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\title{\LARGE \bf
Uniqueness results for the value function via direct
trajectory-construction methods
}

\author{H\'ector J. Sussmann}







\thispagestyle{empty}
%\pagestyle{empty}

%\notitlepage

\begin{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
\begin{abstract}
We present some new \mbox{results},
together with a number of particularly \mbox{simple} and user-friendly 
versions of results obtained in \mbox{recent} years by the \mbox{author} 
and \mbox{M.~Malisoff}, on the \mbox{uniqueness} of \mbox{solutions} of the 
Hamilton-Jacobi-Bellman \mbox{equation} (HJBE) 
for \mbox{deterministic} finite-dimensional \mbox{optimal} control
\mbox{problems} under \mbox{non-standard hypotheses}. Our
\mbox{approach} is completely control-theoretic and totally
self-contained, using the systematic \mbox{construction} of 
\mbox{special} \mbox{trajectories} of \mbox{various} kinds, and not
\mbox{involving} any PDE methods.  We do%definitely 
not assume that the \mbox{Lagrangian} is positive, or 
that the \mbox{dynamics} is Lipschitz-continuous.
\end{abstract}
\end{comment}

\thispagestyle{empty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\LARGE \bf
Uniqueness results\\

\vskip-2mm
for the value function\\

\vskip-2mm
via direct\\

\vskip-2mm
trajectory-construction\\

\vskip1mm
methods
}

$\quad$\\
{\Large H\'ector J. Sussmann}\\

$\quad$\\
\vskip-5mm
Department of Mathematics\\
$\quad$\\
\vskip-10mm
Rutgers University\\

$\quad$\\
$\quad$\\
\vskip-12mm
{\tt\small $\qquad$sussmann@hilbert.rutgers.edu}
$\quad$\\
$\quad$\\
$\quad$\\


\vskip-5mm
{\sl The 42nd IEEE Conference\\
on Decision and Control}\\

$\quad$\\
\vskip-4mm
Maui, Hawaii\\
December 9-12, 2003
\end{center}

%\vskip4mm
\noindent\begin{minipage}{16.3cm}
\baselineskip=12pt
{\small{\it  Supported in part by {\bf\em NSF} Grant DMS01-03901. 
Written while the author was a guest of the
Institut Mittag-Leffler, in Sweden, during the Spring
of 2003. Very special thanks to {\bf\em Anders Lindquist}.}}
\end{minipage}




\thispagestyle{empty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Setting}

\nin Consider an autonomous {\em Lagrangian \mbox{optimization} problem}
with  dynamics 
\begin{displaymath} \dot x=f(x,u)
 \end{displaymath}
and cost functional 
\begin{displaymath}\label{lagra}
J=\int_{\tau_-(\xi)}^{\tau_+(\xi)}L(\xi(t),\eta(t))\,dt.
\end{displaymath}
($\tau_-(\xi)$, $\tau_+(\xi)$ are
the initial and \mbox{terminal} times of $\xi$.)

\nin An {\em augmented trajectory} is a pair $\Xi=(\xi,\xi_0)$ such that
for some control $\eta:(\tau_-(\xi), \tau_+(\xi))\to U$,
\[\left\{\begin{split}\dot \xi&=f(\xi, \eta) \\ \dot\xi_0&=L(\xi,\eta)
\end{split}\right. ,\]
so that $\xi_0$ is a {\em running cost}.

\begin{comment}
 , because  of our \mbox{system}, 
 \mbox{because} it really is a trajectory of the ``\mbox{augmented} 
 \mbox{control} system''  
 \vspace{-2mm}
 \begin{displaymath}\label{dynaaug}
 \dot x=f(x,u)\,,\qquad\dot x_0=L(x,u)
 \end{displaymath}
 \noindent obtained from (\ref{dyna}) by ``adding the cost as
 an extra \mbox{variable}'' in a well known way.
\end{comment}


\begin{comment}
 the \mbox{minimization} is over the set $\axts$ of all pairs
 \mbox{$\Xi=(\xi,\xi_0)$} such that
 \begin{itemize}
 \item[(i)]  \mbox{$\Xi$} \mbox{consists}  of a trajectory
 $\xi$ of (\ref{dyna}) (i.e., a \mbox{locally} \mbox{absolutely} 
 continuous (LAC) function  $\xi$ that satisfies $\dot\xi(t)=f(\xi(t),\eta(t))$
 for almost all $t$) \mbox{corresponding} to some $U$-valued control 
 $\eta$, and a ``\mbox{running} cost'' function $\xi_0$ corresponding to 
 $\xi$ and $\eta$ (i.e., a LAC
 function  $\xi_0$ such that \mbox{$\dot\xi_0(t)=L(\xi(t),\eta(t))$}
 for almost all $t$);
 \item[(ii)]  \mbox{$\xi$} starts at $x$,  and ``ends at the target'' in a 
 sense  to be defined precisely later.
 \end{itemize}
\end{comment}

\begin{comment}
 We will write
 $J(\Xi)$, rather than just $J$, for the left-hand side of
 (\ref{lagra}), \mbox{because} it is easy to see that the natural
 arguments for our cost functional $J$ are really augmented
 \mbox{trajectories}, \mbox{since} (i)~one cannot just regard $J$ as a
 functional of $\xi$ \mbox{only}, because the integral of (\ref{lagra})
 involves the control $\eta$ as well as $\xi$, but on the other hand
 (ii)~the cost is \mbox{completely} \mbox{determined} once we know
 $\xi$ and a \mbox{running} cost $\xi_0$, \mbox{because} in that case
 $L(\xi(t),\eta(t))=\dot\xi_0(t)$. 
\end{comment}
 
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main result}

\nin The state space is $\Omega$, open in $\R^n$. The {\em target space} $\cal
T$ is a closed subset of the closure of $\Omega$, disjoint from $\Omega$.

\nin Write $\axts$ for the set of augmented trajectories 
$(\xi,\xi_0)$ such that $\xi$ starts at $x$ and ``ends at the target''.


\nin The cost $J$ is not a functional of $\xi$, but is determined by the running
cost $\xi_0$ and so it is a functional of the augmented trajectories.


\nin For each \mbox{$x\in\Omega$}, minimize $J$ over all $\Xi\in\axts$.

\nin The infimum $V(x)$ of the costs $J(\Xi)$ of all 
\mbox{augmented} trajectories
$\Xi\in\axts$ is the {\em value} at $x$.

%(If 
%$\axts$ is empty, then  of course 
%\mbox{$V(x)=+\infty$}.)

\nin
$V:\Omega\mapsto\rr\cup\{-\infty,+\infty\}$ is the {\em value
function}.

\nin {\bf Main result:}  under weak regularity conditions on $L$ and $f$,
if a ``Malisoff-type'' condition and a ``LaSalle-type'' 
condition hold, then 
a viscosity solution of the HJBE that satisfies an
\mbox{appropriate} boundary condition and is continuous and bounded below) 
must be the value \mbox{function}.


%``\mbox{Uniqueness}'' is to be understood as ``\mbox{uniqueness}
%within a class defined by some additional properties,'' such as that
%of all functions that are continuous and bounded below.

%We will work with a class of systems which is \mbox{sufficiently} general to
%capture some new and  \mbox{interesting} \mbox{phenomena},
%and restricted \mbox{enough}
%to make it \mbox{possible} to prove strong \mbox{theorems}.

%In  particular,  we will asume that


\newpage
\section{Regularity Assumptions}

\begin{itemize}

\item Each set \[\fsh(x,U)=\{(f(x,u),L(x,u)):u\in U\}\]
is closed and convex (but not necessarily \mbox{compact}).

\item  A {\em coerciveness} condition holds locally:
\[\|f(x,u)\|^r\le L(x,u)+C,\]
with $C>0$ and $r>1$, uniformly with respect to $u$.

\item $f(x,u)$ and $L(x,u)$ are continuous
with respect to $x$, uniformly with respect to $u$ locally in $x$. 

\end{itemize}


{\bf \nin Note: $f(x,u)$ need not be  Lipschitz-continuous with respect to $x$.}

%since one of the main purposes of this work is to clarify the exact
%role of the Lipschitz-continuity assumptions often made in the viscosity
%literature.


\newpage
\section{Uniqueness and \\ Uniquely Limiting Trajectories}
\begin{itemize}
\item[(a)] If  $V$ is a continuous viscosity solution
of the HJBE, then
for each $x\in \Omega$ there exist a 
$\tau\in\,]0,\infty]$ and a forward maximally defined 
$\Xi=(\xi,\xi_0)$ with domain $[0,\tau[$ such that $\xi(0)=x$ and 
\begin{displaymath}\label{steepdesc}
V(\xi(t_0))+\xi_0(t_0)\ge V(\xi(t_1))+\xi_0(t_1)
\end{displaymath}
whenever $0\le t_0\le t_1<\tau$.

\nin The existence of these augmented trajectories
of {\em steepest descent} is proved without Lipschitz-continuity
hypothesis.
\item[(b)] By applying result (a) to $-V$ and $-L$ we can
derive 
the existence of ``Dynamical Programming Inequality'' 
(DPI) trajectories,
along which

\vskip-18mm
\begin{displaymath}\label{steepasc}
V(\xi(t_0))+\xi_0(t_0)\le V(\xi(t_1))+\xi_0(t_1),
\end{displaymath}
\nin whenever $t_0\le t_1$.
\end{itemize}
\newpage
\begin{itemize}
\item[(c)] From the existence of DPI trajectories, conclude that
for every $x$ and every 
``\mbox{sufficiently} nice'' (e.g., piecewise constant) \mbox{control}
$\eta$ {\em there exists a DPI trajectory of $\eta$ 
starting at $x$.}
\end{itemize}

\vspace{-83mm}
\begin{itemize}
\item[(d)] To show that $V$ is bounded above by the value function,
one needs to show DPI holds for {\em every} trajectory corresponding to
every control.

\item[(e)] There is no gap from the conditions in $(c)$ to those in $(d)$
when uniqueness of trajectories holds for every control and initial
condition, for example when for every control $\eta$, locally, a bound

\vskip-18mm
\begin{displaymath}
\langle f(x,\eta(t))-f(y,\eta(t)),x-y\rangle 
\le k(t)\|x-y\|^2\,,
\end{displaymath}

\vskip-9mm
\nin holds, with $k$ integrable.

\item[(h)] $\Xi$ is called {\em uniquely \mbox{limiting}} if
there exists a sequence $\{\eta_j\}_{j=1}$ of piecewise
constant controls with the property that 
\begin{itemize}
\item[(*)] {\em no matter how we choose trajectories
$\Xi_j$ for the $\eta_j$ with the same initial condition
as $\Xi$, the $\Xi_j$ converge to $\Xi$.}
\end{itemize}
\end{itemize}

\begin{comment}
 \newpage
 \begin{remark}
 Our approach owes a great deal to the book
 by A.I. Subbotin.  Subbotin, however, \mbox{only} studies
 \mbox{viscosity} \mbox{solutions} of PDEs of the form $F(x,u(x),Du(x))=0$,
 where the Hamiltonian $F(x,u,p)$ is required to be globally
 \mbox{Lipschitz} with respect to $p$ (cf. \mbox{Equation} (2.2) in
 that book.  A somewhat weaker \mbox{hypothesis} is also
 considered later but, even with the \mbox{weakened}
 \mbox{requirement}, the results do not cover, for example, coercive
 problems of the kind discussed here, such as linear quadratic
 \mbox{optimal} \mbox{control}. \proofend
 \end{remark}
\end{comment}

%\section{The main theorem}\label{sec-standard}
\section{Systems and augmented systems} 
\noindent An {\em $n$-dimensional control \mbox{system}} is
a triple

\vskip-12mm
$$
\Sigma=(\Omega,U,f)
$$

\vskip-8mm
\noindent such that\\
$\bullet$\ $\Omega$ (the {\em \mbox{state} space})
is open in $\R^n$,\\  
$\bullet$\ $U$ (the {\em control \mbox{space}}) is nonempty,\\ 
$\bullet$\ $f$ (the {\em dynamics}) is a map 
$$
\Omega\times U\ni(x,u)\mapsto f(x,u)\in\R^n\,.
$$ 
\nin An {\em $n$-dimensional augmented 
control system} is a 4-tuple

\vskip-8mm
$$
\hat\Sigma=(\Omega,U,f,L)
$$ 
such that\\
$\bullet$\ $\Sigma=(\Omega,U,f,L)$ is an $n$-dimensional control system\\ 
$\bullet$\ $L$ (the {\em Lagrangian}) is a map 

\vskip-11mm
$$
\Omega\times U\ni(x,u)\mapsto L(x,u)\in\R\,.
$$ 

\nin An augmented system \nin $\hat\Sigma=(\Omega,U,f,L)$ is {\em \mbox{continuous}}
if the maps $x\mapsto f(x,u)$ and $x\mapsto L(x,u)$ are continuous for each
fixed $u$.

\section{Uniform continuity}

\nin A system \nin $\hat\Sigma$ is {\em \mbox{uniformly} continuous} on
$S\subset \Omega$ if there is $\omega:\,]0,\infty[\to[0,\infty]$ with
$\lim_{s\downarrow 0}\omega(s)=0$ and
$$
\|f(x,u)-f(y,u)\|+|L(x,u)-L(y,u)|\le\omega(\|x-y\|)
$$
whenever $x,y\in S$, $u\in U$.

\nin A system $\hat\Sigma$ {\em \mbox{locally} \mbox{uniformly} \mbox{continuous}} if it is 
uniformly \mbox{continuous} on compact subsets of $\Omega$, and
{\em globally  \mbox{uniformly} \mbox{continuous}} if it is 
uniformly continuous on $\Omega$.


\section{Lipschitz continuity}

\nin A system $\hat\Sigma$
{\em Lipschitz continuous} if 
\mbox{$x\mapsto f(x,u)\in\R^n$} and 
\mbox{$x\mapsto L(x,u)\in\R$} 
are Lipschitz continuous for each fixed $u$.


\nin A system $\hat\Sigma$
{\em \mbox{uniformly} Lipschitz continuous} on a subset $S$ of $\Omega$
if there exists a positive constant $C$ such that
$$
\|f(x,u)-f(y,u)\|+|L(x,u)-L(y,u)|\le C\|x-y\|
$$
whenever $x,y\in S$ and $u\in U$. We call $\hat\Sigma$
{\em \mbox{locally} \mbox{uniformly} Lipschitz continuous} if it is 
uniformly \mbox{Lipschitz} continuous on every compact subset of $\Omega$, and
{\em globally  uniformly \mbox{Lipschitz} continuous} if it is 
uniformly Lipschitz continuous on $\Omega$. 

\section{Coercivity}

\nin The control system $\hat\Sigma=(\Omega,U,f,L)$ is {\em
\mbox{coercive}} on a subset $S$ of $\Omega$ if there exist real constants
$r$, $A$, $C$, such that $A>0$, $C>0$, $r>1$, and 
\begin{displaymath}\label{iscoer}
\|f(x,u)\|^{r}\le AL(x,u)+C\,\,{\rm for\,\,all}\,\,
x\in S\,,\,u\in U\,.
\end{displaymath}


\nin The system $\hat\Sigma$ is {\em locally coercive} if it is coercive on
every \mbox{compact} subset of $\Omega$, and {\em globally coercive} if it is
\mbox{coercive} on $\Omega$.


\nin If $\hat\Sigma$ is coercive on $S$, then it is always
possible to choose $A=1$ above.


\section{Convexity and semi-continuity}

\nin The {\em augmented dynamics} of the $\hat\Sigma=(\Omega,U,f,L)$
is the map
$$
\fsh:\Om\times U\mapsto\R^{n+1}
$$
such that
\[\fsh(x,u)=(f(x,u),L(x,u)).\]% for $x\in\Omega$, $u\in U$.
 
\nin The augmented system  $\hat\Sigma$ {\em satisfies the convexity and 
upper semicontinuity condition} if, for each $x\in\Omega$, 
\begin{displaymath}\label{convus}
\fsh(x,U)=\bigcap_{\eps>0}\overline{{\rm co}}\Big(\bigcup\{\fsh(x',U):
\|x'-x\|\le\eps\}\Big)\,,
\end{displaymath}
where ``$\overline{{\rm co}}$'' stands for ``closed convex hull,''
and $\fsh(x,U)\equaldef\{(f(x,u),L(x,u)):u\in U\}$.


\nin If $\hat\Sigma=(\Omega,U,f,L)$ is locally 
uniformly continuous, then $\hat\Sigma$ satisfies the \mbox{convexity} and 
upper semicontinuity condition iff
the set  $\fsh(x,U)$ is closed and convex for every $x\in\Omega$.


\section{Trajectories}

A {\em trajectory} of $\hat\Sigma=(\Omega,U,f,L)$ 
is a locally absolutely continuous (LAC) curve 
\[
I\ni t\mapsto\xi(t)\in\Omega
\]
\nin defined on a nonempty subinterval $I$ of $\R$, such that 
\mbox{$\dot\xi(t)\in f(\xi(t),U)$} for almost every \mbox{$t\in I$}. 

\nin The {\em starting time} of a trajectory $\xi$ with domain $I$ 
is $\tau_-(\xi)\equaldef\min\,I$, if the \mbox{minimum} exists.
If the starting time of $\xi$ exists, 
then\\
(i)~the point $x_-(\xi)\equaldef\xi(\tau_-(\xi))$
is the {\em starting point} of\\
\null$\quad\,\,\xi$,\\
(ii)~the ordered pair 
$\partial_-(\xi)\equaldef(\tau_-(\xi),x_-(\xi))$  is the
{\em initial\\
\null$\qquad\,\,$condition} of $\xi$,\\
(iii)~If $\partial_-(\xi)=(t,x)$ (resp. 
$\partial_-(\Xi)\!=\!(t,x)$), we say\\
\null $\qquad\,\,$ that $\xi$ {\em starts at $x$ \mbox{at time $t$}}. 

\nin {\em Augmented trajectories} and related concepts are defined similarly.

\begin{comment}
 An {\em augmented trajectory} of $\hat\Sigma$ is a LAC curve 
 \begin{displaymath}\label{augmtrajo}
 I\ni t\mapsto\Xi(t)=(\xi(t),\xi_0(t))\in\Omega\times\R\,,
 \end{displaymath}
 defined on a subinterval $I$ of $\R$,
 having the property that $\dot\Xi(t)\in\fsh(\xi(t),U)$ for
 almost every $t\in I$.  
\end{comment}


\newpage

\section{Admissible trajectories and\\the value function}

\nin A {\em target} for \mbox{$\hat\Sigma=(\Omega,U,f,L)$} is a closed subset
${\cal T}$ of $\R^n$ such that \mbox{${\cal T}\subseteq{\rm Closure}\,\Omega$}
and ${\cal T}\cap\Omega=\emptyset$.

\nin An augmented trajectory $\Xi=(\xi,\xi_0)$ with domain $I$ {\em ends at
 ${\cal T}$} if  $\xi(\uparrow) \equaldef
\lim_{t\uparrow\sup\,I}\xi(t)$ exists and is in ${\cal T}$.


\nin For each $x\in\Omega$, let $\axts$ be the set of \mbox{all}
$\,\Xi=(\xi,\xi_0)$ 
such that $\xi$ starts at $x$ at time $0$, and ends at $\cal T$,
and $\xi_0$ starts at zero, and moreover the following exists:
\[\xi_0(\uparrow)\equaldef\lim_{t\uparrow\sup\,{\rm domain}\,\Xi}\xi_0(t).\]

\vskip-11mm
\nin The {\em cost} of $\Xi=(\xi,\xi_0)\in\axts$ is 
%\begin{displaymath}\label{thecost}
$J(\Xi)\equaldef \xi_0(\uparrow).$
%\end{displaymath}

\nin The {\em value function} of the optimal control \mbox{problem}
\mbox{defined}
by $\hat\Sigma$ and ${\cal T}$ 
is
\mbox{$\vts:\Omega\cup{\cal T}\mapsto\R\cup\{-\infty,+\infty\}$}
given by
$$
\vts(x)=\left\{\begin{array}{lll}
\inf\{J(\Xi):\Xi\in\axts\}&{\rm if}&x\in\Omega\\
0&{\rm if}&x\in{\cal T}\,.\end{array}\right.
$$

\def\sdsv{{\cal SD}_{\hat\Sigma,V}}
\def\sdsvx{{\cal SD}_{\hat\Sigma,V,x}}
\def\msdsv{{\cal MSD}_{\hat\Sigma,V}}
\def\msdsvx{{\cal MSD}_{\hat\Sigma,V,x}}

\section{Steepest descent}


\nin An \mbox{augmented} trajectory $\Xi=(\xi,\xi_0)$ of $\hat\Sigma$ 
is {\em of steepest descent with respect to the function $V$} 
if
\[
\xi_0(s)+V(\xi(s))\ge\xi_0(t)+V(\xi(t))
\]
whenever $s\le t$ and $s,t\in{\rm domain}\,\Xi$.

\nin Write $\sdsv$ for the set of all  augmented 
\mbox{trajectories} of $\hat\Sigma$ of steepest descent with respect to $V$.

\nin Write $\sdsvx$ for the set of all  $\Xi\in\sdsv$ 
that start at $(x,0)$. 

\nin Write $\msdsvx$ for the set of all
maximal augmented \mbox{trajectories} of $\hat\Sigma$ from $x$ of steepest
descent with respect to $V$, in the sense that these have no extension
to $\tilde \Xi\in\sdsvx$ defined on a strictly larger interval.

\nin It follows from Zorn's lemma that $\msdsvx$ is always non-empty.

\begin{comment}
\nin A {\em maximal augmented trajectory of
$\hat\Sigma$ from $x$ of steepest \mbox{descent} with respect to $V$}
is a $\Xi\in\sdsvx$ that \mbox{cannot} be \mbox{extended} to a
$\tilde\Xi\in\sdsvx$ defined on an interval which is strictly larger
that the domain of $\Xi$.
\end{comment}

\begin{comment}
 \medbreak The following observation is a completely trivial
 \mbox{consequence} of Zorn's lemma, \mbox{given} our 
 definitions, but we state it explicitly as a
 \mbox{separate} \mbox{result} for future \mbox{reference}. 
\end{comment}

\begin{comment}
 \begin{proposition}\label{trivialprop}
 {\em If $\hat\Sigma=(\Omega,U,f,L)$ is an \mbox{augmented} control system,
 $V:\Omega\mapsto\R$ is a function, 
 and $x\in\Omega$, then the set $\msdsvx$ of maximal \mbox{augmented}
 \mbox{trajectories} of $\hat\Sigma$ from $x$ of steepest \mbox{descent}
 with respect to $V$ is nonempty.}\proofend
 \end{proposition}
\end{comment}

\section{Viscosity solutions}

\nin Let $\Omega\subset \R^n$ be open. A {\em subdifferential} of
$V:\Omega\mapsto\R$ at $\bar x\in\Omega$ is a vector $p\in\R^n$ such that
$$
\lim\inf_{x\to\bar x}\frac{V(x)-V(\bar x)-p\cdot(x-\bar x)}{
\|x-\bar x\|}\ge 0\,.\,
$$

\nin The function \mbox{$V:\Omega\mapsto\R$} {\em satisfies the inequality}
\begin{displaymath}
\sup\{-\langle \nabla V(x),f(x,u)\rangle-L(x,u):u\in U\}\ge 0
\end{displaymath}
\nin on $\Omega$ {\em in the viscosity sense} if 
$$
\sup\{-\langle p, f(x,u)\rangle-L(x,u):u\in U\}\ge 0,
$$
for all $x\in\Omega$ and any subdifferential $p$ of $V$
at $x$.


\section{Viscosity solutions (cont'd)}

\nin A {\em superdifferential} of $V$ at $x_*$ is a vector $p$ 
such that $-p$ is a subdifferential of $-V$ at $x_*$.

\nin Similarly, $V$ {\em satisfies the inequality}
\begin{displaymath}
\sup\{-\langle \nabla V(x),f(x,u)\rangle-L(x,u):u\in U\}\le 0
\end{displaymath}
\nin on $\Omega$ {\em in the viscosity sense} if 
$$
\sup\{-\langle p, f(x,u)\rangle-L(x,u):u\in U\}\le 0\,.
$$
for any $x\in\Omega$ and any superdifferential $p\in\R^n$ of
$V$ at $x$.

\nin Finally, $V$ {\em satisfies the equation}
\begin{displaymath}\label{eqeq}
\sup\{-\langle \nabla V(x),f(x,u)\rangle-L(x,u):u\in U\}=0
\end{displaymath}
on $\Omega$ {\em in the viscosity sense} if 
it satisfies both inequalities in the viscosity 
sense. 

\begin{comment}
 \begin{remark}
 The definition of ``viscosity solution'' given here 
 is known to be equivalent to the more \mbox{common} one involving 
 test functions.\proofend
 \end{remark}
\end{comment}

\section{Uniquely limiting arcs}

\vskip-10mm
\nin An {\em augmented arc} is an augmented trajectory whose 
\mbox{domain} is a compact interval.

\vskip-2mm
\nin An augmented arc $\Xi=(\xi,\xi_0)$ of
$\hat\Sigma=(\Omega,U,f,L)$ with \mbox{domain} $[a,b]$ is 
{\em uniquely limiting} if there exists a sequence 
$\{\eta_j\}_{j=1}^\infty$ of piecewise constant functions
$\eta_j:[a,b]\mapsto U$ such that 
\begin{itemize}
\item[(*)] {\em if $\{\Xi_j\}_{j=1}^\infty$ is an arbitrary sequence
of \mbox{maximally} defined augmented trajectories of $\hat\Sigma$ 
such that \mbox{$a\in{\rm domain}(\Xi_j)$} and $\Xi_j(a)=\Xi(a)$
for every $j$, then $[a,b]\subseteq{\rm domain}(\Xi_j)$ if $j$ is large enough,
and $\Xi_j\to\Xi$ uniformly on $[a,b]$ as $j\to\infty$.}
\end{itemize}


\vskip-3mm
\nin An augmented trajectory $\Xi=(\xi,\xi_0)$ with domain $I$ is 
{\em locally uniquely limiting} if for every compact \mbox{subinterval} 
$I'$  of $I$ the restriction of $\Xi$ to 
$I'$ is uniquely limiting.

\nin An augmented trajectory $\Xi=(\xi,\xi_0)$ with domain $I$ is {\em almost
uniquely limiting} if there exists 
a finite subset $B$ of $I$ such that the restriction of
$\Xi$ to every subinterval of $I\backslash B$ is 
locally uniquely limiting.


\section{Example} \begin{itemize}\item[] Let $\Xi=(\xi,\xi_0)$  be an augmented arc  of
$\hat\Sigma$ with \mbox{domain} $[a,b]$. Suppose there is $\delta>0$,
$\eta:[a,b]\mapsto U$, and an integrable $\varphi:[a,b]\mapsto[0,\infty]$, such
that
\begin{itemize}
\item[(i)] $\dot\Xi(t)=\Big(f(x,\eta(t)),L(x,\eta(t))\Big)\,\,$ for ae
$t\in[a,b]$, 
\item[(ii)] the map $t\mapsto\Big(f(x,\eta(t)),L(x,\eta(t))\Big)\,\,$, on the 
\mbox{compact} set 
$I_x\equaldef\{t:a\le t\le b\wedge \|x-\xi(t)\|\le\delta\}$, 
%\item[] $\qquad I_x\equaldef\{t:a\le t\le b\wedge \|x-\xi(t)\|\le\delta\}$, 
is measurable for each $x\in\Omega$, 
\item[(iii)] the map $x\mapsto\Big(f(x,\eta(t)),L(x,\eta(t))\Big)\,\,$, on the 
\mbox{compact} set $I^t\equaldef\{x\in\Omega:\|x-\xi(t)\|\le\delta\}$, 
%\item[] $\qquad I^t\equaldef\{x\in\Omega:\|x-\xi(t)\|\le\delta\}$, 
is continuous for each $t\in[a,b]$, 
\item[(iv)] \mbox{$\Big\langle f(x,\eta(t))-f(x',\eta(t)),x-x'\Big\rangle
\le\varphi(t)\|x-x'\|^2$}
whenever $t\in[a,b]$, $\|x-\xi(t)\|\le\delta$, and
$\|x'-\xi(t)\|\le\delta$,
\item[(v)] the inequality $|L(x,\eta(t))|\le\varphi(t)$ 
holds \mbox{whenever} $t\in[a,b]$ and $\|x-\xi(t)\|\le\delta$.
\end{itemize}
\end{itemize}

\medbreak\noindent Then $\Xi$ is uniquely limiting. \proofend
%\end{example}


\newpage
\section{The Malisoff-type condition}

\nin{\bf Improvements:} 
An {\em improvement} of the \mbox{augmented} arc $\Xi=(\xi,\xi_0)$ with
domain $[a,b]$ is any $\Xi'=(\xi',\xi'_0)$, with domain $[a',b']$, such
that
$$
\xi'(a')=\xi(a)\,,
$$
$$
\xi'(b')=\xi(b)\,,
$$
\[\xi_0'(b')-\xi_0'(a')\le\xi_0(b)-\xi_0(a).\]

\nin{\bf The Malisoff-type condition:}
Every augmented arc has an almost locally uniquely 
limiting improvement. 



%\section{Right-unbounded trajectories}
\section{The LaSalle-type condition}


\nin{\bf Right-unbounded trajectories:}
If $\Omega$ is an open subset of $\R^n$, a
curve \mbox{$\xi\!:\!I\!\mapsto\!\Omega$} is \mbox{{\em right-unbounded} if}
\begin{itemize}
\item[(i)] the interval $I$ is open on the right (that is, 
if \mbox{$\tau=\sup\,I$}, then either (a)~$\tau=+\infty$ or
(b)~$\tau$ is finite and does not belong to $I$),
\item[(ii)] if $\tau$ is finite, then for every compact subset $K$
of $\Omega$ there exists a $\tau_K$ such that $0\le\tau_K<\tau$ and
\mbox{$\xi(t)\notin K$} whenever $\tau_K<t<\tau$. 
\end{itemize}


\nin{\bf The LaSalle-type condition:}
Whenever $x\in\Omega$, $\Xi=(\xi,\xi_0)\in\msdsvx$, 
and $\xi$ is right-unbounded, it follows that $\Xi\in\axts$.

%%%Our main result is the following theorem:

\section{Main result}

%\begin{theorem}\label{mainres}
%{\em
\nin Let $\hat\Sigma=(\Omega,U,f,L)$ be an \mbox{augmented} 
system, let ${\cal T}$ be a target for $\hat\Sigma$, and
let \mbox{$V:\Omega\cup{\cal T}\mapsto\R$}. Suppose

\begin{itemize}\item[]
\begin{itemize}
\item[(1)] $\hat\Sigma$ is locally uniformly continuous, locally coercive,
and such that $F_{\hat\Sigma}(x,U)$ is closed and convex for \mbox{every} 
\mbox{$x\in\Omega$}. 
\item[(2)] $V$ is \mbox{continuous}.
\item[(3)] $V$ satisfies \[\sup\{-\langle \nabla V(x),f(x,u)\rangle-L(x,u):u\in U\}=0\]
on $\Omega$ in the
\mbox{viscosity} sense.
\item[(4)] $V$ vanishes on ${\cal T}$.
\item[(5)] The Malisoff-type condition holds.
\item[(6)] The LaSalle-type condition holds.
\end{itemize} \end{itemize}
Then $V$ is the value function.
%}
%\end{theorem}

%\begin{remark}
%Note: condition (6) was essentially introduced by M. Malisoff.
%\end{remark}


\section{Example: LQ optimal control}

\nin The dynamical law is $\displaystyle \dot x=Ax+Bu$,
in which  $x$ and $u$ take values in $\R^n$ and $\R^m$, \mbox{respectively}.
The pair $(A,B)$ is stabilizable.

\nin The Lagrangian is given by 
$$
L(x,u)=x^\dagger Rx+u^\dagger Su\,,
$$
where the  matrices $R$, $S$ are strictly positive \mbox{definite}.

\nin We take the \mbox{target} set ${\cal T}$ to be the origin of $\R^n$.

\nin Introduce a new scalar nonnegative control $v$ and replace the Lagrangian
by $\tilde L(x,u,v)\equaldef L(x,u)+v$, keeping the same dynamics. This ensures
convexity of $F(x,U)=\{(f(x,u),L(x,u))\}$ for \mbox{every} $x\in\Omega$,

\nin If $V$ is bounded below, then all hypothesis of the main theorem
are easily verified, except for the last.

\section{LQ optimal control (cont'd)}

%\nin The crucial \mbox{technical} issue here is the
%fact that the Lagrangian is not bounded away from zero. 

%\nin The \mbox{hypotheses} of our main \mbox{theorem} (including the coerciveness,
%which \mbox{follows} from the \mbox{positive} \mbox{definiteness} of $S$) are easily 
%verified as long as $V$ is bounded below.

\nin To prove the last hypothesis, let $\Xi=(\xi,\xi_0)$ be
a right-unbounded \mbox{maximal} \mbox{trajectory} of steepest descent with
respect to $V$, with domain $[0,\tau[$, that does not end at the target.

\nin If $\tau$ were \mbox{finite} then the boundedness of the cost would
imply an $L^2$ bound on the control and therefore $\Xi$ could be extended
to \mbox{interval} $[0,\tau]$ and then further, using the assumption that
that $\Xi$ does not end at the \mbox{target}. So $\tau$ is \mbox{infinite}.

\nin On the other hand, the fact that
$V$ is bounded \mbox{below} and $\Xi$ is of \mbox{steepest} descent implies
that the integral 
$$
\int_0^\infty\Big(\xi(t)^\dagger R\xi(t)+
\eta(t)^\dagger S\eta(t)\Big)dt
$$
is finite for some $\eta$. But then $\xi$ and $\eta$ are square-integrable
and thus \mbox{$\dot\xi=A\xi+B\eta$}
implies that $\xi$ is square-integrable and has
a square-integrable derivative. Then Barbalat's lemma
implies that  $\xi$  ends at the \mbox{target}. \proofend
%\end{example}



\section{Example: Fuller's problem}

\nin This is the optimal control problem for the dynamics
$$
\dot x=y\,,\qquad\dot y=u\,,
$$
with control constraint $-1\le u\le 1$.

\nin The Lagrangian is $L(x,y,u)=x^2$ and the target is the origin of $\R^2$.

\nin The Lagrangian is not bounded away from zero, and in fact has a whole line
of zeros.  All but the last \mbox{hypotheses} of our main theorem are easily 
verified as long as $V$ is bounded below.


\section{Fuller's problem (cont'd)}

\nin Let $\Xi=(\xi,\xi_0):[0,\tau\,[\,\mapsto\R^3$
be a right-unbounded \mbox{maximal} \mbox{trajectory} of steepest descent with
respect to $V$ that does not end at the target.


\nin If $\tau$ were \mbox{finite} then the boundedness of the control would
\mbox{trivially} imply that $\Xi$ can be extended to the closed \mbox{interval}
$[0,\tau]$, and even further, using the assumption that $\Xi$ does not end at
the target.  So $\tau$ is \mbox{infinite}.

\nin Since $V$ is bounded \mbox{below} and $\Xi$ is of \mbox{steepest} descent,
$\int_0^\infty x(t)^2dt$ is finite, if we write
\mbox{$\xi(t)=(x(t),y(t))$}.

\nin Then $x(\cdot)$ is square-integrable on $[0,\infty\,[\,$ and its second
derivative is bounded.� By a \mbox{generalization} of Barbalat's lemma, both
$x(\cdot)$ and $y(\cdot)$ go to zero, as desired. \proofend
%\end{example} 

\newpage

\section{Example: reflected brachistochrone}

%\begin{example}
\nin This is
the minimum time problem for 
\begin{eqnarray*}
\dot x&=&u\sqrt{|y|}\,,\\
\dot y&=&v\sqrt{|y|}\,,
\end{eqnarray*}
with constraint $u^2+v^2\le 1$.
The target set ${\cal T}$ consists 
of a point $B\in\R^2$.

\nin The crucial issue is that the dynamical law is not Lipschitz-continuous
with respect to the  state.

\nin All but one of the \mbox{hypotheses} of the main theorem are easily
verified.  The non-trivial check is to verify that every augmented arc has an
almost locally uniquely limiting improvement.

 
\section{Reflected brachistochrone (cont'd)}

\nin Let $\xi:[a,b]\mapsto\R^2$ be any arc. If $\xi(t)$ is never in the $x$-axis
$X$, then $\xi$ satisfies the conditions of the example above, so $\xi$ is
uniquely \mbox{limiting}.

\nin If $\xi(t)$ touches $X$ at a single time $\bar t$, then the
\mbox{restriction} of $\xi$ to each of the \mbox{intervals} $[a,\bar t[$,
$]\bar t,b]$, is \mbox{locally} uniquely \mbox{limiting}, so $\xi$ is
\mbox{almost} locally uniquely \mbox{limiting}.

\nin If $\xi(t)$ touches $X$ at more than one time instant, there exist
$t_-<t_+ \in [a,b]$ such that $\xi(t_-)\in X$, $\xi(t_+)\in X$, and
\mbox{$\xi(t)\notin X$} for $t\in [a,t_-[\cup ]t_+b,b]$.  The restriction
$\tilde\xi$ of $\xi$ to $[t_-,t_+]$ is such that
\mbox{$S=\{t\in[a,b]:\xi(t)\notin X\}$} is the union of a finite or countable
infinite collection ${\cal I}$ of pairwise disjoint relatively open subintervals
of $[a,b]$.  Let $\tilde\xi_I$ be the restriction of $\tilde\xi$ to $I\in {\cal
I}$.

\nin By reflecting, if necessary, $\tilde \xi_I$ is contained in the closed upper
half-plane, so that each $\xi_I$ may be replaced by a cycloid, the time-optimal
solution to the brachistochrone problem.


\begin{comment}
By reflecting 
$\tilde\xi_I$ with respect to $X$, if necessary, we may 
assume that $\tilde\xi_I$ is entirely contained in the
open upper half-plane for every $I\in{\cal I}$. Then 
$\tilde\xi$ is a trajectory of our system entirely 
\mbox{contained}
in the closed upper half-plane. It is well known that the problem 
in the closed upper half-plane $H_+$ 
is the famous ``brachistochrone problem,''
whose time-optimal \mbox{trajectories} $\zeta:[\alpha,\beta]\mapsto H_+$
are cycloids such that $\zeta(t)$ is an interior point of $H_+$ 
whenever $\alpha<t<\beta$.  It follows that we can always replace 
$\tilde\xi$ by a cycloid $\zeta$, thereby \mbox{obtaining} an
almost locally uniquely limiting \mbox{improvement} of $\xi$. \proofend
%\end{example} 
\end{comment}

\newpage

\section{Example: non-uniqueness of trajectories}

%\begin{example} 
\nin Let $\varphi:[0,1]\mapsto\rr$ be a continuous, nonnegative function that
vanishes exactly on the Cantor set, and such that
$\int_0^1\frac{dx}{\varphi(x)}<\infty$.

\nin Extend $\varphi$ to $\rr$ by periodicity.
to a function defined on $\rr$ by making it periodic of
period $1$.

\nin Consider the time-optimal control problem on $\rr$ whose
dynamics is $\dot x=u\varphi(x)$, $|u|\le 1$, and where the target is the origin.

\nin Optimal trajectories from each point $x$ exists and are obtained by using
the control $u=-1$ as long as we are to the right of the origin, and we use
$u=1$ if we are to the left. However, this does not specify the optimal
trajectories, due to non-uniqueness of solutions.

\section{Example cont'd:\\explicit construction of $\varphi$}
 \nin For example, take
 $\varphi$ to be given by \[\varphi(x)={\rm dist}(x,C)^\rho,\] where $C$
 is the Cantor set and $\rho$ is a positive number such that
 $\rho<1-{\rm log}_{_3}2$.


\nin An explicit calculation shows that
 $\int_0^1\frac{dx}{\varphi(x)}=(1-\rho)^{-1}2^\rho\sum_{j=1}^{\infty}
 \theta^j$, where $\theta=\frac{2}{3}\times 3^\rho$. Our choice of
 $\rho$ guarantees that $\theta<1$, so the integral is finite.



\section{Example cont'd: value function}

\nin Find an optimal trajectory starting at $\bar x<0$:

\nin Let $\tau(x)=\int_{\bar x}^x\frac{dy}{\varphi(y)}$. This is strictly
increasing and maps $[\bar x,0]$ onto $[0,\tau(0)]$.

\nin The inverse $\xi=\tau^{-1}:[0,\tau(0)]\to [\bar x,0]$ satisfies
$\dot\xi(t)=\varphi(\xi(t))$ for ae $t\in[0,\tau(0)]$.

\nin $\xi$ is a time-optimal trajectory starting at $\bar
x$ and ending at the target in time $\tau(0) = \int_{\bar
x}^0\frac{dy}{\varphi(y)}$.

\nin A similar construction can be made for  $\bar x>0$, so that the value
 function $\bar V$ for this problem is
$$
\bar V(x)=\int_{\min(x,0)}^{\max(x,0)}\frac{dy}{\varphi(y)}\,.
$$



\section{Example cont'd: \\ the value function $\bar V$ solves the HJBE}

\nin The HJBE for this problem is
%\begin{displaymath}\label{isasol}
$\displaystyle |V'(x)|\varphi(x)-1=0\,.$
%\end{displaymath}

\nin The value function $\bar V(x)=\int_{\min(x,0)}^{\max(x,0)}
\frac{dy}{\varphi(y)}$ solves the HJBE on $\rr\backslash\{0\}$ in
the \mbox{viscosity} sense:

\nin At points $x$ such that $\varphi(x)>0$, $\bar V$ is smooth and
\[  \frac{d\bar V}{dx}=-\frac{1}{\varphi}  \mbox{ if }x<0
\mbox{ and }  \frac{d\bar V}{dx}=\frac{1}{\varphi} \mbox{ if }x>0.\]

\nin  At $x$ where $\varphi(x)=0$, it must be shown that $-1\ge 0$
for any subdifferential of $\bar V$ at $x$, and
$-1\le 0$ for any superdifferential of $\bar V$ at $x$. This
last condition is trivially true. To verify the first, one needs to 
show that there are no subdifferentials of $\bar V$ at such $x$.


\section{Example cont'd:\\ the value function has no subdifferentials}

\nin Recall that $p\in \R$ is a subdifferential of $\bar V$ at $x$ iff 
\[ \liminf_{h\to 0} \frac{V(x+h)-V(x)-ph}{h} \ge 0.\]

\nin Suppose $x<0$ is a zero of $\varphi$. For $h>0$,
\[ V(x+h)-V(x) = -\int_x^{x+h} \frac{1}{\varphi}dx \le
-\frac{h}{\max_{y\in[x,x+h]} \varphi(y)}.\]

\nin Since $\varphi(x)=0$, $\max_{y\in[x,x+h]} \varphi(y) \to 0^+$ as $h\to 0^+$
and so the right derivative of $\bar V$ at $x$ is $- \infty$. Thus there are no
subdifferentials of $\bar V$ at such $x$.

\nin By symmetry, if $x>0$ is a zero of $\varphi$, then the left derivative of
$\bar V$ at $x$ is $+\infty$, and $\bar V$ has  no subdifferentials at $x$.




\section{Example cont'd: \\another solution of the HJBE}

\nin Claim: there exists a nonnegative \mbox{continuous} $\hat V:\rr\mapsto\rr$ other
than $\bar V$ that satisfies the HJBE on $\rr\backslash\{0\}$ and the 
boundary condition $\hat V(0)=0$.

\nin Let $W:[0,+\infty[\to\rr$ be the Cantor function on $[0,1]$.

\nin Extend $W$ to $[0,+\infty[$ by $W(k+x)=k+W(x)$ for any positive integer $k$
and $0\le x <1$, and obtain a continuous nondecreasing function with $W(0)=0$.

\nin $W$ is constant on each connected component of \mbox{$\{x:\varphi(x)>0\}$},
and $W(x)<W(y)$ \mbox{whenever} $[x,y]$ contains a zero of $\varphi$.

\nin Extend $W$ to an even function of $\rr$.% by $W(x)=W(-x)$.

\nin Let $\hat V=\bar V+W$.

\nin Then $\hat V$ is \mbox{continuous}, \mbox{$\hat V(0)=0$}, and $\hat
V(x)>\bar V(x)$ whenever $x\neq 0$.




\section{Example cont'd:\\ $\hat V=\bar V +  W$ solves the HJBE}

\nin Near points $x$ such that $\varphi(x)>0$, $\bar V$ and $\hat
V$ differ by a constant, so either both or none satisfy the HJBE.

\nin In the same way as was argued before, $\bar V$ solves the HJBE
\[\left|V'(x)\right|\varphi(x)-1=0\]
in the viscosity sense in $\rr-\{0\}$ iff it has no subdifferentials
at any zero $x\neq 0$ of $\varphi$.

\nin Suppose $x<0$ is a zero of $\varphi$. It was shown above that the right
derivative of $\bar V$ at $x$ is $-\infty$. Since $W$ is monotone non-increasing
at negative $x$, the right derivative of $\hat V$ at $x$ must also be $-\infty$
and therefore $\hat V$ cannot have a subdifferential at $x$.

\begin{comment}
 \nin If $x\neq 0$ but $\varphi(x)=0$, the \mbox{viscosity} \mbox{solution}
 requirements say that $-1\ge 0$ whenever $p$ is a \mbox{subdifferential} of
 $\hat V$ at $x$, and $-1\le 0$ whenever $p$ is a superdifferential of $\hat V$
 at $x$, and the second one of these conditions is trivially true.  As for the
 first \mbox{condition}, if $x<0$ then we have already shown that the right
 \mbox{derivative} of $\bar V$ at $x$ is equal to $�\infty$, and this clearly
 \mbox{implies} that the right derivative of $\hat V$ at $x$ is equal to
 $�\infty$ as well, since $\hat V=\bar V+W$ and $W$ is \mbox{monotonically}
 \mbox{nonincreasing} near $x$. Hence there \mbox{exist} no
 \mbox{subdifferentials} of $\hat V$ at $x$.
\end{comment}

\nin A similar argument applies if $x>0$.


\section{Example cont'd: conclusion}

\nin It this example, the solution to the HJBE is non-unique.

\nin This is related to non-uniqueness of trajectories in the control system.

\nin The  spurious value \mbox{function} $\hat V$ is bounded below by the true value function.
What fails is the inequality $\hat V\le \bar V$, related to the 
\mbox{dynamic} programming inequality (DPI).

\nin From general results, given any control $u(\cdot)$ and any initial
\mbox{condition} $x_0$, there is a maximal trajectory $\xi$ for $u(\cdot)$
starting at $x_0$ along which the DPI holds -- namely, stopping at the first
zero of $\varphi$.

\nin However, the fact that the DPI holds along $\xi$
does not imply that that the DPI holds for \mbox{every} such trajectory.
In this example, it must fail for the optimal trajectory, for otherwise
$\hat V(x)\le \bar V(x)$ for all $x$.
 \proofend
%\end{example} 

\end{document}

