\def \today {\ifcase \month \or January\or February\or  
  March\or April\or May\or June\or July\or August\or 
  September\or October\or November\or December\fi\   
  \number \day, \number \year} 
\def\comment#1{}
\def\pmb#1{\setbox0=\hbox{$#1$}%
  \kern-.025em\copy0\kern-\wd0
  \kern.05em\copy0\kern-\wd0
  \kern-.025em\raise.0433em\box0 }
\def\equaldef{{\buildrel \rm def \over =}}
\def\steero{$\sim$\kern -0.1cm$\sim$\kern -0.1cm$\sim$\kern-0.2cm$>$\kern-0.28cm$>$\kern-0.28cm$>$\kern-0.28cm$>$\kern-0.28cm$>$\kern-0.28cm$>$\kern-0.28cm$>$\kern0.1cm}
\def\steer#1#2{$#1$\steero$#2$}
\def\steert#1#2#3{\hbox{$#1$\hbox{\lower0.4ex\hbox{\steero}}\kern-0.59cm\raise0.53ex\hbox{$#2$}}\kern0.35cm$#3$}
\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
\newcount\corono%the corollary number
\newcount\propno%the corollary number
\newcount\defino%the definition number
\newcount\defidno%the D definition number
\newcount\equano%the equation number
\newcount\equanonosec%the equation number without the section
\newcount\remano%the remark number
\newcount\examno%the example number
\newcount\figuno%the figure number
\newcount\footno%the footnote number
\def\restr{\lceil}
\sectno=0
\subsectno=0
\equanonosec=1
\footno=0
%BLACKBOARD BOLD
\def\vbar{\mathchoice{\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height4.1ptdepth-.35ptwidth.6pt\kern-.6pt}
   {\vrule height3.1ptdepth-.25ptwidth.5pt\kern-.5pt}}
\def\fudge{\mathchoice{}{}{\mkern.5mu}{\mkern.8mu}}
\def\bbc#1#2{{\rm \mkern#2mu\vbar\mkern-#2mu#1}}
\def\bbb#1{{\rm I\mkern-3.5mu #1}}
\def\bba#1#2{{\rm #1\mkern-#2mu\fudge #1}}
\def\bb#1{{\count4=`#1 \advance\count4by-64 \ifcase\count4\or\bba A{11.5}\or
   \bbb B\or\bbc C{5}\or\bbb D\or\bbb E\or\bbb F \or\bbc G{5}\or\bbb H\or
   \bbb I\or\bbc J{3}\or\bbb K\or\bbb L \or\bbb M\or\bbb N\or\bbc O{5} \or
   \bbb P\or\bbc Q{5}\or\bbb R\or\bbc S{4.2}\or\bba T{10.5}\or\bbc U{5}\or
   \bba V{12}\or\bba W{16.5}\or\bba X{11}\or\bba Y{11.7}\or\bba Z{7.5}\fi}}
%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{
\vskip 0.8true in\sighjsprof\vfil#1\vfil\pleft{#2}\vfil\eject}
\def\finhjsprofito#1#2{
\vskip 0.5true in\sighjsprof\vfil#1\vfil\eject}
\def\thedefaultannotation{\line{\noindent{hjs$\backslash$\TeX}\hfil}}
\def\thedefaultps{ }
%END OF MACROS FOR LETTER
\def\clearmyitemno{\global\myitemno=0}  
\def\clearsubsectno{\global\subsectno=0}  
\def\cleartheono{\global\theono=1}
\def\cleartheotno{\global\theotno=1}
\def\cleartheononosec{\global\theononosec=1}
\def\clearlemmno{\global\lemmno=1}
\def\clearcorono{\global\corono=1}
\def\clearpropno{\global\propno=1}
\def\cleardefino{\global\defino=1}
\def\cleardefidno{\global\defidno=1}
\def\clearremano{\global\remano=1}
\def\clearexamno{\global\examno=1}
\def\clearequano{\global\equano=1}
\def\clearequanonosec{\global\equanonosec=1}
\def\clearfiguno{\global\figuno=1}
\def\figcap#1#2{\medbreak
\centerline{ Figure {#1}:\ #2}
\vbox to 0.3true in{\vfill}}
\def\mybigbreak{\vskip 0.3true in}
\def\mynotsobigbreak{\vskip 0.2true in}
\def\equanosectno{\the\sectno .\the\equano}
\def\figunosectno{\the\sectno .\the\figuno}
\def\adequano{\global\advance\equano by 1} 
\def\adequanonosec{\global\advance\equanonosec by 1} 
\def\adfiguno{\global\advance\figuno by 1} 
\def\numeq{\eqno(\equanosectno\adequano)}
\def\numeqnosec{\eqno(\the\equanonosec\adequanonosec)}
\def\numeqa{\eqno({\rm A}.\the\equano)\adequano}
\def\numeqap{\eqno({\rm \appenlet}.\the\equano)\adequano}
\def\numeqb{\eqno({\rm B}.\the\equano)\adequano}
\def\numeqalign{(\equanosectno)\adequano}
\def\numeqnoa{(\equanosectno.{\rm a})}
\def\numeqnob{(\equanosectno.{\rm b})}
\def\numeqnoc{(\equanosectno.{\rm c})}
\def\numeqcons{\eqno(\the\equano)\adequano}
%use \numeq for displayed eqs. (Includes when you are in \eqalign, but
%NOT \eqalignno.)  Use \numeqalign within \eqalignno.
\def\clearallcounters{\clearequano \cleartheono \clearlemmno \clearcorono 
     \clearremano \clearexamno \cleardefino \clearfiguno \clearsubsectno}
\def\section#1\par{\global\advance\sectno by 1
    \bigbreak \noindent {\bf \S \the\sectno
    .\enspace#1.}\medbreak\clearallcounters}
\def\subsectionnoparnodotsbf#1\par{\global\advance\subsectno by 1
    \bigbreak \noindent {\bf \the\sectno.\the\subsectno
    $\,\,$#1}\medbreak}
\def\sectionnopar#1\par{\global\advance\sectno by 1
    \bigbreak \noindent {\bf \the\sectno
    .\enspace\enspace#1.}\medbreak\clearallcounters}
\def\sectionnoparnodotsmagonebf#1\par{\global\advance\sectno by 1
    \bigbreak \noindent {\mymagonebf \the\sectno
    \enspace$\,\,$#1}\medbreak\clearallcounters}
\def\appen#1#2\par{
    \bigbreak \noindent {\bf Appendix #1
    .\enspace#2.}\medbreak\clearallcounters
     \xdef\appenlet{#1}}
\def\sectiondek#1\par{\global\advance\sectno by 1
    \vskip 1.22true cm \noindent {\the\sectno 
    \enspace\enspace#1}\vskip 0.67true cm\clearallcounters}
\def\sectionnobreak#1{\global\advance\sectno by 1
    \noindent {\bf \S \the\sectno
    .\enspace#1.}\clearallcounters}
\def\nitem{{\global\advance\myitemno by 1}\item{\the\myitemno .}}
\def\nitembra{\item{
\hbox to 1true cm{\hfill[\the\myitemno]\enspace}}}
\def\nitemstart{\clearmyitemno\nitem}
\def\nitembf{{\global\advance\myitemno by 1}\item{{\bf\the\myitemno .}}}
\def\nitembfnoadv{\item{{\bf\the\myitemno .}}}
\def\theo#1\par{\medbreak \noindent {\bf Theorem \enspace 
   \the\sectno .\the\theono}. \enspace 
   {\sl#1}\medbreak \global\advance\theono by 1} 
\def\theot#1\par{\medbreak \noindent {\bf Theorem \enspace 
   T\the\theotno}. \enspace 
   {\sl#1}\medbreak \global\advance\theotno by 1} 
\def\theonosec#1\par{\medbreak \noindent {\bf Theorem\enspace 
   \the\theononosec}. \enspace 
   {\sl#1}\medbreak \global\advance\theononosec by 1} 
\def\propnosec#1\par{\medbreak \noindent {\bf Proposition\enspace 
   \the\theononosec}. \enspace 
   {\sl#1}\medbreak \global\advance\theononosec by 1} 
\def\lemm#1\par{\medbreak \noindent {\bf Lemma \enspace 
   \the\sectno .\the\lemmno}.\enspace 
   {\sl#1}\medbreak \global\advance\lemmno by 1} 
\def\coro#1\par{\medbreak \noindent {\bf Corollary \enspace 
   \the\sectno .\the\corono}. \enspace 
   {\sl#1}\medbreak \global\advance\corono by 1} 
\def\prop#1\par{\medbreak \noindent {\bf Proposition \enspace 
   \the\sectno .\the\propno}. \enspace 
   {\sl#1}\medbreak \global\advance\propno by 1} 
\def\lemmtheo#1\par{\medbreak \noindent {\bf Lemma \enspace 
   \the\sectno .\the\theono}.\enspace 
   {\sl#1}\medbreak \global\advance\theono by 1} 
\def\corotheo#1\par{\medbreak \noindent {\bf Corollary \enspace 
   \the\sectno .\the\theono}. \enspace 
   {\sl#1}\medbreak \global\advance\theono by 1} 
\def\proptheo#1\par{\medbreak \noindent {\bf Proposition \enspace 
   \the\sectno .\the\theono}. \enspace 
   {\sl#1}\medbreak \global\advance\theono by 1} 
\def\defi#1\par{\medbreak \noindent {\bf Definition \enspace 
   \the\sectno .\the\defino}. \enspace 
   {#1}\medbreak \global\advance\defino by 1} 
\def\defid#1\par{\medbreak \noindent {\bf Definition \enspace 
    D\the\defidno}. \enspace 
   {#1}\medbreak \global\advance\defidno by 1} 
\def\rema#1\par{\medbreak \noindent {\bf Remark \enspace 
   \the\sectno .\the\remano}. \enspace 
   {#1}\medbreak \global\advance\remano by 1} 
\def\remap#1\par{\medbreak \noindent {\bf Remark \enspace 
   \appenlet .\the\remano}. \enspace 
   {#1}\medbreak \global\advance\remano by 1} 
\def\exam#1\par{\medbreak \noindent {\bf Example \enspace 
   \the\sectno .\the\examno}. \enspace 
   {#1}\medbreak \global\advance\examno by 1} 
\def\theoieeeinfo#1\par{\medbreak {\it Theorem \enspace 
   \the\theono:} \enspace 
   {#1}\medbreak \global\advance\theono by 1} 
\def\lemmieeeinfo#1\par{\medbreak {\it Lemma \enspace 
   \the\lemmno:}\enspace 
   {#1}\medbreak \global\advance\lemmno by 1} 
\def\coroieeeinfo#1\par{\medbreak {\it Corollary \enspace 
   \the\corono:} \enspace 
   {#1}\medbreak \global\advance\corono by 1} 
\def\remaieeeinfo#1\par{\medbreak {\it Remark:} \enspace 
   {#1}\medbreak }
\def\exam#1\par{\medbreak \noindent {\bf Example \enspace 
   \the\sectno .\the\examno}. \enspace 
   {#1}\medbreak \global\advance\examno by 1}
\def\smalltype{\baselineskip=8pt}
\def\normaltype{\baselineskip=11pt}
\def\footo#1{\openup -1.7\jot
\global\advance\footno by 1\footnote{$^{\the\footno}$}{
\mysmall #1}\openup 1.7\jot}

\def\bigitem#1. #2\par{\parindent=36pt. \item{#1\enspace\enspace} #2}
%WARNING: this last macro hasn't been tested yet!
\def\beginproof{\medbreak\noindent{\sl PROOF.\enspace}}
\def\mybox{$\lower3.5pt\vbox{\hrule height2.6pt depth-2pt\hbox{\vrule width0.6pt\kern1pt\vbox{\kern1pt\phantom{nj}\kern1pt}\kern1.5pt\vrule width0.6pt}\hrule height2.6pt depth-2pt}$\medbreak}
\def\proofend{\enspace\hfill \mybox\bigskip}
%THE FOLLOWING MACROS WERE USED FOR THE 1987 CDC PAPER
\def\rarr{\rightarrow}
\def\rint{\enspace\rightarrow\!\!\!\!\!\!\int}
\def\lint{\enspace\leftarrow\!\!\!\!\!\int}
\def\pvarieta{{\cal  P}({\bf P},\eta)}
\def\pvdata{{\bf P}=(m,{\bf t},{\bf r},{\bf E})}
\def\sre{S_{\rho}(\eta)}
\def\vari{E=\{\eta_{\varepsilon}:\varepsilon\in C\}}
\def\varii{E_{i}=\{\eta_{i,\varepsilon_{i}}:\varepsilon_{i}\in C_{i}\}}
\def\tou{trajectory of uniqueness}
\def\uzsig{{\cal U}_{0}(\Sigma)}
\def\uosig{{\cal U}_{1}(\Sigma)}
\def\uksig{{\cal U}_{k}(\Sigma)}
\def\sigisz{$\Sigma$ is a $C^{0}$ system}
\def\sigiso{$\Sigma$ is a $C^{1}$ system}
\def\sigisk{$\Sigma$ is a $C^{k}$ system}
\def\czad{$C^{0}$-admissible}
\def\coad{$C^{1}$-admissible}
\def\ckad{$C^{k}$-admissible}
\def\mysys{\Sigma=(M,U,f)}
\def\vk{(V,\kappa)}
\def\vkk{(V,\kappa,K)}
\def\chm{{\cal CH}(M)}
\def\km{{\cal K}(M)}
\def\kz{{\cal K}_{0}}
\def\kk{\kappa(K)}
\def\kv{\kappa(V)}
\def\rr{{\rm I}\kern-.2em{\rm R}}
\def\rrsmall{{\mysmall I}\kern-.2em{\mysmall R}}
\def\R{{\rm I}\kern-.2em{\rm R}}
\def\emath{{\rm I}\kern-.2em{\rm E}}
\def\pmath{{\rm I}\kern-.2em{\rm P}}
\def\lmath{{\rm I}\kern-.2em{\rm L}}
\def\ro{{\rr}}
\def\rn{{\rr}^{n}}
\def\rnn{{\rr}^{n\times n}}
\def\rnlow{{\rr}_{n}}
\def\vkinchm{(V,\kappa)\in {\cal CH}(M)}
\def\vkkinkm{(V,\kappa,K)\in {\cal K}(M)}
\def\vvkkinkm{{\cal V}=(V,\kappa,K)\in {\cal K}(M)}
\def\vvkkinkz{{\cal V}=(V,\kappa,K)\in {\cal K}_{0}}
\def\vinkm{{\cal V}\in {\cal K}(M)}
\def\vinkz{{\cal V}\in {\cal K}_{0}}
\def\xinkv{x\in\kappa(V)}
\def\xinkk{x\in\kappa(K)}
\def\fk{f^{\kappa}}
\def\fke{f_{\eta}^{\kappa}}
\def\fkeo{f_{\eta_{1}}^{\kappa}}
\def\fket{f_{\eta_{2}}^{\kappa}}
\def\fkei{f_{\eta_{i}}^{\kappa}}
\def\szve{\sigma_{{\cal V},\eta}^{0}}
\def\sove{\sigma_{{\cal V},\eta}^{1}}
\def\skve{\sigma_{{\cal V},\eta}^{k}}
\def\rzveoet{\rho_{{\cal V},\eta_{1},\eta_{2}}^{0}}
\def\roveoet{\rho_{{\cal V},\eta_{1},\eta_{2}}^{1}}
\def\rtildeoveoet{\tilde \rho_{{\cal V},\eta_{1},\eta_{2}}^{1}}
\def\rkveoet{\rho_{{\cal V},\eta_{1},\eta_{2}}^{k}}
\def\delzveoet{\delta_{\cal V}^{0}(\eta_{1},\eta_{2})}
\def\deloveoet{\delta_{\cal V}^{1}(\eta_{1},\eta_{2})}
\def\delkveoet{\delta_{\cal V}^{k}(\eta_{1},\eta_{2})}
\def\delkvee{\delta_{\cal V}^{k}(\eta,\eta)}
\def\delkv{\delta_{\cal V}^{k}}
\def\delkvj{\delta_{{\cal V}_{j}}^{k}}
\def\dome{{\rm Dom}(\eta)}
\def\domei{{\rm Dom}(\eta_{i})}
\def\domej{{\rm Dom}(\eta_{j})}
\def\domeo{{\rm Dom}(\eta_{1})}
\def\domet{{\rm Dom}(\eta_{2})}
