%From piccoli@sissa.it Wed Aug 20 10:52:59 1997
%To: "Sussmann Hector J." <sussmann@hamilton.rutgers.edu>
%
%Dear Hector,
%
%in the following you will find the Plain tex file of our paper.
%In the next two messages I will send you the ps files of the two picture
%you need to tex the file (you need also the psfig.sty file, if you do not 
%have it I will send it to you).
%
%How was the trip to South America?
%
%I want to thank you for the meeting in Colorado: it was a very nice
%place and a very interesting meeting.
%
%   Regards
%  Benedetto
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\newcount\myitemno%the item number
\newcount\sectno%the section number
\newcount\subsectno%the subsection number
\newcount\theono%the theorem number
\newcount\theotno%the T theorem number
\newcount\theononosec%the theorem number without changing at sections
\newcount\lemmno%the lemma number
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%MACROS FOR SYCON PAPER
\def\thegoodheadline{{\it H.J. Sussmann\hfill\myrunninghead}\hfill{\bf\folio}}
\def\thegoodheadlinewe{
{\it Sontag--Sussmann\hfill\myrunninghead}\hfill{\bf\folio}}
\def\thegoodheadlinewk{
{\it Kokotovic--Sussmann\hfill\myrunninghead}\hfill{\bf\folio}}
\def\thegoodheadlinewkinv{
{\it Sussmann--Kokotovic\hfill\myrunninghead}\hfill{\bf\folio}}
\def\myownsyconheadline
{\ifnum\pageno>0\thegoodheadline\else \hfill\fi}
\def\myownsyconheadlinewitheduardo
{\ifnum\pageno>0\thegoodheadlinewe\else \hfill\fi}
\def\myownsyconheadlinewithkokotovic
{\ifnum\pageno>0\thegoodheadlinewk\else \hfill\fi}
\def\myownsyconheadlinewithkokotovicinv
{\ifnum\pageno>0\thegoodheadlinewkinv\else \hfill\fi}
\def\startsycona{\nopagenumbers\pageno=0
\line{\hfill Report {\sl SYCON-\theshortyear-\thesyconreportnumber}}
\vfill
\centerline{\bf \thetitleone}
\centerline{\bf \thetitletwo}
\vskip 1true cm}
\def\centermathdept{
\centerline{Department of Mathematics}
\centerline{Rutgers University}
\centerline{New Brunswick, NJ 08903}
}
\def\mynameandaddress{
\centerline{H. J. Sussmann$^{*}$}
\centermathdept
\centerline{(908)\ 445-5407}
\centerline{sussmann@hamilton.rutgers.edu\ \ {\it or}\ \ sussmann@pisces.bitnet}}
\def\mynameandaddresswithkokotovic{
\centerline{P. V. Kokotovic$^{+}$}
\centerline{Coordinated Science Laboratory}
\centerline{University of Illinois}
\centerline{1101 W. Springfield Ave.}
\centerline{Urbana, Illinois 61801}
\vskip 0.3true in
\centerline{H. J. Sussmann$^{*}$}
\centermathdept
\centerline{(908)\ 445-5407}
\centerline{sussmann@hamilton.rutgers.edu\ \ {\it or}\ \ sussmann@pisces.bitnet}}
\def\mynameandaddresswithkokotovicinv{
\centerline{H. J. Sussmann$^{*}$}
\centermathdept
\centerline{(908)\ 445-5407}
\centerline{sussmann@hamilton.rutgers.edu\ \ {\it or}\ \ sussmann@pisces.bitnet}
\vskip 0.3true in
\centerline{P. V. Kokotovic$^{+}$}
\centerline{Coordinated Science Laboratory}
\centerline{University of Illinois}
\centerline{1101 W. Springfield Ave.}
\centerline{Urbana, Illinois 61801}}
\def\mynameandaddresswitheduardo{
\centerline{E.D. Sontag$^{+}$ and H.J. Sussmann$^{*}$}
\centermathdept
\centerline{(908)\ 445-3072\ \ ---\ \ (908)\ 445-5407}
\centerline{sontag@fermat.rutgers.edu\ \ {\it or}\ \ sontag@pisces.rutgers.edu}
\centerline{sussmann@hamilton.rutgers.edu\ \ {\it or}\ \ sussmann@pisces.bitnet}}
\def\hjsacknowledgement{\noindent{\mysmall
$^{*}$\ \ This author's work was supported in part by the National
Science Foundation under NSF Grant DMS-8902994.} 
\vskip 0.05cm}
\def\hjsacknowledgementnoauthor{\noindent{\mysmall
$^{*}$\ \ This work was supported in part by the National
Science Foundation under NSF Grant DMS-8902994.} 
\vskip 0.05cm}
\def\hjsacknowledgementcaip{
\noindent{\mysmall $^{*}$\ \ This author's work was supported in part
by the National Science Foundation under Grant DMS-8902994, and by
the CAIP Center, 
Rutgers University, with funds provided by the New Jersey Commission on
Science and Technology and by CAIP's industrial members.}
\vskip 0.05cm}
\def\edsacknowledgement{
\noindent{\mysmall $^{+}$\ \ This author's work was supported in part 
by the National Science Foundation under Grant DMS-8803396, and by 
the U.S. Air Force under Grants AFOSR-85-0247 and AFOSR-88-0235}
\vskip 0.05cm}
\def\kokoacknowledgement{
\noindent{\mysmall $^{+}$\ \ This author's work was supported in part
by the National Science Foundation under  Grant ECS 87-15811 and
by the U.S. Department of Energy under Grant DE-FGD2-88 ER 13939}
\vskip 0.05cm}
\def\edsacknowledgementcaip{
\noindent{\mysmall $^{+}$\ \ This author's work was supported in part 
by the National Science Foundation under Grant DMS-8803396, by the
U.S. Air Force under Grants AFOSR-85-0247 and AFOSR-88-0235, and 
by the CAIP Center,
Rutgers University, with funds provided by the New Jersey Commission on
Science and Technology and by CAIP's industrial members.}
\vskip 0.05cm}
\def\startsyconb{
\vskip 1true cm
\centerline{\bf ABSTRACT}
\vskip 0.15true in
\noindent\theabstract
\vfill
\centerline{\sl Rutgers Center for Systems and Control,
\themonth\ \thelongyear}
\vfill}
\def\startsyconpaper{\startsycona
\mynameandaddress
\startsyconb
{\noindent {\mysmall\theextrafootnote}}
\vskip 0.05true cm
\hjsacknowledgement
\eject}
\def\startsyconpaperwitheduardo{\startsycona
\mynameandaddresswitheduardo
\startsyconb
\edsacknowledgement
\hjsacknowledgement
{\noindent {\mysmall\theextrafootnote}}
\eject}
\def\startsyconpaperwithkokotovic{\startsycona
\mynameandaddresswithkokotovic
\startsyconb
\kokoacknowledgement
\hjsacknowledgement
{\noindent {\mysmall\theextrafootnote}}
\eject}
\def\startsyconpaperwithkokotovicinv{\startsycona
\mynameandaddresswithkokotovicinv
\startsyconb
\hjsacknowledgement
\kokoacknowledgement
{\noindent {\mysmall\theextrafootnote}}
\eject}
\def\startsyconpapercaipwitheduardo{\startsycona
\mynameandaddresswitheduardo
\startsyconb
\edsacknowledgementcaip
\hjsacknowledgementcaip
{\noindent {\mysmall\theextrafootnote}}
\eject}
%MACROS FOR BOLTZMANN MACHINES PAPER
\def\rnbdw{NB\bigl(\rho,\check\cw(\cdot)\bigr)}
\def\ctrw{\ct_{\rho}\bigl(\bar\cw(\cdot)\bigr)}
\def\eps{\varepsilon}
\def\ffi{\varphi}
\def\pmath{{\rm I}\kern-.2em{\rm P}}
\def\emath{{\rm I}\kern-.2em{\rm E}}
\def\jk{{\bf J}_{k}}
\def\jkl{{\bf J}_{k}^{L}}
\def\jklr{{\bf J}_{k}^{L,r}}
\def\jkh{{\bf J}_{k}^{H}}
\def\tk{t_{k}}
\def\tkr{t_{k}^{r}}
\def\tkh{t_{k}^{H}}
\def\neqngwt{{\bf N}=({\cal N},{\cal G},\hat{\cal W},\theta)}
\def\pn{{\bf P}(\cn)}
%\def\pm{{\bf P}(\cm)}
\def\ppn{{\bf P}_{+}(\cn)}
\def\ppm{{\bf P}_{+}(\cm)}
\def\confn{I^{{\cal N}}}
\def\confh{I^{{\cal H}}}
\def\confm{I^{{\cal M}}}
\def\rrn{\rr^{\cal N}}
\def\rrg{\rr^{\cal G}}
\def\rrsg{\rr_{s}^{\cal G}}
\def\rrgn{\rr^{\cal G}\times\rr^{\cal N}}
\def\rrsgn{\rr_{s}^{\cal G}\times\rr^{\cal N}}   
\def\rrgpr{\rr^{{\cal G}'}}
\def\cg{{\cal G}}
\def\cl{{\cal L}}
\def\cn{{\cal N}}
\def\cc{{\cal C}}
\def\cm{{\cal M}}
\def\ch{{\cal H}}
\def\ck{{\cal K}}
\def\cw{{\cal W}}
\def\ct{{\cal T}}
\def\ca{{\cal A}}
\def\calat{{\cal A}_{t}}
\def\calatk{{\cal A}_{t_{k}}}
\def\calatkmin{{\cal A}_{t_{k}}^{-}}
\def\calatkr{{\cal A}_{t_{k}^{r}}}
\def\calatkh{{\cal A}_{t_{k}^{H}}}
\def\cz{{\cal Z}}
\def\cb{{\cal B}}
\def\cwbar{\bar{\cal W}}
\def\cwtil{\tilde{\cal W}}
\def\cwt{({\cal W},\theta)}
\def\cwthat{(\hat{\cal W},\theta)}
\def\gng{{\bf G}(\cn,\cg)}
\def\gngtwo{{\bf G}_{2}(\cn,\cg)}
\def\gmhg{{\bf G}(\cm,\ch,\cg)}
\def\gmhgtwo{{\bf G}_{2}(\cm,\ch,\cg)}
\def\ldata{(\bar P,\delta,\kappa,\nu,\cwtil,\tilde\tau)}
\def\rarr{\rightarrow}
\def\dcwk{\Delta\cw(k)}
\def\eijkone{E_{ij,k}^{1}}
\def\eijkthree{E_{ij,k}^{3}}
\def\eijkfour{E_{ij,k}^{4}}
\def\eijkfive{E_{ij,k}^{5}}
\def\eijkthreece{E_{ij,k}^{3,ce}}
\def\eijkfource{E_{ij,k}^{4,ce}}
\def\eijkfivece{E_{ij,k}^{5,ce}}
\def\eijkthreefl{E_{ij,k}^{3,fl}}
\def\eijkfourfl{E_{ij,k}^{4,fl}}
\def\eijkfivefl{E_{ij,k}^{5,fl}}
\def\eijkrtwo{E_{ij,k,r}^{2}}
\def\dwijk{\Delta w_{ij}(k)}
\def\dwijhk{\Delta w_{ij,H}(k)}
\def\dwijlk{\Delta w_{ij,L}(k)}
\def\dwijlrk{\Delta w_{ij,L,r}(k)}
\def\dwijlrkce{\Delta w_{ij,L,r}^{ce}(k)}
\def\dwijlkce{\Delta w_{ij,L}^{ce}(k)}
\def\dwijkce{\Delta w_{ij}^{ce}(k)}
\def\dwijlrkfl{\Delta w_{ij,L,r}^{fl}(k)}
\def\dwijlkfl{\Delta w_{ij,L}^{fl}(k)}
\def\dwijhkce{\Delta w_{ij,H}^{ce}(k)}
\def\dwijhkfl{\Delta w_{ij,H}^{fl}(k)}
\def\dwijkfl{\Delta w_{ij}^{fl}(k)}
\def\dcwkone{(\Delta\cw(k))^{1}}
\def\dwijkone{(\Delta w_{ij}(k))^{1}}
%MACROS FOR LETTER
\font\mymagfivebf=cmbx10 scaled\magstep 5
\font\mymagfiverm=cmr10 scaled\magstep 5
\font\mymagfourbf=cmbx10 scaled\magstep 4
\font\mymagfourrm=cmr10 scaled\magstep 4
\font\mymagthreebf=cmbx10 scaled\magstep 3
\font\mymagthreerm=cmr10 scaled\magstep 3
\font\mymagtwobf=cmbx10 scaled\magstep 2
\font\mymagtworm=cmr10 scaled\magstep 2
\font\mymagonebf=cmbx10 scaled\magstep 1
\font\mymagonerm=cmr10 scaled\magstep 1
\font\mymagzerobf=cmbx10 scaled\magstep 0
\font\mymagzerorm=cmr10 scaled\magstep 0
\font\mymagfivesl=cmsl10 scaled\magstep 5
\font\mymagfiveit=cmti10 scaled\magstep 5
\font\mymagfoursl=cmsl10 scaled\magstep 4
\font\mymagfourit=cmti10 scaled\magstep 4
\font\mymagthreesl=cmsl10 scaled\magstep 3
\font\mymagthreeit=cmti10 scaled\magstep 3
\font\mymagtwosl=cmsl10 scaled\magstep 2
\font\mymagtwoit=cmti10 scaled\magstep 2
\font\mymagonesl=cmsl10 scaled\magstep 1
\font\mymagoneit=cmti10 scaled\magstep 1
\font\mymagzerosl=cmsl10 scaled\magstep 0
\font\mymagzeroit=cmti10 scaled\magstep 0
\font\mynine=cmr9
\font\myninebf=cmbx9
\font\mysmall=cmr7
\font\mysmallit=cmti7
\font\myverysmall=cmr5
\def\compo{\hbox{$\,${\myverysmall o}$\,$}}
\newcount\lettermarker
%it's 0 while writing a letter and 1 for endfile and labels
\def\myownletterheadline
{\ifnum\pageno>1 \hfil\thedate\hfil P.\folio\else \hfil\fi}
\def\thephone{Phone: (908)445-5407[office]; (908)463-0667[home]}
\def\thephoneson{Phone: (908)445-3072}
\def\theemail{sussmann@hamilton.rutgers.edu\enspace---
sussmann@pisces.bitnet}
\def\theemailson{sontag@hilbert.rutgers.edu\enspace}
\def\thefax{FAX: (908)445-5530}
\def\plno#1{\hbox to 0.45true cm{\hfill #1}}
\def\pleft#1{\line{\noindent #1\hfil}}
\def\pleftskip#1{\line{\noindent #1\hfil}\vskip 0.1true cm}
\def\pleftre#1{\line{\noindent\hbox to 1.2true in{Re:\hfil}{\it #1}\hfil}}
\def\pleftrerm#1{\line{\noindent\hbox to 1.2true in{Re:\hfil}{#1}\hfil}}
\def\plefto#1{\line{\noindent\hbox to 1.2true in{\hfill}{#1}\hfil}}
\def\plright#1{\line{\noindent\hbox to 4.3true in{\hfil}#1\hfil}}
\def\plrightneural#1{\line{\noindent\hbox to 3.7true in{\hfil}#1\hfil}}
\def\pline#1{\line{\ \ \ \ \ \ #1\hfill}}
\def\pltwo#1#2{\line{\noindent\hbox to 3.8true in{#1\hfil}#2\hfil}}
\def\pltwoito#1#2{\line{\noindent\hbox to 1.8true in{#1\hfil}#2\hfil}}
\def\pltwoclose#1#2{\line{\noindent\hbox to 1.4true in{#1\hfil}#2\hfil}}
\def\toplogoa{\pageno=1 
\vglue-\voffset \vskip -1cm
\line{\noindent\mymagfivebf RUTGERS\hfill}
\line{\noindent\sl THE STATE UNIVERSITY OF NEW JERSEY\hfill}
\line{\leaders \hrule width16.5cm height10pt depth-9.6pt\hfil}}
\def\toplogob{\pageno=1 \vglue-\voffset \vskip -1cm
\centerline{\hbox to 5.2cm{\leaders\hrule height-1.2pt depth1.6pt\hfill}}
\centerline{\hbox to 5.2cm{\sl T\hfil H\hfil E\hfil\hfil\hfil\hfil\hfil
S\hfil T\hfil 
A\hfil T\hfil E\hfil\hfil\hfil\hfil\hfil\hfil U\hfil N\hfil I\hfil V\hfil
E\hfil R\hfil S\hfil I\hfil T\hfil Y}}
\vskip 0.4pt
\centerline{\underbar{\hbox to 5.2cm{\mymagfivebf  R\hfil U\hfil T\hfil G\hfil 
E\hfil R\hfil S}}}}
\def\toplogoc{\pageno=1 \vglue-\voffset \vskip -1cm
\centerline{\hbox to 5.9cm{\leaders\hrule height-1.2pt depth1.6pt\hfill}}
\vskip -0.1true cm
\centerline{\hbox to 5.9cm{\mysmall \hfill THE STATE UNIVERSITY OF NEW JERSEY
\hfill}}
\vskip 0.5pt
\centerline{{\hbox to 5.9cm{\mymagfivebf \hfil R\hfill U\hfill T\hfill G\hfill 
E\hfill R\hfill S\hfil}}}
\vskip -0.15true cm
\centerline{{\hbox to 5.9cm{\mysmall \hfill Campus at New Brunswick\hfill}}}
\vskip -0.45true cm
\centerline{\hbox to 5.9cm{\leaders\hrule height-1.2pt depth1.6pt\hfill}}}
\def\toplogom{\pageno=1 
\vglue-\voffset \vskip -1cm
\line{\noindent\mymagthreebf Mathematics of Control, Signals and Systems\hfill}
\line{\leaders \hrule width16.5cm height10pt depth-9.6pt\hfil}}
\def\nextlogoa{\vskip-9pt
\pltwo{\mysmall H. J. Sussmann}{\it Department of Mathematics}
\pltwo{\mysmall\thephone}{\it Hill Center\ -\ Busch Campus}
\pltwo{\mysmall\theemail}{\it New Brunswick, NJ 08903, USA}
\pltwo{\mysmall\thefax}{\it Telephone: (908)445-2390}
\vskip 0.2true in
\plright{\thedate}}
\def\nextlogoason{\vskip-9pt
\pltwo{\mysmall E. D. Sontag}{\it Department of Mathematics}
\pltwo{\mysmall\thephoneson}{\it Hill Center\ -\ Busch Campus}
\pltwo{\mysmall\theemailson}{\it New Brunswick, NJ 08903, USA}
\pltwo{\mysmall\thefax}{\it Telephone: (908)445-2390}
\vskip 0.2true in
\plright{\thedate}}
\def\nextlogoawithout{\vskip-9pt
\pltwo{\ \ }{\it Department of Mathematics}
\pltwo{\ \ }{\it Hill Center\ -\ Busch Campus}
\pltwo{\ \ }{\it New Brunswick, NJ 08903, USA}
\pltwo{\ \ }{\it Telephone: (908)445-2390}
\vskip 0.2true in
\plright{\thedate}}
\def\nextlogob{\vskip 0.1true cm
\centerline{\it Department of Mathematics \enspace$\bullet$\enspace Hill Center for the Mathematical Sciences} 
\centerline{\it Busch Campus\enspace$\bullet$\enspace New Brunswick, New Jersey 08903, 
USA \enspace$\bullet$\enspace Telephone: (908)445-2390}
\vskip 0.1 true cm
\centerline{\mysmall H. J. Sussmann\enspace --\enspace\thephone}
\vskip-3pt
\centerline{\mysmall Electronic Mail:\theemail}
\vskip-3pt
\centerline{\thefax}
\vskip 0.3true cm
\plright{\thedate}}
\def\nextlogom{\vskip-9pt
\pltwo{\mysmall H. J. Sussmann}{\it Department of Mathematics}
\pltwo{\mysmall\thephone}{\it Rutgers University}
\pltwo{\mysmall Electronic Mail:}{\it New Brunswick, NJ 08903, USA}
\pltwo{\mysmall\theemail}{\it Telephone: (908)445-2390}
\vskip 0.2true in
\plright{\thedate}}
\def\logoa{\toplogoa\nextlogoa}
\def\logoason{\toplogoa\nextlogoason}
\def\logoawithout{\toplogoa\nextlogoawithout}
\def\logob{\toplogob\nextlogob}
\def\logoc{\toplogoc\nextlogob}
\def\logom{\toplogom\nextlogom}
\def\sighjs{\plright{H. J. Sussmann}}
\def\sighjsneru{\pltwo{Mahesh Nerurkar}{H. J. Sussmann}}
\def\sighjsneural{\plrightneural{H. J. Sussmann}\nobreak
\plrightneural{Associate Editor, {\it Neural Networks}}}
\def\sighjswithsontag{\pltwo{E. D. Sontag}{H. J. Sussmann}}
\def\sighjswithsontagindent{\pltwo{\ \ \ \ \ \ E. D. Sontag}{H. J. Sussmann}}
\def\sighjswithmcclam{
\pltwo{\hbox to 5true cm{\hfill N. Harris McClamroch \hfill}}
{\hbox to 5true cm{\hfill H\'ector J. Sussmann \hfill}}}
\def\sighejs{\plright{H\'ector J. Sussmann}}
\def\sighjsprof{
\plright{H. J. Sussmann}\nobreak
\plright{\it Professor of Mathematics}}
\def\finhjsneural#1#2{
\vskip 0.5true in\sighjsneural\vfil#1\vfil\vfil\pleft{#2}\vfil\eject}
\def\finhjs#1#2{
\vskip 0.5true in\sighjs\vfil#1\vfil\vfil\pleft{#2}\vfil\eject}
\def\finhjsneru{
\vskip 0.3true in\sighjsneru\vfil\eject}
\def\finhjswithsontag#1#2{
\vskip 0.8true in\sighjswithsontag\vfil#1\vfil\vfil\pleft{#2}\vfil\eject}
\def\finhjswithsontagshort#1#2{
\vskip 0.4true in\sighjswithsontagindent\vfil#1\vfil\vfil\pleft{#2}\vfil\eject}
\def\finhjsson#1#2{
\vskip 0.8true in\sighjswithsontag\vfil#1\vfil\vfil\pleft{#2}\vfil\eject}
\def\finhjswithmcclam#1#2{
\vskip 0.8true in
\pltwo{\hbox to 5true cm{\hrulefill}}{\hbox to 5true cm{\hrulefill}}
\sighjswithmcclam\vfil#1\vfil\vfil\pleft{#2}\vfil\eject}
\def\finhejs#1#2{
\vskip 0.8true in\sighejs\vfil#1\vfil\vfil\pleft{#2}\vfil\eject}
\def\finhjsprof#1#2{
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Control and Optimization}.
}}}



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}}}


%draft: 22 October 1996



\vskip 2cm

\centerline{\bf REGULAR SYNTHESIS} 
\centerline{\bf AND SUFFICIENCY CONDITIONS FOR OPTIMALITY}

\vskip 4em

\centerline{Benedetto Piccoli}
\centerline{\sl SISSA-ISAS}
\centerline{\sl via Beirut 2-4, 34014 Trieste, Italy}

\vskip 1em

\centerline{H\'ector J. Sussmann}
\centerline{\sl Department of Mathematics, Rutgers University}
\centerline{\sl New Brunswick, New Jersey, NJ 08903}

\vskip 21em

\noindent
{\bf Abstract.} The aim of this paper is to analyse the different definitions
of synthesis given in the literature and compare them. In particular, we are
interested in optimal synthesis. We prove the strongest sufficiency theorem 
available in the literature (the proof was only sketched in [27]). 
It is not difficult to check that the synthesis for the famous Fuller
problem [12], satisfies the hypotheses of this theorem, but not of other
sufficiency theorems [2], [6].

\vskip 1em

\noindent
{\bf Keywords.} Optimal control, regular synthesis, sufficient conditions.


\vskip 5em
\centerline{Ref. SISSA 68/96/M}
\vfill\eject


\parnormal


\section{Introduction}

\noindent The aim of this paper is to analyse the different definitions of
synthesis given in the literature and compare them.
In particular we are interested in optimal synthesis. 
Usually the definitions of synthesis are given for two different
purposes. The first one is to treat the problem of existence, the
second one is to study the problem of sufficiency for optimality.
In the first case, see \brunacount, \brunbcount, \piccccount,
\piccdcount, \sussacount, the
definition of synthesis should be
general enough to prove the existence of syntheses for large class of
problems but at the same time a synthesis must be regular enough to
obtain optimality. 
Indeed the general approach is to construct an
extremal synthesis satisfying the regularity assumptions of an
optimality theorem.
The sufficiency theorems ensures the optimality of every extremal
synthesis that is sufficiently regular.
The use of subanalytic sets has been the most successful approach
for proving the existence of optimal syntheses \brunacount,
\brunbcount, \sussacount, \sussbcount,
\sussccount, \sussecount, \sussfcount, \susslcount, \sussmcount.
In this method,
the concept of stratification of a manifold plays a key role and the
syntheses constructed in this way present a good regularity.
In fact, one first proves regularity for extremal trajectories
(more precisely one proves that every extremal trajectories can be
replaced by a regular trajectory having the same cost) and then, using
this, constructs a regular extremal synthesis.
It is clear that the strongest is the optimality theorem, the weakest
are the assumptions of regularity and the largest is the class of problems
for which one can prove the existence of an optimal synthesis.
Hence, in general the definition of synthesis given for sufficiency
theorems is less restrictive than the definition of synthesis given in
existence theorems.

We start by proving the strongest sufficiency theorem available in the
literature (the proof was only sketched in \sussmcount, indeed we prove
a mild generalization of it).
It is not difficult to
check that the synthesis for the famous Fuller problem \fullcount, satisfies
the hypotheses of this theorem, but not of other sufficiency
theorems \boltcount, \brunbcount. The main technical problem is the 
infinite number of
switchings and the optimality was originally proved using 
the symmetry of the problem.
This example shows how a synthesis can be {\sl nicer} than the
trajectories forming it. In particular, it suggests that there should
be an existence theory not strictly related to the regularity
properties of trajectories. The subanalytic sets approach requires
extremal trajectories with a finite number of switchings in order to
obtain a finite dimensional reduction.

There are two essentially different ways of defining a synthesis. In
\boltcount, \brunacount, \cesaricount, \flemcount,
\sussacount, 
a synthesis is defined as a collection of trajectories generated by a
piecewise regular feedback control. 
Indeed the set of interest (that is the set of points for which
there exists an optimal trajectory) is partitioned in a locally finite
collection of 
submanifolds of various dimensions. More precisely the set is stratified and
for each stratum there exists a regular feedback control.
While in \piccacount, \sussmcount, the synthesis is just a collection
of trajectories not necessarily arising from a feedback control.
The only assignment of the feedback control is not sufficient for
describing a synthesis. Indeed a lot of trajectories may be
generated by this control and some of them can be not optimal as it is
shown in Example~1 of section~5. The same example shows how it is
possible to have an optimal synthesis matching the definition given in
\sussmcount, but not the definitions given in \boltcount, \brunbcount. The only point of the
definition given in \brunbcount, not satisfied by this synthesis, is the
continuity of the value function. But it is clear from the same
example that there are nice syntheses presenting a discontinuous value
function, hence the existence theory should be enlarged to cover these cases.
Example~2 shows another synthesis not matching the
definition given in \boltcount.

One interesting open problem is to find the best way of defining a
synthesis. A piecewise regular feedback control gives a good
description of the known optimal syntheses (\brunbcount, \sussmcount), but it presents
the problems illustrated by Example~1.
Therefore one has to add more structure in the definition restricting
the class of systems for which a synthesis exists.
A collection of trajectories is
a more general object, but it has not enough structure to conceptualize
regularity properties shared by many known optimal syntheses and an
existence theory based on this weakly regular object seems impossible.

Another interesting point, that should be further investigated, is the relation
between the geometrical approaches of \brunacount, \brunbcount,
\sussacount, \sussbcount,
\sussccount, \sussecount, \sussfcount, \susslcount, \sussmcount, 
and the theory of
viscosity solutions \bardicount, \flemsonercount. 
The latter is based on the study of the value
function as a solution of the Hamilton--Jacobi equation.

It is worth it to notice that the optimality theorem we prove
is valid for presynthesis (that is a synthesis not necessarily
memoryless), but it is not clear if it is possible to construct an
existence theory for presynthesis. Moreover, to our knowledge, there
are not interesting
examples of problems admitting only presyntheses but not syntheses.

Finally, the contents of this paper concerns optimal syntheses, but
the discussion about the convenient definitions and the structure of
syntheses is interesting also for different problems like asymptotic
stabilization problems.

\vs

\def\nin{\noindent}
\def\vs{\vskip 1em}
\def\R{\rr}
\def\C{{\cal C}}
\def\tf{\tilde f}
\def\xb{\bar x}
\def\g{\gamma}
\def\gx{\gamma_x}
\def\gxb{\gamma_{\bar x}}
\def\e{\eta}
\def\ex{\eta_x}
\def\exb{\eta_{\bar x}}
\def\m{\mu}
\def\mx{\mu_x}
\def\mxb{\mu_{\bar x}}
\def\n{\nu}
\def\nx{\nu_x}
\def\nxb{\nu_{\bar x}}
\def\vp{\varphi}
\def\ve{\varepsilon}
\def\veb{\bar\varepsilon}
\def\pe{\psi_{\ve}}
\def\la{{\lambda}}
\def\l0{\lambda^0}
\def\tx{T_x}
\def\txb{T_{\bar x}}


% \input draft.tex

\section{A sufficiency theorem}

\noindent We start with some definitions that will enable us to state and
prove a mild generalization of
the theorem whose proof was outlined in \sussmcount.  The
assumptions made here will be minimal, so as to achieve maximum generality.
In subsequent sections we will impose more restrictive conditions and apply
our result to other definitions of synthesis given in the literature only
for more regular control systems.

We consider a control system:
\xdef\system{\equanosectno}
$$\dot x=f(x,u)\,,\qquad x\in\Omega\,,\qquad u\in U\,,\numeq$$
and the family $\{{\cal P}(f,L,{\cal T},\tau,x)\}_{x\in\Omega}$ of
minimization problems, where  ${\cal P}(f,L,{\cal T},\tau,x)$ is the 
problem: 
\xdef\minim{\equanosectno}
$${\rm minimize\,\,\,\,\,}\int_a^b L(\g(t),\e(t))dt+\tau(\gamma(b))
\qquad{\rm subject\,\,\,to}\,\,\,(\system)\,,\,\,\,\g(a)=x\,,\,\,\,
{\rm and}\,\,\,\g(b)\in{\cal T}\,.\numeq$$
(A more precise formulation of the problems ${\cal P}(f,L,{\cal T},\tau,x)$,
carefully specifying the space of of curves to be considered, will be given
below. Since both $f$ and $L$ have no explicit dependence on $t$, 
we can always take $b=0$.)

We assume that 

\medbreak
{\sl 
{\pari
\item{(A1)} $\Omega$ is an open subset of $\R^n$,

\item{(A2)} $U$ is a set, 

\item{(A3)} $f:\Omega\times U\rightarrow \R^n$ and $L:\Omega\times
U\rightarrow \R$ are maps,

\item{(A4)} $f(\cdot,u)$ and $L(\cdot,u)$ are
maps of class $C^1$  for every fixed $u\in U$,

\item{(A5)} ${\cal T}$ is a nonempty subset of $\Omega$,

\item{(A6)} $\tau:\Omega\to\R$ is a function of class $C^1$.
}
}

\medbreak\noindent
We use $T_x^cS$ to denote the {\sl contingent tangent}
cone to a set $S$ at a point 
$x\in S$, defined as follows: a vector $v$ is in $T_x^cS$ iff there exists
a sequence $\{x_j\}$ of points of $S$ and a sequence $\{h_j\}$ of strictly
positive numbers such that $h_j\to 0$ and $x_j=x+h_jv+o(h_j)$ as $j\to\infty$.
We write 
$$
\tilde T_xS=T_x^cS\cap(-T_x^cS)=\{v:v\in T_x^cS\quad{\rm and\quad}-v\in 
T_x^cS\}\,.\numeq
$$
 

\medbreak\noindent
As in Sussmann \sussmcount, we use $\tf$ to denote the $\R^{n+1}$ valued map
$(x,u)\to (f(x,u),L(x,u))$.

If $\mu:A\to B$ is a map, we use $\Dom(\mu)$ to indicate the domain
of $\mu$, i.e. the set $A$. 

A {\sl  control} is a $U$-valued map $\eta$ such that $\Dom(\eta)$
is a compact subinterval $[a,b]$ of $\R$.
A {\sl trajectory} $\gamma$ for a control $\eta$ is an absolutely
continuous map $\gamma:\Dom(\eta)\to\Omega$ that satisfies
$\dot\gamma(t)=f(\gamma(t),\eta(t))$ for almost every $t\in \Dom(\eta)$.  If
$\Dom(\eta)=[a,b]$ (so that $\Dom(\gamma)=[a,b]$ as well), and $x=\gamma(a)$,
$y=\gamma(b)$, we say that $\gamma$ (or the pair $(\gamma,\eta)$)
{\sl goes from $x$ to $y$}, and that $\eta$ {\sl steers $x$ to $y$},
and we write $\gamma^-=x$, $\gamma^+=y$.

A control $\eta:[a,b]\rightarrow U$ is {\sl admissible} if the
map $\Omega\times[a,b]\ni(x,t)\to\tf_{\eta}(x,t)=\tf(x,\eta(t))
\in\R^{n+1}$ satisfies the following $C^1$ Carath\'eodory conditions:

\medbreak
{\pariim

\item{(C.i)} {\sl $t\mapsto \tf_{\eta}(x,t)$ is measurable for each $x\in\Omega$,}
\vs
\item{(C.ii)} {\sl for every compact $K\subseteq\Omega$ there exists an
integrable function $\vp_K$ such that for every $x\in K$ and $t\in[a,b]$:}
\xdef\intbound{\equanosectno}
$$\|\tf_{\eta}(x,t)\|+\|D_x\tf_{\eta}(x,t)\|\leq \vp_K(t)\,.\numeq$$

}

\medbreak\noindent If $\eta$ is an admissible control and $\gamma$ a
trajectory corresponding 
to $\eta$, then we say that $(\gamma,\eta)$ is an {\sl admissible pair} for
$\tf$.  We use $\Adm(\tf)$ to denote the set of all admissible pairs for
$\tf$, and write $\Admo(\tf)$, $\Admo(\tf,{\cal T})$, $\Admo_x(\tf,{\cal
T})$, to denote, respectively, the set of all $(\g,\e)\in\Adm(\tf)$ such
that $\Dom(\e)$ is an interval of the form $[T,0]$ with $T\le 0$, the set
of those $(\g,\e)\in\Admo(\tf)$ such that $\gamma(0)\in{\cal T}$, and the
set of those $(\g,\e)\in\Admo(\tf,{\cal T})$ such that $\g^-=x$.  For an
admissible pair $(\gamma,\eta)\in\Admo(\tf,{\cal T})$, we use
$J(\gamma,\eta)$ to denote the {\sl cost} of $(\gamma,\eta)$, so that
$$
J(\g,\e)=\int_{T}^{0} L(\g(t),\e(t))\,dt+\tau(\g^+)\,.
$$
If $\Gamma\!\subseteq\!\Admo(\tf,{\cal T})$ is an arbitrary set of admissible
pairs, we 
define a function $V_{\Gamma}\!:\!\Omega\!\mapsto\!\R\cup\{+\infty\}$,
called the 
{\sl value function of $\Gamma$}, by letting 
$$
V_{\Gamma}(x)=\inf\{J(\gamma,\eta):(\gamma,\eta)\in\Gamma,\,\gamma^-=x\}\numeq
$$ 
for $x\in\Omega$.  
(In particular, $V_{\Gamma}(x)=+\infty$ iff there is no
$(\gamma,\eta)\in\Gamma$ starting at $x$.)  If $\Gamma=\Admo(\tf,{\cal T})$,
then the function $V_{\Gamma}$ is the {\sl value function} of our problem,
and in that case we will write $V_{\tf,{\cal T}}$ rather than
$V_{\Admo(\tf,{\cal T})}$.  

We can now formulate ${\cal P}(f,L,{\cal T},\tau,x)$ precisely: 
${\cal P}(f,L,{\cal T},\tau,x)$ is the problem of finding a 
$(\g,\e)\in\Admo_x(\tf,{\cal T})$ such that 
$J(\g,\e)=V_{\tf,{\cal T}}(x)$.   Any such pair is said to be a
{\sl solution} of ${\cal P}(f,L,{\cal T},\tau,x)$.    A pair 
$(\g,\e)\in\Admo(\tf,{\cal T})$ is {\sl optimal} if it is a solution
of ${\cal P}(f,L,{\cal T},\tau,\g^-)$.

Given $\la\in\R_n$ (the space of row $n$ vectors), 
$\l0\in\R$, $x\in\Omega$ and $u\in U$, we define:
\xdef\hamil{\equanosectno}
$${\cal H}(\la,\l0,x,u)=\langle \la , f(x,u)\rangle +\l0 L(x,u)\,,\numeq$$
\xdef\minhamil{\equanosectno}
$$H(\la,\l0,x)=\inf\{{\cal H}(\la,\l0,x,u):u\in U\}\,.\numeq$$
The functions ${\cal H}:\R_n\times\R\times\Omega\times U\to\R$
and $H:\R_n\times\R\times\Omega\to\R\cup\{-\infty\}$
are known as the {\sl Hamiltonian} and the {\sl minimized Hamiltonian}
of $\tf$.

We say that the pair $(\gamma,\eta)\in \Admo(\tf,{\cal T})$ 
is {\sl extremal} if there exist an
absolutely continuous map $\la:{\rm Dom}(\g)\mapsto \R_n$
and a constant $\l0\geq 0$, that satisfy:

\medbreak


{\pariim
\item{(EX1)} {\sl $(\l0,\la(t))\neq (0,0)$ for some (and hence every) $t$.}

\midbreak

\item{(EX2)} {\sl $\dot\la = -{\partial {\cal H}\over \partial x}
(\la(t),\l0,\gamma(t),\eta(t))$ for a.e. $t\in\Dom(\e)$.}

\midbreak

\item{(EX3)} $H(\gamma(t),\la(t),\l0)={\cal H}(\gamma(t),\la(t),\l0,\eta(t))=0$
{\sl for a.e.}  $t\in \Dom(\eta)$, 

\midbreak

\item{(EX4)} $\langle\la(0),v\rangle=\l0\langle \nabla\tau(\g(0)),v\rangle
\,\,\,{\sl for\,\,\,all}\,\,\,v\in \tilde T_{\g(0)}{\cal T}$.  

}

\medbreak\noindent
Conditions (EX1), (EX2), (EX3), (EX4) are, respectively,
the {\sl nontriviality condition}, the {\sl adjoint equation},
the {\sl Hamiltonian minimization condition}, and the {\sl transversality
condition}. A pair $(\la,\l0)$ such that (EX2) holds is called
an {\sl adjoint vector for} $(\g,\e)$.  An adjoint vector for which (EX3) 
holds is a {\sl minimizing adjoint vector for} $(\g,\e)$.  

The {\sl maximum principle} says that, under some special conditions
on the target set ${\cal T}$, {\sl every optimal pair $(\g,\e)$
is extremal.}
(This is true, for example, if ${\cal T}$ is a submanifold of $\Omega$ 
of class $C^1$, or a closed convex subset of $\Omega$, or
a set which is locally equivalent to a convex set via a $C^1$
diffeomorphism.  More generally, it is true if the cone 
$\tilde T_{\g(0)}{\cal T}$ is convex ---in which case it is of course a 
subspace, since $v\in\tilde T_{\g(0)}{\cal T}\Rightarrow
-v\in\tilde T_{\g(0)}{\cal T}$---  and is an approximating cone 
to ${\cal T}$ at $\g(0)$.   Recall that a closed convex cone $C$ is an
{\sl approximating cone} to a set $S$ at a point $s\in S$ if there exist a
neighborhood $U$ of $0$, and a continuous map $F:U\cap C\to S$,  
such that $F(x)=s+x+o(||x||)$ as $x\to 0$ via values in $C$.) 

\medbreak

A {\sl presynthesis} for our family of problems is a subset
$\Gamma$ of $\Admo(\tf,{\cal T})$ such that,

{\pariim

\item{(PS)} {\sl Whenever $(\gamma_1,\eta_1)\in\Gamma$,
$(\gamma_2,\eta_2)\in\Gamma$, 
and $\g_1^-=\g_2^-$, it follows that $(\gamma_1,\eta_1)=(\gamma_2,\eta_2)$.}

}

The set $\Dom(\Gamma)\equaldef\{\g^-:(\g,\e)\in\Gamma\}$ is called the {\sl
domain} of $\Gamma$.  We say that $\Gamma$ is a {\sl presynthesis on} $S$ 
if $\Gamma$ is a presynthesis and $S=\Dom(\Gamma)$.  Clearly, 
giving a presynthesis on $S$  amounts to choosing, 
for each $x\in S$, a $(\g,\e)\in \Admo(\tf,{\cal T})$ such that
$\g^-=x$.  

If $\Gamma$ is a presynthesis such that $\Dom(\Gamma)$ consists of all
points that can be steered to the origin in finite time by an admissible
pair, then we say that $\Gamma$ is {\sl total}.  

Given a presynthesis $\Gamma$ and an $x\in\Dom(\Gamma)$ 
we will always use $(\gx,\ex)$ to denote the unique pair ($\g,\e)\in\Gamma$ 
such that $\g^-=x$, and will write  $T_x=\min\,\Dom(\g_x)$, so 
$T_x\le 0$ and $\Dom(\g_x)=\Dom(\e_x)=[T_x,0]$.

A presynthesis is {\sl memoryless} if
whenever $x\in S$ and $t\in \Dom(\ex)$ it follows that $y=\gx(t)$ belongs to
$S$ and $\eta_y$ is the restriction of $\ex$ to the interval $[t,0]$.  A
{\sl synthesis} is a memoryless presynthesis.  If each pair of a
presynthesis $\Gamma$ is optimal (resp. extremal) then we say that $\Gamma$
is {\sl optimal} (resp. {\sl extremal}).  In particular, a presynthesis
$\Gamma$ is a total optimal presynthesis if and only if $V_{\Gamma}\equiv
V_{\tf}$.

Our goal is to prove that, under suitable regularity assumptions on
$\Gamma$ and $V_{\Gamma}$, a total extremal presynthesis is optimal.  We
first describe these regularity assumptions in detail.

We say that a function $V:\Omega\rightarrow \R\cup \{-\infty,+\infty\}$ is
{\sl weakly upper semicontinuous} (w.u.s.c) at a point $x\in\Omega$ if
$$
\lim\inf_{y\to x} \lim\sup_{z\to y}V(z)\le V(x)\,.\numeq
$$

Equivalently, $V$ is w.u.s.c. at $x$ if 
there exist sequences $x_j$, $\ve_j$ and $\delta_j$ such that
$x_j\in\Omega$, $x_j\rightarrow x$, $\ve_j>0$, $\delta_j>0$,
$\delta_j\rightarrow 0$, having the property that $V(y)\le V(x)+\delta_j$
whenever $\|y-x_j\|<\ve_j$.

Given a locally Lipschitz vector field $X$ on $\Omega$, we say that $V$ has
the {\sl NDJ (``no downward jumps'') property along $X$} if the following
holds:

{\pariim

\medbreak

\item{(NDJ)}  {\sl For every $\gamma$ integral curve of $X$, 
if $\Dom(\gamma)=[a,b]$ then for every $t\in ]a,b]$,
$$\lim\sup_{h\downarrow 0} V(\gamma(t)) - V(\gamma(t-h))\geq 0$$.}
%if $\{t_j^-\}$, $\{t_j^+\}$ are sequences such that
%$t_j^-\leq t\leq t_j^+$, $t_j^-,t_j^+\rightarrow t$, and 
%$L=\lim_j [V(\gamma(t_j^+))-V(\gamma(t_j^-))]$, then $L\geq 0$.}


\medbreak
}


Finally we say that $V$ satisfies the {\sl weak continuity conditions} for
the control problem (\system), (\minim) if $V$ is lower semicontinuous and
weakly upper semicontinuous, and has the NDJ property along the vector
field $x\to f(x,u)$ for every $u\in U$.

We now define the ``weak differentiability conditions'' for $\Gamma$.  In
this definition it is understood that $\tf(y,\ex(t))=0$ for every $t\notin
\Dom(\ex)$.  A presynthesis $\Gamma=\{(\gamma_x,\eta_x):x\in S\}$ is {\sl
$(f,L)$--differentiable} at a point $\xb\in\Omega$ if the following 
condtion is satisfied:

\medbreak

{\pari

\item{(DC)} {\sl there exist a neighborhood $N$ of $\xb$ (in $\Omega$)
such that $N\subseteq \Dom(\Gamma)$, and an interval $[T,0]$ such that 
$\Dom(\ex)\subseteq [T,0]$ for all $x\in N$,  and an $\bar\ve>0$, 
with the property that}

}

\vs

{\pariim

\item{(DC1)} {\sl there exist, for $\ve\in]0,\bar\ve]$, integrable
functions $\pe :[T,0]\rightarrow \R$ such that
$$
\lim_{\ve\to 0}\int_T^0 \pe(t)\, dt=0\,,\numeq
$$
and the inequality}

}

$$\biggl\|\tf(y,\ex(t))-\tf(\gx(t),\ex(t))-{\partial \tf\over \partial
x}\Bigl(\gxb(t),\exb(t)\Bigr)\cdot(y-\gxb(t))\biggr\|\leq\pe(t).
(||y-\gxb(t)||+||x-\bar x||)\numeq
$$

{\pariim

\item{} {\sl holds for every $y\in\Omega$, $x\in N$,  $t\in[T,0]$ such that 
$\|y-\gxb(t)\|\leq\ve$ and $||x-\bar x||\le\ve$.}

\vs
\item{(DC2)} {\sl the map $v\rightarrow\rho_v$, where $\rho_v$ is the
integrable $\R^{n+1}$ valued function on $[T,0]$ given by}
$$\rho_v(t)=\tf(\gxb(t),\eta_{\xb+v}(t))-\tf(\gxb(t),\exb(t))$$

\item{} {\sl is weak$^*$-differentiable at $v=0$, regarded as a map into the
space of $\R^{n+1}$--valued Borel measures. That is: for every continuous
function $\alpha:[T,0]\rightarrow\R$, the map 
$\R^n\ni v\to \int_T^0
\rho_v(t)\alpha(t) \, \in\R^{n+1}dt$ is differentiable at $v=0$.}

\vs
}

A set $A\subset\Omega$ is {\sl thin} if there exist $A_1$, $A_2$, such that
$A=A_1\cup A_2$, $A_1$ is a finite or countable union of connected embedded
$C^1$ submanifolds of positive codimension and $A_2$ has Hausdorff
dimension strictly less than $n-1$.

We can now state the main theorem (see \sussmcount):
\vs
\xdef\maintheo{\the\sectno.\the\theono}
\theo let $\Omega$, $U$, $f$, $L$ be such that
Assumptions (A1) -- (A6) hold.
Let $\Gamma$ be a total extremal
presynthesis. Assume that 

{\pariim
{\sl 
\item{(\maintheo.a)} the associated value function $V_{\Gamma}$
satisfies the weak continuity conditions, 

\midbreak

\item{(\maintheo.b)} $V_{\Gamma}(x)\le\tau(x)$ for every $x\in{\cal T}$,

\midbreak

\item{(\maintheo.c)}  $\Gamma$ is
$(f,L)$--differentiable at all points in the complement of a thin set $A$.
}

}

\midbreak

\noindent {\sl Then $\Gamma$ is optimal.}

\medbreak

\beginproof
We let $S={\rm Dom}(\Gamma)$, so $S$ is exactly the set of 
all $x\in\Omega$ that can be steered to the target ${\cal T}$ in
finite time by an admissible control.

We first establish the relation between the differential of the value
function $V_{\Gamma}$ and the adjoint covector fields along the
extremal trajectories of the presynthesis.   This result is important in
its own right, so we state it as a separate theorem:

\xdef\maintheodiff{\the\sectno.\the\theono}
\theo Let $\Omega$, $U$, $f$, $L$ be such that
Assumptions (A1) -- (A6) hold.
Let $\Gamma=\{(\gamma_x,\eta_x):x\in S\}$ be a 
presynthesis. Let $\bar x\in S$ be such that 
$\Gamma$ is
$(f,L)$--differentiable at $\bar x$.  Then 

{\pariim
{\sl 
\item{(\maintheodiff.a)} the function $V_{\Gamma}$ is differentiable 
at $\bar x$;

\midbreak

\item{(\maintheodiff.b)} if the pair $(\gamma_{\bar x},\eta_{\bar x})$ is
extremal, then every minimizing adjoint vector $(\la,\l0)$ for 
$(\gamma_{\bar x},\eta_{\bar x})$ satisfies the identity
$\l0 D_y V_{\Gamma}(\xb)= \la(\txb)$. In particular, 
$\l0\neq 0$, and $(\la,\l0)$ is unique up to multiplication by a positive 
constant. 

}

}


\medbreak\noindent{\sl PROOF OF THEOREM \maintheodiff.\enspace}
We use Theorem A.5 in Appendix A with $P=\R^n$, 
letting $f_x(y,t)=f(y,\ex(t))$, $L_x(y,t)=L(y,\ex(t))$, for 
$y\in\Omega$, $x\in N$, $t\in[T_x,0]$, and defining 
$f_x(y,t)=0$, $L_x(y,t)=0$, for $T\le t<T_x$.  
Let $\hat V_\Gamma(x)=\int_{T_x}^0L(\gx(t),\ex(t))\,dt$, so 
$\hat V_\Gamma$ is the ``Lagrangian part'' of our cost functional.

The map $p\mapsto \bar x_p$ of Theorem A.5 is the identity map,
which is certainly differentiable at $\bar x$.   So
Theorem A.5 tells us that the ``endpoint map''$x\mapsto
{\cal E}(x)\equaldef \gx(0)$ and the function
$\hat V_\Gamma$ are differentiable at $\bar x$,  and the identity 
$$
\lambda(0).D_y{\cal E}(\bar x).v+\lambda^0 D_y\hat V_{\Gamma}(\bar x).v
-\lambda(T_{\bar x})
\cdot v\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
\xdef\identi{\equanosectno}

\vskip -1.2cm
$$
\phantom{aaaaaaaaaaaaaa}=\lim_{\ve\downarrow 0}\,\,\ve^{-1}\!\int_T^0\!\Big(
H(\la(t),\l0,\gxb(t),\eta_{\bar x +\ve v}(t))-
H(\la(t),\l0,\gxb(t),\exb(t))\Big)\,dt.\numeq
$$
holds for every vector $v$ if $(\la,\l0)$ is an adjoint vector
for $(\gxb,\exb)$. (Here the function $\la$ is extended 
to $[T,0]$ by letting it equal zero on $[T,T_{\bar x}[$.
The Hamiltonian $H(\la(t),\l0,y,\ex(t))$ is 
also defined for all $t\in[T,0]$, by letting 
$H((\la(t),\l0,y,\ex(t))=\la(t).f(y,\ex(t))+\l0 L(y,\ex(t))$, 
and recalling that this expression vanishes for $t\in[T,T_x[$,
because $f(y,\ex(t))$ and $L(y,\ex(t))$ have been defined to be 
zero for $t\in[T,T_x[$.)

Now, if $(\la,\l0)$ is minimizing, then the right-hand side
of the inequality (\identi) is nonnegative, because
$H(\la(t),\l0,\gxb(t),\exb(t))=0$ and 
$H(\la(t),\l0,\gxb(t),\eta_{\bar x +\ve v}(t))\ge 0$ 
for almost all $t\in[T,0]$. So
$$
\lambda(0).D_y{\cal E}(\bar x).v+\lambda^0 D_y\hat V_{\Gamma}(\bar x).v
-\lambda(T_{\bar x})
\cdot v\ge 0\qquad{\rm for\,\,\,all\,\,\,}v\in\R^n\,.\numeq
$$
Given $v$, the curve $r\mapsto \gamma_{\bar x+rv}(0)$ is defined
for $r$ in a neighborhood of $0$ in $\R$, and is differentiable 
at $0$ with derivative $D_y{\cal E}(\bar x).v$.  On the other hand, this
curve is contained in ${\cal T}$, so the vector 
$w=D_y{\cal E}(\bar x).v$ belongs to  $\tilde T_{\g_{\bar x}(0)}{\cal T}$.  
The transversality condition (EX4) then says that 
$\langle\la(0),w\rangle=\l0\langle \nabla\tau(\g(0)),w\rangle$.
Therefore 
$$
\l0\biggl(\nabla\tau(\g(0)).D_y{\cal E}(\bar x).v+D_y\hat V_{\Gamma}(\bar
x).v\biggr) 
-\lambda(T_{\bar x})
\cdot v\ge 0\qquad{\rm for\,\,\,all\,\,\,}v\in\R^n\,.\numeq
$$
Since the left-hand side of this inequality is linear in $v$,
it has to be equal to zero. So 
$$
\l0\biggl(\nabla\tau(\g(0)).D_y{\cal E}(\bar x).v+D_y\hat V_{\Gamma}(\bar
x).v\biggr)=\lambda(T_{\bar x})\cdot v\qquad
{\rm for\,\,\,all\,\,\,}v\in\R^n\,.\numeq
$$
Clearly, the left-hand side of this equation is $\l0 D_yV_\Gamma(\bar x).v$.
Since the equality holds for all $v$, we have 
$\l0 D_yV_\Gamma(\bar x)=\lambda(T_{\bar x})$. 

In particular, $\l0$ cannot vanish, because $\l0=0$ would imply
$\lambda(T_{\bar x})=0$, contradicting (EX1). 
So we may, after multiplication by a positive constant,  
assume that $\l0=1$, and then 
$$ 
D_y V_{\Gamma}(\xb)= \la(\txb).\numeq
$$
This proves, in particular, the uniqueness of $(\lambda(\cdot),\l0)$
up to multiplication by a positive constant, since the
Cauchy problem for the adjoint equation has unique solutions. \proofend

\medbreak\noindent{\sl CONTINUATION OF THE PROOF OF THEOREM \maintheo.\enspace}
The minimization condition (EX4) of the Pontryagin Maximum Principle
implies that, for a given $x\in S\setminus A$, $u\in U$, the inequality
$$\langle \la(t),f(\gx(t),u)\rangle +L(\gx(t),u)\geq 0$$
holds for almost all $t\in [T_x,0]$, where $(\la,\l0)$ is a minimizing
adpoint vector for $(\gx,\ex)$, normalized so that $\l0=1$. 
Since $t\mapsto \langle \la(t),f(\gx(t),u)\rangle +L(\gx(t),u)$
is continuous on $[T_x,0]$, the 
inequality is valid at time $\tx$. In view of Theorem
\maintheodiff, $\la(T_x)=dV_\Gamma(x)$, so 
\xdef\difvalf{\equanosectno}
$$
\langle dV_{\Gamma}(x),f(x,u)\rangle +L(x,u)\geq 0\,.
\numeq
$$

Now consider an arbitrary $(\g,\eta)\in\Admo(\tf,{\cal T})$, with domain
$[T,0]$.   We want to prove that $J(\g,\e)\ge V_\Gamma(x)$, if $x=\g^-$. 
It follows from Lemma~4.1 of \sussncount that
there exists a sequence of admissible
pairs $(\gamma_j,\eta_j)$, with domain $[T,0]$, 
such that $\gamma_j(0)=\g(0)$,
$\eta_j$ is piecewise constant, $\gamma_j\rightarrow\g$ uniformly 
on $[T,0]$, and
$J(\gamma_j,\eta_j)\rightarrow J(\g,\eta)$. 
Since the function $V_{\Gamma}$ is lower semicontinuous, it follows
that, if 
$V_{\Gamma}(\gamma_j(T))\leq J(\gamma_j,\eta_j)$, then 
$V_{\Gamma}(x)\leq J(\gamma,\eta)$.
Hence it suffices to assume that  the control $\eta$ is piecewise
constant, and we make this assumption from now on. 

We now need a transversality argument to prove that for every open set
$W\subset S$ there exists a point $y\in W$ such that the trajectory
$\gamma^y$, corresponding to the control $\eta$ and ending at $y$,
intersects the thin set $A$ a finite or countable number of times.
We have that $A=A_1\cup A_2$, and $A_1$ is union of a finite or
countable number of positive codimension manifolds $M_j$, while ${\cal
H}^{n-1}(A_2)=0$ where ${\cal H}^{n-1}$ denotes the $n-1$ dimensional
Hausdorff measure.
For every $y\in W$ we have that either the
trajectory $\gamma^y$ is defined on the whole interval $[T,0]$, or
there exists $t_y$, $T\leq t_y<0$, such that the
trajectory $\gamma^y$ is defined on the interval $(t_y,0]$ but can not
be prolonged backward in time.

We start considering the case of a constant control $\bar\eta$.

Let consider first the set $A_1$.
Each manifold $M_j$ is parametrized by some charts $\phi^j_k$ 
defined on open sets $Z^j_k\subset {\R}^{n_k}$, $n_k\leq n-1$.
Here the index $k$ takes value into a subset of the natural numbers.
Given $z\in Z^j_k$, let $a_z<0<b_z$ (with $a_z$ possibly equal
to $-\infty$ and $b_z$ possibly equal to $+\infty$)
be such that  $(a_z,b_z)$ is the maximal interval of definition for
the trajectory $\tilde\gamma_z$, corresponding to the control $\bar\eta$ 
and satisfying $\tilde\gamma_z(0)=\phi^j_k(z)$.
Define $\tilde Z^j_k=\{(z,t):z\in Z^j_k, t\in (a_z,b_z)\}$.
and the map $\Psi:\tilde Z_k\rightarrow \R^n$ by
$\Psi(z,t) = \tilde\gamma_z(t)$. 
Since the control $\bar\eta$ is constant, the
map $\Psi$ is $\C^1$. Moreover
$$
D\Psi (z,t)= \left(M(t,0)
\cdot ({\partial\phi^j_k\over\partial z_1} (z),\dots,
{\partial\phi^j_k\over\partial z_n} (z)),
f(\tilde\gamma_z(t),\bar \eta)\right)=M(t,0)
\left({\partial\phi^j_k\over\partial z_i} (z),
f(\phi^j_k(z),\bar\eta)\right),
$$
where the $M$ is the fundamental matrix solution for the linear system
$$\dot v(t)= D_y f(\tilde\gamma_z(t),\bar\eta)\cdot v(t).$$
Hence $det(D\Psi(z,t))=0$ if and only if 
the vectors ${\partial\phi^j_k\over\partial z_1}(z),\ldots,
{\partial\phi^j_k\over\partial z_{n_k}}(z)$, $f(\phi^j_k(z),\bar\eta)$ are
linearly dependent. Otherwise stated if and only if
the trajectory $\tilde\gamma_z$ intersects the manifold $M_j$ 
(more precisely $\phi^j_k(Z^j_k)$) tangentially. 
By Sard's theorem, the set $N_{jk}\subset\R^n$ of values on which this happens
has measure zero. 
Since $\tilde\gamma_z(t)=y$ if and only if $\gamma^y(-t)=z$, this means that
for every $y\notin N_{jk}$ the trajectory $\gamma^y$ intersects
the set $\phi^j_k(Z^j_k)$ transversally.
Fix $y\notin N_{jk}$. If $\gamma^y$ is defined on $[T,0]$ then
from the compactness of $[T,0]$ we obtain that $\gamma^y$ intersects
$\phi^j_k(Z^j_k)$ a finite number of times.
Otherwise, for every compact subset $K\subset (t_y,0]$,
it is clear that the trajectory $\gamma^y$ restricted to $K$ intersects 
the set $\phi^j_k(Z^j_k)$ a finite number of times.
Since there is an increasing sequence of compact sets $K_n$ such that
$\cup_n K_n=(t_y,0]$ we have that $\gamma^y$ intersects $\phi^j_k (Z^j_k)$
at most a countable number of times.
Let $N=\cup_{jk} N_{jk}$. 
Since $j$ and $k$ takes values inside countable sets, we have that
$meas(N)=0$ and for every
$y\notin N$ the trajectory $\gamma^y$ intersects $A_1$ a finite
or countable number of times.

Again let $\Psi$ be the flow map associated to the vector field
$f(x,\bar\eta)$, and let $\tilde A_2$ be the subset of $A_2\times [T,0]$
on which $\Psi$ is defined.
We have that ${\cal H}^n(A_2\times [T,0])=0$ and,
since $\Psi$ is Lipschitz continuous, $meas(\Psi(\tilde A_2))=
{\cal H}^n(\Psi(\tilde A_2))=0$ 
(see for example \fedcount). 
Hence for almost every $y$ the trajectory $\gamma^y$ does
not intersect the set $A_2$.

Therefore the conclusion holds for the constant control $\bar\eta$.
Now we can repeat the argument considering separately the time sets on
which the control $\eta$ is constant. We conclude using again the
Lipschitz continuity of the flows associated to constant controls.

{}From the weak upper semicontinuity of $V_{\Gamma}$, there exist $x_i$,
$\ve_i>0$, $\delta_i>0$ such that $x_i\rightarrow \gamma(0)$,
$\delta_i\rightarrow 0$ and $V_{\Gamma}(y)\leq
V_{\Gamma}(\gamma(0))+\delta_i$
whenever $\|y-x_i\|\leq \ve_i$.
By the transversality argument,
there exists $y_i$ such that $\|y_i-x_i\|\leq \ve_i$ and the
trajectories $\gamma^{y_i}$ intersect the set $A$ a finite 
or countable number of times.
Since $\Gamma$ is total and $V_{\Gamma}(\gamma^{y_j}(0))<+\infty$,
it is clear that, for every $j$, $\gamma^{y_j}(t)\in S$ for every
$t\in {\rm Dom}(\gamma^{y_j})$.
Hence from (\difvalf) we have that the function:
$$
\varphi(t)=
V_{\Gamma}(\gamma^{y_i}(t))+\int_T^t L(\gamma^{y_i}(s),\eta(s)) ds
$$
is differentiable for every $t$, except a finite or countable number 
(including the switching times of $\eta$), with positive derivative.
Thus, $\varphi$ verifies assumption (B.1.a) of Lemma~B.1 (Appendix~B).
Moreover, the lower semicontinuity of $V_{\Gamma}$ and Remark~B.1
ensure assumption (B.1.b) of Lemma~B.1 for $\varphi$, and the (NDJ) condition
ensures assumption (B.1.c). Hence we can apply Lemma~B.1 to $\varphi$.
Therefore its
value at time $T$ is less or equal its value at time $0$, that is
$$
V_{\Gamma}(\gamma^{y_i}(T))\leq \int_T^0
L(\gamma^{y_i}(t),\eta(t))dt +V_{\Gamma}(y_i)\leq
\int_T^0 L(\gamma^{y_i}(t),\eta(t))dt +V_{\Gamma}(\gamma(0))+\delta_i.
$$
Since $V_{\Gamma}(\gamma(0))\leq \tau(\gamma(0))$ and
$\gamma^{y_i}\rightarrow \gamma$ uniformly (and in particular
$\gamma^{y_i}(T)\rightarrow \gamma(T)=x$), 
using the  lower semicontinuity of $V_{\Gamma}$, we conclude $V_{\Gamma}(x)\leq
J(\gamma,\eta)$. This argument can be repeated for every admissible
pair $(\gamma,\eta)$, thus we conclude that $\Gamma$ is optimal.

\vs

{\bf Remark 1.}
Notice that
the weakly upper semicontinuity of the function
$V_{\Gamma}$ is equivalent to the weakly upper semicontinuity 
at the points of the target.
Indeed consider a point $x$ and its associated admissible pair $(\gx,\ex)$
(in the case $V_{\Gamma}(x)=+\infty$, (2.8) is immediately granted), 
and assume that $V_{\Gamma}$ is weakly upper semicontinuous 
at $\gamma(0)$. Let
$x_j$, $\ve_j$, $\delta_j$ be as in the definition of 
weakly upper semicontinuity.  
Define $\gamma^y$
to be the trajectory corresponding to the control $\ex$ and verifying
$\gamma^y(0)=y$, and let $y_j=\gamma^{x_j}(\tx)$. Now
it is easy to conclude with $y_j$'s playing the role of the $x_j$'s. 

\vs


\section{The Fuller phenomenon}

\medbreak

\noindent The aim of this section is to prove that the famous control
system, considered 
by Fuller in \fullcount, satisfies the hypothesis of Theorem 2.1.


\def\r{\rho}
\def\a{\alpha}


Consider the system:
$$\cases{ \dot x_1 =  x_2   &\cr
                        & $|u|\leq 1$\cr
          \dot x_2 =  u   &\cr}\eqno(3.1)$$
and the minimization problem:
$$\min \int_0^{+\infty} x_1^2(t) dt.\eqno(3.2)$$

In \fullcount{} Fuller showed an optimal synthesis for this problem.
Let define $\zeta_{\pm}=\{(x_1,x_2):x_1=\pm\xi x_2^2,\pm x_2\leq 0\}$
where $\xi$ is the unique positive root of the polynomial
$x^4+x^2/12-1/18$. These two curves divide the plane in two regions
$A^{\pm}$ (respectively below and above $\zeta_+\cup \zeta_-$). For
every point $x$ there exists a unique optimal trajectory $\gamma_x$
that corresponds to control $+1$ when $\gx(t)\in A^+$, to control $-1$
when $\gx(t)\in A^-$ and switches along the curves $\zeta_{\pm}$
reaching the origin in finite time (and then staying at the origin).
See Fig.1.

\vskip 1em


\centerline{\hbox{\psfig{figure=fuller.ps,width=3in,height=3in}}}

\medbreak
\centerline{Figure 1}

\vskip 1em


Therefore we can consider it as a problem of reaching the origin in
finite time for (3.1) minimizing the cost (3.2).
However the sufficiency theorems of \boltcount{} and \brunbcount{} do not apply.
Indeed, this synthesis is formed by bang--bang 
trajectories having an infinite number of 
switchings and the optimality is proved thanks to the symmetry of the
problem. 
There exists a one parameter family of dilations that send optimal
trajectories onto optimal trajectories.
Given a solution $(x_1(t),x_2(t))$ of this problem,
the trajectory 
$$x_{\r}(t)=(x_{1,\r}(t),x_{2,\r}(t))\dot= 
\left(\r^2 x_1\left( { t\over \r}\right), \r
 x_2\left( { t\over \r}\right)\right)\eqno(3.3)$$
is also an optimal solution corresponding to the control
$$u_{\r}(t)=u\left( {t\over \r}\right).\eqno(3.4)$$
Notice that from (3.3), $x_{\r}-x=(\r-1)\, (2x,y)+(\r-1)^2\, (x,0)$.

We want to prove the $(f,L)$--differentiability of the optimal
synthesis at every point in the complement of the switching curves.
It is easy to check the property (DC1), then let consider the property (DC2).

Fix a point $\xb\in\R^2$. There exists $s_1$ such that:
$$\exb=\sum_{k=1}^{+\infty} (\pm 1)^k \chi_{I_k}$$
where:
$$I_k=[s_k,s_{k+1}],\qquad s_k=\theta s_{k-1},\qquad
\theta=\sqrt{ {{1\over 2}-\xi \over {1\over 2}+\xi}},$$
$\chi$ is the indicator function 
and the sign $\pm$ depends on the position of the point in the
plane. More precisely it is $+1$ if $\xb\in A^+$ and $-1$ if $\xb\in A^-$.
Moreover from (3.4):
$$\eta_{x_{\rho}}=\sum_{k=1}^{+\infty} (\pm 1)^k \chi_{I_k^\rho},\qquad
I_k^\rho=[\r s_k,\r s_{k+1}]$$
and
$$\tilde f(\gxb(t),\eta_{x_{\r}}(t))-\tilde f(\gxb(t),\exb(t))=
\left( \matrix{0\cr \eta_{x_{\r}}(t)-\exb(t) \cr 0\cr} \right).\eqno(3.5)$$

Let $[T,0]$ be a set containing $\Dom(\ex)$ for $x$ near $\bar x$,
fix a continuous function $\a:[T,0]\rightarrow\R$ and let $w$ be its module of
continuity. The only nonzero component of (3.5) is the second one.
Hence we have to estimate only the integral:
$$\int_T^0 (\eta_{x_{\r}}(t)-\exb(t)) \a(t) dt=
\int_T^0 \sum_{k=1}^{+\infty} (\pm 1)^k (\chi_{I_k^\rho}(t)-\chi_{I_k}(t))
\a(t)dt=$$
$$=\int_T^0 \mp\chi_{[s_1,\r s_1]}(t)\a(t)dt+
\int_T^0 \sum_{k=2}^{+\infty} ((\pm 1)^{k-1}-(\pm1)^k) 
\chi_{[s_k,\r s_k]}(t) \a(t) dt=$$
$$=\pm (1-\r) s_1\a(s_1)\pm 2
\sum_{k=2}^{+\infty} (-1)^k (\r-1) s_k \a(s_k)+$$
$$\pm\int_{s_1}^{\r s_1} (\a(s)-\a(s_1)) ds
\pm 2\sum_{k=2}^{+\infty} (-1)^k (\r-1) s_k
\int_{s_k}^{\r s_k} (\a(s)-\a(s_k)) ds \leq$$
$$\leq\mp (\r-1) s_1\a(s_1)+(\pm 2)
\sum_{k=2}^{+\infty} (-1)^k (\r-1) s_k \a(s_k)$$
$$\pm (1-\r) s_1 w((1-\r)s_1)\pm 2
\sum_{k=2}^{+\infty} (-1)^k (\r-1) s_k w((\r-1)s_k).$$
Therefore
$$\lim_{\r\to 1} {1\over \r-1}\int_T^0  (\eta_{x_{\r}}(t)-\exb(t)) \a(t)
dt= \mp s_1\a(s_1)+(\pm 2)
\sum_{k=2}^{+\infty} (-1)^k \theta^k s_1 \a(s_k)\eqno(3.6)$$
being the sum convergent since $\a$ is continuous on a compact set,
hence bounded, and $\theta<1$.

This establishes the existence of the derivative in the direction of the
vector $(2x,y)$. It is easy to verify the existence of the
derivative in the direction $f(\gxb(\txb),\exb(\txb))$.
Let $x_{\sigma}=\gxb(\sigma \txb))$.
In this case we have:
$$\lim_{\sigma\to 1} {1\over \sigma-1}\int_T^0 (\eta_{x_{\sigma}}(t)-
\exb(t)) \a(t) dt=\txb \exb(\txb)\a(\txb).\eqno(3.7)$$
Now the two vectors $(2x,y)$, $f(\gx(\tx),\ex(\tx))$ form a basis for
every $x$ in a neighborhood of $\xb$ and are smooth functions of $x$.
Moreover, from (3.6),(3.7) the derivatives in these two
directions are continuous in $x$, hence we obtain the conclusion.

The other assumptions are easily checked and then Theorem 2.1
applies ensuring the optimality of the synthesis described above.

\vs

\section{The Boltianskii and Brunovsky definitions of synthesis}

\medbreak\noindent
We start by reviewing the definition of synthesis proposed by Brunowsky
\brunbcount{} 
and then we recall some further restrictions existing in the
definition given by Boltianskii \boltcount.

We consider the system (\system), (\minim), with the restriction 
that $f$ is $\C^1$ as a function of $(x,u)$.

%Consider the control system:
%$$\dot x=f(x,u),\qquad x\in\R^n,\qquad u\in U\subset\R^m,$$
%$f:\R^n\times\R^m\rightarrow\R^n$ being ${\C}^1$, and the cost
%function
%$$J=\int_0^T f^0(x,u)dt$$
%where $f_0:\R^n\times\R^m\rightarrow\R$ is ${\C}^1$. Given an initial
%point $x_0$ and target point $\hat x$, by an admissible control we
%understand a piecewise continuous function $u:[0,T]\rightarrow U$ such
%that the solution $x(t)$ of the equation
%$$\dot x=f(x,u(t))$$
%satisfying $x(0)=x_0$ exists on $[0,T]$.

%definition of stratification 

Given a manifold $M$, a stratification of $M$ is a locally finite
collection $\P$ of disjoint regular submanifolds such that the following
holds. If $P_1,P_2\in\P$ and $P_1\cap clos(P_2)\not=\emptyset$
then $P_1\subset clos(P_2)$ and $dim(P_1)<dim(P_2)$ (here {\sl clos}
denotes the topological closure and {\sl dim} the topological
dimension of a manifold).
A set for which there exists a stratification is said to be {\sl stratified}.


Let $S\subset\Omega$ be an open domain such that 
%$\hat x\in\bar G$. 
the origin belongs to $S$.
A {\sl regular synthesis} $\Gamma$ in $S$ 
for the control problem (1.1),(1.2) with target
%$\hx$ 
the origin
is a 6--tuple $(\P,\P_1,\P_2,\Pi,\Sigma,v)$ where

\vs
\item{1)} $\P$ is disjoint union of the 
origin
%$\{\hx\}$ 
and a locally finite (in $S$) partition of 
$S\setminus \{
%\hx
0
\}$ into ${\C}^1$ connected submanifolds (called cells),

\vs
\item{2)} $\P$ is a disjoint union of $\P_1$ (cells of type I) and
$\P_2$ (cells of type II), $\Pi:\P_1\rightarrow\P$,
$\Sigma:\P_2\rightarrow\P_1$ and $v:S\rightarrow U$ (the closed loop
control) are maps 
\vs

\nin such that the following properties are satisfied:

\vs
\item{A.} The set $S'=\bigcup \{P\in\P:{\rm dim}\;P<n\}\setminus\{
%\hx
0
\}$ is
a stratified subset of $S$ of dimension $<n$ 

\vs
\item{B.} The function $v$ is $\C^1$ on each cell. If $P_1\in\P$ and
$x\in P_1$, then $f(x,v(x))\in T_x P_1$ (the tangent space to $P_1$ at $x$)
and there exists a time $t_x$
such that the trajectory $\xi_x$ of the equation
$$\dot x=f(x,v(x))$$
with $\xi_x(0)=x$ satisfies $\xi_x(t)\in P_1$ for every $t\in
[0,t_x[$ and $\lim_{t\to t_x} \xi_x(t)\in\Pi(P_1)$. The control
$v$ is continuous on $P_2\cup \Sigma(P_2)$ for every cell $P_2\in
\P_2$.
For each $x\in P_2$ there exists a unique trajectory $\xi_x$ such that
$\xi_x(0)=x$ and $\xi_x(t)\in\Sigma(P_2)$ for $t>0$ small. In this
case $t_x$ is the time in which $\xi_x$ reach $\Pi(\Sigma(P_2))$.
On every cell $P$, $t_x$, $\xi_x$, $u_x(t)=v(\xi_x(t))$ are
continuously differentiable functions of $t,x$ for every $x\in P$,
$t\in[0,t_x]$ (that is can be prolonged to $\C^1$ functions on
$[0,t_x+\ve_x)$ for some $\ve_x>0$)
\vs
\item{C.} The trajectory $\xi_x$ (obtained piecing together the
trajectories on every single cell) is an admissible extremal trajectory
ending at the origin and passing through a finite number of cells
\vs
\item{D.} The value function $V_{\Gamma}$ is continuous.

\vs
\noindent
The definition given by Boltianskii \boltcount{} has the following restrictions:

\vs
\item{i)} Every trajectory $\xi_x $ leaving a first type cell of dimension $k$ 
must enter a cell of dimension $k-1$. 
Every trajectory $\xi_x$ leaving a second type cell of dimension $k$
must enter a first type cell of dimension $k+1$.

\vs
\item{ii)} The only zero dimensional cell must be the point target. The
only cell of dimension one must be a trajectory ending at the target.

\vs
\item{iii)} Every trajectory $\xi_x$ must enter a new cell 
in a transversal way.

\vs

{\bf Remark 2.}
Another definition was given in \sussacount{} using a feedback
control. This definition is more general but strictly related to the
definition given by Brunovsky.
Also in this case the definition was given to prove an existence
theorem for stabilizing synthesis.
While the definition in \sussmcount{} is more general since it is not based on the
existence of any feedback control generating the trajectories. 

\vs

The examples of the following section show optimal syntheses
that do not match the definitions above. Let $\Gamma$ be the
synthesis of Example~1. We have that $V_{\Gamma}$ is discontinuous, hence
$\Gamma$  does not satisfy the point $D$ of the Brunowsky definition. 

The same example show another important feature. It shows the necessity
of the definition of the map $\Sigma$.
Indeed, in this case if
we do not define the map $\Sigma$ then some not optimal
trajectories would be admissible solutions of the discontinuous
feedback. Hence the feedback alone is not sufficient to describe the synthesis.
The hypothesis of the uniqueness of the $\xi_x$ entering
$\Sigma(P_2)$ can be relaxed after adding a specification of $\xi_x$
on an initial interval of time. This was done in \sussacount{}, \susslcount{},
where for every
$x\in P_2\in\P_2$ we have an assigned trajectory $\xi_x$, defined for
an interval of time of length $\ve_x$, that enters $\Sigma(P_2)$.

In any case it is important to specify the cell the trajectory $\xi_x$ is
supposed to enter. If we allow all the trajectories of the discontinuous
feedback then we obtain some not optimal trajectories. For example,
starting from the point $(-1,0)$ we can follow the control $+1$ for
time $2\pi$, reaching again the same point, and then go to the origin
using the control $-1$. This trajectory $\gamma_1$ is admissible indeed at the
point $(-1,0)$ the two controls $\pm 1$ determine the same
velocity vector. However, $\gamma_1$ is clearly not time optimal.
We can also define the trajectory $\gamma_2$,
with domain $[0,+\infty[$,
corresponding to the constant control $+1$ and starting from $(-1,0)$. 
Also this trajectory is
admissible and it neither reaches the origin in finite time nor
approaches it as time goes to infinity.
Notice that the trajectories $\gamma_1$ and $ \gamma_2$ satisfy the equation
$\dot\gamma(t)=f(\gamma(t),v(\gamma(t)))$ for every $t$ in their
domain. Hence they satisfy the equation corresponding to the
discontinuous feedback in any reasonable {\sl weak} sense. 
For definition of
solutions to a discontinuous ODE refer to \filicount.
This phenomenon happens for a single point because the example is in two
dimensions.

The Fuller example itself does not match the definitions above but it
presents a quite regular synthesis. Since every trajectory has an
infinite number of switchings the synthesis does not satisfy point C.

More dramatic is the situation for the definition given by
Boltianskii. The synthesis of Example~1 does not match the restriction
ii). There are many cells of dimension one and zero. This restriction is
not satisfied even by very simple examples in which $V_{\Gamma}$ is
continuous. In \piccacount, \piccbcount, \piccccount, \piccdcount,
\sussdcount, \sussecount, \susshcount, \sussicount, \susslcount, it was
studied the synthesis problem and the optimal trajectories properties
for control affine single--input two dimensional systems. There are
some singular trajectories called turnpike. In general these turnpikes
are one dimensional cells of the time optimal synthesis with
target the origin and they are not necessarily ending at the
origin. For some explicit examples see \piccbcount.

The synthesis of Example~2 does not match the restriction i) as it is
easily shown, since some trajectories leave a three dimensional cell
entering a one dimensional cell. 

Finally, in \brunbcount{} it was shown an example not satisfying
condition iii).



%
%\section{Introduction}
%
%The purpose of this note is to present an example of an optimal synthesis
%that can be computed explicitly and happens to exhibit a peculiar feature,
%not often encountered in such problems.  The fact that this feature can
%occur has deep implications regarding the proper way to define the concept
%of an ``optimal feedback,'' as we now explain.
%
%Consider an optimal control problem arising from a controlled dynamics
%of the form
%\xdef\system{\equanosectno}
%$$
%\dot x=f(x,u)\,,\,\,\,x\in\R^n\,,\,\,\,u\in U\,,\numeq
%$$
%where $U$ is a compact convex subset of $\R^m$,
%and $f:\R^n\times U\to\R^n$ is smooth.  Assume a nonvoid {\it target set}
%$S\subseteq\R^n$ is specified, and we want to study the ``time optimal
%synthesis for (\system) with target $S$.''  (More generally, the state 
%space could be an open subset $\Omega$ of $\R^n$ rather
%than $\R^n$ itself, and one could even take $\Omega$ to be 
%be a smooth manifold. The control space $U$ could be a completely abstract
%set, and very minimal regularity requirements on $f$ would suffice.
%In addition, the cost functional to be minimized could be much more general
%than time, for example a functional of the Lagrangian type.  However,
%since the point of the present note is simply to show that a phenomenon can
%happen by giving an example, and the example happens to be very 
%well behaved, there is no need here to discuss the possibility 
%of more general technical conditions.)
%

\section{Examples}

\noindent{\bf Example 1.}

\noindent We study the time-optimal synthesis for the planar system:
\xdef\eqoo{\equanosectno} 
$$
\dot q= F(q)+ u\,G(q)\,,\qquad |u|\leq 1\,, \numeq
$$
where
\xdef\eqot{\equanosectno} 
$$
q\dot= \left(\matrix{x\cr \cr y\cr}\right)\,\,\in\,\,\R^2\,,\qquad
F(q)\,\dot= \left(\matrix{1-{y\over 2}\cr \cr {x+1\over 2}\cr}\right)\,,\qquad
G(q)\,\dot= \left(\matrix{-{y\over 2}\cr \cr {x+1\over 2}\cr}\right)\,,
\numeq
$$
and the target is the origin of $\R^2$.

The trajectories of (\eqoo) corresponding to the constant control $u\equiv
-1$ are straight horizontal lines going from left to right, while those
corresponding to $u\equiv +1$ are circles centered at the point (-1,1),
running counterclockwise.  The optimal synthesis, to be determined below,
is shown in Fig.2.

\vskip 1em


\centerline{\hbox{\psfig{figure=benny.ps,width=3in,height=3in}}}

\medbreak
\centerline{Figure 2}

\vskip 1em

%A {\it control} is a measurable function $u:[a,b]\rightarrow [-1,1]$ for
%some $a,b\in \R$ such that $a\le b$.  (As usual, we identify measurable
%functions that are equal almost everywhere.)  A {\it trajectory} for a
%control $u$ is an absolutely continuous function
%$\gamma:[a,b]\rightarrow\R^2$ verifying
%$\dot\gamma(t)=F(\gamma(t))+u(t)G(\gamma(t))$ for almost every $t\in[a,b]$.
%A trajectory $\gamma:[a,b]\to\R^2$ is {\it optimal} if there is no
%trajectory $\gamma':[a',b']\to \R^2$ such that $\gamma'(a')=\gamma(T)$,
%$\gamma'(b')=\gamma(b)$, and $b'-a'<b-a$.



An arc $\gamma:[a,b]\to\R^2$ is {\it simple} if $\gamma$ is an injective
map. It is clear that an optimal trajectory is simple. 

Since $G(q)=0$ only for $q=(-1,0)$, it is clear that a trajectory
$\gamma:[a,b]\to\R^2$ uniquely determines the corresponding control
$u(\cdot)$, unless the set $\{t:\gamma(t)=(-1,0)\}$ is of positive
measure. In particular, for simple trajectories $\gamma$ ---the only ones
that will be of interest to us--- the control is uniquely determined by
$\gamma$.

We use $\R^k_*$ to denote the space of real row vectors with $k$ entries,
while $\R^k$ denotes the space of column vectors. If $\Omega$ is an open
subset of $\R^k$ and $X:\Omega\to\R^k$ is a vector field of class $C^1$,
then $\nabla X$ will denote the Jacobian matrix of $X$, i.e.  the $k\times
k$-matrix-valued function on $\Omega$ whose rows are the gradients 
of the components of $X$.  If $X,Y$ are vector fields of class $C^1$,
then $[X,Y]$ will denote the Lie bracket of $X$ and $Y$, so 
$[X,Y]=(\nabla Y).X-(\nabla X).Y$.

%A {\it variational covector field} (or ``adjoint vector'') along a
%trajectory $\gamma$ of (\eqoo) ---corresponding to a control $u(\cdot)$---
%is an absolutely continuous function$\lambda:[a,b]\rightarrow\R^2_*$ 
%that verifies the ``adjoint equation''
%$$
%\dot\lambda(t)=-\lambda(t)\cdot\Bigl(\nabla F(\gamma(t))+
%u(t)\nabla G(\gamma(t))\Bigr) \numeq
%$$
%for almost every $t\in[a,b]$. We call $\lambda$ {\it nontrivial}
%if $\lambda(t)\not\equiv 0$. We define the {\it Hamiltonian}
%${\cal H}:\R^2_*\times\R^2\times\R\rightarrow \R$ by:
%$$
%{\cal H}(\lambda,q,u)=\lambda\cdot\Bigl(F(q)+uG(q)\Bigr)\,.\numeq
%$$
%A covector field $\lambda$ along $\gamma$ is said to be {\it minimizing}
%if:
%\xdef\mineq{\equanosectno} 
%$$
%{\cal H}\Bigl(\lambda(t),\gamma(t),u(t)\Bigr)=\min\{{\cal
%H}(\lambda(t),\gamma(t),w):|w|\leq 1\}\,\,\,\,{\rm for\,\,\,
%a.e.}\,\,\,t\in[a,b]\,.\numeq
%$$
%If a trajectory $\gamma$ is optimal, then $\gamma$ is simple, and has to be
%{\it extremal}, meaning that, if $u(\cdot)$ is the corresponding control,
%then the pair $(\gamma,u(\cdot))$ satisfies the necessary condition given
%by the Pontryagin Maximum Principle:
%
%\medbreak
%\item{(E)} {\it there exist a nontrivial
%minimizing variational covector field
%$\lambda$ and a nonnegative constant $\lambda_0$ such that}
%$$
%\lambda_0+{\cal H}(\lambda(t),\gamma(t),u(t))=0
%\,\,\,{\rm for\,\,\,almost\,\,\,every}\,\,\,t\in[a,b]\,.\numeq
%$$



\medbreak\noindent
If $X,Y$ are $\R^2$-valued functions on a subset of $\R^2$, then 
$(X,Y)$ will denote the $2\times 2$ matrix with columns $X,Y$.
In particular, we can define the functions, see \susshcount:
$$
\Delta_{A}=det\,(F,G)\qquad\Delta_{B}=det\,(G,[F,G])\numeq
$$
where ``{\it det\/}'' stands for determinant.
{}From the results of \susshcount, we know that every extremal trajectory 
of (\eqoo) is a finite concatenation of ``bang'' and ``singular'' arcs.
A bang arc is a trajectory for a constant control with values in
the set $\{-1,1\}$ of endpoints of the control interval. 
Singular arcs lie in the set of zeros of the function $\Delta_{B}$.
In our case,
\xdef\eqdelab{\equanosectno} 
$$
\Delta_{A}={x+1\over 2}\,\,\,\,\,{\rm \,\,\,and\,\,\,}\,\,\,\,\,
\Delta_{B}=-{y\over 4}\numeq
$$
so the only singular arcs are those entirely contained in the $x$ axis,
and these correspond to the control $-1$, so they actually are  bang-bang.
Therefore {\it all the extremals of our problem are bang-bang}.

If $\Delta_A(q)\not= 0$ then the vectors $F(q)$ and $G(q)$ are linearly
independent, and we can define the numbers $f(q)$ and $g(q)$ as the
coefficients of the linear combination:
$$
[F,G](q)=f(q)F(q)+g(q)G(q).\numeq
$$
Let $\Omega_A$,  $\Omega_B$ be, respectively, the sets 
$\{q:\Delta_A(q)\neq 0\}$, $\{q:\Delta_B(q)\neq 0\}$, and define
$\Omega_{AB}=\Omega_A\cap\Omega_B$. 
As was proved in \susshcount, if $\gamma$ is an extremal trajectory and
$\gamma(t)\in\Omega_{AB}$ for every $t$, then $\gamma$ is bang-bang with at
most one switching, and this switching can 
only be from the control value $-1$ to the value $+1$ if $f>0$,
and from $+1$ to $-1$ if $f<0$. In our case, it follows from (\eqdelab) that
$$
\Omega_A=\{(x,y):x\neq-1\}\,,\,\,\,\,\,\,\Omega_B=\{(x,y):y\neq 0\}\,,\numeq
$$
and
$$
[F,G]= \left(\matrix{0\cr {1\over 2}\cr}\right)=
{y\over 2(x+1)}F+{2-y\over 2(x+1)}G\,,\numeq
$$
so $\Omega_{AB}$  is the union of four open quadrants $\Omega_{AB,i}$,
$i=1,2,3,4$, defined by 
$$\eqalign{
\Omega_{AB,1}&=\{(x,y):x>-1\,,\,y>0\}\,,\cr
\,\,\,\Omega_{AB,2}&=\{(x,y):x<-1\,,\,y>0\}\,,\cr
\,\,\,\Omega_{AB,3}&=\{(x,y):(x,-y)\in\Omega_{AB,2}\}\,,\cr
\,\,\,\Omega_{AB,4}&=\{(x,y):(x,-y)\in\Omega_{AB,1}\}\,.\cr}\numeq
$$
Clearly,  $f>0$ on $\Omega_{AB,1}\cup\Omega_{AB,3}$ and 
$f<0$ on $\Omega_{AB,2}\cup\Omega_{AB,4}$.  So the control switchings
are from $-1$ to $1$ on $\Omega_{AB,1}\cup\Omega_{AB,3}$ and 
{}from $1$ to $-1$ on $\Omega_{AB,2}\cup\Omega_{AB,4}$.

%Assume now to have two trajectories $\gamma_1$, $\gamma_2$, having the
%same initial point and the same terminal point. Following [], we have
%that if $\gamma_2^{-1}*\gamma_1$, that is the concatenation of
%$\gamma_1$ with $\gamma_2$ run backwards, is a simple curve enclosing
%the region $\cal R$ then:
%$$T(\gamma_1)-T(\gamma_2)=\int \int_{\cal R}
%{\Delta_B\over(\Delta_A)^2}$$
%where $T$ is the time along a trajectory.

To construct an optimal synthesis, 
we will
actually construct a collection ${\cal F}$ of simple
trajectories such that every
optimal trajectory ending at the target is in ${\cal F}$, and
for every initial point $q$ we will find a $\gamma_q\in{\cal F}$
such that if $\delta\in{\cal F}$ steers $Q$ to $0$ and $\delta\neq\gamma_q$
then $\delta$ is not optimal. Since the existence of optimal trajectories
{}from any initial point follows by elementary arguments, we will be able to
conclude that $\gamma_q$ is optimal.

Suppose that $\gamma:[a,b]\to\R^2$ is time-optimal, $\gamma(b)=(0,0)$,
$a<b$, and $\gamma$ arises from an open-loop control $t\to u(t)$.
Then $\gamma$is simple, and 
is a concatenation of finitely many maximal bang arcs.
Since $\gamma$ is an extremal, we can fix 
a nontrivial minimizing adjoint vector $\lambda$  along $\gamma$
and a constant  $\lambda_0\ge 0$ for which (EX4) holds. Define
the {\it switching function}
$$\varphi(t)=\lambda(t)\cdot G(\gamma(t))\,.$$
{}The minimization condition implies that
$$
\varphi(t)\not= 0\,\, \Longrightarrow \,\,u(t)=-sgn(\varphi(t))\,.\numeq
$$
Let $\delta:[c,d]\to\R^2$ be a maximal $u=-1$ piece of $\gamma$, so
$c<d$ and $\delta$ is of the form: 
$$
\delta(t)=(x_0+t,y_0)\,.\numeq
$$
%Since $\delta$ is maximal,
%
%\medbreak
%\item{(i)}~either 
%\itemitem{(i.1)} $c=a$,
%\item{}~or 
%\itemitem{(i.2)} $c>a$ and a control switching from $1$ to $-1$ occurs at $c$,
%\item{}~and
%\item{(ii)}~either 
%\itemitem{(ii.1)} $d=b$,
%\item{}~or 
%\itemitem{(i.2)}$d<b$ and a control switching from $-1$ to $1$ occurs at $c$.

%\medbreak
\noindent For a.e. $t\in[c,d]$, we have
$\dot \lambda=-\lambda\cdot \nabla (F-G)=0$.
So $\lambda=(\lambda_1,\lambda_2)$ is constant and:
$$
\varphi(t)=-\lambda_1 {y_0\over 2}+\lambda_2 {x_0+t+1\over 2}\numeq
$$
is a linear function of $t$.  Since a linear function cannot have more than
one zero unless it vanishes identically, we conclude that
$a<c<d<b$ can only be true if $y_0=0$. 
If $d=b$, then of course $y_0=0$, since $\gamma(b)=(0,0)$. 
So one of the following two possibilities occurs:

\medbreak
\item{(I)}~$\delta$ is contained in the $x$ axis,
\item{(II)}~$a=c$. 
\medbreak

This shows that all the maximal $u=-1$ pieces of $\gamma$ are entirely 
contained in the $x$ axis, with the only possible exception that 
$\gamma$ may contain an initial horizontal segment $[a,d]$ for which $y\neq
0$. 

%In particular there are extremal trajectories that switch from
%control $+1$ to control $-1$ at a point $(-L,0)$ with $L<0$, and then
%reach the origin with control $-1$.

Let us now analyze the adjoint equation for a maximal $u=+1$ piece
$\delta:[c,d]\to\R^2$ of $\gamma$, such that $c<d$. Clearly, $\delta$ is of
the form: 
\xdef\eqxycircle{\equanosectno} 
$$
x(t)=-1+r \cos(t+\theta)\,,\qquad y(t)=1+r \sin(t+\theta)
\,\qquad c\le t\le d\,,\numeq
$$
where $r>0$ is a constant and $\theta\in [0,2\pi]$. (The possibility
that $r=0$ is obviously excluded, because $\gamma$ is time-optimal and
$c<d$.)  The  components $\lambda_1,\lambda_2$ of $\lambda$ satisfy
$\dot\lambda_1=-\lambda_2$ and $\dot\lambda_2=-\lambda_1$, 
so there exist constants $A>0$, $\theta_1\in[0,2\pi]$, such that
$$
\lambda_1(t)=A\cos(t+\theta_1)\,,\qquad\lambda_2(t)=A\sin(t+\theta_1)
\,\qquad c\le t\le d\,.\numeq
$$
Therefore, up to multiplication by a positive constant,
$$
\varphi(t)= -\cos(t+\theta_1)+r\sin(\theta_1-\theta)\,.\numeq
$$
Now suppose that one of the two endpoints $c,d$ of the domain 
of $\delta$ is a switching time and the corresponding point
in $\R^2$ lies in the $x$ axis.  That is, we assume that
$h\in\{c,d\}$ is such that $\varphi(h)=0$ and $y(h)=0$.  Then 
\xdef\rsintheta{\equanosectno} 
$$
r\sin(h+\theta)=-1\,,\numeq
$$
so $\varphi(t)=-\cos(t+\theta_1)
+r\Bigl(\sin(h+\theta_1)\cos(h+\theta)
-\cos(h+\theta_1)\sin(h+\theta)\Bigr)$, and then
$$\varphi(t)=-\cos(t+\theta_1)+\cos(h+\theta_1)+r\sin(h+\theta_1)\cos(h+\theta)
\,.\numeq$$
In particular, 
\xdef\iszero{\equanosectno} 
$$
0=\varphi(h)=r\sin(h+\theta_1)\cos(h+\theta)\,,\numeq
$$
so $\varphi(t)= -\cos(t+\theta_1)+\cos(h+\theta)$. If $r\neq 1$
then (\rsintheta) implies $0\ge \sin(h+\theta)\neq -1$, so 
$\cos(h+\theta)\neq 0$.  But then (\iszero) implies 
$\sin(h+\theta_1)=0$, so 
$\cos(h+\theta_1)=\pm 1$.  The possibility that 
$\cos(h+\theta_1)=1$ is excluded, for in that case
$\varphi$ would be $>0$ throughout $(c,d)$, contradicting the minimization
condition. So 
$\varphi(t)= -1-\cos(t+\theta_1)$. This shows that 
$\varphi(t)<0$ for $t\in[c,d]-\{h\}$, unless $d-c=2\pi$.  But 
if $d-c=2\pi$ then $\gamma$ would not be simple.
So $d-c<2\pi$, and we have shown that the other endpoint 
$h'$ of $[c,d]$ cannot be a switching point, and therefore $h'\in\{a,b\}$.

It then follows that one of the following possibilities must occur:

\medbreak
\item{(i)} $d=b\,$;
\item{(ii)} $a=c<d<b$ and $\gamma(d)$ lies in the $x$ axis;
\item{(iii)} $a<c<d<b$ and $\gamma(d)=(-1,0)$.

\medbreak\noindent
Indeed, if $d<b$ but $\gamma(d)$ is not in the $x$ axis, then we could let
$\delta'$ be the maximal $u=-1$ piece of $\gamma$ that starts at time $d$,
and our analysis of the $u=-1$ pieces of $\gamma$, applied to $\delta'$,
would show that $\delta'$ cannot in fact occur, since $\delta'$ is not an
initial segment of $\gamma$ and is not contained in the $x$ axis.  So, if
$d<b$ then $\gamma(d)$ lies in the $x$ axis, and $d$ is a switching time of
$\gamma$. Our previous argument shows that $c=a$, unless $r=1$.  But if
$r=1$, and $\gamma(d)$ is in the $x$ axis, then $\gamma(d)=(-1,0)$.  So
$d<b$ implies that either (ii) or (iii) must hold.

We now combine the results about the two types of pieces and provide a
complete description of the optimal trajectories ending at the origin. 
Let us use $L$, $L_0$, $C$ to denote, respectively, the trajectory
types ``$u=-1$ and not contained in the $x$ axis,''
``$u=-1$ and contained in the $x$ axis,'' and ``$u=1$.'' (The symbols
$L$ and $C$ stand for ``line'' and ``circle.'')  Let us 
use $T_1T_2\ldots T_m$ to denote the trajectory type 
``$T_1$ followed by $T_2\,\,\ldots\,\,\,$followed by $T_m$''.

With this notation, we have established that a nontrivial optimal
trajectory $\gamma$ ending at the origin is of one of the following seven
types: $L_0$, $C$, $L_0C$, $LC$, $CL_0$, $LCL_0$, $CL_0C$.  
(Concatenations
of six or more types are ruled out because any such concatenation would
have to contain two different $C$ pieces for which (iii) above holds, so
$\gamma$ would go at least twice through the point $(-1,0)$, and would not
be simple.  excluded for the same reason.  The type $L_0CL_0$ is excluded
because the last switching would have to happen at $(-1,0)$, so $C$ would
have to be the circle with center $(-1,1)$ and radius $1$, which intersects
the $x$ axis only at $(-1,0)$.  So the other switching would also happen at
$(-1,0)$, and $\gamma$ would not be simple.  The exclusion of $L_0CL_0$
implies in turn that the types $CL_0CL_0$, $LCL_0CL_0$ and $L_0CL_0CL_0$
are ruled out as well. The types $L_0CL_0C$ and $LCL_0C$ are excluded
because the switching from $C$ to $L_0$ would have to happen at $(-1,0)$,
so the $L_0$ piece following the switching has to be a segment going from
$(-1,0)$ to a point $(\al,0)$ and $\al$ must satisfy $-1<\al<0$, for if
$\al$ was $\ge 0$ then the $L_0$ piece would already reach the origin
before switching to the last $C$. But if $-1<\al<0$ then the $C$ arc
starting at $(\al,0)$ does not go through the origin. Finally, having
excluded $L_0CL_0C$ we are also forced to exclude $CL_0CL_0C$.)
The $LCL_0$ trajectories are further restricted by the condition that the
switching from $C$ to $L_0$ must happen at the point $(-1,0)$.  Let us call
those $LCL_0$ trajectories that satisfy this extra condition ``good.''

It turns out that the optimal synthesis involves all seven types listed
above, but not all trajectories of one of these types are optimal, so we
need a finer analysis.  We let ${\cal F}$ be the set of all simple
trajectories that end at the origin and are either of one of the 
six types $L_0$, $C$, $L_0C$, $LC$, $CL_0$, $CL_0C$, or of type
$LCL_0$ and good.

We now observe that $\dot x=1-y$ along a $u=1$ arc, so $\dot x>1$ as long
as $y<0$, $u=-1$. Therefore, if $q_1=(x_1,y_1)$ and $q_2=(x_2,y_2)$ satisfy
$y_1=y_2\le 0$, $x_1<x_2$, $x_1+x_2=-2$, so that $q_1$ and $q_2$ can be
joined both by a $C$ arc and an $L$ (or $L_0$) arc, then the $C$ arc is
contained in the lower half-plane, and is faster than the $L$ or $L_0$ arc.
In particular, if we let $L_0(\beta)$ denote, for $\beta>0$, the $L_0$ arc
going from $(-\beta,0)$ to $(0,0)$, we see that {\it $L_0(\beta)$ is not
optimal if $\beta>1$}, because the piece of $L_0$ going from $(-\beta,0)$
to $(-2+\beta,0)$ can be replaced by a faster type $C$ arc.

Next we show that {\it $L_0(1)$ is optimal}. It suffices
to notice that there is no trajectory from $(-1,0)$ to $(0,0)$ of any of
the types $C$, $LC$, $LCL_0$, and exactly one trajectory of each of the
types $L_0$, $L_0C$, $CL_0$, and $CL_0C$, but the last three are not
simple, so $L_0(1)$ is left as the only possible candidate.

We now study the optimal trajectories that are of one of the types $C$,
$L_0C$, $LC$, and $CL_0C$.  Let $\gamma:[a,b]\to\R^2$ be such a trajectory,
and take $b=0$, so $\gamma(0)=(0,0)$.  

If $\gamma$ is a $C$ arc, then $a>-2\pi$, for otherwise $\gamma$ would
contain a full loop. For $0<a<2\pi$, we let $C(a)$ be the $C$ arc
$\gamma:[-a,0]\to\R^2$ such that $\gamma(0)=(0,0)$.  It will be shown below
that the arcs $C(a)$ are all optimal.

Next suppose that $\gamma$ is $LC$.  Since the $C$ piece satisfies
(\eqxycircle), we must have $r=\sqrt{2}$ and $\theta=-{\pi\over 4}$. Let 
$c$ be the time when the switching from $L$ to $C$ occurs. On
$[c,0]$, the switching function $\varphi$ is given, up to multiplication 
by a positive constant, by:
$$
\varphi(t)=\sqrt{2}\sin\Bigl(\theta_1-{\pi\over 4}\Bigr)-\cos (t+\theta_1)
=\sin{\theta_1}-\cos{\theta_1}-\cos (t+\theta_1).
$$
So
$$
\varphi(t)=(1+\sin t)\sin{\theta_1}-(1+\cos t).\cos{\theta_1}
$$
In addition, $c$ cannot be $-\pi$, because  $c=-\pi$ would correspond
to $\gamma(c)=(-2,2)\in Q_{AB,2}$,
and we know that a switching from $u=-1$ to $u=1$ is not permitted
at points in $Q_{AB,2}$.

If $\sin{\theta_1}=0$, then $\cos{\theta_1}=\pm 1$, so 
$\cos{\theta_1}=1$, because $\varphi(0)\le 0$.  
So $\varphi(t)=1+\cos t$, and $\varphi(c)=0$ would imply
$c=-\pi$. Since this possibility has been ruled out,
we conclude that $\sin{\theta_1}\neq 0$.  We can then define
$$
\psi(t)={1+\sin t\over 1+\cos
t}\,,\,\,\,\,\nu={\cos{\theta_1}\over\sin\theta_1}\,.
$$
Then $\psi$ is well defined and smooth except at values of 
$t$ such that $t=\pi + 2k\pi$, $k$ an integer.  In our case, $\psi$ is 
defined on the whole interval $[c,0]$, except at $t=-\pi$ (if
$-\pi\in[c,0]$).   Since $c\neq-\pi$, $\psi(c)$ is defined,
and the switching condition $\varphi(c)=0$ says that $\psi(c)=\nu$.  If
$\sin{\theta_1}>0$, then $\varphi(t)<0$ iff $\psi(t)<\nu$ and 
$\varphi(t)=0$ iff $\psi(t)=\nu$. Then $\psi(0)\le \nu$ and,
since $\lim_{t\to -\pi}\psi(t)=+\infty$, we see that $c$ must 
satisfy $-\pi<c<0$. If
$\sin{\theta_1}<0$, then $\varphi(t)<0$ iff $\psi(t)>\nu$. So
${1\over 2}=\psi(0)\ge\nu$. 
A straightforward
computation shows that ${d\psi\over dt}(t)>0$ for
$-2\pi\le t<-\pi$. Since $\psi(-2\pi)=\psi(0)\ge\nu$, we conclude 
that $\psi(t)>\nu$ for $-2\pi\le t<-\pi$. So, once again, the switching
cannot occur at a point in $[-2\pi,-\pi)$, and then 
$c\in(-\pi,0)$.

So we have shown in all possible cases that $-\pi<c<0$. If
$-\pi<c<-{\pi\over 2}$, then $\gamma(c)\in Q_{AB,2}$, so a switching
{}from $L$ to $C$ is not permitted. If $c=-{\pi\over 2}$, then 
$\gamma(c)$ lies in the $x$ axis, so the previous piece is $L_0$ rather
than $L$.   So $-{\pi\over 2}<c<0$.   If ${\pi\over 4}<c$, then 
$\gamma(c)\in Q_{AB,4}$, and the switching is not permitted either.
So the only possibility left to us is $-{\pi\over 2}<c\le-{\pi\over 4}$.

For $0<\sigma<2\pi$, $s>0$, let $LC(s,\sigma)$ denote the trajectory 
that consists of a $u=-1$ arc during time $s$, followed by a $u=1$ arc
during time $\sigma$, and ending at the origin.   We have shown that 
{\it $LC(s,\sigma)$ is not optimal unless 
${\pi\over 4}\le\sigma<{\pi\over 2}$.}

We now show that {\it when ${\pi\over 4}\le\sigma<{\pi\over 2}$ then
$LC(s,\sigma)$ is optimal.}  To see this, let $q$ be the starting point of
$LC(s,\sigma)$, for $s>0$, ${\pi\over 4}\le\sigma<{\pi\over 2}$.  It is
clear that $LC(s,\sigma)$ is the only $LC$ trajectory from $q$ to $(0,0)$.
Also, it is easy to see that there are no trajectories from $q$ to $(0,0)$
of the types $L_0$, $C$, $L_0C$, or good $LCL_0$. 
There is exactly one $CL_0$ trajectory, but it is easy to see that the time
$\hat\tau$ along this trajectory is larger than the time
$\tau$ along $LC(s,\sigma)$.  (Actually, $\tau<|x|<\hat\tau-{3\pi\over
2}$.) Finally, there is exactly one $CL_0C$ trajectory $\gamma$, 
which can also be ruled out. (If $\bar x=-1-\sqrt{2-(y+1)^2}$, so 
$\bar q=(\bar x,y)$ is the point where 
$LC(s,\sigma)$switches from $L$ to $C$, then both 
$LC(s,\sigma)$ and $\gamma$ go through $\bar q$ and coincide from that point
on. So it suffices to compare the parts of 
$LC(s,\sigma)$ and $\gamma$ up to $\bar q$. The time along 
$LC(s,\sigma)$ is $\bar x-x$, and that along $\gamma$ is 
$>\sqrt{(1+x)^2+(1-y)^2-1}-1+{3\pi\over 2}$, so it suffices to notice 
that $\sqrt{(1+x)^2+(1-y)^2-1}+{3\pi\over 2}+x-\sqrt{2-(y+1)^2}>0$.)

For $s>0$, we let $L_0C(s)$ denote the trajectory that consists of a $u=-1$
arc during time $s$ followed by a $u=1$ arc during time ${\pi\over 2}$, and
ending at the origin.  Then is is easily shown that {\it $L_0C(s)$ is
optimal for every $s>0$.}  (Indeed, if $q=(x,0)$ is the starting point of
$L_0C(s)$, for $s>0$, then $L_0C(s)$ is the only $L_0C$ trajectory from $q$
to $(0,0)$, and there are no simple 
trajectories from $q$ to $(0,0)$ of the types
$C$,  $LC$, $CL_0$, $LCL_0$ or $CL_0C$.  There is an $L_0$ arc, namely,
$L_0(|x|)$, but we know that this arc is not optimal, because $|x|>1$.)

Now define $R_1=\{(0,0)\}$, $R_2=(-1,0)\times\{0\}$, $R_3=\{(-1,0)\}$. 
For $q=(0,0)$, let $\Gamma_q$ be the trajectory $\gamma$ with domain $\{0\}$
such that $\gamma(0)=(0,0)$. For $q=(x,0)\in R_2\cup R_3$, let 
$\Gamma_q=L_0(|x|)$.  Then the $\Gamma_q$, $q\in R_1\cup R_3\cup R_3$ are
optimal. 

For $r>0$, let $D(r)$ be the open disc with center $(-1,1)$ and radius $r$,
and let $\partial D(r)$ denote the boundary of $D(r)$.  Let $R_4=D(1)$,
$R_5=\partial D(1)\backslash\{(-1,0)\}$. Then a point $q\in\R^2$ can be
joined to $(0,0)$ by a good $LCL_0$ trajectory if and only if $q\in R_4$,
and in that case the good trajectory is unique, and we will use $LCL_0(q)$
to denote it.  Then {\it the arcs $LCL_0(q)$, for $q\in R_4$, are
optimal}. To see this, observe that if $q\in R_4$ then the only
trajectories in ${\cal F}$ other than $LCL_0(q)$ that go from $q$ to the
origin are either $LCL_0$ but not good, or $LC$ with a switching from $L$
to $C$ at a time $c$ that violates the requirement that $-{\pi\over
2}<c$. So all the alternative candidates are excluded, and $LCL_0(q)$ must
be the optimal trajectory.

A similar argument shows that if $q\in R_5$ then the unique $CL_0$
trajectory going from $q$ to $(0,0)$ ---henceforth labeled $CL_0(q)$--- is
optimal.

Next, let $R_6=D(\sqrt{2})\backslash (R_2\cup D(1))$.  
If $q\in R_6$, then the obvious
candidate for optimality is the arc $\Gamma_q$ of type $CL_0$ obtained by
following the $C$ arc through $q$ until $R_2$ is reached, and then
continuing along $R_2$ towards the origin.  We now prove that {\it the
arcs $\Gamma_q$, $q\in R_6$, are optimal}.  To see this, observe that if
$q=(x,y)\in R_6$ and $y\neq 0$ then there are no simple 
arcs from $q$ to $(0,0)$
of the types $L_0$, $C$, $L_0C$, $CL_0C$.  There is an $LC$ arc, but (i)~if
$y<0$ then this arc switches from $L$ to $C$ in $Q_{AB,4}$, so it is not
optimal; (ii)~if $y>0$, then the arc is $LC(s,\sigma)$ for some
$\sigma\in\pi,2\pi)$, so it is not optimal either.  If $y<0$ or $y>2$ or
$x>-1$, then there is no good $LCL_0$ arc. If $0<y<2$ and $x<-1$ then there
is a good $LCL_0$ arc, but it switches from $L$ to $C$ at a point in
$Q_{AB,2}$, so this arc is not optimal.  Finally, if $y<0$ then $\Gamma_q$
is the only $CL_0$ arc, and if $y>0$ then there are two such arcs, but the
other one is not optimal, because it contains $L_0(\beta)$ for some
$\beta>1$.  This proves the optimality of the arcs $\Gamma_q$ for $q\in
R_6$, $y\neq 0$. A similar analysis works for $y=0$.  (Alternatively, if
$q=(x,0)$ then $\Gamma_q$ is a cofinal piece of $\Gamma_{q'}$ for some
$q'=(x',y')\in R_6$ such that $y'\neq 0$, so $\Gamma_q$ is optimal.)

Next, we let $R_7=\partial D(\sqrt{2})\backslash\{(0,0)\}$.  For $q\in R_7$
there is a unique trajectory $\Gamma_q$ of type $C$ going from $q$ to
$(0,0)$. The optimality of $\Gamma_q$ for $q\in R_7$ is proved by
considerations similar to those used for $R_6$.

We let $R_8=(-\infty,-2)\times\{0\}$, $R_9=\{(x,y):1-\sqrt{2}<y<0\,,\,
x<-1\,,\,(x+1)^2+(1-y)^2>2\}$, $R_{10}=(-\infty,-1)\times\{ 1-\sqrt{2}\}$.
Then $R_9\cup R_{10}$ is exactly the set of starting points $q$ of the
trajectories $LC(s,\sigma)$, for $s>0$, ${\pi\over 4}\le\sigma<{\pi\over
2}$, and we know that for each $q\in R_9\cup R_{10}$
the $LC(s,\sigma)$ trajectory is unique and optimal. We define $\Gamma_q$ 
to be this trajectory. A similar fact is true for $R_8$, with 
$L_0C(s)$ rather than $LC(s,\sigma)$, and $\Gamma_q$ is then defined 
to be the optimal trajectory.   

Finally, we let $R_{11}=\R^2\backslash(R_1\cup\ldots\cup R_{10})$.  For
$q\in R_{11}$, we let $\Gamma_q$ be the unique simple 
$CL_0C$ trajectory from $q$
to $(0,0)$. We show that {\it $\Gamma_q$ is optimal for all $q\in R_{11}$}.
Let $\gamma\in {\cal F}$ be an extremal trajectory starting at $q$ and
verifying $\gamma(0)=0$. There is no such trajectory of type $L_0$,$C$
or $L_0C$. The case $LC$ is impossible, otherwise $\gamma$ would have
a switching at time $c\in [-5\pi/4,-\pi/2[$. 
If $\gamma$ is of type $CL_0$ then it contains a not optimal
trajectory $L_0(\beta)$, $\beta>1$.
Finally, if $\gamma$ is of type $LCL_0$ good then $\gamma(t)=p\in
R_6$ for some $t$, but $\Gamma_p$ is not the restriction of $\gamma$
to $[t,0]$. Hence, if $\gamma$ is optimal then it is of type $CL_0C$.


Since we have considered all the extremal trajectories 
this concludes the construction.


This ends the analysis and proves that the synthesis of Fig.1 is the
optimal one.

Let define the feedback control $u(x,y)$ to be equal to $0$ on 
$R_1$, to $-1$ on the set
$R_2\cup R_3\cup R_4\cup R_8\cup R_9\cup R_{10}$,  
and to $+1$ on $R_5\cup R_6\cup R_7\cup R_{11}$. 
Then the $+1$ trajectory starting from the point $(-1,0)$ is an admissible
trajectory for this feedback control but it is not optimal.

\bigbreak


%%%%%%%%%%%%%%SECOND EXAMPLE


\noindent{\bf Example 2.}

\noindent The following example provides a nice synthesis that is optimal but
does not match the restriction i) of the definition of
Boltianskii. Consider the system:
$$\cases{\dot x = u {\sqrt{x^2+y^2}\over \sqrt{1+x^2+y^2}} &\cr 
         \dot y = v {\sqrt{x^2+y^2}\over \sqrt{1+x^2+y^2}} & 
                                           $u^2+v^2+w^2\leq 1$ \cr
         \dot z = w {1\over \sqrt{1+x^2+y^2}} &\cr}.$$
and the problem of stabilization to the origin in minimum time.
Consider the following regions:
$$R_0=\{(0,0,0)\},\qquad  R_1=\{(x,y,z): x=y=0, z>0\},\qquad
R_2=\{(x,y,z):x=y=0,z<0\},$$
$$R_3=\{(x,y,z):x^2+y^2>0\}.$$
We assign a feedback control $(u_i(x,y,z),v_i(x,y,z),w_i(x,y,z))$ for
every region $R_i$ in the following way:
$$u_0=v_0=w_0=0,\qquad u_1=v_1=0,\;w_1=-1\qquad
u_2=v_2=0,\;w_2=1,$$
$$u_3=-{x\over\sqrt{x^2+y^2}},\;v_3=-{y\over\sqrt{x^2+y^2}},
\;w_3=0.$$
Hence starting from the point $(x,y,z)$ we first reach the $z$ axis
using control $(u_3,v_3,w_3)$, and then we reach the origin running on
the $z$ axis. In this way we have constructed a synthesis $\Gamma$.
It is possible to prove that this synthesis $\Gamma$ is optimal applying
Theorem~2.1. However, in this case it is immediate to verify that the
associated value function
$$V_{\Gamma}=|z|+\sqrt{x^2+y^2}$$
satisfies the Hamilton--Jacobi equation on the set
$\{(x,y,z):x^2+y^2>0,z\not=0\}$. Indeed:
$$\inf_{u^2+v^2+z^2\leq 1}\left(
{\partial V\over \partial x}\,u\,{\sqrt{x^2+y^2}\over
\sqrt{1+x^2+y^2}}+
{\partial V\over \partial y}\,v\,{\sqrt{x^2+y^2}\over
\sqrt{1+x^2+y^2}}+
{\partial V\over \partial z}\,w\,{1 \over
\sqrt{1+x^2+y^2}} \right)=$$
$$=-\left\|\left({x\over \sqrt{1+x^2+y^2}},
{y\over \sqrt{1+x^2+y^2}},
{1\over \sqrt{1+x^2+y^2}} \right)\right\|=-1.$$

Notice that every trajectory starting from a point of the three
dimensional cell $R_3$ either reaches a one dimensional cell,
i.e. $R_1,R_2$, or reaches the zero dimensional cell $R_0$. Hence
$\Gamma$ does not match the restriction i) of section~4.


\bigbreak
\appen{A}{Differentiation of trajectories with respect to a parameter} 

\noindent The purpose of this appendix is to prove a general theorem on
differentiation with respect to a parameter $p$, taking values in a normed
space  $P$, of a family of trajectories $x_p:[a,b]\mapsto\R^n$
of vector fields $f_p$.  We will state and prove the theorem under
minimal hypotheses, much weaker than what is actually needed for the main 
results of this paper, because the theorem has other applications, such as
the theory of envelopes, where the more general statement is useful.

If $\dot x_p(t)=f_p(x_p(t),t)$, then formal differentiation with respect
to $p$ at $p=p_0$ in some direction $v$ yields the variational equation
\xdef\varformal{\equanoapno}
$$
\dot y_{p_0,v}(t)={\partial f_{p_0}\over \partial{x}\,\,\,}
\Bigl(x^{p_0}(t),t\Bigr).y_{p_0,v}(t) 
+\lim_{\ve\downarrow 0}{w_{p_0+\ve v}(t)-w_{p_0}(t)\over \ve}\,,\numeqap
$$
where $y_{p_0,v}(t)$ is the directional derivative of $x_p(t)$ at $p_0$ in
the $v$-direction, and 
$w_p(t)=f_p(x_{p_0}(t),t)$.   The technical problem that will concern
us here is to make (\varformal) rigorous even when the limit in the
right-hand side only exists in a weak sense. For example, suppose 
that $n=1$ and we are looking at the  control system $\dot x=u$, and 
a family of ``bang-bang'' controls $u_s$ depending on $s\in[0,\infty[$,
such that $u_s(t)=1$ for $t<s$, and $u_s(t)=-1$ for $t\ge s$.  Let 
$x_s:[0,\infty[\,\mapsto\R$ be the solution of $\dot x=u_s$, $x(0)=0$. Then 
$x_s(t)=t$ for $t\le s$, and $x_s(t)=2s-t$ for $t\ge s$. Given any
$T>0$, the limit $\lim_{\ve\to 0}{u_{s+\ve}^T-u_s^T\over\ve}$
---where $u_s^T$ is the restriction of $u_s$ to $[0,T]$---
does not exist pointwise, but exists in the weak$^*$ sense 
---regarding the $u_s$ as members of the dual of the space of continuous
functions on $[0,T]$--- and equals $2\delta_s$, where $\delta_s$ is the 
Delta function at $s$.  If we let $y_s(t)$ be the derivative of $x_s(t)$
with respect to $s$ ---so $y_{s}(t)=y_{s,1}(t)$--- then $y_s$ 
satisfies the differential equation $\dot y_s(t)=2\delta_s(t)$, which makes
better sense in integrated form: $y_s(t)=2\chi_{[s,\infty[}(t)$.
This says, in particular, that $y_s(T)=2$ for $s<T$, and $y_s(T)=0$
for $s>T$.    

As this example shows, the variational equation that arises
when the limit of (\varformal) exists in a weak sense is a linear equation
$\dot y(t)=A(t)y(t)+w(t)$ where the ``input'' $w$ is the formal derivative
of a function $W$ which is not necessarily absolutely continuous or even
continuous. So our first task will be to study the solutions of linear 
systems with ``generalized inputs'' $w$.

\medbreak
Let $-\infty<a<b<\infty$, and let $n$ be a positive integer.  
If $A\in L^1([a,b];\R^{n\times n})$, and $\bar Y\in\R^n$,
then it is well known that the initial value problem
\xdef\inhom{\equanoapno}
$$
\dot Y(t)=A(t)\cdot Y(t)+w(t)\,,\,\,\,\,Y(a)=\bar Y\numeqap
$$
has a unique solution for every initial condition and every map 
$w\in L^1([a,b],\R^n)$.  Because (\inhom) can also be written formally as
\xdef\inhomfor{\equanoapno}
$$
dY(t)=A(t)\cdot Y(t)\,dt+w(t)\,dt\,,\,\,\,\,Y(a)=\bar Y\numeqap
$$
we will use $Y_{A,w\,dt,\bar Y}$ to denote the unique solution of (\inhom). 

As explained before, it will be important for us to be able to solve
(\inhom) for ``inputs'' $w$ more general than integrable functions. The
appropriate class of inputs turns out to be that of {\sl formal derivatives
of bounded measurable functions}.  Precisely, we let $BM([a,b],\R^n)$
denote the space of all bounded Lebesgue measurable functions
$W:[a,b]\to\R^n$, and use $BM_0([a,b],\R^n)$ to denote the set of those
$W\in BM([a,b],\R^n)$ such that $W(a)=0$.  (In $BM([a,b],\R^n)$ we do {\it
not} identify two functions that are equal almost everywhere.)  We endow
$BM([a,b],\R^n)$ with the norm 
$$||W||_{BM}=\sup\{||W(t)||:t\in[a,b]\}\,.\numeqap
$$
Then $BM([a,b],\R^n)$ is a Banach space, and $BM_0([a,b],\R^n)$ is a closed
subspace of $BM([a,b],\R^n)$.  The space $C^0([a,b],\R^n)$ of continuous
maps $W:[a,b]\mapsto\R^n$ is of course a closed subspace of $BM([a,b],\R^n)$,
but $L^\infty([a,b],\R^n)$ is {\sl not} a subspace of $BM([a,b],\R^n)$,
because in $L^\infty([a,b],\R^n)$ we identify functions that are equal
a.e., but in $BM([a,b],\R^n)$ we do not.

If $W\in BM_0([a,b],\R^n)$ is absolutely continuous, and $w=\dot W$, 
so that, formally,  $dW=w\,dt$,  then we can rewrite (\inhom) formally as
\xdef\inhomint{\equanoapno}
$$
dY(t)=A(t)\cdot Y(t)\, dt+dW(t)\,\,\,\,\,\,\,\,\,\,Y(a)=\bar Y\,.\numeqap
$$
A solution of (\inhomint) is then a continuous map $t\to Y(t)$ such 
that 
\xdef\inhomintsol{\equanoapno}
$$
Y(t)=\bar Y+\int_a^tA(s)\cdot Y(s)\,ds+W(t)\,\qquad{\rm for
\quad all\quad} t\in[a,b]\,.\numeqap
$$ 
Now (\inhomintsol) makes sense for arbitrary $Y\in BM([a,b],\R^n)$,
$W\in BM_0([a,b],\R^n)$, even if $W$ is not absolutely continuous.
So we turn it into a definition:  for a $W\in BM_0([a,b],\R^n)$, a 
{\sl solution} of (\inhomint) is a map
$Y\in BM([a,b],R^n)$ such that (\inhomintsol) holds.     It then follows that 

\medbreak


{\pariim

\item{(EU)} {\sl for every $W\!\!\in\!\!BM_0([a,b],\R^n)$  and every 
$\bar Y\!\!\in\!\!\R^n$ there exists a unique solution $Y$ of (\inhomint).}

}

\medbreak\noindent
Indeed, uniqueness is trivial, since the difference $Y^1-Y^2$ of two solutions
is a solution of the homogeneous problem (i.e. of (\inhom) with $w\equiv 0$
and $\bar Y=0$),  so $Y^1-Y^2\equiv 0$.  To prove existence, we observe that 
(\inhomintsol) is equivalent to 
\xdef\inhomintsolequiv{\equanoapno}
$$
Y(t)-W(t)=\bar Y+\int_a^tA(s)\cdot (Y(s)-W(s))\,ds+\int_a^tA(s)\cdot W(s)
\,ds\,\qquad{\rm for\quad all\quad} t\in[a,b]\,,\numeqap
$$ 
i.e. to $dZ=AZ\,dt+AW\,dt$, $Z(a)=\bar Y$, with $Z=Y-W$.  Since $AW\in
L^1$, because  
$A\in L^1$ and $W$ is bounded and measurable, the equation
$\dot Z(t)=A(t)\cdot Z(t)+A(t)W(t)$ has a unique solution $Z^*$ for which 
$Z^*(a)=\bar Y$.  If we let $Y(t)=Z^*(t)+W(t)$,  and recall that 
$W(a)=0$,  we see that $Y$ is the desired solution of (\inhomint).

We will use $Y_{A, dW,\bar Y}$ to denote the solution of (\inhomint),
whose existence  and uniqueness has just been established.   
In addition to proving (EU), the preceding argument also gives an explicit 
formula for $Y_{A,dW,\bar Y}$, namely:
\xdef\inhomintsolfla{\equanoapno}
$$
Y_{A,dW,\bar Y}=W+Y_{A,AW\,dt,\bar Y}\,.\numeqap
$$
Clearly, the function $Y_{A,AW\,dt,\bar Y}$ is absolutely continuous,
so the regularity properties of $Y_{A,dW,\bar Y}$ ---modulo absolutely
continuous  functions--- are exactly the same as those of $W$. 
(For example: $Y$ is continuous,  right-continuous, or left-continuous 
at a point $t\in[a,b]$ if and only if $W$ is; $Y$ is of bounded variation 
iff $W$ is; $Y$ is absolutely continuous iff $W$ is.)

Gronwall's inequality, applied to (\inhomintsolequiv), tells us that
$$
||Y_{A,dW,\bar Y}(t)-W(t)||\le e^{||A||_{L^1}}\Bigl(||\bar Y||+
||A||_{L^1}||W||_{BM}\Bigr)\,.\numeqap
$$
Therefore
\xdef\ineqyw{\equanoapno}
$$
||Y_{A,dW,\bar Y}(t)||_{BM}\le ||W||_{BM}+e^{||A||_{L^1}}\Bigl(||\bar Y||+
||A||_{L^1}||W||_{BM}\Bigr)\,.\numeqap
$$
On the other hand, (\inhomintsol) also implies that
\xdef\ineqwy{\equanoapno}
$$
||W||_{BM}\le||\bar Y||+||Y_{A,dW,\bar Y}||_{BM}\Bigl(1+||A||_{L^1}\Bigr)\,.\numeqap
$$   
\xdef\theolin{{\rm \appenlet}.\the\theono}
\theoap
Let $\{W_j\}$,  $\{\bar Y_j\}$, $\{A_j\}$ be sequences in 
$BM_0([a,b],\R^n)$, $\R^n$, $L^1([a,b],\R^{n\times n})$, respectively.
Assume that $\{A_j\}$ is bounded in $L^1([a,b],\R^{n\times n})$. 
Let $Y_j=Y_{A_j,dW_j,\bar Y_j}$. Then

\itemitem{(\theolin.1)}  {\sl $\{Y_j\}$ is bounded in $BM([a,b],\R^{n})$ if 
and only if $\{W_j\}$ is bounded in $BM([a,b],\R^{n})$ and $\{\bar Y_j\}$ 
is bounded in $\R^{n}$.}

\itemitem{(\theolin.2)} {\sl if $\{A_j\}$ converges in $L^1([a,b],\R^n)$
to a limit  $A$, $\bar Y\in\R^n$, $\bar Y_j\to\bar Y$, 
$\{W_j\}$ is bounded in $BM([a,b],\R^n)$, 
$W\in BM_0([a,b],\R^n)$, 
and $Y=Y_{A,dW,\bar Y}$, then $Y_j\to Y$ almost everywhere iff $W_j\to W$ 
almost everywhere.  In that case,  $Y_j-W_j$ converges to $Y-W$ uniformly on 
$[a,b]$. In particular, for every $t\in[a,b]$, $Y_j(t)\to Y(t)$ iff 
$W_j(t)\to W(t)$.} 

\beginproof
Statement (\theolin.1) follows from the bounds (\ineqyw) and (\ineqwy).  To
prove (\theolin.2), we first observe that under the hypotheses of
(\theolin.2) we can apply (\theolin.1) and conclude that $\{Y_j\}$ is
bounded in $BM([a,b],\R^n)$.

Let $\tilde Y_j=Y_j-W_j=Y_{A_j,A_jW_j\,dt,\bar Y_j}$, 
$\tilde Y=Y-W=Y_{A,AW\,dt,\bar Y_j}$.  Then 
$$
\tilde Y_j(t)=\bar Y_j+\int_a^tA_j(s)Y_j(s)\,ds
=\bar Y_j+\int_a^tA_j(s)\tilde Y_j(s)\,ds+\int_a^tA_j(s)W_j(s)\,ds\,,\numeqap
$$ 
so
\xdef\forgrony{\equanoapno}
$$
\tilde Y_j(t)-\tilde Y(t)=\int_a^t A(s)\Bigl(\tilde Y_j(s)-\tilde Y(s))\,ds+
R_j(t)\,,\numeqap
$$
where 
$$
R_j(t)=\bar Y_j-\bar Y+\int_a^t\Bigl(A_j(s)-A(s)\Bigr)\tilde Y_j(s)\,ds
-\int_a^t A(s)\Bigl(W_j(s)-W(s)\Bigr)\,ds\,,\numeqap
$$
and also
\xdef\forgronw{\equanoapno}
$$
\tilde Y_j(t)-\tilde Y(t)=\bar Y_j-\bar Y+\int_a^t\Bigl(A_j(s)-A(s)\Bigr)
Y_j(s)\,ds
+\int_a^t A(s)\Bigl(Y_j(s)-Y(s)\Big)\,ds\,.\numeqap
$$
If $W_j-W\to 0$ a.e., then, since the sequences $\{\tilde Y_j\}$, $\{Y_j\}$
and $\{W_j\}$ are uniformly bounded, and $A_j\to A$ in $L^1$, it is clear
that the first two terms of the sum defining $R_j(t)$ go to zero uniformly
as $j\to\infty$.  Also,
$$
\biggl\|\int_a^t A(s)\Bigl(W_j(s)-W(s))\,ds\biggr\|\le \int_a^b||A(s)||\cdot
||W_j(s)-W(s)||\,ds\,,\numeqap
$$
which goes to zero because
$s\to ||A(s)||$ is integrable and the sequence $\{W_j-W\}$ is uniformly
bounded and goes to zero almost everywhere.  So, if we let 
$K_j=\sup\{||R_j(t)||:a\le t\le b\}$, we see that $K_j\to 0$ as
$j\to\infty$. Then Gronwall's inequality, applied to (\forgrony), gives 
$$
||\tilde Y_j(t)-\tilde Y(t)||\le K_je^{||A||_{L^1}}\,.\numeqap
$$
So $\tilde Y_j-\tilde Y\to 0$  uniformly.

If $Y_j-Y\to 0$ a.e., then the first two terms of the right-hand side of (\forgronw) 
go to zero uniformly, and the third one  is bounded by 
$\int_a^b||A(s)||\cdot||Y_j(s)-Y(s)||\,ds$, which also goes to zero. So
$\tilde Y_j-\tilde Y\to 0$  uniformly.

We have thus shown that $\tilde Y_j-\tilde Y\to 0$  uniformly if either
$W_j\to W$ a.e. or $Y_j\to Y$ a.e. This clearly implies  all the conclusions
of (\theolin.2).\proofend
 
A very important subspace of $BM([a,b],\R^n)$ is $BV([a,b],\R^n)$, the space of
all functions $W:[a,b]\mapsto\R^n$ such that (a)~$W$ is of bounded
variation, (b)~$W$ is right-continuous at every point of $]a,b]$, and
(c)~$W(a)=0$.  There is a canonical correspondence between $BV([a,b],\R^n)$ and
the dual space $C^0([a,b],\R_n)^*$ of the Banach space $C^0([a,b],\R_n)$ of
continuous functions $[a,b]\mapsto\R_n$. (We write $R_n$ rather than $R^n$
because we want to think of the members of $C^0([a,b],\R_n)$ as {\sl
row-vector-valued functions}, cf. below.)

The members of $C^0([a,b],\R_n)^*$ are the {\sl $\R^n$-valued Borel
measures on $[a,b]$}.  When $n=1$, they are the {\sl finite signed Borel
measures on $[a,b]$}.  Clearly, $C^0([a,b],\R_n)^*$ is canonically
identified with $\Bigl(C^0([a,b],\R)^*\Bigr)^n$, so the members of
$C^0([a,b],\R_n)^*$ can be thought of as $n$-tuples of finite signed Borel
measures.  The identification map from $BV([a,b],\R^n)$ to
$C^0([a,b],\R_n)^*$ is the one that assigns to a function $W\in
BV([a,b],\R^n)$ the unique Borel measure $\mu_W$ such that
$\mu_W([a,t])=W(t)$ for $a<t\le b$. (The measure of a single-point set
$\{t\}$ is then $\mu_W(t+)-\mu_W(t-)$, which is equal to
$\mu_W(t)-\mu_W(t-)$ if $t>a$ and to $\mu_W(a+)$ if $t=a$.  In particular,
$\mu_W(\{t\})=0$ iff $\mu_W$ is continuous at $t$.)

{}From now on we identify a function $W\in BV([a,b],\R^n)$ with its
corresponding measure $\mu_W$, and write $\int\alpha\cdot dW$ for
$\int\alpha\cdot d\mu_W$.  As explained above, it is convenient to think of
the members of $C^0([a,b],\R_n)$ as row-vector-valued functions
$\alpha=(\alpha_1,\ldots,\alpha_n)$, and of those of $BV([a,b],\R^n)$ as
column-vector-valued functions $W=(W^1,\ldots,W^n)$, so the expression
$\alpha\cdot dW$ stands for the inner product $\sum_k\alpha_k\,dW^k$.
Clearly, $\int\alpha\cdot dW$ is defined, more generally, for an arbitrary
bounded Borel measurable $\R_n$-valued map $\alpha$ on $[a,b]$.

Naturally,  the integral $\int\alpha\,dW$ of a scalar bounded Borel measurable
function $\alpha:[a,b]\mapsto\R$ is also well defined, and is a
vector in $\R^n$.   We emphasize that, if
$W$ is not continuous, then for an integral such as $\int_s^tdW$ to be well
defined we have to be careful to specify whether the integral is to be
interpeted as $\int \chi_{[s,t]}\,dW$ or $\int \chi_{]s,t]}\,dW$ or $\int
\chi_{[s,t[}\,dW$ or $\int \chi_{]s,t[}\,dW$, so we will never write
$\int_s^t\,dW$ unless $W$ is continuous at $s$ and $t$.
 
On $BV([a,b],R^n)$ there are at least three important topologies , namely,
(a)~the one induced by the norm $||\ldots||_{BM}$, (b)~the strong topology
of $BV([a,b],\R^n)$ as the dual of the Banach space $C^0([a,b],\R_n)$,
(c)~the weak$^*$ topology, also corresponding to the duality with
$C^0([a,b],\R_n)$.  The first one will not be of direct interest to us,
except for the fact that, in order to apply Theorem \theolin{}, we will
often need to know that a set is bounded in the norm $||\ldots||_{BM}$. The
second one is induced by the {\sl total variation norm}:
$$
||W||_{TV}=\sup\biggl\{\Bigl|\int \alpha\cdot dW\Bigr|:\alpha\in
C^0([a,b],\R_n)\,,\,||\alpha||_{sup} \le 1\biggr\}\,.\numeqap
$$ 
It is well known that $||W||_{TV}$ is the supremum of the sums
$S_{\pi}(W)$, ranging over all partitions
$\pi\in\Pi([a,b])=\cup_{m}\Pi_m([a,b])$, where $\Pi_m([a,b])$ is the set of
all $m+1$-tuples $(t_0,t_1,\ldots,t_m)$ of members of $[a,b]$ such that
$a=t_0<t_1<\ldots<t_m=b$ and, for $\pi=(t_0,t_1,\ldots,t_m)\in \Pi([a,b])$,
$S_{\pi}(W)\equaldef\sum_{i=1}^m||W(t_i)-W(t_{i-1})||$.  In particular, it
is clear that $||W||_{BM}\le ||W||_{TV}$ for all $W$ in $BV([a,b],\R^n)$.
Therefore, {\sl a subset of $BV([a,b],\R^n)$ which is bounded in $TV$ norm
is also bounded in $BM$ norm}.

We will need the following
\xdef\lemweakconv{{\rm \appenlet}.\the\lemmno}
\lemmap
If a sequence $\{W_j\}$ in $BV([a,b],\R^n)$ weak$^*$-converges to a $W\in
BV([a,b],\R^n)$, then (a)~$\{W_j\}$ is bounded in $TV$ norm, (b)~$W_j(a)\to
W(a)$ and $W_j(b)\to W(b)$, (c)~every subsequence of $\{W_j\}$ has a
subsequence that converges to $W$ at all but at most countably many points
of $[a,b]$, (d)~$\int_a^b\beta(s)||W_j(t)-W(t)||\,dt\to 0$ for every
function $\beta\in L^1([a,b],\R)$.

\beginproof
Statement (a) follows from the uniform boundedness theorem, and Statement
(b) is trivial, since $W_j(a)=0=W(a)$, $W_j(b)=\int 1\,dW_j$, and
$W(b)=\int 1\,dW$.  Statement (d) clearly follows from (c), using the
dominated convergence theorem together with the fact that boundedness in
$TV$ norm implies boundedness in $BM$ norm.

So all we need to do is prove (c).  If $Z^i$ is the $i$-th component of $Z$
for $Z\in BM([a,b],\R^n)$, $i=1,\ldots,n$, then $\{W_j\}$
weak$^*$-converges to $W$ iff $\{W_j^i\}$ weak$^*$-converges to $W^i$ for
each $i$.  From this it clearly follows that it suffices to prove (c) for
$n=1$.

If $n=1$, then every finite signed Borel measure $\mu$ on $[a,b]$ 
admits a unique decomposition $\mu=\mu^+-\mu^-$,  where $\mu^+$ and $\mu^-$
are nonnegative Borel measures that are mutually singular (that is:
there exist two $\mu$-measurable sets $E$ and $F$, such that
$[a,b]=E\cup F$, $E\cap F=\emptyset$, $\mu^+(E)=\mu^-(F)=0$).
This gives, using the identification of $BV([a,b],R)$ with
$C^0([a,b],R)^*$, a decomposition $Z=Z^+-Z^-$ of every $Z\in BV([a,b],\R)$
into a difference of two members $Z^+$, $Z^-$ of  
${\cal M}(a,b)$, the space of all monotonically
nondecreasing functions $W:[a,b]\mapsto\R$ such that $W(a)=0$ and $W$ is
right-continuous at every $t\in]a,b]$.    It is well known that
$||Z||_{TV}=Z^+(b)+Z^-(b)$.   

Now let $\{W_{j(k)}\}_{k=1}^\infty$ be an arbitrary subsequence of $\{W_j\}$, 
and write  $Z_k=W_{j(k)}$. Letting
$Z_k=Z_k^+-Z_k^-$, we have $||Z_k^+||_{TV}=Z_k^+(b)\le||Z_k||_{TV}$, so 
the sequence  $\{Z_k^+\}$ is bounded in $BV([a,b],\R)$. Similarly,  
$\{Z_k^-\}$ is bounded in $BV([a,b],\R)$.   Since the closed balls
of $BV([a,b],R)$ are weak$^*$-compact and metrizable,  there exists
a subsequence $\{Z_{k(\ell)}\}$ of $\{Z_k\}$ such that  
$Z_{k(\ell)}^+$ and $Z_{k(\ell)}^-$ weak$^*$-converge to 
limits $U$, $V$. Then of course  $Z_{k(\ell)}=Z_{k(\ell)}^+-Z_{k(\ell)}^-$ 
weak$^*$-converges  to  $U-V$, and also to $W$,  so $U-V=W$.  (But this does
{\sl not} mean that $U=W^+$ and $V=W^-$.) 

Since the $Z_{k(\ell)}^+$ are in ${\cal M}(a,b)$, it is well known (and
very easy to prove directly) that $\{Z_{k(\ell)}^+(t)\}$ converges 
to $U(t)$ at every point
of continuity of $U$.  Since $U$ is monotonically nondecreasing,
$Z_{k(\ell)}^+(t)\to U(t)$ at all but countably many $t$'s in $[a,b]$.  A
similar conclusion holds for $Z_{k(\ell)}^-$ and $V$. So $Z_{k(\ell)}(t)\to
W(t)$ for all but countably many $t$'s, concluding our proof. \proofend


As a corollary of Lemma \lemweakconv{}, we get the integration by parts 
formula:
\xdef\lemparts{{\rm \appenlet}.\the\lemmno}
\lemmap
If $\lambda:[a,b]\mapsto\R_n$ is absolutely continuous, and $W\in
BV([a,b],\R^n)$, then 
\xdef\forparts{\equanoapno}
$$
\lambda(b)\cdot W(b)=\int \lambda\cdot dW+\int_a^b \dot\lambda(t)
\cdot W(t)\,dt\,.\numeqap
$$

\beginproof
The result is obviously true, from the standard product rule for
derivatives, if $W$ is also absolutely continuous. In the general case, we
use the fact that an arbitrary $W\in BV([a,b],\R^n)$ is weak$^*$-limit of a
sequence $\{W_j\}$ of absolutely continuous functions.  Then
(i)~$\lambda(b)\cdot W_j(b)\to \lambda(b)\cdot W(b)$, because $W_j(b)\to
W(b)$, (ii)~$\int \lambda\cdot dW_j\to\int \lambda\cdot dW$, because
$\{W_j\}$ weak$^*$-converges to $W$, and $\lambda$ is continuous, and
(iii)~$\int_a^b \dot\lambda(t)\cdot W_j(t)\,dt\to
\int_a^b \dot\lambda(t)\cdot W(t)\,dt$, because Part (d) of Lemma
\lemweakconv{} implies that 
$\int_a^b ||\dot\lambda(t)||.||W_j(t)-W(t)||\,dt\to 0$,  since
$\dot\lambda\in L^1$.   \proofend

An important consequence of Lemma \lemparts{} is the following
\xdef\lemadjo{{\rm \appenlet}.\the\lemmno}
\lemmap
Let $W\in BV([a,b],\R^n)$, $A\in L^1([a,b],R^{n\times n})$,
$\bar Y\in\R^n$, and write $Y=Y_{A,dW,\bar Y}$. Suppose
$\lambda:[a,b]\to\R_n$ is absolutely continuous and satisfies
$\dot\lambda(t)=-\lambda(t)A(t)$ for almost all $t$.  Then
\xdef\foradjo{\equanoapno}
$$
\lambda(b)\cdot Y(b)-\lambda(a)\cdot Y(a)=\int\lambda\cdot dW\,.\numeqap
$$

\beginproof
Write $Y(t)=V(t)+W(t)$, where $V(t)=Y(a)+\int_a^tA(s).Y(s)\,ds$.  
Then 
$$
\lambda(b)\cdot Y(b)-\lambda(a)\cdot Y(a)=
\lambda(b)\cdot V(b)-\lambda(a)\cdot V(a)+\lambda(b)\cdot W(b)\,.\numeqap
$$
The function $t\to\lambda(t)\cdot V(t)$ is absolutely continuous
and its derivative is $\dot\lambda(t)\cdot V(t)+\lambda(t)\cdot\dot V(t)$,
which equals $-\lambda(t)A(t)Y(t)+\lambda(t) A(t)Y(t)-\dot\lambda(t)\cdot W(t)$,
i.e. $-\dot\lambda(t)\cdot W(t)$. So 
$$
\lambda(b)\cdot Y(b)-\lambda(a)\cdot Y(a)=\lambda(b)\cdot W(b)
-\int_a^b\dot\lambda(t)\cdot W(t)\,dt\,.\numeqap
$$
Equation (\foradjo) follows from this and the integration by parts formula
(\forparts). \proofend

It is clear that, if $A\in L^1([a,b],\R^{n\times n})$, $\bar Y\in\R^n$, and 
$W\in BM_0([a,b],\R^n)$, then $Y_{A,dW,\bar Y}-W$ is absolutely continuous,
so $W\in BV([a,b],\R^n)$ iff $Y_{A,dW,\bar Y}$ is of bounded variation,
which happens iff $Y_{A,dW,\bar Y}-\bar Y\in BV([a,b],\R^n)$.

\xdef\theolinbv{{\rm \appenlet}.\the\theono}
\theoap
Let $\{W_j\}$,  $\{\bar Y_j\}$, $\{A_j\}$ be sequences in 
$BV([a,b],\R^n)$, $\R^n$, $L^1([a,b],\R^{n\times n})$, respectively.
Assume that $\{A_j\}$ is bounded in $L^1([a,b],\R^{n\times n})$  and 
$\{\bar Y_j\}$ is bounded in $\R^{n}$. Let $Y_j=Y_{A_j,dW_j,\bar Y_j}$ . Then

\itemitem{(\theolinbv.1)}  {\sl $\{Y_j-\bar Y_j\}$ is bounded in 
$BV([a,b],\R^{n})$ if and only if $\{W_j\}$ is bounded in
$BV([a,b],\R^{n})$.}

\itemitem{(\theolinbv.2)} {\sl If $\{A_j\}$ converges in $L^1([a,b],\R^n)$
to a limit $A$, $\bar Y_j\to\bar Y$, $W\in BV([a,b],\R^n)$, and we write
$Y=Y_{A,dW,\bar Y}$, then $\{W_j\}$ weak$^*$-converges to $W$ iff
$\{Y_j-\bar Y_j\}$ weak$^*$-converges to $Y-\bar Y$.  In that case,
$Y_j-W_j-\bar Y_j$ converges to $Y-W-\bar Y$ strongly in
$BV([a,b],\R^n)$. In particular, (a)~$W_j(b)\to W(b)$, (b)~$Y_j(b)\to
Y(b)$, (c)~for every $t\in[a,b]$, $Y_j(t)\to Y(t)$ iff $W_j(t)\to W(t)$.}



\beginproof
We know from theorem \theolin{} that $\{Y_j-\bar Y_j\}$ is bounded in $BM$
norm iff $\{W_j\}$ is.  Since boundedness in $BV$ norm implies boundedness
in $BM$ norm, we see that boundedness of one of the two sequences in $BV$
norm implies that there is a constant $C$ such that $||Y_j(t)||\le C$ for
all $j,t$, and then the integral $I_j=\int_a^b||A_j(t)Y_j(t)||\,dt$ is
bounded by a fixed constant $K$.  But $I_j=||Y_j-\bar Y_j-W_j||_{BV}$.  So
boundedness in $BV$ of one of the sequences $\{W_j\}$, $\{Y_j\}$ implies
boundedness in $BV$ of the other one.

If either $W_j$ weak$^*$-converges to $W$, or $Y_j-\bar Y_j$
weak$^*$-converges to $Y-\bar Y$, then the uniform boundedness theorem
implies that either $\{W_j\}$ or $\{Y_j-\bar Y_j\}$ is bounded in $BV$
norm, so both sequences are bounded in $BV$ norm ---and hence also in
$BM$--- norm by a constant $C$.  Writing $Z_j=Y_j-\bar Y_j-W_j$, $Z=Y-\bar
Y-W$, we have then $Z_j-Z=R_j^1+R_j^2$, where
$$
R_j^1(t)=\int_a^t\Bigl(A_j(s)-A(s)\Bigr)Y_j(s)\,ds\,,\qquad
R_j^2(t)=\int_a^tA(s)\Bigl(Y_j(s)-Y(s)\Bigr)\,ds\,.\numeqap
$$
Then $||R_j^1||_{BV}\le C||A_j-A||_{L^1}$, so $||R_j^1||_{BV}\to 0$.  Also,
$$||R_j^2||_{BV}\le||A||_{L^1}.||\bar Y_j-\bar Y||+
\int_a^b||A(t)||.||Y_j(t)-\bar Y_j-(Y(t)-\bar Y)||\,dt\,.\numeqap
$$  

If $\{Y_j-\bar Y_j\}$ weak$^*$-converges to $Y-\bar Y$, then Lemma
\lemweakconv{} implies that $||R_j^2||_{BV}\to 0$, so $Z_j\to Z$ in
$BV([a,b],\R^n)$.

If $\{W_j\}$ weak$^*$-converges to $W$, then we rewrite $R_j^2$ as
$$
R_j^2(t)=\int_a^tA(s)\Bigl(Z_j(s)-Z(s)\Bigr)\,ds+R_j^3(t)+R_j^4(t)\,,\numeqap
$$
where  $R_j^3(t)=\int_a^tA(s)\Bigl(\bar Y_j-\bar Y\Bigr)\,ds$ and
$R_j^4(t)=\int_a^tA(s)\Bigl(W_j(s)-W(s)\Bigr)\,ds$.  We then 
have $||R_j^3||_{BV}\le||A||_{L^1}.||\bar Y_j-\bar Y||$ and
$||R_j^4||_{BV}\le\int_a^b||A(s)||.||W_j(s)-W(s)||\,ds$.  Lemma
\lemweakconv{} then imples that $||R_j^4||_{BV}\to 0$.  So
$Z_j(t)-Z(t)=\int_a^tA(s).(Z_j(s)-Z(s))\,ds+R_j(t)$, where
$R_j=R_j^1+R_j^3+R_j^4$, so $||R_j||_{BV}\to 0$.  Then Gronwall's
inequality implies that $||Z_j-Z||_{sup}\le e^{||A||_{L^1}}||R_j||_{BV}$.
If $Q_j(t)=\int_a^tA(s).(Z_j(s)-Z(s))\,ds$, then $||Q_j||_{BV}\le
||A||_{L^1}||Z_j-Z||_{sup}$, so $||Q_j||_{BV}\to 0$.  Since
$Z_j-Z=Q_j+R_j$, we see that $||Z_j-Z||_{BV}\to 0$.

So we have shown that if one of the two sequences $\{W_j-W\}$,
$\{Y_j-\bar{Y}_j-(Y-\bar{Y})\}$ weak$^*$-converges to $0$, then
$Y_j-\bar{Y}_j-W_j\to Y-\bar{Y}-W$ strongly in $BV([a,b],\R^n)$.  This
implies all the conclusions of (\theolinbv.2).\proofend
  
\medbreak
Recall that, if $P,Q$ are normed spaces, and $F:\Omega\to Q$ is a map
defined on a neighborhood $\Omega$ of a point $p$ of $P$, then $F$ is said
to be {\sl Fr\'echet differentiable} at $p$ if there exists a bounded
linear map $L$ ---called the {\sl Fr\'echet differential} of $F$ at $p$---
such that $F(p')=F(p)+L(p'-p)+o(||p'-p||)$ as $p'\to p$.  If $X$ is a Banach
space and $Q=X^*$, then we call $F$ {\sl weak$^*$ Fr\'echet differentiable}
at $p$ if for every $x\in X$ the map $\Omega\ni p'\mapsto\langle
F(p'),x\rangle\in\R$ is Fr\'echet differentiable at $p$.  (We use $\langle
x^*,x\rangle$, for $x\in X$, $x^*\in X^*$, as an alternative notation for
$x^*(x)$.)

It is easy to see that in general weak$^*$ Fr\'echet differentiability does
not imply Fr\'echet differentiability.  (For example, if $X=C^0([0,1],\R)$,
$P=\R$, $\Omega=[0,1]$, and $F(t)=\chi_{[0,t]}$ for $t\in\Omega$, then for
every $\alpha\in X$ the map $t\mapsto \langle F(t),\alpha\rangle$ is just
the map $t\mapsto\int_a^t\alpha(s)\,ds$, which is differentiable at every
$t\in]0,1[$.  The weak$^*$-limit $\lim_{h\to 0}h^{-1}(F(t+h)-F(t))$ is
$\delta_t$, the delta function at $t$ (i.e. the functional
$\alpha\to\alpha(t)$), but $||h^{-1}(F(t+h)-F(t))-\delta_t||_{BV}=2$ if
$h\neq 0$ and $t,t+h\in[0,1]$.)

Suppose $P$ is a normed space, $X$ is a Banach space, $p\in P$,
$F:\Omega\to X^*$ is a map defined on a neighborhood $\Omega$ of $p$, and
$F$ is weak$^*$ Fr\'echet differentiable at $p$.  Then for each $x\in X$
there is a bounded linear functional $\theta_x$ on $P$ such that $\langle
F(p+v),x\rangle=\langle F(p),x\rangle+\theta_x(v)+o(||v||)$ as $v\to
0$ in $P$. Clearly, $\theta_x$ is unique, and is given by the formula
$$
\theta_x(v)=\lim_{h\to 0}h^{-1}\langle F(p+hv)-F(p),x\rangle\,.\numeqap
$$
For any given $v\in P$, use $\theta^v$ to denote the map
$x\to\theta_x(v)$. Then $\theta^v$ is a limit of linear maps, so it is
linear.  If $\{h_j\}$ is any sequence going to $0$, then the sequence of
linear functionals $\nu_j$ on $X$ given by $\nu_j(x)=h_j^{-1}\langle
F(p+h_jv)-F(p),x\rangle$ converges pointwise to $\theta^v$. Since each
$\nu_j$ is a bounded linear functional on $X$, it follows for the uniform
boundedness theorem that $\theta^v$ is bounded.  Since
$\theta^v(x)=\theta_x(v)$, which is linear with respect to $v$, we can
conclude that $v\mapsto \theta_v$ is a linear map from $P$ to $X^*$. We use
$D^w\!F(p)$ to denote this map, and $D^w\!F(p).v$ to denote its value for a
particular $v$, so $D^w\!F(p).v=\theta^v$, and 
\xdef\defofdwf{\equanoapno}
$$
\langle D^w\!F(p).v,x\rangle=\lim_{h\to 0}h^{-1}\langle
F(p+hv)-F(p),x\rangle\,.\numeqap
$$
\xdef\isbounded{{\rm \appenlet}.\the\lemmno}
\lemmap
If $P$ is a normed space, $X$ is a Banach space, $p\in P$,
$F:\Omega\to X^*$ is a map defined on a neighborhood $\Omega$ of $p$,
and $F$ is weak$^*$ Fr\'echet differentiable at $p$.  Then: 

{\pari
\item{(\isbounded.i)} {\sl there is a $C>0$ such that} 
\xdef\bdforf{\equanoapno}
$$
||F(p+v)-F(p)||\le C||v|| \qquad{\sl for \,\,\,all\,\,\,sufficiently
\,\,\,small \qquad} v\in P\,,\numeqap
$$

\item{(\isbounded.ii)} {\sl the linear map $D^w\!F(p):P\mapsto X^*$ 
defined by (\defofdwf) is bounded.} 

}


\beginproof
Suppose (\isbounded.i) is not true. Then there exists a sequence
$\{v_j\}$ in $P$, converging to $0$ and such that
$||F(p+v_j)-F(p)||>j||v_j||$. Define, for each $j$, a  
member $\sigma_j$ of $X^*$,  by 
$$
\sigma_j={F(p+v_j)-F(p)-\theta^{v_j}\over ||v_j||}\,.\numeqap
$$
Then $\sigma_j(x)\to 0$ for each $x$, so the sequence $\{\sigma_j\}$ is
pointwise bounded and  hence uniformly bounded.   
    
On the other hand, for each $x\in X$, the numbers ${\theta^{v_j}(x)\over
||v_j||}$ are bounded, because $\theta^{v_j}(x)=\theta_x(v_j)$, and
$\theta_x$ is a bounded linear functional on $P$,  so
$|\theta^{v_j}(x)|\le||\theta_x||.||v_j||$. So, using the uniform
boundedness theorem once again, we conclude that the sequence
$\Bigl\{{\theta^{v_j}\over ||v_j||}\Bigr\}$ is bounded in $X^*$.  Since
$\{\sigma_j\}$ is 
bounded, we conclude that the sequence
$\Bigl\{{F(p+v_j)-F(p)\over ||v_j||}\Bigr\}$ is
bounded, contradicting the fact that $||F(p+v_j)-F(p)||>j||v_j||$.
So (\bdforf) holds for some $C>0$, and we have proved (\isbounded.i). 

It follows from  (\isbounded.i) that $|\langle
F(p+v)-F(p),x\rangle|\le C||x||.||v||$ for all $x\in X$, if $v\in P$ is
small enough.  Applying this with $hv$ in the role of $v$, and letting
$h\downarrow 0$, we find that
$$
|\langle D^w\!F(p).v,x\rangle|\le C||x||.||v||\qquad{\rm for\,\,\,all\qquad}
x\in X\,,\,\,\,v\in P\,.\numeqap
$$
Therefore $||D^w\!F(p).v||\le C||v||$, so $D^w\!F(p)$ is a bounded linear 
operator from $P$ to $X^*$.  \proofend


We are now ready to study the differentiability properties with respect to
a parameter $p$, belonging to a normed space $P$, of a family of
trajectories $x_p:[a,b]\to\R^n$ of time-varying vector fields $f_p$.  We
will prove, under minimal technical hypotheses, that if the initial
condition map $p\mapsto x_p(a)$ is differentiable at $p=0$, and the map
$p\mapsto f_p$ satisfies a ``weak differentiability condition'' at the
trajectory $x_0$, then the map $p\mapsto x_p$, is weak$^*$- differentiable
at $p=0$, as a map into $BV([a,b],\R^n)$, and in particular the endpoint
map $p\mapsto x_p(b)$ is differentiable at $p=0$.

A {\sl curve in $\R^n$} is a continuous map $\xi:I\mapsto\R^n$, whose
domain $I={\rm Dom}(\xi)$ is a nonempty interval. A curve $\xi$ is an {\sl
arc} if ${\rm Dom}(\xi)$ is compact.  The {\sl graph} $G(\xi)$ of a curve
$\xi$ is the set $G(\xi)=\{(\xi(t),t):t\in{\rm Dom}(\xi)\}$.  If
$\xi:[a,b]\mapsto\R^n$ is an arc in $\R^n$, and $\ve>0$, then the
$\ve$-{\sl tube about $\xi$} is the set ${\cal
T}(\xi,\ve)=\{(x,t):x\in\R^n,a\le t\le b,||x-\xi(t)||\le\ve\}$.

A {\sl time-varying vector field} on $\R^n$ is an $\R^n$-valued map $f$
whose domain ${\rm Dom}(f)$ is a ---possibly empty--- subset $S$ of the
product $\R^n\times\R$. We will use $TVVF(\R^n)$ to denote the set of all
time-varying vector fields on $\R^n$.  A {\sl trajectory}, or {\sl integral
curve}, of an $f\in TVVF(\R^n)$ is a locally absolutely continuous curve
$\xi$ in $\R^n$ such that $G(\xi)\subseteq{\rm Dom}(f)$ and
$\dot\xi(t)=f(\xi(t),t)$ for almost all $t\in{\rm Dom}(\xi)$.  We use
$\traj(f)$ to denote the set of all trajectories of $f$, and $\traj_c(f)$
to denote the set of all $\xi\in\traj(f)$ such that ${\rm Dom}(\xi)$ is
compact.

If $f\in TVVF(\R^n)$ and $\xi\in\traj(f)$, we say that $\xi$ is a {\sl
maximal} trajectory of $f$ if it cannot be extended to a trajectory
$\tilde\xi:\tilde I\mapsto\R^n$ of $f$ defined on an interval $\tilde I$
such that $I\subseteq\tilde I$ but $I\neq\tilde I$. We use $\mtraj(f)$ to
denote the set of all maximal trajectories of $f$.  Given $\bar x\in\R^n$,
$\bar t\in\R$, we use $\traj(f,\bar x,\bar t)$, $\mtraj(f,\bar x,\bar t)$
to denote, respectively, the set of all $\xi\in\traj(f)$ such that
$\xi(\bar t)=\bar x$, and the set $\traj(f,\bar x,\bar t)\cap\mtraj(f)$.
Zorn's lemma implies that every $\xi\in\traj(f,\bar x,\bar t)$ can be
extended to a $\tilde\xi\in\mtraj(f,\bar x,\bar t)$, so
$\traj(f,\bar{x},\bar t)\neq\emptyset$ iff
$\mtraj(f,\bar{x},\bar{t})\neq\emptyset$. Clearly,
$\traj(f,\bar{x},\bar{t})\neq\emptyset$ iff
$(\bar{x},\bar{t})\in{\rm{}Dom}(f)$.  (Indeed, if
$(\bar{x},\bar{t})\in{\rm{}Dom}(f)$, then the map $\xi$ with domain
$\{\bar{t}\}$ such that $\xi(\bar{t})=\bar x$ is in $\traj(f)$.)
 


We say that an $f\in TVVF(\R^n)$ has the {\sl forward existence property}
at a point $(\bar x,\bar t)$ if for some $\ve>0$ there exists a
$\xi\in\traj(f,\bar x,\bar t)$ with domain $[\bar t,\bar t+\ve]$.  We say
that $f$ has the {\sl forward limit property} on a subset $S$ of
${\rm{}Dom}(f)$ if, whenever $\xi:[a,b[\,\mapsto\R^n$ is a trajectory of $f$
which is contained in a compact subset of $S$, it follows that
$\lim_{t\uparrow b}\xi(t)$ exists.

Suppose we are given the following data:

\medbreak

{\pari

\item{(D.1)} a  normed space $P$,

\item{(D.2)} a family ${\bf f}=\{f_p\}_{p\in P}$ of
time-varying vector fields on $\R^n$, depending on a parameter $p\in P$,

\item{(D.3)} a $p_0\in P$, 

\item{(D.4)} a trajectory $\xi:[a,b]\mapsto\R^n$ of $f_{p_0}$.

}

\medbreak\noindent We then define functions $w_p:[a,b]\mapsto\R^n$  by 
\xdef\defofwp{\equanoapno}
$$
w_p(t)=f_p(\xi(t),t)-f_{p_0}(\xi(t),t)\quad{\rm for\quad}t\in[a,b]\,,\numeqap
$$
so $w_p$ is defined on $[a,b]$ whenever $G(\xi)\subseteq{\rm Dom}(f_p)$.
If $w_p$ is integrable, we let 
\xdef\defofWp{\equanoapno}
$$
W_p(t)=\int_a^tw_p(s)\,ds\qquad{\rm for\qquad}t\in[a,b]\,.\numeqap
$$
We then let ${\cal W}$ be the map  that sends $p$ to ${\cal W}(p)=W_p$,
with domain  
\xdef\defofdomw{\equanoapno}
$$
{\rm Dom}({\cal W})=\{p\in P: G(\xi)\subseteq{\rm Dom}(f_p),
\,\,w_p\in L^1([a,b],\R^n)\}\,.\numeqap
$$
We regard ${\cal W}$ as a map from ${\rm Dom}({\cal W})$ to 
$BV([a,b],\R^n)$. 

\medbreak\noindent  We then say that ${\bf f}$ is {\sl weakly
differentiable at $p_0$ along $\xi$} if there exists $\bar\ve>0$ such that

\medbreak
{\pariim 

\item{(WD1)} {\sl ${\cal T}(\xi,\bar\ve)\subseteq{\rm Dom}(f_p)$ for every 
$p$ such that $||p-p_0||\le\bar\ve$.}

\item{(WD2)} {\sl For every $p$ such that $||p-p_0||\le\bar\ve$,  $f_p$ has
the forward limit property on ${\cal T}(\xi,\bar\ve)$ and the forward
existence property at every $(x,t)$ such that $a\le t<b$ and
$||x-\xi(t)||<\bar\ve$.}

}

{\pariim

\item{(WD3)}  {\sl There exist $A\!\in\!L^1([a,b],\R^{n\times n})$, 
and functions $\psi_\ve\!\in\!L^1([a,b],\R)$,  for 
$\ve\!\in ]0,\bar\ve]$, such that}
\xdef\fpxt{\equanoapno}
$$
||f_p(x,t)-f_p(\xi(t),t)-A(t).(x-\xi(t))||\le\psi_\ve(t)(||x-\xi(t)||
+||p-p_0||)\numeqap
$$
{\sl whenever $a\le t\le b$, $||x-\xi(t)||\le\ve$,   $||p-p_0||\le\ve$,  and} 
\xdef\limpsive{\equanoapno}
$$
\lim_{\ve\downarrow 0}\int_a^b\psi_\ve(t)\,dt=0\,.\numeqap
$$


\item{(WD4)}  {\sl The function $w_p$ defined by (\defofwp)
is integrable for every $p$ such that $||p-p_0||\le\bar\ve$ and, if
${\cal{}W}$ is defined as above, then ${\cal W}$ is weak$^*$ Fr\'echet
differentiable at $p_0$.}

}


\parnormal

\medbreak\noindent 
The second part of condition (WD4) says that for each $\alpha\in
C^0([a,b],\R_n)$ the map $p\to \int\alpha\cdot dW_p$ is Fr\'echet
differentiable at $p_0$.  Lemma \isbounded{} then implies 
---since $W_{p_0}=0$--- that there exists a bounded linear map $D^w{\cal
W}(p_0):P\mapsto BV([a,b],\R^n)$ having the property that
\xdef\flafordifwp{\equanoapno}
$$
\int\alpha\cdot dW_p=\langle{\cal W}(p)-{\cal W}(p_0),\alpha\rangle=
\int\alpha\cdot\Bigl(D^w{\cal W}(p_0).(p-p_0)\Bigr)+o(||p-p_0||)
\qquad{\rm as}\qquad p\to p_0\numeqap
$$
for each $\alpha\in C^0([a,b],\R_n)$, and also that there is a constant
$C$ such that
\xdef\constantc{\equanoapno}
$$
||W_p||_{TV}\le C||p-p_0||\qquad{\rm for\,\,\,all\,\,\,}p\in P{\rm
\,\,\,such\,\,\,that\,\,\,}\|p-p_0\|{\rm \,\,\,is\,\,\,
sufficiently \,\,\,small}.\numeqap
$$
\xdef\theodiftraj{{\rm  \appenlet}.\the\theono}
\theoap
Let $P$, ${\bf f}$, $p_0$, $\xi$ be data as in (D.1,2,3,4). 
Assume that ${\bf f}$ is weakly differentiable at $p_0$ along $\xi$.  Let
$\bar\ve>0$ be such that (WD1) to (WD4) hold.  Then

{\pariim

\item{(\theodiftraj.a)} {\sl  For every $\ve\in]0,\bar\ve]$ there exists an 
$\delta\in]0,\ve[$ such that, whenever $||p-p_0||\le\delta$, $||\bar
x-\xi(a)||\le\delta$, and $\zeta\in\mtraj(f_p,\bar x,a)$, it follows that
$[a,b]\subseteq{\rm Dom}(\zeta)$ and $||\zeta(t)-\xi(t)||\le\ve$ for all
$t\in[a,b]$.}

\item{(\theodiftraj.b)} {\sl If $\{\bar x_p\}_{||p-p_0||\le\bar\ve}$ is a
family of points of $\R^n$ such that the map $p\mapsto \bar x_p$ is
Fr\'echet differentiable at $p_0$ and $\bar x_{p_0}=\xi(a)$, then, if
$\{\xi_p\}_{||p-p_0||\le\bar\ve}$ is a family of curves such that
$\xi_p\in\mtraj(f_p,\bar x_p,a)$, and $x_p$ denotes the restriction of
$\xi_p$ to $[a,b]$ ---which is well defined for small enough $||p-p_0||$ by
(\theodiftraj.a)--- then the map ${\cal X}$ that sends each $p$ to the
function $x_p$ is weak$^*$ Fr\'echet differentiable at $p_0$ as a map from
$P$ to $BV([a,b],\R^n)$.  Moreover, $D^w{\cal X}(p_0)$ can be computed as
follows.  For $v\in P$, 
$$
D^w{\cal{}X}(p_0).v=Y_{A,\,d(D^w{\cal{}W}(0).v),
\,L.(p-p_0)}\,,\numeqap
$$
where $L:P\to\R^n$ is the differential at $p_0$
of the map $p\to \bar x_p$.} 

}

\beginproof
We will assume, as we clearly may without loss of generality, that
$p_0=0$. Also, we know that there is a $C>0$ such that the bound
(\constantc) holds.  So we can asssume, by making $\bar\ve$ smaller, if
necessary, that $||W_p||_{TV}\le C||p||$ whenever $||p||\le\bar\ve$.  By
making $C$ larger and $\bar\ve$ smaller, if necessary, we may also assume
that $C\ge 1$, and
\xdef\alsoassume{\equanoapno}
$$
8e^{||A||_{L^1}}||\psi_\ve||_{L^1}<1 \qquad{\rm for\qquad}
\ve\in]0,\bar\ve]\,. \numeqap
$$ 
Pick $\ve\in]0,\bar\ve[$, and choose $\delta>0$ such that  
\xdef\choicea{\equanoapno}
$$
C\delta<\ve\,.\numeqap
$$  
Notice that $\delta<\ve$, since $C\ge 1$. 

Let $p$, $\bar x$ be such that $||p||\le\delta$ and
$||\bar{x}-\xi(a)||\le\delta$.  Let $\zeta\in\mtraj(f_p,\bar x,a)$, and let
$I={\rm Dom}(\zeta)$.  Then $a\in I$. Let $c$ be the supremum of those $t$
such that $a\le t\le b$ and $||\zeta(s)-\xi(s)||\le\ve$ for $a\le s\le t$.
Then $a<c\le b$, $c\in I$, and either $c=b$ or $||\zeta(c)-\xi(c)||=\ve$.
(The fact that $a<c$ follows from the local existence property, since
$||\bar x-\xi(a)||\le\delta<\ve<\bar\ve$, and $\zeta$ is continuous.  The
fact that $c\in I$ follows from the forward limit property, since the graph
of the restriction of $\zeta$ to $[a,c[$ is contained in ${\cal
T}(\xi,\ve)$.  If $c<b$ and $||\zeta(c)-\xi(c)||<\ve$, then the forward
existence property implies that $\zeta$ is not maximal.)

We will estimate $||\zeta(t)-\xi(t)||$ for $t\in[a,c]$, and show that ---if
$\ve$ is small enough, and $\delta$ is suitably chosen---
$||\zeta(t)-\xi(t)||<\ve$ for all such $t$, including $t=c$.  This will
imply that $c=b$,
 
{}From now on, we work on the interval $[a,c]$.  We have  
$$
\zeta(t)-\xi(t)=\bar x-\xi(a)+\int_a^t\Bigl(f_p(\zeta(s),s)-f_0(\xi(s),s)\Bigr)
\,ds\,,\numeqap
$$
so 
$$
\zeta(t)-\xi(t)=\bar x-\xi(a)+W_p(t)+
\int_a^t\Bigl(f_p(\zeta(s),s)-f_p(\xi(s),s)\Bigr)\,ds\,.\numeqap
$$
Let $r_p(t)=f_p(\zeta(t),t)-f_p(\xi(t),t)-A(t).(\zeta(t)-\xi(t))$.
Then 
$$
\zeta(t)-\xi(t)=\bar x-\xi(a)+W_p(t)+\int_a^tA(s).(\zeta(s)-\xi(s))\,ds
+\int_a^tr_p(s)\,ds\,,\numeqap
$$
so
$$
||\zeta(t)-\xi(t)||\le||\bar x-\xi(a)||+
||W_p||_{sup}+\int_a^t||A(s)||.||\zeta(s)-\xi(s)||\,ds+\int_a^t
r_p(s)\,ds\,,\numeqap
$$
and then 
$$
||\zeta(t)-\xi(t)||\le||\bar x-\xi(a)||+||W_p||_{sup}+
2\ve||\psi_\ve||_{L^1}+\int_a^t||A(s)||.||\zeta(s)-\xi(s)||\,ds\,,\numeqap
$$
because $||r_p(s)||\le 2\ve\psi_{\ve}(s)$ for $s\in[a,c]$, since 
$||\zeta(s)-\xi(s)||\le\ve$ and $||p||\le\ve$.

Since $||\bar x-\xi(a)||\le\delta$, and  
$||W_p||_{sup}\le||W||_{TV}\le C||p||\le C\delta$, Gronwall's inequality
implies  
$$
||\zeta(t)-\xi(t)||\le
e^{||A||_{L^1}}\Bigl((C+1)\delta+2\ve||\psi_\ve||_{L^1}\Bigr)\,.\numeqap
$$
In view of (\alsoassume),we have  
$$
2\ve e^{||A||_{L^1}}||\psi_\ve||_{L^1}<{\ve\over 4}\,.\numeqap
$$
Now suppose that 
\xdef\choiceb{\equanoapno}
$$
\delta\le\kappa\ve\,,\qquad{\rm where}\qquad\kappa=(4C+4)^{-1}e^{-||A||_{L^1}}
\,.\numeqap
$$
Then  
$$
||\zeta(t)-\xi(t)||\le{\ve\over 2}\qquad{\rm for\qquad all}\qquad
t\in[a,c]\,.\numeqap
$$   
This is our desired estimate, from which it follows, as explained above, 
that $c=b$, and then $||\zeta(t)-\xi(t)||\le\ve$ for all $t\in[a,b]$.  This 
completes the proof of (\theodiftraj.a).  

Besides proving (\theodiftraj.a), we have also shown that, if $\kappa$ is
the constant given by (\choiceb), then the conclusion of (\theodiftraj.a)
holds for every $\delta$ such that $0<\delta\le\kappa\ve$.  (Notice that if
$\delta\le\kappa\ve$ then (\choicea) holds.)

Now suppose we are given a map $P\ni p\mapsto\bar x_p\in\R^n$ which is
defined for $||p||\le\bar\delta$ for some $\bar\delta>0$.  Assume that this
map is Fr\'echet differentiable at $0$, with differential $L$, and is such
that $\bar x_0=\xi(a)$.  Write $z_p=Y_{A,dV_p,L.p}$, where $V_p=D^w{\cal
W}(0).p$.

It then follows that there exist a constant $C'\ge 1$ and a 
$\bar\delta'\in]0,\bar\delta]$ such that  (i)~$C'\bar\delta'\le\kappa\bar\ve$,
and (ii)~$||\bar x_p-\xi(a)||\le C'||p||$ for  $||p||\le\bar\delta'$. 

Pick in an arbitrary fashion $\xi_p\in\mtraj(f_p,\bar x_p,a)$ for
$||p||\le\bar\delta'$. Then, if $||p||\le\bar\delta'$, it follows that
(i)~$\xi_p$ is defined, (ii)~$\xi_p\in\mtraj(f_p,\bar x,a)$ for an $\bar x$
---namely, $\bar x_p$--- such that $||\bar x-\xi(a)||\le C'\bar\delta'
\le\kappa\bar\ve$.    Since $C'\ge 1$, we also have 
$||p||\le\kappa\bar\ve$. Therefore we can conclude ---using $\ve=\bar\ve$ and
$\delta=\kappa\bar\ve$,  that $[a,b]\subseteq{\rm Dom}(\xi_p)$ 
and $(\xi_p(t),t)\in{\cal T}(\xi,\bar\ve)$ for $t\in[a,b]$. 

Let $x_p$ be the restriction of $\xi_p$ to $[a,b]$.  For
$||p||\le\bar\delta'$, we have
$$
x_p(t)-\xi(t)-z_p(t)=\bar
x_p-\xi(a)-L.p+\int_a^tA(s).(x_p(s)-\xi(s)-z_p(s))\,ds 
+W_p(t)-V_p(t)+R_p(t)\,,\numeqap
$$ 
where $R_p(t)=\int_a^tr_p(s)\,ds$.  If $p\neq 0$, let 
$$
Y_p=||p||^{-1}(x_p-x_0-z_p)\,,\qquad U_p=||p||^{-1}(W_p-V_p+R_p)\,,
\qquad \bar Y_p=||p||^{-1}(\bar x_p-x_0(a)-L.p)\,.\numeqap
$$
Then $Y_p=Y_{A,dU_p,\bar Y_p}$.

We have to prove that $Y_p$ weak$^*$ converges to $0$ as $p\to 0$.  Since
$\bar Y_p\to 0$, our conclusion will follow if we show that $U_p$ weak$^*$
converges to $0$.  Since $V_p=D^w{\cal W}(0).p$, it is clear that
$||p||^{-1}(W_p-V_p)$ weak$^*$-converges to $0$.  So all we need is to
prove that $||p||^{-1}R_p$ weak$^*$-converges to $0$.  If $||p||$ is
sufficiently small, then we can apply the previous estimates using
$\delta=||p||$, $\ve={\delta\over\kappa}$.  We conclude that $||r_p(s)||\le
2\ve\psi_{\ve}(s)$ for all $s\in[a,b]$. This implies that
$$
||r_p||_{L^1}\le{2||p||\over K}||\psi_{\kappa^{-1}||p||}||_{L^1}\,.\numeqap
$$
This shows that $||R_p||_{TV}=||r_p||_{L^1}=o(||p||)$ as $p\to 0$.
So $||p||^{-1}R_p$ converges to $0$ strongly ---and 
{\it a fortiori} weakly$^*$--- in $BV([a,b],\R^n)$. This completes 
our proof. \proofend

If $f\in TVVF(\R^n)$ and  $\xi:[a,b]\mapsto\R^n$ is a trajectory of $f$, 
we say that {\sl $f$ is differentiable along $\xi$} if the map 
$x\mapsto f(x,t)$ is differentiable at $\xi(t)$ for almost every 
$t$.  In that case, we write 
$D_{\xi}f(t)\equaldef{\partial f\over\partial x}(\xi(t),t)$, and refer 
to the map $[a,b]\!\ni\! t\!\mapsto\!D_\xi f(t)\!\in\!\R^{n\times n}$ as the {\sl differential
of $f$ along $\xi$.}

\xdef\lemdifisadj{{\rm \appenlet}.\the\lemmno}
\lemmap
If ${\bf f}$ is weakly differentiable at $p_0$ along a trajectory 
$\xi:[a,b]\to\R^n$ of $f_{p_0}$, then $f_{p_0}$ is differentiable along 
$\xi$ and the map $A$ of (WD3) is such that $A(t)=D_\xi f_{p_0}(t)$ 
for almost all $t\in[a,b]$. 

\beginproof
We begin by showing that the functions $\psi_\ve$ for which (WD3) holds can
be chosen so that $\psi_\ve(t)$ is a monotonically nonincreasing function
of $\ve$ for each $t$.  To see this, start with some family
$\{\psi_\ve\}_{\ve\in]0,\bar\ve]}$ of functions for which (WD3) holds, and
define $\hat\psi_{\ve}=\psi_{2^{-k}\bar\ve}$ whenever
$2^{-k-1}\bar\ve<\ve\le 2^{-k}\bar\ve$ and $k$ is a nonnegative integer.
Then (WD3) also holds if the $\hat\psi_\ve$ are substituted for the
$\psi_\ve$.  Next define
$\bar\psi_\ve(t)=\min\{\hat\psi_{\ve'}(t):\ve\le\ve'\le\bar\ve\}$.  Then
the $\bar\psi_\ve$ are well defined integrable functions of $t$, because
for each $\ve$ the set of functions
$\{\hat\psi_{\ve'}:\ve\le\ve'\le\bar\ve\}$ is finite.  It is clear that
(WD3) also holds with the $\bar\psi_\ve$ istead of the $\hat\psi_\ve$, and
in addition $\hat\psi_\ve(t)$ is nonincreasing as a function of $\ve$.

Now, if we pick the $\psi_\ve$ to be monotonically nonincreasing with
respect to $\ve$, it follows that the limit $\rho(t)=\lim_{\ve\downarrow
0}\psi_\ve(t)$ exists for every $t$.  By the dominated convergence theorem,
$\int_a^b\rho(t)dt=0$.  Since $\rho(t)\ge 0$ for all $t$, it follows that
$\rho(t)=0$ almost everywhere.

Let $t$ be such that $\rho(t)=0$.  Let $\theta(x)=\psi_{||x-\xi(t)||}(t)$.
Then $\lim_{x\to\xi(t)}\theta(x)=0$.  On the other hand, (WD3) implies that
\xdef\saysprec{\equanoapno}
$$
||f_{p_0}(x,t)-f_{p_0}(\xi(t),t)-A(t).(x-\xi(t))||\le\theta(x)
||x-\xi(t)||\,.\numeqap
$$
Since $\theta(x)\to 0$ as $x\to\xi(t)$,  (\saysprec) says precisely
that $A(t)$ is the differential at $\xi(t)$ of the map 
$x\mapsto f_{p_0}(x,t)$, completing our proof. \proofend
 
If $f\in TVVF(\R^n)$ is differentiable along a trajectory 
$\xi:[a,b]\to\R^n$ of $f$, and the matrix-valued function $D_\xi f$ 
is integrable, then the ordinary differential equation 
\xdef\adjequa{\equanoapno}
$$
\dot\lambda(t)=-\lambda(t).D_{\xi}f(t)\,,\numeqap
$$
for an $\R_n$-valued function $\lambda$ on $[a,b]$, is the {\sl adjoint
variational equation} ---or, simply, the {\sl adjoint equation}--- for $f$
along $\xi$.  A solution of the adjoint equation (i.e. an absolutely
continuous function $\lambda:[a,b]\mapsto\R_n$ such that (\adjequa) holds
for almost all $t$) is known as an {\sl adjoint vector for $f$ along
$\xi$.}

\xdef\theodifwithlambda{{\rm \appenlet}.\the\theono}
\theoap  
Let $P$, ${\bf f}=\{f_p\}_{p\in P}$, $p_0\in P$, $\xi:[a,b]\to\R^n$ be as
in Theorem \theodiftraj. Let $P\ni p\mapsto\bar x_p\i\R^n$ be a map which
is Fr\'echet differentiable at $p_0$ and such that $\bar x_{p_0}=\xi(a)$,
and let $L_a$ be its differential at $p_0$.  Let $\xi_p$ be, for $p$ close
enough to $p_0$, maximal trajectories of $f_p$ such that
$\xi_p(a)=\bar{x}_p$, and let $x_p$ be the restriction of $\xi_p$ to
$[a,b]$. Then the map $p\mapsto x_p(b)$ is differentiable at $p_0$. Let
$L_b$ be its differential at $p_0$.  Let $\lambda:[a,b]\to\R^n$ be an
adjoint vector for $f_{p_0}$ along $\xi$.  Then
$$
\lambda(b)\cdot L_b(v)-\lambda(a)\cdot L_a(v)
=\lim_{\ve\downarrow 0}\,\ve^{-1}\!\int_a^b\Bigl\langle\lambda(t),f_{p_0+\ve v}
(\xi(t),t)-f_{p_0}(\xi(t),t)\Bigr\rangle dt\,.\numeqap
$$

\beginproof
Assume, without loss of generality, that $p_0=0$. According to Theorem 
\theodiftraj{} and Lemma \lemdifisadj, the map $p\mapsto x_p$ is weak$^*$
Fr\'echet differentiable at $0$ ---as a map from $P$ to $BV([a,b],\R^n)$---
and its differential at $0$ is the map that sends $v\in P$ to
$Y_{A,d(U.v),L_a(v)}$, where $U=D^w{\cal W}(0)$, and ${\cal W}$ is the map
$p\mapsto W_p$, where
$W_p(t)=\int_a^t\Bigl(f_{p}(\xi(s),s)-f_{0}(\xi(s),s)\Bigr)dt$, and
$A=D_{\xi}f$.  Then $p\mapsto x_p(b)$ is Fr\'echet differentiable at $0$,
and the differential $L_b$ is given by $L_b(v)=Y_{A,d(U.v),L_a(v)}(b)$.
Clearly, $L_a(v)=Y_{A,d(U.v),L_a(v)}(a)$.  Since $\lambda$ satisfies
$\dot\lambda=-\lambda.A$, Lemma \lemadjo{} tells us that
$$
\lambda(b)\cdot L_b(v)-\lambda(a)\cdot L_a(v)=\int_a^b\lambda.d(U.v)\,.
$$
Since $U.v$ is the weak$^*$-limit of $\ve^{-1}W_{\ve v}$ as $\ve\downarrow
0$, our conclusion follows. \proofend

We now turn to the differentiation of a Lagrangian cost functional.  

If $f\in TVVF(\R^n)$, a {\sl Lagrangian} for $f$ in $\R^n$ is a real-valued 
function $L$  such that

\medbreak

{\pari

\item{(L.1)}  ${\rm Dom}(f)\subseteq{\rm Dom}(L)$;

\item{(L.2)} the function $t\mapsto L(\xi(t),t)$ 
is locally integrable for every $\xi\in\traj(f)$.

}

\medbreak\noindent
If $f\in TVVF(\R^n)$ has components $f^1,\ldots,f^n$, and $L$ is a
Lagrangian for $f$, then we can consider the time-varying vector field
$[L;f]$ in $\R^{n+1}$ whose components are $(f^0,f^1,\ldots,f^n)$, where
$f^0\equiv L$.  A trajectory of $[L;f]$ can then be regarded a pair
$(\xi^0,\xi)$, where $\xi\in\traj(f)$ and $\xi^0$ is a real-valued locally
absolutely continuous function on ${\rm Dom}(\xi)$ such that
$\dot\xi^0(t)=L(\xi(t),t)$ for a.e. $t$.  Any such function $\xi^0$ will be
called a {\sl running $L$-cost along $\xi$}.  Our definition of a
Lagrangian implies that a running $L$-cost along $\xi$ exists for all
$\xi\in\traj(f)$, and it is clear that any two running $L$-costs along
$\xi$ must differ by a constant. If $\xi$ is such that ${\rm Dom}(\xi)$ is
of the form $[a,b]$ or $[a,b[$ or $[a,\infty[$, then there is a canonical
choice of running $L$-cost along $\xi$, obtained by letting
$\xi^0(t)=\int_a^tL(\xi(s),s)\,ds$.  In that case, we will write $\xi^L$ to
denote the curve $t\mapsto(\xi^0(t),\xi(t))$, where $\xi^0$ is the canonical
running $L$-cost.

Now suppose we are given data
$P$, ${\bf f}$, $p_0$, $\xi$  as in (D.1,2,3,4), as well as  

\medbreak

{\pari

\item{(D.5)} a family ${\bf L}=\{L_p\}_{p\in P}$ such that each $L_p$
is a Lagrangian for $f_p$.  

}

\medbreak\noindent
Given a trajectory $\xi:[a,b]\mapsto\R^n$ of
$f_{p_0}$, we say that the pair $({\bf f},{\bf L})$ is {\sl weakly
differentiable at $p_0$ along $\xi$} if the family $\{[L_p;f_p]\}_{p\in P}$
is weakly differentiable along $\xi^{L_{p_0}}$.

It is easy to see that $({\bf f},{\bf L})$ is weakly differentiable at
$p_0$ along $\xi$ iff there exists $\bar\ve>0$ such that  (WD1,2,3,4)
hold, and in addition

\medbreak
{\pariim

\item{(WD5)}  {\sl There exists $h\in L^1([a,b],\R_n)$ such that
\xdef\lpxt{\equanoapno}
$$
||L_p(x,t)-L_p(\xi(t),t)-h(t).(x-\xi(t))||\le\psi_\ve(t)(||x-\xi(t)||
+||p||)\numeqap
$$
whenever $a\le t\le b$, $||x-\xi(t)||\le\ve$ and
$||p||\le\ve$.}  

\item{(WD6)}  {\sl If we let $w_p^0(t)=L_p(\xi(t),t)-L_{p_0}(\xi(t),t)$,
$W_p^0(t)=\int_a^tw_p^0(s)\,ds$, then the map
${\cal W}^0$ that sends $p$ to ${\cal W}^0(p)=W_p^0\in BV([a,b],\R)$ is
weak$^*$ Fr\'echet differentiable at $p_0$.} 

}

\medbreak\noindent
If $f\in TVVF(\R^n)$, $\xi:[a,b]\mapsto\R^n$ is a trajectory of $f$, and
$L$ is a Lagrangian for $f$, we say that {\sl $L$ is differentiable along
$\xi$} if the map $x\mapsto L(x,t)$ is differentiable at $\xi(t)$ for almost
every $t$.  In that case, we write $\nabla_{\xi}L(t)\equaldef
\nabla_xL(\xi(t),t)$, and refer to the map $[a,b]\ni t\to \nabla_\xi
L(t)\in\R^{n\times n}$ as the {\sl differential of $L$ along $\xi$.}
 
Lemma \lemdifisadj{}, applied to the curve $\xi^{L_{p_0}}$,  yields

\xdef\lemdifisadjwithl{{\rm \appenlet}.\the\lemmno}
\lemmap
If $({\bf f},{\bf L})$ is weakly differentiable at $p_0$ along a trajectory 
$\xi:[a,b]\to\R^n$ of $f_{p_0}$, then $f_{p_0}$ and $L_{p_0}$ are
differentiable along $\xi$ and the maps $A$, $h$ of (WD3,5) are such that 
$A(t)=D_\xi f(t)$ and $h(t)=\nabla_\xi L_{p_0}(\xi(t),t)$ for almost all 
$t\in[a,b]$. \ \proofendno

Under the hypothesis of Lemma \lemdifisadjwithl{}, we can apply Theorem
\theodifwithlambda{} to the family $\{[L_p;f_p]\}_{p\in P}$ and the curve
$\xi^{L_{p_0}}$.   Define
$$
J_p(\zeta)=\int_a^b L_p(\zeta(t),t)\,dt\,,\numeqap
$$
if $\zeta:[a,b]\to\R^n$ is a curve such that the integral exists.  (We will
only be using this when $\zeta\in\traj(f_p)$, in which case $J_p(\zeta)$ is
well defined.)  An adjoint vector for $[L_{p_0};f_{p_0}]$ along
$\xi^{L_{p_0}}$ is then a pair $(\lambda^0,\lambda)$ such that
$\lambda^0\in\R$, $\lambda:[a,b]\mapsto\R_n$ is absolutely continuous, and
\xdef\adjeqwithlzero{\equanoapno}
$$
\dot\lambda(t)=-\lambda(t).A(t)-\lambda^0h(t)\qquad{\rm for\qquad a.e.\qquad}
t\in[a,b]\,.\numeqap
$$
Theorem \theodifwithlambda{} then gives:
\xdef\theodifwithlambdalag{{\rm \appenlet}.\the\theono}
\theoap
Assume that we are given data 
$P$, ${\bf f}$, $p_0$, $\xi$, ${\bf L}$  as in (D.1,2,3,4,5).
Assume that $({\bf f},{\bf L})$ is weakly differentiable at
$p_0$ along $\xi$.  Let $P\ni p\mapsto\bar x_p\i\R^n$ be a map which is
Fr\'echet differentiable at $p_0$ and such that $\bar x_{p_0}=\xi(a)$, and
let $L_a$ be its differential at $p_0$.  Let $\xi_p$ be, for $p$ close
enough to $p_0$, maximal trajectories of $f_p$ such that
$\xi_p(a)=\bar{x}_p$, and let $x_p$ be the restriction of $\xi_p$ to
$[a,b]$. Then the maps $p\mapsto x_p(b)$ and $p\mapsto J_p(x_p)$ are
differentiable at $p_0$. Let $L_b$, $L_b^0$ be their differentials at
$p_0$.  Then, if $(\lambda^0,\lambda)$ is an adjoint vector for
$[L_{p_0},f_{p_0}]$ along $\xi$, it follows that 
$$
\lambda(b)\cdot L_b(v)+\lambda^0 L_b^0(v)
-\lambda(a)\cdot L_a(v)=\lim_{\ve\downarrow 0}\ve^{-1}\!\int_a^b\!\Big(
H_{p_0+\ve v}(\xi(t),\lambda(t),\lambda_0,t)-
H_{p_0}(\xi(t),\lambda(t),\lambda_0,t)\Big)\,dt\,,\numeqap
$$
where 
$$
H_p(x,z,z^0,t)=\langle z,f_{p}(x,t)\rangle+z^0 L_p(x,t)\numeqap
$$
for $x\in\R^n$, $z\in\R_n$, $z^0\in\R$, $t\in[a,b]$. \proofendno

\appen{B}{A lemma}

\xdef\lemincreasing{{\rm \appenlet}.\the\lemmno}
\lemmap
Let $f$ be a real-valued function on a compact interval $[a,b]$. Assume that
there exists a finite or countable subset $E$ of $[a,b]$ with the following 
properties:

{\pariim

{\sl

\item{(\lemincreasing.a)} $\lim\inf_{h\downarrow 0}{f(x+h)-f(x)\over h}\ge 0$
for all $x\in[a,b[\setminus E$, 

\item{(\lemincreasing.b)} $\lim\inf_{h\downarrow 0}{f(x+h)-f(x)}\ge 0$
for all $x\in[a,b[$,

\item{(\lemincreasing.c)} $\lim\sup_{h\downarrow 0}{f(x)-f(x-h)}\ge 0$
for all $x\in]a,b]$.

}

}

\noindent{\sl Then $f(b)\ge f(a)$.}

\beginproof  Let $\ve>0$ be arbitrary. We will prove that $f(b)\ge f(a)-\ve$. 
Let $\ve'={\ve\over 1+b-a}$ and $E=\{x_k\}_{k\in N}$, $N\subset {I\!\! N}$. 

For each $x\in[a,b]$, let $I(x)=\{k:x_k<x\}$. Define $g:[a,b]\mapsto\R$ by 
$$
g(x)=\sum_{k\in I(x)}2^{-k}\ve'\,.
$$
Then $g$ is left-continuous and monotonically nondecreasing. 

Let $S$ be the set of all $x\in[a,b]$ such that 
$f(y)\ge f(a)-(y-a)\ve'-g(y)$ for all $y\in[a,x]$.   It is clear that 
$a\in S$, and also that $S$ is an interval. So $S$ is of the form 
$[a,c[$ or $[a,c]$, with $c\in[a,b]$.   

Suppose $S=[a,c[$.   Then $c>a$.  By  (\lemincreasing.c), given 
$\delta>0$ there is a sequence
$\{z_j\}$ in $[a,b]$ such that $z_j<c$, $z_j\to c$, and 
$f(c)-f(z_j)\ge -\delta$.  Then $z_j\in S$, so 
$f(z_j)\ge f(a)-(z_j-a)\ve'-g(z_j)\ge f(a)-(c-a)\ve'-g(c)$. It then follows that
$f(c)\ge f(a)-(c-a)\ve'-g(c)-\delta$. Since $\delta$ is arbitrary,  we have
$f(c)\ge f(a)-(c-a)\ve'-g(c)$, so $c\in S$, which is a contradiction.

Therefore $S=[a,c]$ for some $c\in[a,b]$.   We now show that $c=b$.
Assume that $c<b$, and consider separately the cases  $c\notin E$,  $c\in E$.

If $c\notin E$, then (\lemincreasing.a) implies that there exists
a $c'>c$ such that $f(x)-f(c)\ge -(x-c)\ve'$ for $x\in[c,c']$.  But then, since
$f(c)\ge f(a)-(c-a)\ve'-g(c)$, we have $f(x)\ge f(a)-(x-c)\ve'-g(c)\ge
f(a)-(x-a)\ve'-g(x)$ for $x\in[c,c']$.  Therefore $c'\in S$, contradicting
the fact that $S=[a,c]$.   
   
Now suppose $c\in E$. Then $c=x_k$ for some $k$, and $x_k\neq b$. By 
(\lemincreasing.b),  there exists $h>0$ such that $c+h\le b$ and 
$f(y)-f(c)\ge-2^{-k}\ve'$ for all $y\in[c,c+h]$.  If $y\in]c,c+h]$, then 
$k\notin I(c)$, and $I(c)\cup\{k\}\subseteq I(y)$. Therefore  
$g(y)\ge g(c)+2^{-k}\ve'$.   Since $f(c)\ge f(a)-(c-a)\ve'-g(c)$,
we have $f(y)\ge f(c)-2^{-k}\ve'\ge f(a)-(c-a)\ve'-2^{-k}\ve'-g(c)\ge
f(a)-(y-a)\ve'-g(y)$. So $c+h\in S$, which is a contradiction. 

We have therefore proved that $c=b$. Then 
$$
f(b)\ge f(a)-(b-a)\ve'-g(b)\ge f(a)-(b-a)\ve'-\ve'=f(a)-\ve\,.
$$
Since $\ve$ is arbitrary, we have shown that $f(b)\ge f(a)$, as 
desired.   \proofend  

\remap
If $f$ is lower semicontinuous then (\lemincreasing.b) automatically 
holds. \proofend   



\bigbreak
\centerline{{\bf REFERENCES}}

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\bye



