sum_ir_i\theta_i({\bf r},\eps))z_i^\omega\,,\nonumber\\
{\cal E}_1(\eps)&=&\sum_i\int_{I_i({\bf r},\eps)}\Big(
\beta_i(t)-\beta_i(t_i)\Big)\,dt\,.\nonumber
\eeqn
It then follows from (\ref{estihamo}) that 
\beq\label{estidif}
I(\xi^{{\bf r},\eps})-I(\xi_*)=
\eps{\bf r}\cdot{\bf z}^\omega-{\cal E}_2(\eps)+
{\cal R}_{\xi^{{\bf r},\eps},\omega}\,.
\eeq
We now estimate the remainder terms of  (\ref{estidif}). Clearly, 
$\alpha(u,t)\!=\!\psi_\omega(t).u\!-\!L(\xi_*(t),u,t)$.   Let
\mbox{$\Lambda_i(t)\!=\!L(\xi_*(t),u_i,t)$} for $i>0$, 
$\Lambda_0(t)=L(\xi_*(t),\eta_*(t),t)$. For $\eps>0$, define 
$\sigma_1(\eps)$ to be the supremum of the numbers 
$|\Lambda_i(t)-\Lambda_i(s)|$, ranging over all 
$i\in\{0,\ldots,m\}$ and all $t,s\in\tilde J_\nu({\bf u})$ such that 
$|t-s|\le\eps$.
Also, let $\sigma_2(\eps)$ be the supremum of the integrals
$\int_t^{t+\eps}k(s)\,ds$, for $a\le t\le b-\eps$.
Let $\sigma_3(\eps)=\sup\{\|\eta_*(t)-\eta_*(s)\|:
t,s\in\tilde J_\nu({\bf u}),|t-s|\le\eps\}$.  Then 
Then, for $\ell=1,2,3$, 
$\sigma_\ell(\eps)\to 0$ as $\eps\downarrow 0$, because 
the  $\Lambda_i$ and $\eta_*$ are continuous on $\tilde J_\nu({\bf u})$,
and $k$ is integrable. Moreover, if $t\in I_i({\bf r},\eps)$, then 
\beqn
&\,\,&\beta_i(t)-\beta_i(t_i)=\Lambda_0(t)-\Lambda_0(t_i)-(
\Lambda_i(t)-\Lambda_i(t_i))\nonumber\\
&\,\,&\,\,+(\psi_\omega(t)-\psi_\omega(t_i))\cdot v_i
+\psi_\omega(t)\cdot(\eta_*(t)-\eta_*(t_i))\,.\nonumber
\eeqn
so $|\beta_i(t)-\beta_i(t_i)|\le\sigma_4(\eps)$, where
\mbox{$\kappa_2=\int_a^bk(t)\,dt$,} and
\mbox{$\sigma_4(\eps)=2\sigma_1(\eps)+\kappa_1\sigma_2(\eps) 
+\kappa_2\sigma_3(\eps)$}.

It then follows that $\|{\cal E}_1(\eps)\|\le\eps\sigma_4(\eps)$. 

On the other hand,
\mbox{$z_i^\omega\!=\!-\!\psi_\omega(t_i)\cdot v_i\!+\!\Lambda_i(t_i)
\!-\!\Lambda_0(t_i)$}. So, if we let
\mbox{$\kappa_3=\max\{\Lambda_i(t_i)-\Lambda_0(t_i):i=1,\ldots,m\}$},
we get the bound \mbox{$|z_i^\omega|\le\kappa_4$}, where 
$\kappa_4=\kappa_1\kappa_2+\kappa_3$. This implies the estimate
$\|{\cal E}_2(\eps)\|\le\eps\sigma_5(\eps)$, where 
we let $\sigma_5(\eps)=\sigma_4(\eps)+\kappa_4\theta(\eps)$. 

We now get a bound for ${\cal{}R}_{\xi^{{\bf r},\eps},\omega}$, using
(\ref{flarem}).  First, it is clear that $S_{\xi^{{\bf r},\eps}}(t)=0$
unless $t$ belongs to $\cup_{i=1}^mI_i({\bf r},\eps)$. Let $\sigma_6(\eps)$
be the supremum of the numbers
$|L(q,\eta_i(t),t)-L(\xi_*(t),\eta_i(t),t)|$, for $i=0,1,\ldots,m$,
$t\in\tilde J_\nu({\bf u})$, and $\|q-\xi_*(t)\|\le(\kappa_1+1)\eps$, where
$\eta_i(t)=u_i$ for $i=1,\ldots,m$, and $\eta_0(t)=\eta_*(t)$.  Then the
bound $\int_a^b\|S_{\xi^{{\bf r},\eps}}(t)\|dt\le\eps\sigma_6(\eps)$ holds
and, moreover, $\sigma_6(\eps)$ goes to $0$ as $\eps\downarrow 0$, because the
$m+1$ functions \mbox{$(q,t)\to L(q,\eta_i(t),t)$} are continuous on ${\cal
T}(\xi_*,\bar\delta_3)$.  So 
\beq\label{estidifo}
I(\xi^{{\bf r},\eps})-I(\xi_*)=
\eps{\bf r}\cdot{\bf z}^\omega+\int_a^bR_{\xi^{{\bf r},\eps},\omega}(t)dt
+{\cal E}_3(\eps)\,,
\eeq
where ${\cal E}_3(\eps)=-{\cal E}_2(\eps)+\int_a^bS_{\xi^{{\bf
r},\eps}}(t)dt$, and then 
$\|{\cal E}_3(\eps)\|\le\eps\sigma_7(\eps)$, where 
$\sigma_7(\eps)=\sigma_5(\eps)+\sigma_6(\eps)$, so 
$\sigma_7(\eps)\to 0$ as $\eps\downarrow 0$. 

Now, if we knew that $\omega\in\Gamma(W_\delta^{\xi^{{\bf r},\eps}})$, we
could apply (\ref{estirem}) and get the estimate 
\beq\label{could}
\int_a^b\|R_{\xi^{{\bf r},\eps},\omega}(t)\|dt\le\eps\sigma_8(\eps)\,,
\eeq
where $\sigma_8(\eps)=(\kappa+1)\int_a^bk_{(\kappa+1)\eps}(t)\,dt$,
so that $\sigma_8(\eps)\to 0$ as $\eps\downarrow 0$, thanks to (A8.c). 
If, in addition, we could guarantee that ${\bf r}\cdot{\bf z}^\omega<0$
then we would know, for small $\eps$, that 
$I(\xi^{{\bf r},\eps})<I(\xi_*)$. If we could achieve this for an 
${\bf r}\in\Sigma_0^m$, then 
$\xi^{{\bf r},\eps}$ would satisfy the same endpoint conditions as
$\xi_*$, up to a small error. Then, if  we could correct the error, 
we would obtain a $\xi\in\aqulep$ for which $I(\xi)<I(\xi_*)$. Finally, if
we could make sure that this $\xi$ is in ${\cal N}$, we would have obtained
the desired contradiction. 

There is, however, no obvious reason for 
$\omega$ to belong to $\Gamma(W_\delta^{\xi^{{\bf r},\eps}})$. Indeed, our
hypotheses guarantee that for every $\omega$ there is an ${\bf
r}\in\Sigma_0^m$ for which  ${\bf r}\cdot{\bf z}^\omega<0$. Then, for such
an ${\bf r}$, and a small enough $\eps$, $\Gamma(W_\delta^{\xi^{{\bf
r},\eps}})$ is nonempty, but this does not imply that 
$\omega\in\Gamma(W_\delta^{\xi^{{\bf r},\eps}})$. To obtain an $\omega$ 
and an ${\bf r}$ with the desired properties, and to correct the error 
on the terminal state, we need a fixed-point  argument. 

First, we fix a positive radius $R$ and a continuous map 
$$
v\to\pmb{\sigma}(v)=(\sigma_1(v),\ldots,\sigma_m(v))
$$ from the closed ball
$\bmath(R)\equaldef\{x\in\rr^n:\|x\|\le R\}$ to $\Sigma^m$, such that
$\sum_{i=1}^m\sigma_i(x)v_i=x$ for every $x\in\bmath(R)$. (The
existence of $R$ and $\pmb{\sigma}$ follows easily from the normality
assumption.)

Next, for each $\omega\in\Gamma(W)$ we choose an $r(\omega)\in\Sigma_0^m$
such that ${\bf r}(\omega)\cdot{\bf z}^\omega<0$. Since $\Gamma(W)$ is a
compact metric space---with the distance $d$ defined by

$d(\omega_1,\omega_2)=\sup\{\|\psi_{\omega_1}(t)
-\psi_{\omega_2}(t)\|:a\le t\le b\}$, 

\noindent which gives rise to the topology of weak convergence---and the
map $\omega\to{\bf z}^\omega$ is continuous, a simple partition of unity
argument enables us to choose a continuous function
$\Gamma(W)\ni\omega\to{\bf r}(\omega)\in\Sigma_0^m$ such that ${\bf
r}(\omega)\cdot{\bf z}^\omega<0$ for all $\omega\in\Gamma(W)$. Then the
compactness of $\Gamma(W)$ implies that the scalar function
$\Gamma(W)\ni\omega\to{\bf r}(\omega)\cdot{\bf z}^\omega$ attains its
maximum value $-\bar\mu$, which is of course negative, so $\bar\mu>0$ and
${\bf r}(\omega)\cdot{\bf z}^\omega\le-\bar\mu$ for all
$\omega\in\Gamma(W)$.  We let $\kappa_3=\sup\{\|{\bf
z}^\omega\|:\omega\in\Gamma(W)\}$, write $\bar\mu=3\hat\mu$, and choose a 
number $s$ such that $0<3s<1$ and $\kappa_3s\le\hat\mu$.  We then define
$$
\hat{\bf r}(\omega,x)=(1-s){\bf r}(\omega)+s\pmb{\sigma}(x)\,,\,\,
{\rm for}\,\,\omega\in\Gamma(W),\,x\in\bmath(R)\,.
$$
Then $\hat{\bf r}(\omega,x)\cdot{\bf z}^\omega\le-\hat\mu$ for all
$(\omega,x)\in\Gamma(W)\times\bmath(R)$.

We now fix an $\eps\in\,]\,0,\bar\eps_2]$, and let
$\delta=(\kappa_1+1)\eps$.  It is then clear that 
the map $\Sigma^m\ni{\bf r}\to\eta^{{\bf r},\eps}\in L^1([a,b],\rr^n)$ 
is continuous. So the map 
$$
\Gamma(W)\times\bmath(R)\ni(\omega,x)\to\eta^{\hat{\bf r}(\omega,x),\eps}
\in L^1([a,b],\rr^n)
$$
is continuous. Since $\delta\le\bar\delta_3$, and the arc 
$\xi^{\hat{\bf r}(\omega,x),\eps}$ belongs to ${\cal
A}^{\delta}(Q,U,L,a,b)$ for every $(\omega,x)$, the set 
${\cal U}_\eps(\omega,x)\equaldef\Gamma\Big(W_\delta^{\xi^{\hat{\bf
r}(\omega,x),\eps}})$ is a weakly compact, convex, nonempty subset 
of $L^1([a,b],\rr^n)$.  Moreover, it is easy to show that the set-valued 
map ${\cal U}_{\eps}$ is upper semicontinuous. (Indeed, suppose 
$\omega_j\to\omega$, $x_j\to x$, $\zeta_j\in{\cal U}_\eps(\omega_j,x_j)$, and
$\zeta_j\to\zeta$. Then, for each $j$, the inequality
\beqn
&\,&L(\xi^{\hat{\bf r}(\omega_j,x_j),\eps}(t),\eta_*(t),t)
\le L(\xi_*(t),\eta_*(t),t)\nonumber\\
&\,&\,\,\,+\zeta_j(t)\cdot(\xi^{\hat{\bf
r}(\omega_j,x),\eps}(t)-\xi_*(t))+\delta k_\delta(t)\nonumber 
\eeqn
holds for almost every $t$, so we can pick a subset $E_0$ of
$[a,b]$ of measure $b-a$ such that the inequality holds for all $t\in E_0$. 
Since $q\to L(q,\eta_*(t),t)$ is continuous, and the arcs 
$\xi^{\hat{\bf r}(\omega_j,x_j),\eps}$ converge uniformly to 
$\xi^{\hat{\bf r}(\omega,x),\eps}$, we find 
\beq\label{estilam}
L(\xi^{\hat{\bf r}(\omega,x),\eps}(t),\eta_*(t),t)
\le L(\xi_*(t),\eta_*(t),t)+\lambda(t)
\eeq
for all $t\in E_0$, where 
$$
\lambda(t)=\delta k_\delta(t)+\lim\inf_{j\to\infty}\Big(\zeta_j(t)
\cdot(\xi^{\hat{\bf r}(\omega_j,x_j),\eps}(t)-\xi_*(t))\Big)\,.
$$
Now, for each $t\in E_0$, the sequence $\{\zeta_j(t)\}_{j=1}^\infty$ is
bounded, and the vectors $\xi^{\hat{\bf r}(\omega_j,x_j),\eps}(t)-\xi_*(t)$
converge to $\xi^{\hat{\bf r}(\omega,x),\eps}(t)-\xi_*(t)$. Therefore, 
$$
\lambda(t)=\delta k_\delta(t)+\lim\inf_{j\to\infty}\Big(\zeta_j(t)
\cdot(\xi^{\hat{\bf r}(\omega,x),\eps}(t)-\xi_*(t))\Big)\,.
$$
Let $C_j$ denote the closed convex hull in $L^1$ of the functions
$\zeta_k$ for $k\ge j$. Then $C_j$ is weakly closed, so 
$\zeta\in C_j$.  So, for each $j$, 
$\zeta$ is a limit in $L^1$, as $k\to\infty$, of convex 
combinations $\tilde\zeta_{j,k}=\sum_{\ell=j}^{j+A(j,k)}
\alpha_{j,k,\ell}\zeta_\ell$. By passing to a subsequence of 
$\{\tilde\zeta_{j,k}\}_{k=1}^\infty$, we may assume that 
$\tilde\zeta_{j,k}(t)\to\zeta(t)$ as $k\to\infty$ for almost all $t$,
i.e. that there is a subset $E_j$ of $[a,b]$ of measure $b-a$ such that 
$\tilde\zeta_{j,k}(t)\to\zeta(t)$ as $k\to\infty$ for all $t\in E_j$. Let 
\mbox{$E=\cap_{j=0}^\infty$ $E_j$}.   If $t\in E$, and 
$\alpha>0$, then  there is a $j$ such that 
$$
\zeta_\ell(t)\cdot(\xi^{\hat{\bf r}(\omega,x),\eps}(t)-\xi_*(t))
\ge\lambda(t)-\delta k_\delta(t)-\alpha
$$
whenever $\ell\ge j$. It then follows that 
$$
\tilde\zeta_{j,k}(t)\cdot(\xi^{\hat{\bf r}(\omega,x),\eps}(t)-\xi_*(t))
\ge\lambda(t)-\delta k_\delta(t)-\alpha
$$
for all $k$, and then 
$$
\zeta(t)\cdot(\xi^{\hat{\bf r}(\omega,x),\eps}(t)-\xi_*(t))\Big)
\ge\lambda(t)-\delta k_\delta(t)-\alpha\,.
$$
Since $\alpha$ was arbitrary, we have shown that 
$$
\zeta(t)\cdot(\xi^{\hat{\bf r}(\omega,x),\eps}(t)-\xi_*(t))
\ge\lambda(t)-\delta k_\delta(t)\,.
$$
It then follows from (\ref{estilam}) that
\beqn
&\,&L(\xi^{\hat{\bf r}(\omega,x),\eps}(t),\eta_*(t),t)\le L(\xi_*(t),\eta_*(t),t)
\nonumber\\
&\,&\,\,+\zeta(t)\cdot(\xi^{{\bf r}(\omega),\eps}(t)-\xi_*(t))
+\delta k_\delta(t)\nonumber
\eeqn
for all $t\in E$. Since $E$ has full measure, and $\zeta$ clearly belongs
to $\Gamma(W)$, we conclude that $\zeta\in
{\cal U}_\eps(\omega,x)$, as desired.)

We now impose the extra condition that $\Theta(\eps)\le Rs$, and 
observe that
\beqn
\xi^{\hat{\bf r}(\omega,x),\eps}(b)&=&\hat q+\eps\hat{\bf
r}(\omega,x)\cdot{\bf v} 
+\zeta^{\hat{\bf r}(\omega,x),\eps}\nonumber\\
&=&\hat q+(1\!-\!s)\eps{\bf r}(\omega)\!+\!\eps s\pmb{\sigma}(x)\cdot{\bf v}
\!+\!\zeta^{\hat{\bf r}(\omega,x),\eps}\nonumber\\
&=&\hat q+\eps s\pmb{\sigma}(x)\cdot{\bf v}
+\zeta^{\hat{\bf r}(\omega,x),\eps}\nonumber\\
&=&\hat q+\eps sx+\zeta^{\hat{\bf r}(\omega,x),\eps}\,,\nonumber
\eeqn
so
$$
\Biggl\|
\frac{1}{\eps s}\Big(\xi^{\hat{\bf r}(\omega,x),\eps}(b)-\hat q\Big)-x
\Biggr\|\le 
\frac{\|\zeta^{\hat{\bf r}(\omega,x),\eps}\|}{\eps s}
\le\frac{\Theta(\eps)}{s}\le R\,.
$$
Therefore, if we define 
$$
{\cal V}_\eps(\omega,x)={\cal U}_\eps(\omega,x)\times\Biggl\{
x-\frac{1}{\eps s}\Big(\xi^{\hat{\bf r}(\omega,x),\eps}(b)-\hat
q\Big)\Biggr\}\,,
$$
we see that ${\cal V}_\eps$ is an upper semicontinuous set-valued 
map on $\Gamma(W)\times\bmath(R)$ whose values are nonempty compact
convex subsets of $\Gamma(W)\times\bmath(R)$. So we can
apply the infinite-dimensional version of the Kakutani fixed-point 
theorem and conclude that there exists a pair $(\omega_\eps,x_\eps)$ such
that  $\omega_\eps\in{\cal U}_\eps(\omega_\eps,x_\eps)$ and 
$x_\eps-\frac{1}{\eps s}\Big(\xi^{\hat{\bf r}(\omega_\eps,x_\eps),\eps}(b)-
\hat q\Big)=x_\eps$. It then follows that
$\xi^{\hat{\bf r}(\omega_\eps,x_\eps),\eps}(b)=\hat q$.

Since $\omega_\eps\in\Gamma\Big(W_\delta
^{\xi^{\hat{\bf r}(\omega_\eps,x_\eps),\eps}}\Big)$, 
we can apply  (\ref{estidifo}) and  (\ref{could})
with ${\bf r}=\hat{\bf r}(\omega_\eps,x_\eps)$ and 
$\omega=\omega_\eps$, and obtain
\beqn
I(\xi^{\hat{\bf r}(\omega_\eps,x_\eps),\eps})-I(\xi_*)
&\le&\eps\hat{\bf r}(\omega_\eps,x_\eps)\cdot{\bf
z}^{\omega_\eps}+\eps\sigma_9(\eps)\nonumber\\
&\le&\eps(\sigma_9(\eps)-\hat\mu)\,,\label{estidifogood}
\eeqn
where $\sigma_9=\sigma_7+\sigma_8$.  

It now suffices to take $\eps$ such that $\sigma_9(\eps)<\hat\mu$, and
we get 
\beq
I(\xi^{\hat{\bf r}(\omega_\eps,x_\eps),\eps})<I(\xi_*)\,.
\eeq
Since $I(\xi^{\hat{\bf r}(\omega_\eps,x_\eps),\eps})\in\aqulep$, 
we have contradicted the assumption that $\xi_*$ was a solution of our 
minimization problem, provided that 
$I(\xi^{\hat{\bf r}(\omega_\eps,x_\eps),\eps})\in{\cal N}$. To guarantee
this, we assume
\bi\bi
\item[(A9)]  {\sl ${\cal N}$ is a variational neighborhood of
$\eta_*$}.
\ei\ei
Condition (A9) means that for every finite subset $U_0$ of 
$U$ there exists an $\eps>0$ such that, if $\eta:[a,b]\to U$
is such that $\eta(t)\in\{\eta_*(t)\}\cup U_0$ for every 
$t\in[a,b]$, $\{t:\eta(t)=u\}$ is measurable for every $u\in U_0$, and 
${\rm meas}\Big(\{t:\eta(t)\neq\eta_*(t)\}\Big)<\eps$, then $\eta\in{\cal
N}$. 

Under assumptions (A1)-(A9), we have shown that, if $\xi_*$ is a 
solution of our minimization problem, then for every $m$ and every
${\bf u}=(u_1,\ldots,u_m)\in U^m$ there exists a subset $G({\bf u})$ of
$[a,b]$ of full measure, such that for every $m$-tuple ${\bf
t}=(t_1,\ldots,t_m)$ of members of $G({\bf u})$ there is a 
nontrivial pair $(\psi,\psi_0)$ that satisfies (\ref{inclu}) and
(\ref{maxi}). Moreover, it is clear from the defintion of 
$G({\bf u})$ that $G({\bf u})$ only depends on the set 
$V=\{u_1,\ldots,u_m\}$, so we can define 
$G(V)$ to be equal to $G({\bf u})$ is $V$ is any finite subset of $U$ 
and $V=\{u_1,\ldots,u_m\}$. Then what we have shown implies that,
if $V$ is any finite subset of $U$, and $T$ is a finite subset of 
$G(V)$, there is a nontrivial pair $(\psi,\psi_0)$ that satisfies
(\ref{inclu}), $\psi_0\ge 0$, and 
\beq\label{maxiset}
{\cal H}_{\xi_*,\psi}^{\psi_0}(u,t)
\!\le\!{\cal H}_{\xi_*,\psi}^{\psi_0}(\eta_*(t),t)\,\,\,\,
{\rm for}\,\,\,\,t\in T\,.
\eeq
where we let ${\cal H}_{\xi_*,\psi}^{\psi_0}(u,t)\equaldef
H^{\psi_0}(\xi_*(t),u,\psi(t),t)$. Let $\Psi(V,T)$ be the set of all pairs 
$(\psi,\psi_0)$ that satisfy (\ref{inclu}), (\ref{maxiset}), $\psi_0\ge 0$,
and the normalization condition $\|\psi(b)\|+\psi_0=1$. Then, if 
$V\subseteq U$ is finite, and $T\subseteq G(V)$ is finite,  
the set $\Psi(V,T)$ is compact in $C^0([a,b],\rr^n)\times\rr$, and we 
have shown that it is nonempty.  Now, 
let ${\bf T}$ be the set of all finite subsets of $G(V)$. Then 
if $T_1\in{\bf T},\ldots,T_k\in{\bf T}$, 
$$
\Psi(V,T_1)\cap\ldots\cap\Psi(V,T_k)=
\Psi(V,T_1\cup\ldots\cup T_k)\neq\emptyset\,.
$$
So the family $\{\Psi(V,T)\}_{T\in{\bf T}}$ has the finite intersection
property. Hence $\bigcap_{T\in{\bf T}}\Psi(V,T)\neq\emptyset$. That is,
there exists, for each finite subset $V$ of $U$, a pair 
$(\psi,\psi_0)$ that satisfies (\ref{inclu}), $\psi_0\ge 0$,
the normalization condition $\|\psi(b)\|+\psi_0=1$, and 
\beq\label{maxisetall}
{\cal H}_{\xi_*,\psi}^{\psi_0}(u,t)
\!\le\!{\cal H}_{\xi_*,\psi}^{\psi_0}(\eta_*(t),t)\,\,\,\,
{\rm for}\,\,\,\,t\in G(V)\,.
\eeq
Now, let ${\bf V}$ be the set of all finite subsets of $U$. For 
each $V\in{\bf V}$, let $\hat\Psi(V)$ be the set of all pairs
$(\psi,\psi_0)$ that satisfy (\ref{inclu}), $\psi_0\ge 0$,
the normalization condition $\|\psi(b)\|+\psi_0=1$, and
\beq\label{maxisetallae}
{\cal H}_{\xi_*,\psi}^{\psi_0}(u,t)
\!\le\!{\cal H}_{\xi_*,\psi}^{\psi_0}(\eta_*(t),t)\,\,\,\,
{\rm for}\,\,\,{\rm a.e.}\,\,\,t\in[a,b]\,.
\eeq
We have shown that $\hat\Psi(V)\neq\emptyset$ for  every 
$V\in{\bf V}$. Moreover, 
$\hat\Psi(V)$ is clearly compact in $C^0([a,b],\rr^n)\times\rr$.
If $V_1\in{\bf V},\ldots,V_k\in{\bf V}$, then 
$$
\hat\Psi(V_1)\cap\ldots\cap\hat\Psi(V_k)=
\hat\Psi(V_1\cup\ldots\cup V_k)\neq\emptyset\,.
$$
So the family $\{\hat\Psi(V)\}_{V\in{\bf V}}$ has the finite intersection
property. Hence $\bigcap_{V\in{\bf V}}\hat\Psi(V)\neq\emptyset$. That is,
there exists a pair 
$(\psi,\psi_0)$ that satisfies (\ref{inclu}), $\psi_0\ge 0$,
and the normalization condition $\|\psi(b)\|+\psi_0=1$, and is such that
for every $u\in U$ the inequality 
\beq\label{maxioneu}
H^{\psi_0}(\xi_*(t),u,\psi(t),t)
\!\le\!H^{\psi_0}(\xi_*(t),\eta_*(t),\psi(t),t)
\eeq
holds for almost all $t\in[a,b]$. 

Finally, if we assume 
\bi\bi
\item[(A10)] {\sl the map $U\ni u\to L(\xi_*(t),u,t)$ is continuous for every 
$t\in[a,b]$,}
\ei\ei
then we can use the fact that $U$ is separable, and conclude that, if 
$(\psi,\psi_0)$ satisfies the above conclusions, then 
\vspace{-4mm}
\beqn
&\,\,&\max\Big\{H^{\psi_0}(\xi_*(t),u,\psi(t),t):u\in U\Big\}
\phantom{aaaaaaaaa}\nonumber\\
&\,\,&\,\,=\!H^{\psi_0}(\xi_*(t),\eta_*(t),\psi(t),t)\label{maxioneuu}
\eeqn
for almost all $t\in[a,b]$. 

So we have proved:
\bt
Assume that conditions (A1)-(A9) hold, and $(\xi_*,\eta_*)$
is a solution of (\ref{standardu}). Then there exists a 
nontrivial pair $(\psi,\psi_0)$ that satisfies (\ref{inclu}), 
$\psi_0\ge 0$, and is such that for every $u\in U$ (\ref{maxioneu})
holds for almost all $t\in[a,b]$.  If, in addition, (A10) holds, then 
(\ref{maxioneuu}) is satisfied for almost all $t\in[a,b]$. \proofend
\et



\vspace{-2mm}
\begin{thebibliography}{99}

%\bibitem{agra1}{Agrachev, A. A. and R. V. Gamkrelidze,
%``A second-order optimality principle for a time-optimal problem,''
%{\it Math. Sbornik} {\bf 100}, 142 (1976).}

%\bibitem{agra2}{Agrachev, A. A. and R. V. Gamkrelidze,
%``The exponential representation of flows and the chronological
%calculus,''  {\it Math. Sbornik} {\bf 109}, 149 (1978).}

\bibitem{berkovitz}{Berkovitz,~L.~D., {\it Optimal Control Theory},
Springer-Verlag, New York, 1974.}

%\bibitem{clarkeminimal}{Clarke,~F.~H., ``The Maximum Principle 
%under minimal hypotheses,'' {\it S.I.A.M. J. Control Optim.} {\bf 14}
%(1976), pp.~1078-1091.}

\bibitem{clarkebook}{Clarke,~ F.~H., {\it Optimization and Nonsmooth
Analysis,} Wiley Interscience, New York, 1983.}

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

\bibitem{sussloja} {Sussmann, H.J., ``A strong version of the Maximum
Principle under weak hypotheses,'' {\it Proc. 33rd IEEE Conference on
Decision and Control}, Orlando, FL, 1994, pp.~1950-1956.}

\bibitem{susspavel} {Sussmann, H.J., ``A strong version of the Lojasiewicz 
Maximum Principle,'' in {\it Optimal Control of Differential Equations,}
Nicolai H. Pavel Ed., M. Dekker Inc., New York, 1994,  pp.~293-309.}

\bibitem{sussrecent} {Sussmann, H.J.,  
``Some recent results on the maximum principle of optimal control theory.''
In {\it Systems and Control in the Twenty-First Century}, 
C.I.~Byrnes, B.N.~Datta, D.S.~Gilliam \&{} C.F.~Martin Eds.,
Birkh\"auser Boston, 1997, pp.~351-372.}

\bibitem{susskobe} {Sussmann, H.J.,  
``A strong maximum principle for systems of differential inclusions.''
In {\it Proceedings of the 35th IEEE Conference on Decision and Control},
Kobe, Japan, Dec. 1996.  IEEE publications, 1996, pp.~1809-1814.}

\bibitem{sussmulti}{Sussmann, H. J., 
``Multidifferential calculus: chain rule,  open mapping
and transversal intersection theorems,'' to appear in 
{\it Optimal  control: theory, algorithms, and applications},
W.W. Hager \&{} P.M. Pardalos Eds., Kluwer Acad. Publishers, 
1998.}

\bibitem{sussbro}{Sussmann, H. J., ``Geometry and optimal control,''
in {\it Mathematical Control Theory}, J. Baillieul and
J. C. Willems, Eds., Springer-Verlag, New York, 1998, pp.~140-198.}

\bibitem{susssanto}{Sussmann, H. J., ``Transversality conditions and a
strong maximum principle for systems of differential inclusions,'' 
in {\it Proc. 37th IEEE Conference on Decision and Control}, Tampa,
FL, Dec. 1998. IEEE publications, 1998, pp.~1-6.}

\end{thebibliography}

\end{document}
