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%              FURTHER RESULTS ON THE BELLMAN EQUATION FOR
%                  EXIT TIME OPTIMAL CONTROL PROBLEMS
%                     WITH NONNEGATIVE LAGRANGIANS
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%                  by M. Malisoff and H. J. Sussmann
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%          A preliminary version of this paper appeared in the
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%                   in Sydney, Australia, Dec. 2000.      
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\title{
Further Results on the Bellman Equation for Exit Time Optimal
Control Problems with Nonnegative Lagrangians \footnote{
This is a preliminary draft which is not meant for circulation and should
not be quoted without permission.  
A summary of this paper was presented at 
the 39th IEEE Conference on Decision and Control.
This document was last revised  on January 3, 2001.}}
%\\{\small Version 2.2} }
%\thanks{\noindent Supported in part by
%DIMACS grant NSF-CCR91-19999 and NSF Grant DMS95-00798.
%This work was carried out between December 1998 and February 1999,
%while the author was a Research Assistant for H\'ector Sussmann.
%The author thanks Dr. Sussmann for his helpful comments
%on early drafts of this paper.  This paper appears in {\em Proc.
%38th IEEE Conf. Decision Control.}}}


\author{Michael Malisoff \footnote{Supported in part by US
Air Force Grant F49620-98-0242 and NSF Grant CCR91-19999.  The final
part of this author's work on this manuscript was done in  January
2001 while the author was a member of the faculty of Texas A \& M
University-Corpus
Christi.}
 \ \ and
\ \ H\'ector J. Sussmann\footnote{Supported in part 
  by NSF Grant DMS95-00798.}\\
Department of Mathematics\\Hill Center-Busch Campus\\ Rutgers
University\\110 Frelinghuysen Road\\ Piscataway, NJ 08854-8019,
USA\\ $\mathtt{\{malisoff, sussmann\}\, @\,
math.rutgers.edu}$}\date{}
\pagestyle{myheadings} \markboth
{Further Results on the Bellman Equation
for Exit Time
Optimal Control}
{Further Results on the Bellman Equation
for Exit Time
Optimal Control}

\begin{document}
\maketitle
%\normalsize
\begin{abstract}
We study the Bellman equation for variable interest rate 
exit time
control problems with nonnegative Lagrangians
using the dynamic programming approach.  We 
characterize the value functions for
these problems as the unique viscosity solutions of the
corresponding Bellman equations that satisfy appropriate side 
conditions.     We allow problems whose dynamics,
interest rates, Lagrangians, control sets, and targets are all
unbounded, including cases where the Lagrangian and interest rate
are not bounded below by positive constants and  the 
value function and exit cost are discontinuous.   These include 
undiscounted problems.  
As a
consequence, we show that the value function of `Fuller's
Problem' is the unique viscosity solution of the corresponding
Bellman equation which is 
 null at the origin and bounded below.  We also apply our results to 
degenerate eikonal equations from geometric optics, including cases
where the target is unbounded, and to shape from shading
equations from image processing.

\vspace{.2em}

\noi {\bf Key Words.} viscosity solutions, dynamical systems, optimal
control

\vspace{.2em}

\noi {\bf AMS Subject Classifications.} 49L25, 93Cxx.
  
\end{abstract}
\section{Introduction}
 This note studies
Hamilton-Jacobi-Bellman equations (HJBE's) for a large class of
variable interest rate unbounded optimal control problems for
fully nonlinear systems 
\begin{equation}
\label{problem} \twoif {y'(t)\; =\; f(y(t),u(t)), \; \; t \ge
0,}{u(t)\in U} {y(0)\; =\; x}{}.
\end{equation}
  The objectives of these problems are the infimization of
\begin{equation}
\label{prob} \displaystyle \int_0^{t_x(\beta)}
\ell\left[y_x(s,\beta),\beta(s)\right]\, e^{-\delta(s,\beta,x)}
 ds\; +\;
g\left[y_x(t_x(\beta),\beta)\right]\, e^{-\delta(t_x(\beta),\beta,x)}
\end{equation}
%e^{-\int_0^th(y_x(s,\beta),\beta(s))\, ds}
over all $\beta\in {\cal U}(x)$ for each $x\in\R^N$, 
where 
\[{\cal U}:=\{\! \text{measurable functions} [0,+\infty)\to
U\},\] 
$y_x(\cdot,u)$ is the solution of (\ref{problem}) for each
$u\in {\cal U}$, 
$\T\subseteq\R^N$ is a closed set which we
will refer to as a target, 
\[t_x(u):=\inf\left\{t\ge 0:
y_x(t,u)\in\T\right\}\in[0,+\infty],\] 
${\cal U}(x)$ is the set of
all $\beta\in\cal U$ for which $t_x(\beta)<\infty$, and
\[\delta(t,\beta,x):=\int_0^th(y_x(r,\beta),\beta(r))\, dr.\]
Functions $\beta\in\calu$ will be referred to as inputs or controls,
$U$ is called the control set, and $\ell$ is called
the Lagrangian.  
 The class of
problems we consider includes the classical linear quadratic (LQ)
case where 
\[\ell(x,u)=x'Rx+u'Qu,
\] including cases where  $Q$ is
positive semidefinite but not positive definite and $\vec 0\in U$
(cf. Remark \ref{LQrk}).
We also allow cases where $f$,$\ell$, $h$,  and  $\T$ are
unbounded, including cases where $h$ and $\ell$ are not bounded
below by positive constants and $g$ is discontinuous (cf. $\S$\ref{defs} 
for
the 
exact hypotheses on the data and $\S$\ref{newsection} for
generalizations).

The value function of (\ref{prob}) will be denoted by $v$, and
$\cal R$ denotes the set of points that can be brought to $\T$ in
finite time using the dynamics (\ref{problem}) and some control
$\beta\in {\cal U}$. Therefore, 
\begin{equation}\label{value}
\ds v(x)=
\twoif{
\inf_{\scr \beta\in {\cal U}(x)}
\left(\int_0^{t_x(\beta)} \ell\left[y_x(t,\beta),\beta(t)\right]\,
e^{-\delta(t,\beta,x)}
 dt\; +\;
g\left[y_x(t_x(\beta),\beta)\right]\,
e^{-\delta(t_x(\beta),\beta,x)}\right),}{x\in\crr}{+\infty,}{x\not\in\crr}
\end{equation}
The value function $v$ will sometimes be denoted by $v[g]$.
Notice that since $U$ is bounded, it need not be the case that
$\cal R$ is all of $\R^N$, even in the classical case where  the
dynamics (\ref{problem}) has the form 
\[\dot x=Ax+Bu\] and $(A,B)$
is a controllable pair (cf. \cite{sontag}).
 The
HJBE of (\ref{prob}) is
\begin{equation}
\label{HJBE} \ds \supu\; \left\{\; -f(x,u)\cdot Dv(x)\; -\;
\ell(x,u)\; +\; h(x,u)\cdot v(x)\; \right\}\; \; =\; \; \;  0\; .
\end{equation}
We will study solutions of (\ref{HJBE})
on sets of the form
$\Omega\setminus\T$, where $\Omega\subseteq\R^N$ is an open set
which may or may not contain $\T$. Since the value
function (\ref{value}) will not in general be continuous (cf.
$\S$IV.3.1 of \cite{BARD} and Example \ref{firstnewex} below), we
will study (\ref{HJBE}) in the framework of discontinuous
viscosity solutions (cf. \cite{BARD} and \cite{FLEM}).  By
discontinuous, we will mean ``not necessarily continuous'' (cf.
Definition \ref{definitionvis} below).

We prove a uniqueness theorem which characterizes $v$ as the
unique free boundary (discontinuous) viscosity solution of (\ref{HJBE}) on
${\cal R}\setminus\T$ that satisfies appropriate boundary
conditions. The result applies to Fuller's Problem (FP) (cf.
\cite{SUSS}, \cite{Su99}, \cite{ZB94}, and $\S$\ref{fullersubsection}
below),
degenerate eikonal and shape from shading equations (cf. \cite{BARD},
\cite{SorCDC2K},
and $\S\S$\ref{eikonalsubsection}-\ref{shape} below), 
as well as problems with 
unbounded control sets.
We also prove a local version of the theorem that shows that any
viscosity solution of (\ref{HJBE}) on a more general open set of
the form $\Omega\setminus\T$ which satisfies appropriate boundary
conditions coincides with the value function on $\Omega$ (cf.
$\S$\ref{extend}).  Our arguments are based on  Zorn's Lemma and
an extension of Barb\v{a}lat's Lemma (cf. \cite{meagre} and Lemma
\ref{lemma1} below).

Value function characterizations of this kind have been studied
and applied by many authors for a large variety of stochastic and
deterministic optimal control problems and for differential games.
Also, uniqueness characterizations are used to establish the convergence
of many recently developed numerical schemes for
approximating value functions.   One proves the
convergence of these schemes by showing that the value
function $v_n$ of the $n$th discretization of the problem
converges uniformly on compact sets to a function $w$ which is a
discontinuous viscosity solution of the HJBE as the discretization
becomes finer, and then one invokes the  characterization of the
value function as the unique discontinuous viscosity solution of
the equation  to show that $w$ is in fact the value function. Recent
accounts of work in these areas may be found in \cite{BARD},
\cite{BFS99}, \cite{FLEM}, and \cite{Sou99}. 

For theorems
characterizing $v$ as the unique nonnegative viscosity solution of
the HJBE (\ref{HJBE}) on $\crr\setminus\T$ that satisfies
appropriate boundary conditions, see \cite{S99a}. The papers
\cite{BS91} and \cite{sorav} give general uniqueness
characterizations of the HJBE for exit time problems whose
Lagrangians $\ell$ satisfy conditions of the form
\begin{equation}
\label{pos}
\left( \forall \eps>0\right)
\; \left(\exists C_\eps>0 \text{s.t.}
 \forall a\in U \text{\&} \forall x\not\in B( \T, \eps)
\; \left(
\ell(x,a)\ge C_ \eps\right)\right),
\end{equation}
where \[B( S, \eps):=\left\{y\in \R^N: \inf\left\{||p-y||: p\in
S\right\}<\eps\right\}\] for all $S\subset\R^N$ and $||\cdot||$ is
the usual norm.  For cases where $\partial \T$ is compact, the papers
\cite{BS91} and \cite{sorav} characterize the value function as the 
unique proper viscosity solution of the HJBE that agrees with the final
cost on $\T$ (where by properness of a function $w$, we mean the 
condition that $w(x)\to +\infty$ as $||x||\to\infty$).
  Since we allow $\ell(\cdot, a)$
to vanish outside $\T$ for some values of $a$, these earlier
results cannot be applied to (\ref{HJBE}).  
Indeed, one easily
finds HJBE's for exit time problems  that
have several proper viscosity solutions when  
the positive lower bound condition (\ref{pos}) fails.
Here is an example where this occurs:
\bex{motivation}
Choose the dynamics
\[
\dot x(t)\; =\; u(t)\; \in\; U:=[-1,1],\] choose
\[
\ell(x,a)\; \; :=\; \; 
(x+2)^2\; (x-2)^2 \; x^2\;  (x+1)^2\;
(x-1)^2,\]
 and set $g\equiv 0$ and $h\equiv 0$. Let $v_1$ and $v_2$ denote
 the value functions for the associated problem (\ref{prob}) with
the targets $\T_1=\{0\}$ and $\T_2=\{0, 2, -2\}$, respectively.
One can easily check that $v_1$ and $v_2$ are both proper 
and that
both are  viscosity solutions of the associated HJBE  
\[
|Dv(x)|\; \; =\; \; 
(x+2)^2\; (x-2)^2 \; x^2\;  (x+1)^2\;
(x-1)^2
\]
on $\R\setminus\T$ with $\T:=\T_1$.  Moreover, with $\T=\{0\}$,
 the problem satisfies the hypotheses of the well-known
theorems which characterize value functions of exit time control
problems as the unique proper viscosity solutions of (\ref{HJBE})
which are zero on $\T$ (cf. $\S$IV.4 of \cite{BARD}) 
save for the fact
that the positive lower
bound requirement (\ref{pos}) on $\ell$ is not satisfied. \eex

\br{introrem}
We will show that the ingredient missing from the preceding example is the
extra condition 
\[
\left(x\not\in\T,\beta\in\calu, \int_0^\infty
\! \! \ell(y_x(s,\beta),\beta(s))\, ds<\infty\right)\implies \left(
\exists p\in\T \text{s.t.} \lim_{s\to\infty}y_x(s,\beta)=p\right).
\]
In Example \ref{motivation}, this condition of course fails, since the
trajectory
$x(t)\equiv 1$ gives zero running costs but does not reach the 
target $\T=\{0\}$.  However,
even in the 
classical case where $f$ and $\ell$ are Lipschitz and $h$ is a positive
constant, this extra condition is not sufficient to guarantee uniqueness
of solutions of the HJBE when $g$ is discontinuous (cf. Example
\ref{discexample}).  
This motivates our study our study of problems with
discontinuous exit costs (cf. $\S$\ref{newsection}).
\er
\br{another}
Another approach to guaranteeing that the value function is the unique
viscosity 
solution of the HJBE is to assume the following condition:
\begin{equation}
\label{alt}
\left(x\not\in\T\right)\implies\left(\int_0^t
\ell(y_x(s,\beta),\beta(s))\, ds>0 \; \forall t>0 \; \&\; 
\forall\beta\in\calu\right)
\end{equation}
For uniqueness characterizations for problems satisfying (\ref{alt}), see
\cite{M99a} and \cite{M99b}.  The results of \cite{M99b} imply as a
special case that the Fuller Problem value function is the unique 
viscosity 
solution
of the corresponding HJBE which vanishes at the origin and satisfies a
generalized version of properness which allows functions which are not
bounded below (cf. Remark \ref{rk4}).  
\er


  Our work is part of a
larger project which generalizes uniqueness characterizations for
HJBE's to versions that cover well-known optimization problems
with non-Lipschitz dynamics and Lagrangians which violate the
usual boundedness assumptions. Earlier work in this area can be
found in \cite{BDL97} (which covers finite horizon LQ problems)
and  \cite{DAL} (which covers infinite horizon LQ problems). For
results on infinite horizon optimal control problems with
non-Lipschitz dynamics, see \cite{I85}. The paper \cite{MALI}
gives a uniqueness characterization for the HJBE for a class of
exit time problems whose dynamics are continuous but not Lipschitz
continuous, including the Reflected Brachystochrone Problem (cf.
\cite{Su-Japan} and \cite{Su-book-article}) and degenerate eikonal
equations with non-Lipschitz speed functions.
  For a uniqueness
characterization for solutions of (\ref{HJBE}) based on 
Clarke's proximal differentials (cf. \cite{C89}) which applies to
problems with bounded dynamics, see \cite{Y00}.  

This paper is organized as follows.  In $\S$\ref{defs}, we
give the main assumptions and main lemmas.  
We state global versions of
our main
results in $\S$\ref{main}, which also explains how our results
extend what was already known about the exit time HJBE (\ref{HJBE}).  In
$\S$\ref{fullersection}, we recall Fuller's Problem and show how
our results apply to this problem's HJBE.  We also show how our
results apply to degenerate eikonal equations from geometric optics
and shape from shading equations from image processing.  In
$\S$\ref{proof},
we prove our main results.    
We prove local
versions of our main results in $\S$\ref{extend}, and we close in
$\S$\ref{newsection} by showing how to extend our results to
problems with unbounded control sets and discontinuous exit costs.

\section{Definitions, Assumptions, and Main Lemmas}
\label{defs}

  The value function $v$ is not in general continuous (cf.
\cite{BARD} and Example \ref{firstnewex} below), so we study
solutions of the HJBE
(\ref{HJBE}) in the discontinuous viscosity sense.  By a discontinuous
viscosity solution, we will mean a viscosity solution which is 
not necessarily continuous.  We use the following `test function'
definition
of viscosity solutions.  
For each $S\subseteq\R^N$ and each locally
bounded function $w:S\to\R$, we define the {\bf upper envelope}
$w^\star$ and {\bf lower envelope} $w_\star$ of $w$ by
\[
\ds w^\star(x)=\limsup_{y\to x}w(y)\; \; \text{and}\; \;
w_\star(x)=\liminf_{y\to x}w(y)
\]
Notice that $w_\star$ is lower semicontinuous and $w^\star$ is
upper semicontinuous.
 Letting $C^1(S)$ denote the set of all real-valued
continuously differentiable functions on any subset $S$
of a Euclidean space, the definition
of viscosity solutions can then be stated as follows:

\bd{definitionvis}

Assume  $\cg\subseteq \R^N$ is open, $S\supseteq\cg$,
$F:\R^N\times\R\times \R^N\to\R$ is continuous,  and 
$w: S\to\R$ is locally bounded. We will say that $w$ is  a {\bf
(discontinuous) viscosity
solution } of \[F(x,w,Dw(x))\; =\; 0\] on $\cg$ provided that the
following
conditions hold:
\begin{itemize}
\item[]\bi
\item[$\; (C_1)$\ \ ]
If $\gamma \in C^1(\cg)$ and $x_o\in\cg$ is a local minimizer of  
$w_\star-\gamma$, then $F(x_o,\, w_\star(x_o),\, D\,
\gamma(x_o))\, \ge\, 0$.
\item[\ $\; (C_2)$\ \ ]
If $\lambda\in  C^1(\cg)$ and $x_1\in\cg$ is a local maximizer 
of 
$w^\star-\lambda$, then $F(x_1,\, w^\star(x_1),\, D\,
\lambda(x_1))\, \le\, 0$.
\end{itemize}\ei\ed
  By a denseness argument, the preceding definition is
unchanged if we replace $C^1(\cg)$ with the set $C^{\infty}(\cg)$
of infinitely differentiable real-valued 
functions on $\cg$ in $(C_1)$ and
$(C_2)$. We also use the following equivalent definition of
viscosity solutions based on the {\bf superdifferentials}
$D^+w^\star(x)$ and {\bf subdifferentials}  $D^-w_\star(x)$ of $w$.  Let
$\cal G$, $S$, $F$, and $w$ be as in Definition
\ref{definitionvis}, and define
\[
\displaystyle D^+w^\star(x)\; :=\; \left\{p\,\in\, \R^N:\; \;
\limsup_{{\cal G}\smni y\to x}
\frac{w^\star(y)-w^\star(x)-p\cdot(y-x)}{||x-y||}\; \le\; \,
0\right\}
\]
\[
\displaystyle D^-w_\star(x)\; :=\; \left\{p\, \in\, \R^N : \; \;
\liminf_{{\cal G}\smni y\to x}
\frac{w_\star(y)-w_\star(x)-p\cdot(y-x)}{||x-y||}\;  \ge\; \,
0\right\}
\]
One checks that conditions $(C_1)$ and $(C_2)$ are  equivalent to
\bi\item[]
\begin{itemize}
\item[$(C'_1)$]
\ \ $F(x,w_\star(x),p)\ge 0$ for all $x\in{\cal G}$ and $p\in
D^-w_\star(x)$
\item[$(C'_2)$]
\ \ $F(x,w^\star(x), p)\le 0$ for all $x\in{\cal G}$ and $p\in
D^+w^\star(x)$
\end{itemize}
\ei


 \noindent respectively, so we equivalently define discontinuous
 viscosity solutions by saying that
 $w$ is a
viscosity solution of $F(x,w(x), Dw(x))=0$ on $\cal G$ provided 
conditions  $(C'_1)$-$(C'_2)$ hold. Set $\R_{\, +}=[0,\infty)$ and
\[B_R(q)\; =\; \{p\in\R^N: ||p-q||<R\}\] for each $R>0$ and $q\in \R^N$.
 For cases where the control set 
$U\subseteq\R^N$ is compact, we let $U^r$ denote the set of Radon
probability measures on $U$ (i.e., the probability
measures which are defined on the smallest $\sigma$-algebra on
$U$  containing all the open subsets of $U$) with the weak-$\star$
topology, ${\cal U}^r:=\left\{\! \! \text{measurable
functions}[0,\infty)\to U^r\right\}$, and
\[
k^r(x,m)\; =\; \int_U k(x,u)dm(u),\; \; k=f,\ell,h,\; \;
x\in\R^N,\; \; m\in U^r.  
\]
We call ${\cal U}^r$ the set of {\bf relaxed controls} on $U$.
Notice that ${\cal U}^r$ includes all controls in $\cal U$, which
can be viewed as Dirac probability measure valued controls, and
that any relaxed control can be approximated by standard controls
in the weak-$\star$ topology (cf. \cite{W72}).  For each relaxed
control $\beta$, we let $y_x(\cdot, \beta)$ denote the unique
solution of the initial value problem $\dot y(t)=f^r(y(t),\beta(t))$,
$y(0)=x$ on $[0,\infty)$
when this problem has a unique solution, 
and we extend the interest rate $\delta$ by setting 
\[
\delta(t,\beta,x)\; :=\; \int_0^th^r(y_x(s,\beta),\beta(s))\, ds
\]
for all $t\ge 0$, $\beta\in {\cal U}^r$, and $x\in \R^N$.  We
define the exit times
\[
t^S_q(\beta):=\inf\{t\ge 0:  y_q(t,\beta)\in S\}
\]
for all $S\subset\R^N$, $q\in \R^N$, and $\beta\in {\cal U}^r$.
We say that a function $\Gamma:\R^N\times U\to G$ into a subset
$G$ of a Euclidean space is {\bf locally Lipschitz} provided that for each
$R>0$, there exists a constant $L_R>0$ such that
\begin{equation}
\label{poundpound}
||\Gamma(x,u)-\Gamma(y,u)||\; \; \le\; \;  L_R\,
||x-y||\; \;  \forall\; \;  x,y\in
B_R(0)\; \; \text{and}\; \;  \forall\; \; u\in U.
\end{equation}
 If we can choose $L_R\equiv L>0$ independently of
$R>0$ in (\ref{poundpound}), then we say that $\Gamma$ is {\bf Lipschitz
(with Lipschitz
constant} $L$).  For cases where $f$ is 
Lipschitz, the trajectories $y_x(\cdot,\beta)$ are defined on
$[0,\infty)$ for all $\beta\in{\cal U}^r$ and all $x\in\R^N$ (cf.
\cite{BARD}). 
In what follows, $STC{\cal T}$ denotes the requirement that  
$\T \subset  {\rm
Interior}\, {\cal R}(\eps)$ for all $\eps>0$, where
${\cal R}(\eps)=
\left\{x\in\R^N:  \exists  t\in[0,\eps) \; \text{and}\;
\beta\in {\cal U} \;  {\rm s.t.}\;  y_x(t,\beta)\in
{\cal T}\right\}$.
In much of what follows, we 
assume the following:\bi\item[]\bi
\item[$(A_0)$]
The control set $U\subseteq \R^M$ is compact and nonempty.
\item[$(A_1)$]
The mappings $h,\ell:\R^N\times U\to\R_{\, +}$ are continuous, and
$f:\R^N\times U\to\R^N$ is Lipschitz  with
Lipschitz constant $L$.
\item[$(A_2)$]
The target $\T\subseteq\R^N$ is closed, $STC\T$ holds, and
$g:\crr\to\R$ is continuous.
\item[$(A_3)$]
If $\beta\in {\cal U}^r$ and $x\in \R^N$, and if \[\int_0^{ \infty}
e^{-\delta(s,\beta,x)} \, \ell^r(y_x(s,\beta),\beta(s))\,
ds<\infty,\] then there exists a point $\bar p\in \T$ such that
$\ds \lim_{s\to\infty} y_x(s,\beta)=\bar p$.
\item[$(A_4)$]
For each $q\in \crr$,  there exists a constant $\bar h_q<\infty$ so that 
\[\int_0^{t^\crr_q}h(y_q(s,\beta),\beta(s))\, ds\le 
\bar h_q\]
for all $\beta\in \calu^r$ satisfying $t^\crr_q(\beta)<\infty$.

 \ei\ei


 \noindent We will also study cases where $U$ is unbounded and $g$
is discontinuous
(cf. $\S$\ref{newsection} below).  In the classical case where
$\crr=\R^N$, $(A_4)$ of course holds vacuously.  Condition $(A_3)$ has the
intuitively appealing interpretation that trajectories which can be
continued indefinitely at finite cost eventually reach the target. 
 Henceforth, we write $t_x(\cdot)$ to mean $t^\T_x(\cdot)$.
 Our hypotheses are weaker than the usual
 ones since we do not require positive
 upper or lower bounds on $h$ or on $\ell$ and since we allow unbounded
 dynamics and targets.    
Conditions $(A_0)$-$(A_4)$ 
are not sufficient for the
continuity of the
value
 function.  The following example of an exit time problem
satisfying $(A_0)$-$(A_4)$ whose value function is discontinuous comes
from $\S$IV.3.1 of \cite{BARD}:
\bex{firstnewex}
Take $N=2$, $U=\{1,2\}$, $f(x,1)=(0,1)$, $f(x,2)=(-1,0)$, $h\equiv 0$, 
and $\ell\equiv 1$.  Choose the target $\T=\{x\in \R^2: x_1\le 0
\text{\&} x_2\ge 0\}$ and the final cost
\[
g(x)=
\left\{
\begin{array}{cl}
-2,    & x_2\ge 1\\
-2x_2, & 0\le x_2\le 1\\
0, & x_2\le 0
\end{array}
\right.
\]
One checks (cf. \cite{BARD}) that $v(x)=-x_2$ for $x_1\le 0$ and $x_2\le 
0$ and that $v$
is discontinuous on  the halfline $\{(0,x_2): x_2\le 0\}$.\eproof\eex

For cases where
 the sets 
\begin{equation}
\label{sets}
\{(f(x,a),\ell(x,a),h(x,a)): a\in U\}\end{equation}
 are convex for
 all $x\in \R^N$ (e.g., Fuller's Example), it follows from the Filippov
 Selection Theorem (cf. \cite{W72}) that each relaxed control
 $\beta\in {\cal U}^r$ admits a standard control $\gamma\in {\cal
 U}$ with the property that
 \[
 \int_0^t k^r(y_x(s,\beta),\beta(s))\, ds\; \; =\; \;
\int_0^t k(y_x(s,\gamma),\gamma(s))\, ds\; \; \forall t,\; \;
k=h,\ell,f
\]
For such cases, the results we give go through if one only
requires the conditions $(A_3)$-$(A_4)$ to hold for standard
controls $\beta\in \cal U$ instead of for all relaxed controls
$\beta\in {\cal U}^r$.  

One easily checks that if $(A_1)$ holds,
then $h^r$ and $\ell^r$ are also continuous and $f^r$ is
Lipschitz with Lipschitz constant $L$.
It follows from assumption $(A_1)$  that for  each control
$u\in{\cal U}^r$, the trajectory $y_x(\cdot,u)$ satisfies the
conditions
\begin{equation}
\label{splus} ||y_x(t,u)-y_z(t,u)||\; \le\; e^{\scr Lt} ||x-z||\;
\; \text{for\ \ all}\; \; t\ge 0
\end{equation}
and
\begin{equation}
\label{sssss} ||y_x(t,u)-x||\; \; \le \; \; M_x\, t\; \;
\text{for\ \ all}\; \; t\; \in\; [0,1/M_x],
\end{equation}
where $M_x=\sup\left\{||f(z,a)||: a\in U,  ||z-x||\le
1\right\}$  if the supremum is nonzero and $M_x=1$ otherwise (cf.
Chapter 3 of \cite{BARD}).  By the estimate (\ref{splus}) and the $STC\T$
assumption, $\crr$ is open.  Note that $\crr$ coincides with the relaxed
reachability set
\[\crr^r:=\left\{p\in\R^N:  t_p(\beta)<\infty \text{for some}
\beta\in {\cal U}^r\right\}\] 
for cases where the sets (\ref{sets}) are
convex (e.g., if $f$, $\ell$, and $h$ are affine in the controls).  


\br{neww} Conditions $(A_0)$-$(A_4)$ hold under standard hypotheses.
The condition $STC{\cal T}$ is satisfied when suitable
assumptions are made on the directions of the vector field $f$ and
its Lie brackets at $\partial{\cal T}$ (cf. \cite{susssymm} for a
sufficient condition for $STC\T$ which generalizes Chow's Theorem
from \cite{J97}). 
Condition $(A_3)$
generalizes the
convergence condition in Barb\v{a}lat's Lemma (cf. \cite{meagre}).
For sufficient conditions for $(A_3)$ to hold with $h\equiv 0$,
see Lemma \ref{lemma1} below. Notice that we allow $\T$ to be
unbounded. Notice too that $(A_4)$ holds vacuously for all continuous
nonnegative interest rates $h$ in the classical case
where 
$\crr$ is all of $\R^N$. In particular,
we do not require the interest rate $h$ to be bounded above, nor
do we require it to be bounded below by a positive constant.  If
conditions $(A_0)$-$(A_2)$ hold, then we can assume that the last
component in (\ref{problem}) has the form $\dot \tau=1$ and that
the target has the form $\T\times[0,\infty)$ (by adding the
integrator $\dot\tau=1$). 
Then $(A_4)$ holds if $h$ has the form
$
h(x,\tau,u)=\phi(\tau)$ with   $\phi\in L^1(\R_\gez,\R_\gez)\cap
C(\R_\gez)$, or, more generally, $h(x,\tau,u)=\phi(\tau
b(x,\tau,u))$ for $b$ bounded, continuous, and nonnegative
if we also assume that $\phi$ is nondecreasing.\eproof\er

We let $v^r$ denote the value
function for the exit time problem for the dynamics $f^r$, the
Lagrangian $\ell^r$, and the interest rate $h^r$, i.e., 
\[\ds
v^r(x):=
\twoif{
\inf_{\scr \beta\in {\cal U}^r(x)}
\left(\int_0^{t_x(\beta)} \ell^r\left[y_x(t,\beta),\beta(t)\right]\,
e^{-\delta(t,\beta,x)}
 dt\; +\;
g\left[y_x(t_x(\beta),\beta)\right]\,
e^{-\delta(t_x(\beta),\beta,x)}\right),}{x\in {\cal
R}^r}{+\infty,}{x\not\in\crr^r}
\]
where for each $x\in\R^N$, we let ${\cal U}^r(x)$ denote the set
of relaxed controls $\beta\in\calu^r$ with the property that  $y_x(\cdot,
\beta)$ reaches $\T$ in finite time. 
  
We will give conditions guaranteeing
that if $\Omega\subseteq\R^N$ is open and $\omega_o\in\R\cup\{+\infty\}$,
then the value function $v^r$
is the unique viscosity solution of (\ref{HJBE}) on
$\Omega\setminus\T$ in the class of functions $w$ satisfying 
\bi
\item[]
\bi
\item[]
\bi
\item[$BC(\Omega,\omega_o)$\ ]
$w: \Omega\to\R$ is locally bounded and bounded below, 
 $w_\star\ge g\ge w^\star$ on $\T\cap \Omega$, 
$w<
\omega_o$ on $\Omega$,
and $\ds \lim_{x\to x_o} w(x)=\omega_o$
for all $x_o\in 
\partial\Omega$. \ei\ei\ei\noindent
Notice that for cases where $\Omega=\R^N$, the limit condition in
$BC(\Omega, \omega_o)$ is satisfied vacuously for any choice of
$\omega_o$.  In particular,
there are no growth restrictions on functions $w$ satisfying
the condition $BC(\R^N, +\infty)$.  We will mainly study
viscosity solutions of (\ref{HJBE}) on $\crr\setminus\T$ which
satisfy $BC(\crr,\omega_o)$, but see $\S$\ref{extend} for
the case of general
open sets $\Omega$.  Set
\[\dist(q,P):=\inf\{||q-p||: p\in P\}\] for all $q\in\R^N$ and
$P\subseteq\R^N$.  Our uniqueness results are based on the
following representation lemma whose proof is a slight generalization of
the proof of Theorem III.2.32 on p.122 of \cite{BARD}:

\bl{lemma0}

Let $(A_0)$-$(A_2)$ hold,  let $B\subseteq \R^N$ be a bounded open
set, and let $w$ be a viscosity solution of the HJBE (\ref{HJBE})
on $B$.  Let $q\in B$, and set 
\[
T_\mu(q)=\inf_{\scr u\in{{\cal U}^r}} \{ t\ge 0: \dist(y_q(t,u),
\partial B)\le \mu \}
\; \; \forall \mu>0.\]
\noindent Then for all 
$\beta\in {\cal U}^r$ and $r\in [0,t^B_q(\beta))$, we have 
\begin{equation}
\label{first} w^\star(q)\; \; \le\; \; \int_0^r
e^{-\delta(s,\beta,q)} \ell^r\left[y_q(s,\beta), \beta(s)\right]\, ds
\; \;
+\; \; e^{-\delta(r,\beta,q)} w^\star\left[y_q(r,\beta)\right].
\end{equation}
\noi 
Moreover, for all 
$\mu\in (0,
\dist(q,\partial B))$, 
and $t\in [0,
T_\mu(q))$, we have 
\begin{equation}
\label{second} w_\star(q)\ge\inf_{\scr \beta\in {\cal
U}^r}\left[\int_0^t e^{-\delta(s,\beta,q)} \ell^r\left[y_q(s,\beta),
\beta(s)\right]\, ds + 
e^{-\delta(t,\beta,q)}w_\star\left[y_q(t,\beta)\right]\right].
\end{equation}
\el

\noindent  We remark for later reference that we can set
$r=t^B_q(\beta)$ in (\ref{first}) for cases where
$t^B_q(\beta)<\infty$. We will be especially interested in cases
where $\ell(x,a)=R(x)$ for continuous functions $R$ having the
 property that if $\phi:[0,+\infty)\to\R$ has a Lipschitz
derivative, then 
$\int_0^{\infty} R(\phi(s))\, ds<\infty$ implies 
$\lim_{s\to\infty}\phi(s)=0$.    
To study these cases,
we
introduce the following definition:

\bd{mk} A continuous function $\alpha:\R\to\R_{\, +}$ is said to be of 
class
$\cal{MK}$ provided that $\alpha(0)=0$ and $\alpha$ is even and strictly
increasing on $[0,+\infty)$.\ed 

\br{rkmk}
The results that follow stay true if we instead define $\cal{MK}$
functions to be the set of continuous, even, positive definite
functions $\alpha:\R\to\R_+$ for which $\alpha(0)=0$ and which 
weakly increase on $[0,\infty)$.  
Notice that class $\cal{MK}$ functions need not be bounded, convex, or
Lipschitz. 
Any 
function $\beta$  of Sontag Class $\cal K$ gives rise to a function
$\alpha$ of
class of $\cal{MK}$ which we define by setting
$\alpha(s)=\beta(|s|)$ (cf. \cite{meagre}). In particular, for
each $q>1$,  the function $x\mapsto |x|^q$ is of class ${\cal
MK}$.\er  
The following lemma is easily shown:

\bl{meagrelemma} Let $\alpha$ be of class $\cal{MK}$ and
let $\phi:[0,+\infty)\to\R$ be a
Lipschitz function for which
$\int_0^{+\infty}\alpha(\phi(s))\, ds<+\infty$.  Then $\ds
\lim_{s\to\infty}\phi(s)=0$. \el We use the following
consequence
of Lemma \ref{meagrelemma}: \bl{lemma1} Let $\alpha$ be of class
$\cal{MK}$, let 
$\phi:[0,+\infty)\to\R$ be differentiable, 
let $\phi'$ be Lipschitz, and 
 let $\int_0^{+\infty}\alpha(\phi(s))\, ds<+\infty$.
Then $\ds\lim_{t\to\infty}\phi(t)=\lim_{t\to\infty}\phi'(t)=0$.\el
\bpr It suffices to check that $\lim_{t\to\infty}\phi'(t)=0$ (since then
$\phi$ is Lipschitz and we can use Lemma \ref{meagrelemma} to
conclude).  
 Suppose
this were not the case.  We could then find 
$1\le t_k\uparrow\infty$ and a constant $\gamma\in(0,1)$ such that for all
$k$,
either (i) $\phi'(t_k)\ge \gamma$ or (ii) $\phi'(t_k)\le
-\gamma$. Passing to a subsequence, we can assume
\begin{equation}\label{dug}t_{k+1}\; -\; t_k\; \; \ge \; \; 1\; \;
\text{for\ \ all}\; \;  k.\end{equation} Let $C>1$ be any
Lipschitz constant for $\phi'$. We can assume 
(i)
holds for all indices $k$ in an infinite index set $I_+$.  
(Otherwise, apply the same argument to $-\phi$.)
Choose
$
\delta\in(0,\; \frac{\gamma}{2C})$, and set 
$I_k=[t_k-\delta, t_k+\delta]$ for all $k\in\N$.   
This gives
\[\phi'(t)\; =\; \phi'(t)\, -\, \phi'(t_k)\, +\, \phi'(t_k)\;
\ge\; -C\, |t-t_k|\, +\, \gamma\; \ge\; -C\, \delta\, +\, \gamma\;
\ge\; \gamma/2\] for all $t\in I_k$
and all $k\in I_+$. For each $k\in I_+$, set 
\[v_k:=\inf\left\{|\phi(t)|:
t\in I_k\right\}\] and pick $s_k\in I_k$
so that 
$\left\vert\phi(s_k)\right\vert =\; v_k$.
We consider
these subcases for $k\in I_+$:
\bi
\item[(1)\ \ ]
If $\phi(s_k)> 0$, then  $s_k=t_k-\delta$ (since $\phi'>0$ on
$I_k$), so \[\phi(t)\; \; \ge
\; \; \gamma/2(t-(t_k-\delta))\; \; \forall t\in I_k.\]
 Since
$\alpha$ is increasing on $(0,\infty)$, we get  
\begin{equation}
\label{dig} \int_{I_k}\alpha[\phi(s)]\, ds
\ge \int_{I_k}
\alpha[\gamma/2(s-t_k+\delta)]\, ds\ge
\int_0^{\frac{\gamma\delta}{4}}2\alpha(u)/\gamma\, du.
\end{equation}
\item[(2)\ \ ]
If $\phi(s_k)< 0$, then since $\phi'>0$ on $I_k$, 
we get $s_k=t_k+\delta$, so
\[-\phi(t)=-\phi(t_k+\delta)+
\int_t^{t_k+\delta}\! \! \phi'(u)du 
\ge\; \gamma/2(t_k+\delta-t)\; \; \forall t\in I_k. 
\] Since $\alpha$ is 
even, this again gives
$\int_{I_k}\alpha(\phi(s))\, ds\; \ge\;
 \frac{2}{\gamma}
\int_0^{\frac{\gamma\delta}{4}}\alpha(u)\, du$.
\item[(3)\ \ ]
Assume $\phi(s_k)=0$.  Since $\phi'>0$ on $I_k$,  we can 
apply the method of 
subcase (1) to $[s_k,t_k+\delta]$ and the method of subcase (2) to
$[t_k-\delta, s_k]$ to get \[\left\vert\phi(t)\right\vert\; \; \ge\;
\; \frac{\gamma}{2}\left\vert t-s_k\right\vert\; \; \text{for\ \ 
all}\; \; t\in I_k.\] Since
$[s_k-(t_k-\delta)]\vee[(t_k+\delta)-s_k]\ge \delta/2$, this gives
\begin{eqnarray}
\int_{I_k}\alpha[\phi(s)]\, ds &\ge&
\frac{2}{\gamma}
\int_0^{\frac{\gamma}{2}[s_k-(t_k-\delta)]}\alpha(u)\, du+
\frac{2}{\gamma}\int_0^{\frac{\gamma}{2}[(t_k+\delta)-s_k]}\alpha(u)\,
du\nonumber\\&\ge& \frac{2}{\gamma}
\int_0^{\frac{\gamma\delta}{4}}\alpha(u)\, du.\end{eqnarray}
 \ei Since $\delta< 1/2$ and (\ref{dug}) holds, the intervals 
$I_k$ are disjoint.
Since
\[\int_0^\infty \alpha(\phi(s))\, ds\; \; <\; \; \infty\;
\;\; \; \;  \text{and}\; \; \; \; \;
\int_0^{\frac{\gamma\delta}{4}}\alpha(u)\, du\;
>\; 0,
\]
 it follows from summing on $k\in I_+$ that $I_+$ is finite.  This
contradiction establishes the result.  
\epr
\br{rkkl}
The preceding proof shows that the statement of Lemma \ref{lemma1}
remains true if the hypothesis
$\int_0^\infty \alpha(\phi(s))\, ds<\infty$
is replaced by the three requirements that 
$h:\R\to\R_+$ is continuous, 
\[\int_0^\infty e^{-\int_0^sh(\phi(r))dr} 
\alpha(\phi(s))\, ds<\infty,\]
and $\phi$ satisfies the integral constraint
$\int_0^\infty h(\phi(s))\, ds<\infty$.\er


\section{Statement of Main Result and Remarks}
\label{main} \noindent We will prove the following theorem:
\bt{thm1}
 Assume that  $(A_0)$-$(A_4)$ hold and that $w$ is a viscosity
solution of 
(\ref{HJBE}) on $\crr\setminus\T$ which satisfies $BC(\crr,
\omega_o)$ 
for some $\omega_o\in\R\cup\{+\infty\}$.  
If $\omega_o\in[0,\infty]$, then 
$w\equiv v$ on 
$\{x:\crr: w(x) < e^{-h_x}\omega_o\}$.  If $\omega_o<0$, then
$w\equiv v$ on $\crr$.  
\et

\br{newt}
\label{newtt}
As we will show in $\S$\ref{proof}, the statement of this theorem
remains true if $v$ and $\crr$ are replaced by $v^r$ and $\crr^r$,
respectively.\er

\br{after}
Notice that the preceding theorem gives a uniqueness
characterization on all of $\crr^r$ for the undiscounted case where
$\bar h_x\equiv 0$ and for discounted cases where
$\omega_o=+\infty$. Moreover, if $\crr=\R^N$, then 
$BC(\crr,\omega_o)$ implies 
$BC(\crr,+\infty)$,
so we again get a uniqueness characterization on all of $\crr$.  
Notice that for undiscounted cases where 
$v$ satisfies $BC({\cal R}^r, \nu_o)$ for
some
$\nu_o\in\R\cup\{+\infty\}$, this theorem becomes a uniqueness
characterization for the value function of (\ref{prob}) which
shows in particular that $v\equiv v^r$ (by Remark \ref{newtt}).   
This
follows from the
fact (cf. \cite{BARD}) that for such cases, $v$ is a viscosity
solution of the HJBE (\ref{HJBE}) on $\crr\setminus\T$.  In
particular, if $\crr=\R^N$, then the theorem shows that
$v$ is the unique viscosity of the corresponding HJBE on
$\crr\setminus\T$ in a class of functions which includes functions
which are not proper and which are not continuous.
The assumptions that $U$ is compact and 
$g$ is continuous can be relaxed (cf. $\S$\ref{newsection}).\er

\br{thm3rm}
Recall that $t_x(\beta):=\inf\{t\ge 0:
y_x(t,\beta)\in\T\}\in [0,\infty]$.  For the classical case where
$\omega_o\ge 0$, our proof of Theorem 
\ref{thm1} will show that if $p\in \crr$ and $k\in[h_p,\infty)$ are such
that $w(p)<e^{-k}\omega_o$, then each sequence $0\le m_j\uparrow+\infty$
admits an input $\alpha_p\in \calu$ for which the following hold
\bi
\item
[(i)]
$w\left[y_p(m_j,\alpha_p)\right]<e^{\scr
-k+\delta(m_j,\alpha_p,p)}\omega_o$ for all $j$
with $m_j<t_p(\alpha_p)$ 
\item[(ii)] $\lim_{\scr s\to
t_p(\alpha_p)} y_p(s,\alpha_p)$ exists and lies in $\T$.\ei
This is an optimal control analog of global  
asymptotical stability from control systems theory
(cf. \cite{sontag}). Indeed, when $h\equiv 0$,
$g\equiv 0$, and $\omega_o\ne \infty$, the theorem gives a trajectory
which approaches  the null set of $w$
without leaving $\{x\in\crr:
w(x)<e^{-k}\omega_o\}$ along integral times, i.e., the equilibrium
is approached with no large excursions.    
\er

\br{rk2}  Using the first part of Lemma \ref{lemma0}, we can
strengthen Theorem \ref{thm1} by proving that if
conditions $(A_0)$-$(A_4)$ hold with $h\equiv 0$,
 if $\Omega\subset\R^N$ is an
open set containing $\T$, and if $w$ is a viscosity solution of
the HJBE (\ref{HJBE}) on $\Omega\setminus\T$ which satisfies
$BC(\Omega,\omega_o)$ for some $\omega_o\in\R_{\, +}
\cup\{+\infty\}$, then $w\equiv v$ on $\Omega$.
In particular,
we conclude that $\Omega\subseteq {\cal R}$, since $v\equiv
+\infty$ outside $\cal R$. This more general version follows 
from 
Theorem \ref{thm2} in $\S$\ref{extend} below,
which also covers cases where $\T\not\subseteq\Omega$. \er

\br{rk3} Notice that if $h\equiv 0$ and if the Lagrangian $\ell$
is such that
\begin{equation}
\label{old} \forall\; \eps>0,\; \; \exists\, C_\eps>0\; \;
\text{s.t.}\; \; \ell(x,u)\; >\; C_\eps\; \; \forall x\, \in\,
\R^N\setminus B(\T,\eps)\; \; \&\; \; \forall u\, \in\,  U,
\end{equation}
then it follows from 
\cite{sorav} that $v$ is the only possible continuous
viscosity solution of
the HJBE (\ref{HJBE}) on ${\cal R}\setminus\T$ which  satisfies
$BC({\cal R}, \omega_o)$.  On the other hand,  in the general case
we consider here, 
condition (\ref{old}) could fail, even in the classical case where
$\ell(x,u)=x'Rx+
u'Qu$ with $Q$ positive definite, since we allow $\vec 0\in U$.
Therefore, our uniqueness characterization does not follow from
known results.\er




\section{Physical Examples}
\label{fullersection}
Before turning to the proof of Theorem \ref{thm1}, we give some
applications of the theorem.
Our results apply to a variety of physical problems.  In particular,
we can use Theorem \ref{thm1} to get uniqueness characterizations for
the HJBE of Fuller's Example (cf. \cite{ZB94}).  Also, the theorem gives
uniqueness
characterizations for the the eikonal equations in geometric optics for
the propogation
of light in media with unbounded speed functions.
Our results also apply to shape from shading equations in image
processing.    
This section
discusses  these applications.  

\subsection{Fuller's Example}
\label{fullersubsection}
 
This subsection shows how
to apply Theorem \ref{thm1} to get a uniqueness
characterization for a class of problems which includes the Fuller
Problem (FP) (cf. \cite{ZB94}). The problems have dynamics
\begin{equation}
\label{dynamics}
 \twoif{\dot x(t)\; =\; y(t),\; \; \dot y(t)\; =\;
u(t)\; \in\; U:=[-1,+1]}{}{(x(0), y(0))'\; =\; p}{}
\end{equation}
and the solution trajectory 
of (\ref{dynamics}) for a given $u\in {\cal
U}$
will be denoted by $t\mapsto (y_{1,p}(t,u), y_{2,p}(t,u))'$. We
choose $g\equiv 0$ and $h\equiv 0$. To each function $\alpha$ of
class $\cal{MK}$, we assign the exit time problem
\[
 \text{For\  each}\;  p\in\R^2,\; 
\text{infimize} \;  \int_0^{t_p(\beta)}\! \! 
\alpha\left[y_{1,p}(s,\beta)\right]\, ds\;  \text{over\ \ all}\; 
\beta\in {\cal U}(p). \; \; \;    \left(P_\alpha\right)
\]
with target $\T=\{0\}\subseteq\R^2$.  
The value function  of $P_\alpha$ will be denoted
by $v_\alpha$. For the special case where $\alpha(s)=s^2$, we get
the Fuller Problem, whose value function we
denote by $v_{\scr FP}$.   By standard arguments, the dynamics
(\ref{dynamics}) satisfies STC$\T$ (cf. \cite{sontag}). The
reachability set $\crr$ for (\ref{dynamics}) is $\R^2$, and
condition $(A_3)$ holds, by Lemma \ref{lemma1}.   Theorem \ref{thm1} and Remark
\ref{after} therefore
give the following:

 \bc{corfp}

For each function $\alpha$ of class $\cal{MK}$,  
$v_\alpha$ 
is the unique
viscosity solution of 
\begin{equation}
\label{hjf}\ds
 -\; \; y(Dw((x,y)'))_1\; \; +\; \; \vert (Dw((x,y)'))_2\vert\; \; -\;
 \;
\alpha(x)\; \; =\; \; 0
\end{equation}
on $\R^2\setminus\{\vec 0\}$
in the class of functions which are 
bounded below 
and null at
$\vec 0\in\R^2$.\ec In particular, there does not exist a
viscosity solution $w$ of (\ref{hjf}) on $\R^2\setminus\{\vec 0\}$
which is bounded below, which vanishes at $\vec 0$, and which
takes negative values.   Notice that Corollary \ref{corfp} extends
a result in \cite{S99a} which characterizes the Fuller Problem
value function as the unique viscosity solution of (\ref{hjf}) in
the class of nonnegative functions which vanish at
the origin. This result can be generalized as follows.  Let
$M=[a_{ij}]$ be an $N\times N$ matrix with 
$a_{ij}=0$ for 
$i\ge j$ and  $a_{ij}\ne 0$ for $i<j$, so $M$ is upper triangular
with a null main diagonal and all entries above the main diagonal
nonzero.  Let $\alpha$ be any function of class ${\cal MK}$. Then the
value function for the problem
\[
\text{Infimize} \; \; \int_0^{t_p(\beta)} \sum_{j=1}^{N-1}
\alpha[x_j(t, \beta)]\, dt \; \; \text{over\ \ all}\; \;  \beta\in{\cal
U}(p)
\]
\[
\text{subject\  \ to}\; \; 
\dot x= Mx+(0,0,\ldots,0,1)' u,\; \; x(0)=p\in \R^N, \; \; u\in U:=[-1,+1]
\]
with the target $\T=\{0\}\subset \R^N$ is the unique viscosity solution
of the corresponding HJBE on $\R^N\setminus\{0\}$ which is bounded below
and null at $0\in\R^N$.  
This
follows from the fact that the appropriate controllability matrix has
full rank (which guarantees $STC\T$) and a repeated application of Lemma
\ref{lemma1} which we
omit.

\br{rkmin} The hypotheses of Corollary \ref{corfp} are minimal,
since for the FP case where $\alpha(x)=x^2$, the
Bellman equation (\ref{hjf}) admits the viscosity solution
$w(x,y)\equiv -v_{\scr FP}(-x,y)$ which is not bounded below.  
One way to check
 that $w$ is a viscosity solution of (\ref{hjf}) on
$\R^2\setminus\{\vec 0\}$ is as follows. First recall
(cf. \cite{ZB94}) that for
each $x\in \R^2$, there exist a point $z\in \R^2$, an input
$\alpha^\star: [0,\infty)\to U:=[-1,1]$, and a $t>0$ such that the
corresponding FP trajectory $s\mapsto (y_{\scr
1,z}(s,\alpha^\star), y_{\scr 2,z}(s,\alpha^\star))$ satisfies
\[
v_{\scr FP}(z)=\int_0^{+\infty}\left[y_{\scr
1,z}(s,\alpha^\star)\right]^2\, ds\; \; \text{and}\; \;
x=y_z(t,\alpha^\star),
\]
i.e., $x$ is an `optimal point'.  Also, since $v_{\scr FP}$ is
continuous, it follows from Soravia's backward
dynamic programming principle (cf. \cite{S93b} or $\S$III.2.3 of
\cite{BARD})
that $v_{\scr FP}$ is a bilateral viscosity solution of the FP Bellman
equation,
i.e.,  for all 
$(p_1, p_2)'\in
D^+v_{\scr FP}(x,y)\cup D^-v_{\scr FP}(x,y)$ and  all 
$(x,y)'\in \R^2\setminus\{\vec 0\}$, we have 
\[
-y\; p_1\; \; +\; \; |p_2|\; \; -\; \; x^2\; \; =\; \; 0
\]
The fact that $w$ is a viscosity solution of (\ref{hjf}) on
$\R^2\setminus\{\vec 0\}$ now follows since
\[
(p_1,p_2)\in D^+w(x,y)\; \; \implies\; \; (p_1,-p_2)\in
D^-v_{\scr FP}(-x,y) \] and \[(p_1,p_2)\in D^-w(x,y)\; \;
\implies\; \; (p_1,-p_2)\in D^+v_{\scr FP}(-x,y)
\]
for all $(x,y)\in\R^2\setminus\{\vec 0\}$.  We leave the details
to the reader.



\er



\br{rk4}

The paper \cite{M99b} proves a local uniqueness result for
(\ref{HJBE}) which implies as a special case that the FP
value function is the unique viscosity solution of the
corresponding HJBE (\ref{hjf}) on $\R^2\setminus\{\vec 0\}$ in the
class of functions $w$ which vanish at $\vec 0$, are continuous on
$\R^2$, and satisfy the following: 

\bi\item[]\bi
\item[$(REG)$] For each $x\in
\R^2\setminus\{\vec 0\}$, there is a bounded open set
$\Omega_x\subseteq \R^2\setminus\{\vec 0\}$ which contains $x$ and
an $\omega_{o,x}\in\R\cup\{+\infty\}$ such that
\bi\item[]\bi\item[]\ben\item[]\ben
\item[${\rm\ \ \ }(REG_1)$\ \ \ ]
$w(x)<\omega_{o,x}$  
\item[\ \ $(REG_2)$\ \ \ ]
$\liminf_{\scr \Omega_x\ni p\to x_o} w(p)\ge\omega_{o,x}$ for all
$x_o\in
\partial(\Omega_x)\setminus\{\vec 0\}$
\item[\ \ $(REG_3)$\ \ \ ]
STC$(\Omega_x^c)$. \een\een\ei\ei\ei\ei
Note that $(REG_2)$ only restricts the values of $w$ near
the boundaries of the $\Omega_x$'s, so functions which are not
bounded below can still satisfy (REG).   
In particular,  \cite{M99b} proves that the FP value function is the
unique
proper viscosity solution of (\ref{hjf}) which is continuous in
the plane and  vanishes at the origin, where by properness we mean
the condition $w(x)\to +\infty$ as $||x||\to\infty$. The main 
argument of \cite{M99b} is a generalization of the uniqueness
arguments for free boundary problems from \cite{BS91} and
\cite{sorav}.  The results of the paper \cite{M99b} also apply to
problems which violate the asymptotical 
condition $(A_3)$ and eikonal equations.   The novelty
of
Corollary \ref{corfp} above is that it 
characterizes the FP value function as a unique solution of the HJBE 
within a class of
functions which includes functions that violate $(REG)$ and which
are not proper.\er 
\br{lLQrk}
\label{LQrk}
The preceding argument generalizes to the following LQ problem on 
$\R^{2N}$.  Partition the state space to $x=(x_L,x_R)\in\R^N\times \R^N$,
and pick $\T=\{0\}\subseteq\R^{2N}$ and any compact control set
$U\subseteq\R^N$.  Take $h\equiv 0$, the LTI system $f(x,u)=Ax+Bu$,
and the quadratic cost $\ell(x,u)=x'_LRx_L+u'Qu$, where
\[
A=
\left(\begin{array}{cc}
0 & A_1\\
0 & 0
\end{array}\right) \; \; \; \text{and}\; \; \; 
B=
\left(\begin{array}{c}
0\\
B_1
\end{array}
\right)
\]
for $A_1$ and $B_1$ of size $N\times N$, $B_1$ having all entries
nonzero, and $R$ and $Q$ positive definite.  (Similar results hold if $Q$
is positive semidefinite.)  Let $f$ be such that $STC\T$ holds
(e.g., $f(x,u)=u$).  Since $\ell(x,u)\ge \lambda_{\scr min}||x_L||^2$,
where $\lambda_{\scr min}>0$ is the smallest eigenvalue of $R$, it follows
that the corresponding exit time value function is the unique
viscosity solution of the HJBE
\[
-Dv(x)Ax-\frac{1}{4}Dv(x)BQ^{-1}B'\left[Dv(x)\right]'-x'_LRx_L=0
\]
on $\R^{2N}\setminus\{0\}$ in the class of functions that vanish at $0$
and are bounded below.  Taking $N=1$, $Q=0$,  and $R=A_1=B_1=1$, we
recover
Corollary \ref{corfp}.\er



\subsection{Degenerate Eikonal Equations}
\label{eikonalsubsection}
This subsection shows how Theorem \ref{thm1} applies to Bellman 
equations for a class of exit time problems from geometric optics.
The problems have the dynamics $f(x,y,a,b)=(a,b)\in\overline{B_1(0)}
\subseteq \R^2$, the interest rate $h\equiv 0$, and the Lagrangians 
\[\ell(x,y,a,b)\; =\; \left[1+\sqrt{||(x,y)||}\right]^{-p},\]
 where $p>0$
will be further
specified below.  (The argument we are about to give also applies if
we instead take the Lagangian 
$(1+\sqrt{|x|})^p$ or 
$(1+\sqrt{|y|})^p$, or 
if the state space and control set  are in $\R^N$ for
$N$ arbitrary.)

We choose any closed target $\T\subseteq\R^2$,
and we let $v_{e,p}$ denote the value function for the exit time
problem we have defined for each $p>0$.  The corresponding HJBE
is
\begin{equation}
\label{eikeq}
||Dv(x,y)||=\frac{1}{\left[1+\sqrt{||(x,y)||}\right]^p},
\end{equation}
which is the eikonal equation of geometric optics for the propogation
of light in a medium of speed
\[c(x,y)=\left[1+\sqrt{||(x,y)||}\right]^p.\]
Viscosity solutions of eikonal equations have been 
studied extensively (cf. \cite{BARD}, which covers cases where the
speed of the medium is bounded and also uniqueness questions for
eikonal equation solutions on bounded sets subject to Dirichlet
conditions, and the paper \cite{SorCDC2K}).  However, (\ref{eikeq})
is not covered by these results since $c$ is unbounded
and $\T$ may be unbounded.    
It is very easy to check that for $0\le p\le 2$, the exit time problems
for these data satisfy $(A_0)$-$(A_4)$.  Indeed, if $q\in\R^2$
and $\phi(s)$ is a trajectory for $f$ starting at $q$, then we can
find a $K>0$ so that, for each $L>K$, we have
\[\ds
\int_0^L
\frac{ds}{\left[1+\sqrt{||\phi(s)||}\right]^p} \; \ge\; 
\int_0^L
\frac{ds}{\left[1+\sqrt{||q||+s}\right]^p} \; \ge\; 
\frac{1}{2}\int_K^L\frac{ds}{s^{p/2}}\; \to\; \infty
\]
as $L\to\infty$, so $(A_3)$ is satisfied vacuously.  By standard results
from \cite{BARD}, $v_{e,p}$ is viscosity solution of (\ref{eikeq}) 
on $\R^2\setminus\T$,  so we
conclude as follows:

\bc{eikcor}
For each $p\in[0,2]$ and closed target $\T\subseteq\R^2$, $v_{e,p}$
is the unique viscosity solution of (\ref{eikeq}) on $\R^2\setminus\T$
in the
class of
functions which are null on $\T$ and bounded below.
\ec

\br{rkkeik}
We emphasize that it was not necessary to assume that the target $\T$
is bounded.  If $p> 2$, then Theorem \ref{thm1} does not apply,
since $(A_3)$ could fail.  For example, if $p=4$,  
and if we take $\T=\{(x,0)\in\R^2: x\le
-1\}$  and $\beta\equiv (1,0)$, then $\int_0^\infty \ell(y_{\scr
(0,0)}(s,\beta),\beta(s))\, ds<\infty$ but the trajectory does not
approach the target.  
Moreover, the standard uniqueness
characterizations for exit time HJBE's (e.g., Corollary
IV.4.3 of \cite{BARD}) would not apply, since the Lagrangian
is not bounded below by a positive constant.  However, using the
results of \cite{M01}, one can show that the statement of
Corollary \ref{eikcor} remains true even without the restriction
$p\in[0,2]$.  This is done by rewriting the HJBE (\ref{eikeq}) as
\begin{equation}
\label{neweik}
\left[1+\sqrt{||(x,y)||}\right]^p||Dv(x,y)||-1=0, \; \; (x,y)\not\in\T
\end{equation}
and then viewing (\ref{neweik}) as the HJBE for the exit time 
problem with the non-Lipschitz dynamics
\[\tilde f(x,y,a,b)=\left[1+\sqrt{||(x,y)||}\right]^p(a,b)\]
 (with
$(a,b)\in\overline{B_1(0)}$
as before) and Lagrangian $\tilde\ell\equiv 1$.  For details, see $\S$6.1
of \cite{M01}.
\er
\subsection{Shape From Shading Equations}
\label{shape}
Our results also apply to equations of the form
\[
I(x)\Psi(Du(x))-b(x)\cdot Du(x)-h^2(x)=0
\]
for $I$ nonnegative and $\Psi$ a convex function with $\Psi(0)=0$.  This
equation is studied in \cite{S99a}.  Taking the Legendre transform
$\Psi^\star$ of $\Psi$, which is nonnegative, we can rewrite this equation
as
\[
\max_{a\in{\rm domain}(\Psi^\star)}
\left\{
-(b(x)-I(x)a)\cdot Du(x)-\left[h^2(x)+I(x)\Psi^\star(a)\right]\right\}=0.
\]
A particular case of this equation is
\begin{equation}
\label{tone}
I(x)\left[1+||Du(x)||^2\right]^{1/2}-1=0,\; \; x\in\Omega\subseteq\R^2
\end{equation}
for open sets $\Omega$, which in fact (cf. \cite{S99a}) can be written as 
\begin{equation}
\label{ttwo}
\ds
\max_{||a||\le 1}\left\{
I(x)a\cdot
Du(x)-\left[1-I(x)\left(1-||a||^2\right)^{1/2}\right]\right\}=0.
\end{equation}
The equation (\ref{ttwo}) arises in shape from shading models in image
processing, where $I(x)\in[0,1)$ is the intensity of light reflected
by an object (cf. \cite{SorCDC2K}).  The objective in image processing is
to 
reconstruct the unknown function $u$, representing the height of the
surface on some subset $\Omega$ of the plane, from the brightness
of a single two-dimensional image of the surface.  For the case of a
Lambertian surface which is not self-shadowing and which is illuminated
by a single distant vertical light source, the height $u$ satisfies
(\ref{ttwo}).  Now pick any closed target 
$\T\subseteq \R^2\setminus\{0\}$ and $\Omega=\R^2\setminus\T$
 and choose the intensity function
\[
I(x)=\frac{||x||}{1+||x||}.
\]
Then (\ref{ttwo}) is a Bellman equation for an exit time problem with
the dynamics \[f(x,u)=-I(x)u,\] the control set $U=\overline{B_1(0)}$, the
Lagrangian
\[
\ell(x,u)=1-\frac{||x||}{1+||x||}(1-||u||^2),
\]
$h\equiv 0$, and $g\equiv 0$.  Since $\ell$ violates the positivity 
condition (\ref{pos}), the known results cannot be used to get
uniqueness characterizations for (\ref{ttwo}).  On the other hand, using
the fact that $||y_q(s,\beta)||\le ||q||+s$ for all $\beta\in \calu^r$,
$s\ge 0$, and $q\in \R^2$, one can easily check that $(A_0)$-$(A_4)$ hold.
Therefore, we conclude that the value function is the unique viscosity
solution of (\ref{ttwo}) on $\R^2\setminus\T$ in the class of functions
which vanish on $\T$ and which are  bounded below.

\section{Proof of Main Result}
\label{proof}
This section proves Theorem \ref{thm1} for the case of
viscosity solutions on $\crr^r\setminus\T$ (cf. Remark \ref{newtt}).  
Let $w$ and $\omega_o\in [0,\infty]$ satisfy the hypotheses, 
and let $\bar x\in {\cal R}^r\setminus\T$ satisfy
\[
w(\bar x)\; <\;  e^{-\bar h_{\bar x}}\omega_o
\]
The case where $\omega_o<0$ is handled by replacing 
$h_{\bar x}$ with $0$ in what follows.  
The fact that $v^r(\bar x)\ge w^\star(\bar x)$ is an easy
consequence of the first part of Lemma \ref{lemma0}.  This
argument is carried out in a more general context in
$\S$\ref{extend} below (cf. Case A).  Since $w_\star(\bar x)\le
w^\star(\bar x)$, Theorem \ref{thm1} will therefore follow 
once we
show that  $w_\star(\bar x)\ge v^r(\bar x)$.  



To simplify the notation, we will assume that $w$ is lower
semicontinuous (so $w_\star\equiv w$), and we drop the superscript
$r$.   Choose $\eps\in(0,1)$ such that
\begin{equation}
\label{choice} w(\bar x)\; \; <\; \; e^{-\bar h_{\bar
x}}\omega_o\; -\; \eps.
\end{equation}
 (We
allow $\omega_o=+\infty$, in which case (\ref{choice}) holds for
all $\eps>0$.)  We drop the subscript on $\bar h$ in the sequel.
We also set
\[
\Delta(t,\gamma,\beta)\; :=\; \; \int_0^t h(\gamma(r),\beta(r))\,
dr
\]
and
\[
J(t,\gamma,\beta)\; :=\; \int_0^t e^{-\Delta(s,\gamma,\beta)}\,
\ell(\gamma(s),\beta(s))\, ds
\]
wherever these quantities are defined. The inequality $w(\bar x)\ge
v(\bar x)$ will evidently follow if we can find an input $\bar\beta\in
{\cal U}$ that drives $\bar x$ to $\T$ in finite time and a
corresponding trajectory $\bar\phi$ which are such that
\begin{equation}
\label{goal} w(\bar x)\; \; \ge\; \; J(t_{\bar x}(\bar \beta),
\bar\phi,\bar\beta)\; +\; e^{-\Delta(t_{\bar x}(\bar
\beta),\bar\phi,\bar\beta)} g\left[\bar\phi(t_{\bar
x}(\bar\beta))\right]  \; \; -\;
\; \eps,
\end{equation}
since that would give \[w(\bar x)\; \; \ge\; \;   v(\bar x)\; -\;
\eps,\] and then we can let $\eps\downarrow 0$. We now prove the
existence of such a $\bar \beta$ by using Zorn's Lemma. In what
follows, we define the functions $E_1, E_2, \ldots$ by
\[E_j(t)\; \; \equiv\; \; \frac{\eps}{8}\, \left[\,
e^{-(j-1)}-e^{-(t+j-1)}\, \right] \; \; \text{for\ \ each}\; \; t
>0.\]
Set
\[
{\cal Z}_1\; \; =\; \; \left\{
\begin{array}{c}
(t,\gamma):  \; \; \; 0\le t\le 1, \; \text{and} \gamma:\,
[0,t]\to \crr\; \text{is a trajectory}\\[.8em]\! \! \! \! {\rm
for}\; \; \dot x\, \, =\, \, f(x,\beta )\; \text{for \ some}\;
\beta\, \in\, {\cal U}\; \, \text{with}\,  \gamma(0)\, \,  =\, \,
\bar x\\[.8em] \! \! \! \! \text{so that} w(\bar x)\, \ge\,
J(t,\gamma,\beta)+e^{-\Delta(t,\gamma,\beta)}w(\gamma(t))- E_1(t).
\end{array}
\right\}.
\]
Then ${\cal Z}_1$ is partially ordered by the relation
\begin{equation}
\label{rel} 
\left((t_1,\gamma_1)\; \; \sim\; \; (t_2, \gamma_2)\right)\; \;
\iff\; \; \left(\;  t_1\le t_2\; \; \&\; \;
\gamma_2\lceil[0,t_1]\equiv \gamma_1\;  \right) \end{equation}
Moreover, one can use standard arguments to check that every
totally ordered subset of ${\cal Z}_1$ has an upper bound in
${\cal Z}_1$.

To do this, let $\{(s_j, \mu_j)\}_j$ be a totally ordered subset
of ${\cal Z}_1$, and set $s_o=0$ and $\bar s=\sup_j s_j$.
Assume $\bar s>0$.  
Define the
trajectory $\bar \mu:[0,\bar s)\to\R^N$ by $\bar\mu(p)=\mu_j(p)$
for any $j$ so that $s_j>p$. (This is valid since the $s_j$'s
increase and $\mu_j$ extends $\mu_k$ for $k\le j$.)  Let
$\beta_j:(s_{j-1}, s_j]\to U$ be a control for the trajectory
$\mu_j\lceil (s_{j-1}, s_j]$, fix $\bar u\in U$, and define
$\beta^\dagger:(0,\infty)\to U$ by
\[
\beta^\dagger(s)= \twoif{\beta_j(s),}{s_{j-1}\le s< s_j, \; \; j\ge
1}{ \bar u,}{s\ge \bar s}.
\]
Then
$\beta^\dagger$ is measurable and therefore admits a corresponding
trajectory $y_{\bar x}(\cdot, \beta^\dagger)$ on $[0,\infty)$.
Moreover, $y_{\bar x}(\cdot, \beta^\dagger)$ coincides with $\bar
\mu$ on $[0,\bar s)$. Setting $\bar\mu(\bar s)=y_{\bar x}(\bar s,
\beta^\dagger)$, it follows that  $(\bar s, \bar \mu)$ is an upper
bound for $\{(s_j, \mu_j)\}_j$ in the relation (\ref{rel}). Also,
the choice of $\eps>0$ guarantees that $\bar \mu(\bar s)\in \crr$,
which implies that $(\bar s, \bar \mu)\in {\cal Z}_1$.

Indeed, if $\bar\mu(\bar s)\not\in\crr$, 
then we would get
\[ \crr\ni\mu_j(s_j)\; \; \to\; \; \bar\mu(\bar
s)\; \in\; \partial\crr\; \; \; \text{and}\; \; \; 
w\left[\mu_j(s_j)\right]\; \; \to\; \; \omega_o.
\]
Let $\alpha_j$
denote the control for $\mu_j$ for all $j$. 
Consider the case where $\omega_o\not =\infty$.  
Fixing an integer $\kappa$ such that
\[ w(\mu_j(s_j))\; \; \ge\; \;  \omega_o\; -\; \eps\, e^{\bar
h}/(2\kappa) \; \; \text{and}\; \;
e^{-\Delta(s_j,\mu_j,\alpha_j)}\; \; <\; \; \kappa\, e^{-\bar h}\;
\; \text{for\ all}\; \; j,\] it would then follow from $(A_4)$
and the nonnegativity  of $\ell$
that \beqn w(\bar x)&\ge &
\liminf_{j\to\infty}\left[J(s_j,\mu_j,\alpha_j) \; +\;
e^{-\Delta(s_j,\mu_j,\alpha_j)}w(\mu_j(s_j))- E_1(s_j)\right]\\&\ge&
\liminf_{j\to\infty}\left[
e^{-\Delta(s_j,\mu_j,\alpha_j)}\left[\omega_o-\frac{\eps e^{\bar 
h}}{2\kappa}\right]- E_1(s_j)\right]\\&\ge& e^{-\bar h}\, \omega_o-\eps
\eeqn for all $j$.
This contradicts  (\ref{choice}), so we conclude
that $\bar \mu(\bar s)\in \crr$.  
Since $0<e^{-\bar h}\le
\lim_{j\to\infty} e^{-\Delta(s_j,\mu_j,\alpha_j)}$, the
contradiction is
clear from the limit in $BC(\crr,\omega_o)$ for cases
where $\omega_o=+\infty$. Since $w$ is lower semicontinuous at
$\bar \mu(\bar s)\in \crr$, it now follows that $(\bar s, \bar \mu)\in
{\cal Z}_1$ is an upper bound for $\{(s_j, \mu_j)\}$.
If $\bar s=0$, then $(s_1,\mu_1)$ is the
desired upper bound.  


It follows from Zorn's Lemma that ${\cal Z}_1$ contains a maximal
element, which we will denote by $(\bar t, \bar \gamma)$. We can 
assume that $\bar \gamma$ does not reach $\T$ on $[0,\bar t\,
]$ (since otherwise (\ref{goal}) is satisfied if we use the input
for $\bar \gamma$). We will now show that $\bar t=1$.  First recall 
that the estimate (\ref{splus}) and the $STC\T$ assumption
guarantee that $\crr$ is open, and that
the target $\T$ is closed.  It follows that there exists an open
set \[B\; \ni\;  \bar\gamma(\bar t\, )\; =:\; \hat x\] whose closure
lies in $\crr\setminus\T$.   Suppose that $\bar t<1$. Taking $x=\hat x$
in the estimate (\ref{sssss}), one easily checks that 
\[
\tau:=\inf_{\beta\in {\cal U}^r}\left\{t: \dist(y_{\hat
x}(t,\beta),\partial B)\le \dist(\hat x,\partial B)/2\right\}\; >\; 0.\]
Indeed, for each $\tilde\delta>0$, we can use (\ref{sssss}) to
find a $\tilde t>0$ so that $y_{\hat x}(t,\beta)\in
B_{\tilde\delta}(\hat x)$ for all $\beta\in \calu$ and all $t\in
[0,\tilde t\, ]$. Choosing \[\tilde\delta\; =\; 1/4\, \dist(\hat
x,\partial B),\] it follows that
\begin{eqnarray}
\dist(y_{\hat x}(t,\beta), \partial B)&\ge & \dist(\hat x,
\partial B)-||y_{\hat x}(t,\beta)-\hat x||\nonumber\\&\ge&
\frac{3}{4}\dist(\hat x, \partial B)\nonumber
\end{eqnarray}
for all $t\in [0,\tilde t\, ]$ and $\beta\in \calu$.  In
particular, $\tau\ge \tilde t>0$.


By conclusion (\ref{second}) of Lemma \ref{lemma0}, it follows
that there are $t\in (0,1-\bar t\, )$ and $\beta\in {\cal U}$ so
that
\begin{eqnarray}
\label{newone}
 w(\hat x)&\ge& 
J(t, y_{\hat x}(\cdot, \beta),\beta)\; +\;
e^{-\delta(t,\beta,\hat x)}
w(y_{\hat x}(t, \beta))\nonumber\\&-& E_1(\bar t+t)\; +\;
E_1(\bar t\, )
\end{eqnarray}
 and so that $y_{\hat x}(\cdot, \beta)$ remains in
$B\subseteq\crr\setminus\T$ on
$[0,t]$. Let $\tilde \alpha$ denote the control for $\bar\gamma$.
Since $(\bar t, \bar \gamma)\in {\cal Z}_1$, it follows that
\begin{equation}
\label{secondnewone}
 w(\bar x)\; -\; e^{-\Delta(\bar t,\bar\gamma,\tilde\alpha)}
 w(\hat x)\; \; \ge\; \; J(\bar t,
\bar\gamma,\tilde\alpha)\; -\; E_1(\bar t\, ).
\end{equation}
Let $\beta^\sharp$ denote the concatenation of 
$\tilde\alpha\lceil [0, \bar t\, ]$ followed by $\beta$. If we multiply
(\ref{newone}) through by
$e^{-\Delta(\bar t, \bar \gamma,\tilde\alpha)}$, change variables,
and add the result to (\ref{secondnewone}), then we get
\begin{eqnarray}
\label{thirdnewone}
 w(\bar x)&\ge&
J\left(\bar t+t, y_{\bar x}(\cdot, \beta^\sharp),
\beta^\sharp\right)\nonumber\\&+&
e^{-\delta(\bar t+t, \beta^\sharp,\bar x)}
w\left(y_{\bar x}(\bar t+t,\beta^\sharp)\right)\; -\;
E_1(\bar t+t).
\end{eqnarray}
Since $t$ was chosen so that $\bar t+t<1$ and $B\subseteq\crr$, we
conclude from (\ref{thirdnewone}) that \[\left(\bar t+t, y_{\bar
x}(\cdot, \beta^\sharp)\right)\; \in\;  {\cal Z}_1.\] This
contradicts the maximality of the pair $(\bar t, \bar \gamma)$.
Therefore, $\bar t=1$.  Moreover, the nonnegativity of $\ell$
guarantees that $w(\bar\gamma(1))<e^{-\bar
h+\Delta(1,\bar\gamma,\tilde\alpha)}\omega_o$.

We will now extend $\bar \gamma:[0,1]\to\crr$ to get the desired
trajectory.  Set
\[
{\cal Z}_2\; \; =\; \; \left\{
\begin{array}{c}
(t,\gamma):  \; \; \; 0\, \le\,  t\, \le\,  1, \; \text{and}
\gamma:\; [0,t]\to \crr\; \text{is a trajectory}\\[.8em]\! \! \!
\! {\rm for}\; \; \dot x\,  \, =\, \,  f(x,\beta )\; \text{for
some}\; \beta\, \in\, {\cal U}\; \, \text{with}\,  \gamma(0)\, \,
=\, \, \bar \gamma(1)\\[.8em] \! \! \! \! \text{so that} w(\bar
\gamma(1))\, \ge\,
J(t,\gamma,\beta)+e^{-\Delta(t,\gamma,\beta)}w(\gamma(t))- E_2(t).
\end{array}
\right\},
\]
and partially order ${\cal Z}_2$ by the relation (\ref{rel})
as before. We can 
assume  that
\begin{equation}
\label{same}
 w(\bar\gamma(1))\; \; <\; \; e^{-\bar
h+\Delta(s,\bar\gamma,\tilde\alpha)}\omega_o-\eps
\end{equation}
If (\ref{same}) does not hold,  
then choose $\eps'>0$ so that
\[w(\bar\gamma(1))\; \; <\; \; e^{-\bar
h\; +\; \Delta(s,\bar\gamma,\tilde\alpha)}\omega_o\; -\; \eps'.\]
Then the construction we are about to give produces a trajectory
$\gamma:[0,1]\to\crr$ and a corresponding input $\beta$ with the
property that
\begin{eqnarray}
w(\bar\gamma(1))&\ge& J(1,\gamma,\beta)+e^{\scr-
\Delta(1,\gamma,\beta)}w(\gamma(1))-\frac{\eps'}{8}(e^{-1}-e^{-2})\nonumber\\&\ge&
J(1,\gamma,\beta)+e^{\scr
-\Delta(1,\gamma,\beta)}w(\gamma(1))-E_2(1)\nonumber,\end{eqnarray}
and then the procedure is iterated to get the desired extension of
$\bar\gamma$.  
The fact that each totally ordered subset of ${\cal Z}_2$ has an
upper bound in ${\cal Z}_2$ follows from the proof of the
corresponding statement for ${\cal Z}_1$ and the definition of
$\bar h$ from assumption $(A_4)$.  Indeed, let $(s_j,\gamma_j)$ be
a totally ordered subset of ${\cal Z}_2$, let $\alpha_j$ be the
control corresponding to the trajectory $\gamma_j$ for all $j$,
and define $\beta^\dagger$, $\bar\mu$, and $\bar s$ as in the
earlier proof. We can assume that
 $\omega_o\in [0,\infty)$, as before.  If
we
suppose that $\bar\mu(\bar s)\in
\partial \crr$, then we get
\begin{eqnarray}
e^{-\bar h+\Delta(1,\bar\gamma,\tilde\alpha)}\omega_o\; -\; \eps&
>&
w(\bar\gamma(1))\nonumber\\&\ge & 
e^{-\Delta(s_j,\mu_j,\alpha_j)}w(\mu_j(s_j))-\eps\nonumber\\&\to&
e^{-\Delta(\bar s,\bar\mu,\beta^\dagger)}\omega_o-\eps,
\end{eqnarray}
so
\[
e^{-\bar h+\Delta(1,\bar\gamma,\tilde\alpha)+\Delta(\bar
s,\bar\mu,\beta^\dagger)}\omega_o\; \; >\; \; \omega_o.
\]
Since the definition of $\bar h$ guarantees that
$e^{-\bar h+\Delta(1,\bar\gamma,\tilde\alpha)+\Delta(\bar
s,\bar\mu,\beta^\dagger)}<1$,
this contradicts the assumption that $\omega_o\ge 0$.  Therefore,
it follows that $\bar\mu(\bar s)\in \crr$, so ${\cal Z}_2$ admits
a maximal element.  By the preceding, we know that the
maximal element in  ${\cal Z}_2$ has the form $(1,\tau_2)$. Let
$u_2:[0,1]\to U$ denote the input for $\tau_2$. We can assume
that $\tau_2$ does not reach $\T$ on $[0,1]$. Indeed, if there is
an $s_2\in [0,1]$ such that $\tau_2(s_2)\in \T$, then we 
can satisfy the requirement (\ref{goal}) by setting
\[
\bar\alpha(t)= \twoif{\tilde\alpha(t),}{0\le t\le 1}
{u_2(t-1),}{1\le t\le 1+s_2},
\]
where $\tilde\alpha$ is the control for $\bar \gamma$ on $[0,1]$.
Let $c_2$ denote the concatenation of $\tilde\alpha\lceil[0,1]$
followed by $u_2\lceil[0,1]$, and let $\gamma_2$ denote the
corresponding concatenated trajectory for $c_2$ from $\bar x$ to
$\tau_2(1)$.

Now reapply the procedure to $\gamma_2(2)$ to get a longer
trajectory that extends $\gamma_2$ and reaches $\T$ or runs for  three
time units without
leaving $\crr$. The procedure is iterated, and it results in a
sequence of terminal points \[p_n\; :=\; \phi_n(n)\; \in\;
\{x\in\crr: w(x)<e^{-\bar h+\Delta(n,\phi_n,c_n)}\omega_o\}\]  for
trajectories $\phi_n$ which begin at $\bar x$, are defined on
$[0,n]$, and remain in $\crr\setminus\T$ without loss of generality. It
follows that for
all $n\in\N$, we have
\begin{equation}
\label{trips} w(\bar x)\; \; \ge \; \; J(n,\phi_n,c_n)\; +\;
e^{-\Delta(n,\phi_n,c_n)}\,  w(p_n)\; -\;
\frac{\eps}{8}(1-e^{-n}).
\end{equation}
 Setting
$\hat \alpha(s) := c_n(s)$ for
 $n-1<s\le n$ and $n\in\N$, 
letting $\hat \phi$ denote
the trajectory for $\hat\alpha$,
and letting $b<0$ denote a lower
bound for $w$, a passage to the limit as $n\to\infty$ in
(\ref{trips}) gives
\[
\int_0^\infty e^{-\delta(s,\hat\alpha,\bar x)}\ell^r(y_{\bar
x}(s,\hat\alpha),\hat\alpha(s))\, ds \; \; \le\; \; w(\bar x)\;
-\; b\; +\; \frac{\eps}{2}\; <\; \; \infty,
\]
since $\hat\phi\equiv \phi_n$ on $[0, n]$.   It follows from the
assumption $(A_3)$ that $\bar p:=\lim_{s\to\infty}\hat\phi(s)$ exists and
lies in the target $\T$.
Since $U$ is compact and $\{p_n\}$ is bounded, and since $\ell$ is
continuous, it follows
from the estimate (\ref{sssss}) that  
\[
\sup\left\{\ell(y_{p_n}(s,\beta),\beta(s)): \; \;  n\in \N, \;
\beta\in {\cal U},\;   0\le s\le 1/M\right\}\; \; \le \; \; M\; \;
<\; \; +\infty
\]
for some $M>0$.  
Since condition $STC\T$ is satisfied, it follows that there is an
input $\tilde \beta$ and an $m\in \N$ so that
 \begin{equation}
 \label{maxx}
 J\left(t_{p_m}(\tilde \beta), y_{p_m}(\cdot,
 \tilde\beta),\tilde\beta\right)\; \; \bigvee\; \;
\left[g\left(y_{p_m}(t_{p_m}(\tilde\beta),\tilde\beta)\right)-w(p_m)\right]
\;
\; <\; \; \eps/4.
\end{equation}
Indeed, the estimates (\ref{splus})-(\ref{sssss}), the fact that
$\{p_n\}$ converges to a point in $\T$, and the continuity of $g$
guarantee the existence of a constant $\kappa\in(0,1/M)$ 
such that $\vert g(p_n)-g(y_{p_n}(t,\beta))\vert <\eps/8$
for all $\beta\in {\cal U}$ and $t\in [0,\kappa)$ and all large
$n$. Also,  the $STC\T$ condition and the estimate (\ref{sssss})
ensure that we can drive $p_n$ to $\T$ in time
$<\eps/[4(M+\kappa)]\wedge 1$ without leaving $\crr$ when $n$ is large
enough.  The estimate (\ref{maxx}) now follows since 
\[\{q\in\crr:
w( q)-g(q)>-\eps/8\}\] is open (by the assumption that $w$ is lower
semicontinuous).
 Fix $m\in\N$ for which there is a $\tilde \beta$ satisfying
 (\ref{maxx})
 and let $\bar \alpha$ denote the concatenation of
 the input $c_m\lceil[0,m]$ followed by the input  $\tilde
 \beta$.
We combine (\ref{trips}) and (\ref{maxx}) to get \beqn w(\bar
x)&\ge& J(m,\phi_m,c_m)\; \; +\; \; e^{-\Delta(m,\phi_m,c_m)}\,
w(p_m)\nonumber\\&+& J\left(t_{p_m}(\tilde\beta), y_{p_m}(\cdot,
\tilde\beta),\tilde\beta\right)\; \; -\; \; \frac{\eps}{8}(1-e^{-m})\;
\; -\; \; \frac{\eps}{4}\nonumber\\ &\ge& J(m,\phi_m,c_m)\; \; +\;
\; J\left(t_{p_m}(\tilde\beta), y_{p_m}(\cdot,
\tilde\beta),\tilde\beta\right)\nonumber\\&+&
e^{-\Delta(m,\phi_m,c_m)}\, g\left(y_{p_m}(t_{p_m}(\tilde
\beta),\tilde\beta)\right) \; \; -\; \;
\frac{\eps}{8}(1-e^{-m})-\frac{\eps}{2}\nonumber\\ &\ge&
\int_0^{t_{\bar x}(\bar \alpha)} e^{-\Delta( s, y_{\bar
x}(\cdot,\bar\alpha),\bar\alpha)}\,  \ell\left(y_{\bar
x}(s,\bar\alpha), \bar\alpha(s)\right)\, ds\nonumber\\&+&
e^{-\Delta(t_{\bar x}(\bar\alpha),y_{\bar
x}(\cdot,\bar\alpha),\bar\alpha)}\, g\left(y_{\bar x}(t_{\bar
x}(\bar\alpha),\bar\alpha)\right)\; \; -\; \;
\frac{\eps}{2}(2-e^{-m}).\eeqn Therefore, we can set
$\bar\beta=\bar\alpha$ to satisfy the requirement (\ref{goal}).
This proves  Theorem \ref{thm1}.  Our proof showed that
\[w(y_{\bar x}(j,\bar\alpha))\; \; <\; \; e^{-\bar
h+\delta(j,\bar\alpha,\bar x)}\omega_o\; \; \text{for\ \  all}\;
\;  \N\ni j<t_{\bar x}(\bar\alpha).\]  Since we could have replaced
$\bar h$ with any positive constant $k>\bar h$ in our proof, 
and since we could also have replaced $[0,1]$ with
$[m_j,m_{j+1}]$ for any $m_j\uparrow+\infty$ in the definitions of the
${\cal Z}_j$'s, the conditions of Remark \ref{thm3rm}
hold also.


\section{Local Uniqueness Characterizations}
\label{extend} 
This section extends Theorem
\ref{thm1} to get a more general  result that
applies to  solutions of (\ref{HJBE}) on $\Omega\setminus\T$ for
general open sets $\Omega\subseteq\R^N$, including cases where
$\T\not\subseteq\Omega$.  Our notation is as before, so
\[\delta(s,\beta,x):=\int_0^th^r(y_x(s,\beta),\beta(s))\, ds\] for
all $s\ge 0$, $\beta\in {\cal U}^r$, and $x\in \R^N$.   We assume this
 local version of $(A_4)$:
\bi\item[]\bi
\item[$(A'_4)$\ \ ]
$\left(\forall q\in \Omega\right)\; \; \left(\left(\exists\bar
h_q<\infty\right)\; 
\left(y_q(t,\beta)\in\partial\Omega\implies
\delta(t,\beta,q)\le \bar h_q\right)\right)$. \ei\ei

\noi We prove the following generalization of Theorem \ref{thm1}:


\bt{thm2} Assume $(A_0)$-$(A_3)$ hold,
$\Omega\subseteq\R^N$ is open, 
$w$ is locally bounded on $\Omega$, and one of
the following:

\bi
\item[1.]
$\T\subseteq\Omega$.
\item[2.]
$\partial\Omega\subseteq\T$, and there is a neighborhood $\cal N$
of $\T$ such that   $w^\star\le g$ on ${\cal N}\cap\Omega$. \ei Assume 
$w$ is a viscosity solution of (\ref{HJBE}) on $\Omega
\setminus\T$ satisfying $BC(\Omega, \omega_o)$.
If $\omega_o\in [0,\infty]$ and $(A'_4)$ holds, then 
$w\equiv v^r$ on $\{x\in\Omega: w(x)<e^{-\bar
h_x}\omega_o\}$. If $\omega_o<0$, then $w\equiv v^r$on $\Omega$.  \et

\br{lastttt} Note that we do not require $\Omega$ to be bounded
in Theorem \ref{thm2}.  If there exist $\Omega$ and $w$ which
satisfy the requirements of this theorem, then it must be the case
that $\Omega\subset{\cal R}$, since $v^r\equiv +\infty$ on
$\R^N\setminus{\cal R}$.  The novelty of Theorem \ref{thm2} is
that it covers cases where $\T\not\subseteq \Omega\not =\crr$.
Note that Theorem \ref{thm2} gives a uniqueness characterization
on all of $\Omega$ for the undiscounted case where one can take
$\bar h\equiv 0$.
Moreover, our proof of Theorem \ref{thm2}
will show that for each $k\in [\bar h,\infty)$ and for each
$p\in\Omega$ with $w(p)<e^{\scr -k}\omega_o$, there exists a
control $\alpha_p\in {\cal U}^r$ such that
\[w(y_p(j,\alpha_p))\; \; < \; \;
e^{-k+\delta(j,\alpha_p,p)}\omega_o\; \; \text{for\ all}\; \; \N\; \ni\;  
j\;
<\; t_p(\alpha_p)\] and such that
\[
t_p(\alpha_p)\; <\; \infty\; \; \text{or}\; \; \lim_{s\to\infty}
y_p(s,\alpha_p)\in\T.
\]
We can also allow $U$ to be closed but not
necessarily compact and $g$ to be discontinuous if we impose more
regularity on the data (cf.
$\S$\ref{newsection}). \er


\br{fpf} Theorem \ref{thm2} gives local  versions of Corollary
\ref{corfp} on the FP and its generalizations
$\left(P_\alpha\right)$, in the following sense. Let $\alpha$ be any
convex 
function of class $\cal{MK}$, let $\xi>0$ be given, and let
$v_\alpha$ denote the value function for 
$\left(P_\alpha\right)$ as before. Pick \[\Omega\; =\; \{x\in\R^2:
v_\alpha(x)<\xi\}.\] Since $v_\alpha$ is convex and therefore continuous
(cf. \cite{R70}, p. 81), $\Omega$ is
open. Also, $v_\alpha$ satisfies the  condition $BC(\Omega, \xi)$.
It follows from Theorem \ref{thm2} that if $w$ is a viscosity
solution of the HJBE (\ref{hjf}) on $\Omega\setminus\{\vec 0\}$,
and if there is a $\hat \xi>0$ so that $\Omega =  \{x\in\R^2:
w(x)<\hat \xi\}$ and $w$ satisfies $BC(\Omega,\hat\xi)$, then $w\equiv
v_\alpha$ on $\Omega$. In this
way, we get a uniqueness characterization for the Bellman equation
(\ref{hjf}) on any sublevel set of $v_\alpha$. \er


We now prove Theorem \ref{thm2}, assuming for notational simplicity
 that
$w$ is 
continuous (so $w\equiv w^\star\equiv w_\star$). 
Also, we assume that $\omega_o\in[0,\infty]$ and $(A'_4)$.  (The
$\omega_o<0$ case obtains by replacing $\bar h$ with $0$ as before.)
We drop the subscripts on
$\bar h$ and the superscripts $r$ as before. We indicate the changes
needed in the proof 
of Theorem \ref{thm1}.   The
proof of the inequality ``$w\ge v$'' is exactly as before except
that we replace $\cal R$ with $\Omega$ in the definitions of
${\cal Z}_1$ and ${\cal Z}_2$. To prove the reverse inequality,
fix $x\in\Omega\setminus \T$ and $\eps>0$ such that
$w(x) < e^{-\bar h}\omega_o-\eps$, and let $B$ be any bounded open
set in $\Omega$ that contains $x$ and is such that $\bar
B\subseteq \Omega\setminus\T$.  Then $w$ is also a viscosity
solution of the HJBE (\ref{HJBE}) on $B$.  It follows
from Lemma \ref{lemma0} that
\begin{equation}
\label{gps} w(x)\; \;  \le\; \;
\int_0^te^{-\delta(s,\beta,x)}\ell(y_x(s,\beta),\beta(s))\, ds\;
+\; e^{-\delta(t,\beta,x)}\, w(y_x(t,\beta))
\end{equation}
for all $\beta\in{\cal U}$ and $t\in [0,t^B_x(\beta))$. Suppose
that $w(x)>v(x)$.   By the infimum definition, we would then be
able to find a relaxed control ${\tilde \beta}\in {\cal U}(x)$
which is such that
\begin{equation}
\label{neweq}
\! \! \!  \int_0^{t_x({\tilde\beta})}\! \! \! 
e^{-\delta(s,\tilde\beta,x)}\,
\ell\left(y_x(s,{\tilde\beta}),{\tilde\beta}(s)\right)ds +
e^{-\delta(t_x(\tilde\beta),\tilde\beta, x)}
g\left(y_x(t_x({\tilde\beta}),{\tilde\beta})\right)<w(x).
\end{equation}
We consider three cases.

\bi\item[]\bi \item[{\bf Case A}\ ]
 If $y_x(s,
\tilde \beta)\in\Omega  \text{for} 0\le s\le t_x(\tilde \beta)$, then
$t_x(\tilde \beta)$ is a limit of exit times from sets $B=B_k$ as
above. For example, take $B_k\subseteq\Omega\setminus\T$ to be a tube
around the restriction
of $y_x(\cdot,\tilde\beta)$ to $[0,\, t_x(\tilde\beta)-1/k]$ for
$k$ large enough. In that case, we arrive at a contradiction 
with (\ref{neweq}) once
we put $\beta=\tilde\beta$ and $t=t_x^{B_k}(\tilde\beta)$  in
(\ref{gps}) and pass to the limit as $k\to \infty$.

\item[{\bf Case B}\ ]
Assume there is an $s\le t_x(\tilde\beta)$ for which
$y_x(\cdot, \tilde \beta)\notin\Omega$ and that
$\T\subseteq\Omega$.  Let $\hat \tau$ denote the supremum of the
times in $(0, t_x(\tilde\beta)]$ at which $y_x(\cdot,\tilde\beta)$
is in $\partial\Omega$.  Since $\T\subseteq\Omega$ and $\T$ is
closed and $\Omega$ is open, it follows that
$\hat\tau<t_x(\tilde\beta)$ and that $y_x(\hat \tau,
\tilde\beta)\in \partial\Omega$. Apply the first part of Lemma
\ref{lemma0} with the choices \[q\;  =\; z_n\;  :=\;
y_x(\hat\tau+1/n, \tilde\beta),\; \;  \; \; \; \beta(\cdot)\; =\;
\alpha_n(\cdot)\;  :=\;  \tilde\beta(\cdot+\hat\tau+1/n)\]
\noi and 
\[
r\;  =\;  t_{z_n}(\tilde\beta(\cdot+\hat
\tau+1/n)),\] with $B$ chosen to be a tube in $\Omega$ which
contains a portion of the trajectory $y_x(\cdot, \tilde\beta)$
that runs from $z_n$ to $y_x(t_x(\tilde\beta),\tilde\beta)\in \partial\T$
for $n$ large. Lemma \ref{lemma0}
would then give \beqn w(x)&>& \int_0^{\hat \tau+1/n}
e^{-\delta(s,\tilde\beta, x )}\,
\ell\left(y_x(s,\tilde\beta),\tilde\beta (s)\right)\; ds\nonumber\\&+&
\left[\int_0^{t_x(\tilde\beta)-\hat \tau-1/n}e^{-\delta(s,
\alpha_n, z_n)}\, \ell\left(y_{z_n}(s,\alpha_n),\alpha_n(s)\right)\;
ds\right.\\[.3em]&+&
\left.e^{-\delta(t_{z_n}(\alpha_n),\alpha_n,z_n)}\,
w\left[y_{z_n}(t_{z_n}(\alpha_n),\alpha_n)\right]
\right]
e^{-\delta(\hat\tau+1/n,\tilde\beta,x)}
\\[.3em]&\ge&
\int_0^{\hat\tau+1/n}e^{-\delta(s,\tilde\beta,x)}\,
\ell\left(y_x(s,\tilde\beta),\tilde\beta(s)\right)\; ds \; +\;
e^{-\delta(\hat\tau+1/n,\tilde\beta,x)}\,  w(z_n). \eeqn Since
$\ell$ is nonnegative, we get \[w(x)>
e^{-\delta(\hat\tau+1/n,\tilde\beta,x)}w(z_n)\] for large $n$. But
\[
\Omega \; \ni\;  z_n\; \; \to\; \;   y_x(\hat \tau,\tilde\beta)\;
\in\;
\partial\Omega
\]
\noi and 
\[
e^{-\delta(\hat\tau+1/n,\tilde\beta,x)}\; \to\;
e^{-\delta(\hat\tau,\tilde\beta,x)}\; \ge\; e^{-\bar h},
\]
so the limit condition in $BC(\Omega,\omega_o)$
implies  \[ e^{-\delta(\hat\tau+1/n,\tilde\beta,x)} w(z_n)\;
\; \to\; \; e^{-\delta(\hat\tau,\tilde\beta,x)} \omega_o \; \;
\ge\; \; e^{-\bar h}\omega_o.\] This contradicts the fact that
$w(x)<e^{-\bar h}\omega_o$.
\item[{\bf Case C}\ ]
Finally, assume that there is an $s\le t_x(\tilde\beta)$ for which
$y_x(\cdot, \tilde \beta)\notin\Omega$ and that Condition 2 of
Theorem \ref{thm2} holds.  Since $\partial\Omega\subseteq\T$, it
follows that \[\inf\left\{t\ge
0: y_x(t,\tilde\beta)\not\in\Omega\right\}\;
=\; t_x(\tilde\beta).\]  
Condition (\ref{neweq}) and the limit condition in
$BC(\Omega,\omega_o)$ guarantee the existence of a sequence
$t_n\uparrow t_x(\tilde\beta)$ and a constant $k$ for which
\begin{eqnarray}
e^{-\bar h}\omega_o-\eps &\le&
\liminf_{n\to\infty}\left\{e^{-\delta(t_n,\tilde\beta,x)}\left[\omega_o-\frac{\eps}{ke^{-\bar
h }}\right]\right\}\nonumber\\&\le & 
\liminf_{n\to\infty}\left\{e^{-\delta(t_n,\tilde\beta,x)}\,
w(y_x(t_n,\tilde\beta))\right\}\nonumber\\&<&w(x),\nonumber
\end{eqnarray}
which is a contradiction.    \ei\ei Therefore,
$w(x)\le v(x)$ for all $x\in\Omega$ such that $w(x)<e^{-\bar
h}\omega_o$.  As pointed out earlier, the case where $\omega_o<0$
is handled by replacing $\bar h$ with $0$ in the preceding
argument.  
 This establishes Theorem \ref{thm2}.

 \section{Unbounded Control Sets and Discontinuous Exit
Costs}
\label{newsection}
We close by showing how to relax our assumptions that the control set $U$
is compact and that the exit cost $g$ is continuous.  
To cover cases
where $g$ is discontinuous, we adapt the backward programming methods of
\cite{B97}.
Roughly speaking, we
extend our results to cases where $U$ is noncompact by replacing $U$ with
a compact set of vector field valued controls and a new dynamics which
gives the same trajectories as the original dynamics $f$.  


\subsection{Discontinuous Exit Costs}
This subsection shows how to generalize 
the arguments of the preceding sections to 
cases where the exit cost $g$ is discontinuous.  
We assume conditions $(A_0)$-$(A_3)$ and $(A'_4)$ hold except that instead
of assuming
that $g$ is continuous, we assume that $g$ is locally bounded and
nonnegative. It follows that
$g_\star:\crr\to\R$ is lower semicontinuous, that $g^\star:\crr\to\R$ is
upper semicontinuous, and that there exist sequences $\{g^-_m\}$ and
$\{g^+_m\}$ of continuous nonnegative functions such that 
\begin{equation}
\label{seq}
\ds
\sup_mg^-_m=g_\star,\; \; \; \; 
\inf_mg^+_m=g^\star,\; \; \; \; 
g^-_k\le g^-_{k+1} \; \forall k,\; \; \; \;  \text{and}\; \; \; \; 
g^+_k\ge g^+_{k+1} \; \forall k 
\end{equation}
pointwise on $\crr$.  Let $\Omega\subset\crr$ and $w$ satisfy the 
hypotheses of 
Theorem \ref{thm2}.
Let $USC(g)$ denote the set of upper semicontinuity points of $g$, 
\[USC_o(g)=USC(g)\cap \{x: g(x)=0\},\] 
and $w$ be a
viscosity solution of (\ref{HJBE}) on $\Omega\setminus\T$ which satisfies
the following relaxed analog of $BC(\Omega, \omega_o)$:
\bi
\item[]
\bi\item[]
\bi
\item[$BC_r(\Omega,\omega_o)$\ \ ]
$w:\Omega\to\R$ is bounded below, 
$\omega_o\in
\R\cup\{+\infty\}$,
 $w_\star\ge
g_\star$ and 
$w^\star\le g^\star$ on
$\T\cap\Omega$,
$w<\omega_o$ on $\Omega$,
and
$\ds\lim_{x\to x_o} w(x)=\omega_o$ $\forall x_o\in \partial\Omega$.  
\ei\ei\ei
Following \cite{B97}, we also assume that there is an outer field at each
point along $\partial\T$, i.e., the conditions

\bi\item[]\bi
\item[(OF1)]\ 
$d(\cdot, \partial\T)$ is $C^{1,1}$ in some neighborhood of $\partial\T$.
\item[(OF2)]\ 
$\forall x\in\partial\T$, $\exists a\in U$ and
$\delta>0$ s.t. $-f(x,a)\cdot
Dd(x,\partial\T)\ge \delta>0$.
\ei\ei
Condition $(OF2)$ says that at each point of $\partial \T$, the system
can choose a direction pointing into $\T$.
Finally, we assume the following variant of $(A_3)$:
\bi\item[]\bi
\item[$(A'_3)$\ ]
If $\beta\in \calu^r$ and  $x\in\R^N$,  and if  
$J(\infty,y_x(\cdot, \beta),\beta)<\infty$, 
then there is a point
$\bar
p\in \partial\T\cap USC_o(g)$ for which $\ds \lim_{s\to\infty}
y^r_x(s,\beta)=\bar p$.
\ei\ei
\br{mins}
A slight variant of what follows works if we replace 
$\bar p\in \partial\T\cap USC_o(g)$ in $(A'_3)$ with the conditions that
$\bar p\in \partial\T\cap USC(g)$ and $g(\bar p)=g_{\scr \min}$,
where $g_{\scr \min}$ is any lower bound on $g$ (cf. Remark
\ref{verylast}).\er

\noi We prove the following consequence of Theorem \ref{thm2}:  

\bc{cordisc}
Let the preceding assumptions hold.  If $\omega_o\in \R_+\cup\{+\infty\}$,
then the following inequalities hold on $\left\{x\in
\Omega\setminus\T: v^r[g](x)<e^{-h_x}\omega_o\right\}$:
\ben
\item
$w_\star\ge \left[v^r[g_\star]\right]_\star$
\item
$w^\star\le v^r[g^\star]$
\een
If $\omega_o<0$, then the previous two inequalities hold on all
of
$\Omega\setminus\T$.\ec

In particular, $w\equiv v^r[g]$ is continuous if $g$ is continuous and
$v^r$ 
is lower semicontinuous.
Note that under the preceding hypotheses, it could be that $w\ne v^r[g]$
at points of discontinuity of $g$, even if $w$ and $v^r$ are both
continuous.  
Here is an example from $\S$3 of \cite{B97} where this occurs:

\bex{discexample}
Take $h\equiv\ell\equiv 1$, $f(x,a)\equiv a$, and 
$\T=U=\overline{B_1(0)}\subseteq\R^N$. Use $\Omega=\crr=\R^N$.
Set $x_o=(1,0)$ and 
\[g=\ind_{\scr
\partial\left(\overline{B_1(0)}\right)\setminus\{x_o\}}.\]  One
checks that 
$v^r[g](x)=1-e^{-||x-x_o||}$ on $\Omega\setminus\T$ and that 
$u_\xi(x)=1-\xi e^{-||x-x_o||}$ is also a 
viscosity solution of the HJBE on $\R^N\setminus\T$
for all $\xi\in ]0,1[$ satisfying the correct boundary condition.\eex

The proof of Corollary \ref{cordisc}
is similar in spirit to the arguments of \cite{B97},
except we use $(A'_3)$ and a weak-$\star$
convergence argument to replace the requirements in \cite{B97} that the
discount factor $h$ is a positive constant and that $f$ and $\ell$ are
globally bounded.  
Define $\tilde J$ by
\[
\tilde J(x,\beta,t,\phi)=\int_0^t \ell^r(y_x(s,\beta),\beta(s))\, 
e^{-\delta(s,\beta,x)}\, 
ds +e^{-\delta(t,\beta,x)}\phi(y_x(t,\beta))
\]
wherever the RHS is defined.  
Our strategy is to first show that $w_\star$ majorizes
the lower envelope of the optional stopping value function 

\[\ds
v_-(x)\; \; :=\; \; 
\twoif{\inf
\left\{\tilde J(x,\beta,\theta,g_\star): \; \beta\in \calu^r, \; 
\theta\in[0,\infty), \; 
y^r_x(\theta,\beta)\in\T
\right\},}{x\in\crr^r}{+\infty,}{x\not\in\crr^r}  
\]
Once this is done, the first condition of Corollary \ref{cordisc} 
will
follow
from the following lemma (whose proof is essentially the same as the proof
of Lemma 2.15 from \cite{B97} except local bounds on $f$ are
used instead of global ones):

\bl{newnew}
Under the previous assumptions, let $\theta\in(0,\infty)$, $\mu\in
\calu^r$, and $x\in \Omega$ be given.  Assume that $y_x(\theta,\mu)\in
\partial\T$.  Then there exist sequences $x_n\to x$ and $\mu_n\to\mu$
weak-$\star$ on $[0,\infty)$ such that $y_{x_n}(t_{x_n}(\mu_n), \mu_n)=
y_x(\theta,\mu)$ and 
$t_{x_n}(\mu_n)=\theta$
for all $n$.  
\el

Let us show how the first conclusion of Corollary \ref{cordisc} would
follow from $w_\star\ge (v_-)_\star$.  
Let $\eps>0$ be given, and choose $x\in\Omega\setminus\T$.
We assume for simplicity that $\omega_o<0$.  (The $\omega_o\ge 0$ 
case follows
easily from the argument we give.)  Using the Lipschitz
continuity of $f$, 
we would then be able to find $\bar\theta<\infty$, $\bar\mu\in \calu^r$,
and sequences $x_n\to x_o$, $\eps_n\downarrow 0$, and $\mu_n\to\bar\mu$
weak-$\star$ such that
\begin{eqnarray}
v_-(x_o)+\eps&\ge& \tilde
J(x_o,\bar\mu,\bar\theta,g_\star)\nonumber\\&\ge&
\tilde J(x_n,\mu_n,t_{x_n}(\mu_n),g_\star)-\eps_n\nonumber\\&\ge&
v^r[g_\star](x_n)-\eps_n,\nonumber
\end{eqnarray}
which gives 
\begin{equation}
\label{digg}
v_-(x_o)\ge \left[v^r[g_\star]\right]_\star(x_o)
\end{equation}
 and the
first inequality of the corollary  
once we take inferior limits.  To prove that $w_\star\ge (v_-)_\star$, we
need the following lemma:

\bl{limlem}
If $\bar x\in \Omega\setminus\T$ is such that 
$v^r(\bar x)<e^{-h_{\bar x}}\omega_o$, then 
\[\ds \liminf_{m\to\infty}v^r[g^-_m](\bar
x)\ge\left[v^r[g_\star]\right]_\star(\bar x)\]  
The limit holds on all of
$\Omega\setminus\T$ if $\omega_o<0$.
\el
\bpr
We again assume that $\omega_o<0$ for simplicity and choose
$\bar
x\in \Omega\setminus\T$.  We can clearly assume that 
$\bar x\in\crr^r$.  We continue to use the notation of $\S$\ref{proof}.  
Choose a sequence $\beta_n\in \calu^r$ and a
sequence of
corresponding trajectories $\gamma_n$ 
for $f^r$ starting at $\bar x$ such
that
\begin{equation}
\label{lbs}
v^r\left[g^-_m\right](\bar x)\ge -\frac{1}{m}+\tilde J(\bar x, \beta_m,
t_{\bar x}(\beta_m), g^-_m)\; \; \forall m\in \N
\end{equation}
By passing to a subsequence and using a diagonal argument
based on the classical compactness lemma (cf. \cite{BARD}, p.354), 
 we can assume that
there is an input
$\beta_o\in\calu^r$ and a corresponding trajectory $\gamma_o$ for
the dynamics $f^r$ starting at $\bar x$ so that
$\beta_m\to\beta_o$ weak-$^\star$ and $\gamma_m\to\gamma_o$ uniformly on
compact intervals.  

If $\{t_{\bar x}(\beta_m)\}$ is bounded, 
we can assume  
that $t_{\bar x}(\beta_m)\to t_o\in [0,\infty)$ and
$\gamma_m(t_{\bar x}(\beta_m))\to\gamma(t_o)\in\partial\T$ (since
$\partial\T$ is
closed).  One easily checks (cf. \cite{B97}) that since the $g^-_m$'s are
continuous, we have 
\[\ds
\liminf_{m\to\infty}g^-_m\left[\gamma_m(t_{\bar x}(\beta_m))\right]\ge
g_\star(\gamma(t_o)).
\]
It follows from (\ref{digg}) and (\ref{lbs}) that 
\[\ds\liminf_{m\to\infty}v^r[g^-_m](\bar x)\ge v_-(\bar x)\ge 
\left[v^r[g_\star]\right]_\star(\bar x).\]  On the other hand, if
$\{t_{\bar x}(\beta_m)\}$ is unbounded, then we can assume  that
$t_{\bar x}(\beta_m)\uparrow+\infty$.  It follows from the
nonnegativity of $\ell$ and $g$, (\ref{seq}), and (\ref{lbs}) that 
$
0\le J(t,\gamma_o,\beta_o)\le 1+v^r[g](\bar x)<\infty$
for all $t\ge 0$ (bu picking $m$ in (\ref{lbs}) so that 
$t_{\bar x}(\beta_m)\ge t$).  
We conclude from the assumption $(A'_3)$ that there is a $\bar p\in
\partial\T\cap USC_o(g)$ for which 
\begin{equation}
\label{pbar}
\ds\lim_{t\to\infty}y_{\bar x}(t,\beta_o)=\bar p.
\end{equation}
Let $\eps>0$ be given.  Using $STC\T$, it follows that there exist $M>0$
and extensions $\gamma^+_o$ and $\beta^+_o$ of the restrictions
$\gamma_o\lceil [0,M]$ and 
$\beta_o\lceil [0,M]$ so that $\gamma_o$ reaches $\partial\T$ at some time
$t^\star\ge M$
and which give 
\begin{equation}
\label{pointfive}
J(M,\gamma_o,\beta_o)\ge J(t^\star, \gamma^+_o,
\beta^+_o)-\frac{\eps}{3} \; \; \text{and}\; \; 
g^\star(\gamma^+_o(t^\star))\le \eps/3,
\end{equation}
where we use the upper semicontinuity of $g$ at $\bar p$  and the estimate
(\ref{sssss}).
On $[0,M]$, $\gamma_m\to\gamma_o$ uniformly and $\beta_m\to\beta_o$
weak-$^\star$, so (\ref{lbs}) gives 
\begin{eqnarray}
v^r[g^-_m](\bar x)&\ge & 
-\frac{1}{m}+J(M,\gamma_m,\beta_m)\nonumber\\&\ge&
-\frac{1}{m}-\frac{\eps}{3}+J(M,\gamma_o,\beta_o)\nonumber\\&\ge&
-\eps+J(t^\star,\gamma^+_o,\beta^+_o)\nonumber\\&\ge&
-2\eps+v_-(\bar x)
\end{eqnarray}
for large $m$, so 
$\liminf_{m\to\infty}v^r[g^-_m](\bar x)\ge v_-(\bar
x)$ and we conclude from (\ref{digg}) as before.
 \epr

We now prove Corollary \ref{cordisc}.  We assume that $\omega_o<0$ and
$x\in
\Omega\setminus\T$ for simplicity as before.
For each $m\in \N$, the argument of $\S$\ref{proof}
and the fact that $w_\star\ge g^-_m$ on $\T$ for each $m$ gives
$w_\star(x)\ge v^r[g^-_m](x)$ for each $m$, so the first inequality of the
corollary follows from Lemma \ref{limlem}.  On the other hand,
since $g\le g^+_m$ for all $m$ and the $g^+_m$'s are continuous, it
follows from the arguments of $\S$\ref{extend} that $w^\star(x)\le
v^r[g^+_m](x)$ for each $m$.  One easily checks that
\[
\limsup_{m\to\infty}v^r[g^+_m](x)\le v^r[g^\star](x),
\]
so we get the second inequality of the lemma.  This completes the
proof of Corollary \ref{cordisc}.  
\br{lippo}
 We remark that if $v^r[g]$
is lower semicontinuous, then the preceding proof gives 
$w_\star=w^\star=v^r[g]$ at all points in $\Omega$ where $g$ is
continuous, so if $g$
is continuous on $\Omega$, then there are in fact no 
viscosity solutions of the HJBE on $\Omega\setminus\T$ satisfying
$BC_r(\Omega,\omega_o)$ which have discontinuities.\er  

\br{verylast}
Similar results hold if we replace the requirement $g(\bar p)=0$ in
$(A'_3)$
with  the requirement that $g(\bar
p)=g_{\scr \min}$, where $g_{\scr \min}$
is any finite lower bound for $g$.  One way to do this is as follows.  We
assume $f=(\tilde f,1)$, $\Omega\subseteq\crr$ is bounded, 
 and that there
is an integrable
function $\phi$ for which 
\begin{equation}
\label{zvi}
|h((x,s),a)-h((y,s),b)|\; \le\;  \phi(s)\left[||x-y||+||a-b||+\bar
R\right]
\end{equation}
for all $x,y\in
\R^{N-1}$ and  
$a,b\in
U$, where $\bar R$ is constant. We also assume  $g\lceil \T$ is
bounded.  
The proof is as before up to the existence of $\bar p\in\partial\T$ in
(\ref{pbar}), except now $g(\bar p)=g_{\scr \min}$.
Choose $M>0$
as before.  We indicate the changes needed to complete the proof.  
The case where $\{t_{\bar x}(\beta_m)\}$ is bounded is handled exactly
as before.  By passing to a subsequence, it remains to consider
the case where $t_{\bar x}(\beta_m)\to\infty$.
First
assume that $\int_0^\infty h(\gamma_o(s),\beta_o(s))\, ds<\infty$.  By a
diagonal argument, we can assume that
\[
a_m(k):=\int_{k-1}^kh(\gamma_m(s),\beta_m(s))\,
ds\; \; \; \to\; \; \;  a(k):= 
\int_{k-1}^kh(\gamma_o(s),\beta_o(s))\, ds \; \; \; \; \forall k\in\N.
\]
Also, the boundedness of $\gamma_o$ and the $\gamma_m$'s in $\Omega$
and the estimate (\ref{zvi}) guarantee that \[\sum_{k=1}^\infty
a_m(k)<\infty\; \; \forall m\in\N.\]
Therefore, we get
\[
\int_0^\infty h(\gamma_m(s),\beta_m(s))\, ds\to\int_0^\infty
h(\gamma_o(s),\beta_o(s))\, ds,\]
by dominated convergence.  Notice that $g^\star(\bar p)=g(\bar p)$.  
For possibly larger $M>0$, (\ref{lbs}) and elementary arguments
which we leave to the reader therefore
give
\begin{eqnarray}
\label{arrays}
v^r[g^-_m](\bar x)&\ge &
-\frac{1}{2m}+J(M, \gamma_o,\beta_o)+e^{-\delta(t_{\bar
x}(\beta_m),\beta_m,\bar x)}g^-_m\left[\gamma_m(t_{\bar
x}(\beta_m))\right]\nonumber\\
&\ge&
-\frac{1}{2m}-\frac{\eps}{3}+J(M,
\gamma_o,\beta_o)+e^{-\delta(M,\beta_o,\bar x)}g(\bar 
p)\nonumber\\
&\ge&
-\eps+J(t^\star, \gamma^+_o,\beta^+_o)+e^{-\delta(t^\star,\beta^+_o,\bar
x)}g^\star(\gamma^+_o(t^\star))\nonumber\\
&\ge&
-\eps+v_-(\bar x)
\end{eqnarray}
for some $t^\star\ge M$ depending on $m$ (where we use the estimate
(\ref{sssss}) to ensure
that $\gamma^+(t^\star)$ is near enough to $\bar p$ and assume that
$g^-_m\ge g_{\scr min}$ for all $m$ for the third inequality). 
We then conclude as before.  On the other hand, if
\[
\int_0^\infty h(\gamma_o(s),\beta_o(s))\, ds=+\infty,\] then one easily
checks that 
$\lim_{m\to\infty}\int_0^{t_{\bar x}(\beta_m)} h(\gamma_m(s),\beta_m(s))\,
ds=+\infty$,
so 
\[
\lim_{m\to\infty}e^{-\delta(t_{\bar x}(\beta_m),\beta_m,\bar x)}
g^-_m\left[\gamma_m(t_{\bar x}(\beta_m))\right]=0
\]
since $g_\star\lceil\T$ is bounded, and
\[
\lim_{M\to\infty} e^{-\delta(M,\beta_o,\bar x)}g(\bar p)=0.
\]
The inequalities in (\ref{arrays}) now follow as before.

\er

\subsection{Unbounded Control Sets}
The assumption that $U$ is compact can be relaxed to the
requirement that $U$ is closed if we add regularity conditions on
that data.  One way to do this is to assume that $f$ and $\ell$
penalize large control set values and consider only viscosity
solutions $w$ of the Bellman equation on bounded sets for which the
semidifferential sets $D^+w^\star(x)$ and $D^-w_\star(x)$ are
locally bounded, in which case the infimization in the Bellman
equation can be taken over bounded subsets of $U$ (cf.
\cite{BDL97}, \cite{DAL}, and \cite{M99b} for details).

Another way to do this is as follows.  For simplicity, we
assume 
 $h\equiv 0$ and 
$\{(f(x,a),\ell(x,a)): a\in A\}$
is convex
 for all $x\in\R^N$ (which implies that all trajectories for
 $f^r$ are also trajectories for $f$, by
 the Filippov Selection Theorem).  We give $C(\R^N,
\R^N\times \R)$ the topology of compact convergence.
 The additional assumptions we
 make are as follows:
\bi\item[]\bi
\item[$(NC_1)$]
\ \ $\sup\{\ell(0,u):u\in U\}\; <\; \infty$.
\item[$(NC_2)$]
\ \ $\{(f(\cdot, u),\ell(\cdot, u)):u\in U\}\; \subseteq \;
C(\R^N, \R^N\times \R)$ is closed.
\item[$(NC_3)$]
\ \ $f$ is bounded on $B_R(0)\times U$ for each $R>0$. \ei\ei 
These guarantee that the supremum in the HJBE is always finite.  
It
follows from the Ascoli-Arzel\'a Theorem that
\[
K\; :=\; \{k_u(\cdot)\; :=\; (f(\cdot,u), \ell(\cdot, u)): u\in
U\}
\]
is a compact subset of the metric space $C(\R^N, \R^N\times \R)$.
Define the projection mappings $\pi_j$ on $K$ by
\[
\pi_j(k_u(\cdot))=\twoif{f(\cdot, u),}{j=1}{ \ell(\cdot,
u),}{j=2}\; \; \; \forall u\in U.
\]
We now apply the method of our proofs to the exit time problem
whose dynamics $\Phi$, Lagrangian $\Lambda$, and control set
$\tilde{\cal U}$
 are
defined by
$
\Phi(x,k) = (\pi_1\circ k)(x)$, 
$\Lambda(x,k)=
(\pi_2\circ k)(x)$, and 
$\tilde{\cal U}\; :=\;
\{[0,\infty)\ni t\mapsto k_{\beta(t)}: \beta\in {\cal U}\}$.
Let $\tilde v$ denote the value function of this problem. Since
the trajectories of $\Phi$ with the controls $\tilde {\cal U}$ are
exactly the trajectories of $f$ with controls in $\cal U$, it
follows that $\tilde v\equiv v \equiv  v^r$.
Moreover, the new problem satisfies conditions $(A_0)$-$(A_3)$
(with the compact metric space $K$ replacing $U$, and so forth).
It follows from the proofs we gave 
that any viscosity solution of the HJBE on $\crr\setminus\T$ which
satisfies $BC(\crr,\omega_o)$ for some $\omega_o\in
\R\cup\{+\infty\}$ coincides with the value function $v$ of the
original problem. Arguing in a similar way, we get analogs of
Theorem \ref{thm2} and Corollary \ref{cordisc} which apply to problems
with noncompact control
sets, since we can take weak-$\star$ limits of relaxed controls on the
compact metric space $K$.   We leave the details to the reader.  

\section{Conclusions}

This paper presented uniqueness characterizations for
discontinuous viscosity solutions for the Bellman equation of
deterministic optimal control problems with
exit times and fully nonlinear systems. We allow problems whose
dynamics, interest rates, Lagrangians, control sets, and targets
are all unbounded.  Also, we allow cases where the Lagrangian and
the interest rate are not bounded below by positive constants,
including the undiscounted case where the interest rate is
identically zero, and we allow discontinuous value functions and
discontinuous exit costs.

We gave conditions under which the value functions of these
problems are the unique viscosity solutions of the corresponding
Bellman equations that vanish on the target, are bounded below,
and satisfy certain growth conditions.  These growth conditions
hold vacuously for cases where each point in the state space can
be brought to the target in finite time. In particular, we showed
that the value function for the Fuller Problem is the unique
viscosity solution of the associated Bellman equation which is
bounded below and which vanishes at the origin. We also gave a
local uniqueness characterization which applies to solutions of
exit time Bellman equations on neighborhoods which may or may not
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