We consider the problem of optimally controlling a system of either ODEs or SDEs with respect to a vector-valued cost functional. Optimisation of the cost is considered with respect to a partial ordering generated by a given proper cone $K$. Since in the vector case minima may not exist, we define vectorial value functions as (Pareto) minimals of the ordering. The main results of this paper are that we demostrate existence of value maps of the system as minimals of the ordering contained in the trade-off manifold of all minimals and also prove that for any direction in the dual cone $K^*$, the projection of the minimal along this direction is a scalar viscosity solution of a certain HJB equation. This allows to construct a family of feedback controls in both the deterministic and stochastic case by using PDE theory.
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\begin{document}

\title[Stochastic Optimal Control with Vectorial Cost]{On Hamilton-Jacobi-Bellman Equations Arising in Deterministic and Stochastic Optimal Control with Vectorial Cost}

%    Information for first author
\author{Nikos Katzourakis and Tristan Pryer}
%    Address of record for the research reported here
\address{Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, Berkshire, UK}
%    Current address
%\curraddr{Department of Mathematics and Statistics, Case Western
%Reserve University, Cleveland, Ohio 43403}
\email{n.katzourakis@reading.ac.uk, t.pryer@reading.ac.uk}
%    \thanks will become a 1st page footnote.


%    General info
\subjclass[2010]{Primary 35F21, 49L20, 49L25, 93E20, 58E17; Secondary 93B52, 35Q93}

\date{}

%\dedicatory{This paper is dedicated to our advisors.}

\keywords{Hamilton-Jacobi-Bellman PDE, Optimal Control, Stochastic Optimal Control, Vectorial Cost Functional, Nash equilibrium, Pareto minimals.}

\begin{abstract} We consider the problem of optimally
  controlling a system of either ODEs or SDEs with respect to a
  vector-valued cost functional. Optimisation of the cost is considered with
  respect to a partial ordering generated by a given proper cone $K$. Since in the vector case minima may not exist, we define vectorial value functions as (Pareto) minimals of the ordering. The main results of this paper are that we demostrate existence of value maps of the system as minimals of the ordering contained in the trade-off manifold of all minimals and also prove that for any direction in the dual cone $K^*$, the projection of the minimal along this direction is a scalar viscosity solution of a certain HJB equation. This allows to construct a family of feedback controls in both the deterministic and stochastic case by using PDE theory.
\end{abstract} 

\maketitle

\section{Introduction} \label{section1}

Let $A\sub \R^m$ be a compact set and $F, \sigma: \R^n \by A \larrow
\R^n$ continuous maps. For $x\in \R^n$, consider the following
initial value problem for a system of stochastic differential
equations (SDEs)
\beq
  \label{1.1}
  \left\{
\begin{array}{l}
    \d {\chi}(s) \,= \, F\big(\chi(s),\al(s)\big) \d s\ +\ \sigma\big(\chi(s),\al(s)\big) \d W(s), \quad t<s<T,\ms
    \\
 \ \ \,   \chi(t) \,=\, x,
  \end{array}
  \right.
\eeq
in the It$\hat{\textrm{o}}$ sense. Here $\al$ is a measurable map is the class $\mA$,
where \beq \label{1.2} \mA\, :=\, \Big\{ \al \in L^\infty(0,T)\ \big|
\ \al(t)\in A, \text{ for a.e. }t\in(0,T)\Big\} \eeq and $W(s)$ is an
$n$ dimensional Wiener process over a stochastic basis. Note that in
the case $\sigma = 0$ the problem reduces to a system of ordinary
differential equations (ODEs)
 \beq \label{1.1a} \left\{
\begin{array}{l}
\dot{\chi}(s)\, = \, F\big(\chi(s),\al(s) \big), \ \ \ t<s<T,\ms\\
\, \chi(t)\, =\, x.
\end{array}
\right.
\eeq

In this paper we consider the problem of optimally controlling the
system \eqref{1.1} with respect to the \emph{vectorial cost
  functional}

\beq \label{1.3}
C_{x,t}[\al]\ :=\ \Exp \qb{
  g\big( \chi(T)\big)\ +\ \int_t^T h\big(\chi(s),\al(s)\big)\, ds
}
\eeq
where 
\beq  \label{1.4}
h\ : \ \R^n \by A \larrow \R^N,\ \ \ g\ : \ \R^n \larrow \R^N,
\eeq
are given maps, called the \emph{running cost} and the \emph{terminal
  cost} respectively and $\Exp$ is the expectation defined with respect to the measure induced by the stochastic basis.
% DEFINED WITH RESPECT TO... (AMEND APPROPRIATELY) .
In \eqref{1.3} $\chi$ denotes the
stochastic (deterministic) flow map of \eqref{1.1} when $\sigma
\not\equiv 0$ ($\sigma \equiv 0$) respectively, having supressed the
dependence in $x,\al$:
\beq  \label{1.5}
\chi(s)\ \equiv\ \chi \big(s,x,\al(s) \big).
\eeq

We will need to assume that $F,g,h,\sigma$ are bounded and Lipschitz continuous with respect to $x\in \R^n$, uniformly with respect to $a\in A$:
\begin{gather} \label{1.6}
  %\begin{split}
  \left.
    \begin{array}{r}
      |F(x,a)|,\ |g(x)|,\ |h(x,a)|,\ \norm{\sigma(x,a)}\ \leq\ C, \ms \\ 
      |F(x,a)-F(y,a)|\ \leq\ C|x-y|, \ms\\
      |g(x)-g(y)|\ \leq\ C|x-y|,\ms\\
      |h(x,a)-h(y,a)|\ \leq\ C|x-y|, \ms\\
      |\sigma(x,a)-\sigma(y,a)|\ \leq\ C|x-y|, \ms
    \end{array}
    \right \}
    \text{ for all }x,y\in \R^n,\ a\in A.
  %\end{split}
\end{gather}

In both the deterministic and stochastic cases of \emph{scalar cost
  functional}, namely when $N=1$, this problem is standard in Control
Theory and there is an extensive literature, see for instance Evans
\cite{E}, Fleming-Soner \cite{FS}, Baldi-Capuzzo Dolcetta \cite{BCD},
Lions \cite{L}, Fleming-Rishel \cite{FR} and Kenneth \cite{Ke}. When $C_{x,t}[\al] \in [0,\infty]$, optimisation with respect to this scalar cost functional is unambiguous: one seeks to
minimise \eqref{1.3} over all admissible controls $\al \in \mA$. Along
the lines of (ordinary) dynamic programming, one may define the
\emph{value function} \beq \label{1.7} u(x,t)\ :=\ \inf_{\al \in \mA}
C_{x,t}[\al], \ \ \ x\in \R^n,\, 0<t<T.  \eeq It then follows by
standard PDE theory (see e.g.\ \cite{E} for the case of vanishing white noise) that under assumption
\eqref{1.6} the value function is Lipschitz continuous and solves an
initial value problem for a Hamilton--Jacobi--Bellman (HJB) PDE
\beq \label{1.8} 
\left\{
\begin{array}{r}
u_t\ +\ H(\cdot,Du,D^2 u)\, =\, 0, \ \text{ in }\R^n\by (0,T),\ms\\
u\, =\, g,\ \text{ on }\R^n\by \{0\},\ \ 
\end{array}
\right.
\eeq
in the Viscosity sense (see e.g. \cite{CIL} for the theory of viscosity solutions, and for a more elementary introduction we refer to \cite{K}). Here the Hamiltonian $H$ is the function defined by 
\beq \label{1.9} 
H(x,p,Z)\, := \ \min_{a\in
  A}\left\{\frac{1}{2}\sigma(x,a)\otimes \sigma(x,a) : Z \ +\ F(x,a)\cdot
p\ +\ h(x,a) \right\}.  
\eeq 
The HJB equation can be utilised to construct a \emph{feedback control
  $\al^*$} which optimally drives the dynamics of the flow generated
by the system \eqref{1.1} and minimises both the running cost and the
terminal cost. Roughly, at points of differentiability of $u$, $\al^*$
is defined by selecting for each $t<s<T$ the $\al(s)$ which realises
the minimum:
\[
  \begin{split}
    H\Big( \chi^*(s), Du\big(\chi^*(s),&\al^*(s)\big), D^2u\big(\chi^*(s),\al^*(s)\big) \Big)
    \\
    &=\
    \frac{1}{2}\sigma \big(\chi^*(s),\al^*(s)\big)\otimes \sigma \big(\chi^*(s),\al^*(s)\big) : D^2 u\big(s,\al^*(s)\big) 
    \ms\\
    &\ \ \ \ +\,
    F\big(\chi^*(s),\al^*(s)\big) \cdot Du\big(\chi^*(s),\al^*(s)\big)\ 
    +
    \ h\big(\chi^*(s),\al^*(s)\big),
  \end{split}
\]
where 
\[
\chi^*(s)\ :=\ \chi \big(s,x,\al^*(s) \big).
\]
Conversely in the deterministic case, given any HJ equation $u_t+H(\cdot,Du)=0$ with $H(x,p)$ concave in $p$, one can always relate it to a scalar optimisation problem for an ODE system of the form \eqref{1.1a} for some deterministic cost functional, by using the fact that concave functions can be written as infima of a family of affine functions.

In this paper we consider instead the case of $N\geq 2$ and we seek to
optimise the vectorial cost functional \eqref{1.3}. Vectorial
%(Deterministic)
Optimal Control and vectorial 
%(Deterministic) 
Dynamic
Programming are extremely important in applications and have been
extensively studied in the last 50 years, mostly in connection to
real-world applications like in Economics/Finance,
Mechanics/Engineering, Aeronautics, Automotive industry etc, see e.g.\
Bellman-Fan \cite{BF}, Chen-Huang-Yang \cite{CHY}, Debreu \cite{D},
Henig \cite{H}, Isii \cite{I}, Luenberger \cite{L1, L2, L3, L4}, Olech
\cite{O}, Salukvadze \cite{S}, Pareto \cite{P}, Boyd-Vanderberghe
\cite{BV} and references therein.

In the vectorial case, ones needs to be very careful regarding the
meaning of ``minimise the vector cost". In (finite-dimensional) Optimisation theory (see e.g.\ \cite{BV}), it is fairly standard to consider minimisation with respect to a \emph{partial ordering ``$\leq_K$" generated by a convex cone $K\sub \R^N$} with
some further properties, that is for $\xi,\eta \in \R^N$, we define
\[
\xi \, \leq_K \eta\ \ \ \Longleftrightarrow\ \ \ \eta-\xi \, \in \, K.
\]
Vector-valued optimisation is extremely important in applications, and there a very active current research on the topic, mostly in connection to multi-criterion optimisation and Nash equilibria, see for instance \cite{DD, M, MGGJ, GC, RBG, BKR, RK}. In the scalar case we typically have $K=[0,\infty)$, in the case of symmetric $N\by N$ matrices one might take 
\[
K\, =\, \mS(N)^+\,=\,\big\{A \in \R^{N\by N}\, | \ A=A^\top \geq 0\big\}.
\]
In the case of $\R^N$, one simple choice of cone could be 
\[
K\,=\,\R^N_+\,=\,\big\{\xi \in \R^N \ | \ \, \xi_i \geq0,\ 1\leq i \leq N \big\},
\]
which results in the \emph{componentwise ordering} of $\R^N$. The
choice of ordering is determined by the priority of objectives in the
case of cost functionals with multi-dimensional range. A typical difficulty of
the vectorial case is that \emph{minima of the partial ordering may
  not exist}, and except for some prominent (but otherwise
ill-behaved) orderings like the lexicographic ordering, this is
usually the case. By minimum with respect to the ordering $\leq_K$
over a set $S \sub \R^N$ we mean a point $\xi \in S$ satisfying
\[
\xi\, \leq_K \eta, \ \ \text{ for all }\eta \in S.
\] 
(We do not define the vectorial ``inf", but this can be done in the obvious way.) The way to overcome this difficulty is to seek instead for \emph{minimals}, usually called \emph{Pareto Minimals} (\cite{P}). A point $\xi \in S$ is a (Pareto) Minimal of the set $S\sub \R^N$ with respect to the ordering $\leq_K$ when 
\[
\text{ For any } \eta \in S \ :\   \eta\, \leq_K \xi \ \ \ \Longrightarrow \ \ \ \ \eta\, =\, \xi.
\] 
Minima and minimals coincide for global (linear) orderings, but in general \emph{they do not}. Unlike minima, minimals always exist and correspont to choices for which ``there is no better available choice", while minima, if they exist, correspond to ``the best available choice".  Once again, this distiction has no bearing in the case of linear orderings. A well known method in order to construct minimals of a partial ordering is the so-called \emph{scalarisation method} which is recalled later. Roughly, the idea of scalarisation is that 

{

%%%%%%%%%% do not erase the gaps, changes the typesetting !

\center{

\emph{a partial ordering can be recovered from a family of scalar orderings along projections on lines generated by the direction in the dual cone $K^*\sub \R^N$}.}

}
\smallskip

\noi By using scalarisation, one can construct a \emph{manifold of
  Pareto minimals} which corresponds to the manifold of ``unimprovable
choices'' and, motivated by the applications in Financial Mathematics,
is usually called \emph{the trade-off manifold} (``manifold'' here is
meant in the loose and not the strict mathematical sense, since it
may lack the usual locally euclidean structure).

In this paper we commence a program which is along the lines of the
scalar theory. Our central goal is to identify the appropriate vectorial extension of the concept of value function and derive the respective vectorial analogues of the HJ and HJB equations which are connected to the deterministic and stochastic control problems. The solutions of these PDEs would allow to construct feedback controls which optimally drive the system \eqref{1.1}.  To the best of our knowledge, this line of development via PDE theory has not been pursued before. 

Interestingly, it turns out that, via the method of scalarisation, instead of a system of HJ or HJB equations as one might expect due to the vectorial nature of the
cost, we obtain a \emph{parametric family of HJ/HJB equations, where the parameters
  $\la$ are the unit directions inside the dual cone $K^*$ relative to
  the selected partial ordering}. The respective Viscosity Solutions of these HJ/HJB equations are projections of the family of vectorial value functions $\{u^\la\}$ along directions normal to certain supporting hyperplanes. The value functions are Pareto minimals with respect to the ordering and form the \emph{trade-off manifold inside the space of maps $\, \R^n\by(0,\infty) \larrow \R^N$}. This manifold gives rise to a respective manifold of feedback
controls $\{\al^\la\}$. The study of the topological structure of
these manifolds of optimal choices seems to be an interesting topic in
itself, but will not be considered in this introductory work.

This paper is organised as follows: In Section \ref{section2} we collect basic facts about cones, orderings, minimals and viscosity solutions, including the scalarisation method and the existence of Pareto minimals in the case of (finite-dimensional) Optimisation theory. 

In Section \ref{section3} we consider the case of optimally controlling the system \eqref{1.1a} without white noise with respect to the deterministic version of the vectorial functional \eqref{1.4}. By introducing the appropriate value functions as (Pareto) minimals of the cost with respect to a fixed ordering (Definition \ref{def-PM}), we prove their existence as a consequence of the scalarisation method (Lemma \ref{lemma-PM}). Next, we derive the analogue of the Hamilton--Jacobi equation which arises in the deterministic vector case and show that appropriate projections of the value functions along lines generated by the dual cone are viscosity solutions of a family of HJ equations (Theorem \ref{theorem1} and Propositions \ref{prop:dynamic-opt-det} and \ref{prop:exist-det}) parameterised by the directions in the dual cone. These solutions allow to construct a family of feedback controls which drive the system \ref{1.1a} in an ``unimprovable" fashion (Remark \ref{feedback-controls}).


 Finally, in Section \ref{section4} we turn our attention to the problem of stochastic optimal control of \eqref{1.1} via PDE theory, and extend the results of Section \ref{section3} to the case of non-trivial white noise. The results of this section are in correspence to those of Section \ref{section3}, the main difference being that here we have a family of 2nd order Hamilton--Jacobi--Bellman equations parameterised by the directions inside the dual cone and whose solutions optimally drive the system \eqref{1.1}.

\ms

%%%%%%% leave some space between the sections, I suffocate!







\section{Cones, Generalised Ordering, Minimals and Viscosity Solutions} \label{section2}

In this section we collect some rudimentary material related to
gereralised ordering, cones, minima, minimals, scalarisation and
viscosity solutions. These notions and results we recall herein can be
found in different guises sparsely distributed inside our references (and mostly proofless).
We recall them here for the sake of completeness of the exposition and
for the convenience of the reader.

\subsection{Generalised Orderings with Respect to Cones.} Let $K\sub \R^N$ be a non-empty set. $K$ is called a \emph{cone} when 
\[
\text{$\xi \in K$ \  implies \ \ $t \xi \in K$,\ \  for any $t\geq 0$,}
\]
that is when 
\[
\text{$tK = K$, \ \ for all $t\geq 0$.}
\]
A cone $K$ is called a \emph{Proper Cone} when
\begin{itemize}
\item $K$ is a closed convex set, \ms

\item the topological interior of $K$ is non-empty: $\inter(K)\neq \emptyset$, \ms

\item $K$ contains no line: $\xi \in K$ and $-\xi \in K$ implies $\xi=0$. 

\end{itemize}

\ms

\noi Some examples of proper cones are the ones given in the introduction, that is

\begin{itemize}
\item $K=[0,\infty)$, in $\R^1$, \ms

\item $K=\mS(N)^+$,  in $\R^{N\by N}$, \ms

\item $K=\R^N_+$, in $\R^N$.

\end{itemize}

\[
\underset{\text{Figure 1.}}{\includegraphics[scale=0.2]{Figure1}}
\]

\noi However, the lexicographic cone $K_{\text{lex}}\sub \R^N$, defined by
\begin{align}
K_{\text{lex}}\, :=\ &\{0\} \bigcup \Big\{\xi \in \R^N\ \Big| \ \xi_i>0,\, \forall i\in\{1,...,N\} \Big\} \bigcup\nonumber\\
&\Big\{ \xi \in \R^N\ \Big| \ \exists k \in\{1,...,N-1\} \, :\, \xi_1=\cdots =\xi_k=0, \, \xi_{k+1}>0 \Big\}
\nonumber
\end{align}
is not a proper cone. Every proper cone $K\sub \R^N$ induces a \emph{partial ordering} ``$\geq_K$", given by
\[
\eta \, \geq_K \xi\ \ \ \Longleftrightarrow\ \ \ \eta-\xi \, \in \, K.
\] 
Obviously, $\eta \, \leq_K \xi$ means $-\eta \, \geq_K -\xi$. The respective \emph{strict ordering ``$>_K$"} is defined analogously:
\[
\eta \, >_K \xi\ \ \ \Longleftrightarrow\ \ \ \eta-\xi \, \in \, \inter(K),
\] 
but will not be used in this work. Properness of the cone implies that the relation $\geq_K\ \sub \R^N\! \by \R^N$ is actually a partial ordering compatible with the topological and linear structure of $\R^N$: 

\begin{itemize}

\item $\xi \, \leq_K \xi$,\ms

\item $\xi \, \leq_K \eta $ and $\eta \, \leq_K \zeta $ imply $\xi \, \leq_K \zeta $, \ms

\item $\xi \, \leq_K \eta $ and $\xi \, \geq_K \eta$ imply $\xi=\eta$, \ms

\item $\xi \, \leq_K \eta $ and $t\geq0$ imply $t\xi \, \leq_K t\eta $, \ms

\item $\xi_m \, \leq_K \eta_m $ and $\xi_m \ri \xi$, $\eta_m \ri \eta$ as $m\ri \infty$, imply $\xi \, \leq_K \eta $, \ms

\item $\xi' \, \leq_K \eta'$ and  $\xi'' \, \leq_K \eta''$ imply $\xi'+\xi'' \, \leq_K \eta'+\eta'' $. \ms

\end{itemize}













\subsection{Mimina, Minimals and their Geometric Interpretation.}

Let $S \sub \R^N$ be a non-empty set and suppose we are given a partial ordering ``$\leq_K$'' generated by a proper cone $K\sub \R^N$. By mimicking the scalar case, one may define the minimum of $S$ with respect to the ordering ``$\leq_K$''  as a point $\xi \in S$ such that
\[
\xi \, =\, \min \, S \ \ \ \Longleftrightarrow  \ \ \  \xi\, \leq_K \eta, \ \ \text{ for all }\eta \in S.
\] 
One may also define the infimum of the set $S$ as the minimum of the closure $\overline{S}$ of $S$, that is as the point $\xi\in \R^N$ such that
\[
\xi \, =\, \inf \, S \ \ \ \Longleftrightarrow  \ \ \  \xi \, =\, \min \, \overline{S}.
\] 
The minimum, if is exists, it is unique. In a more compact form, its definition reads
\[
\xi \, =\, \min \, S \ \ \ \Longleftrightarrow  \ \ \  S\, \sub \, \xi\, +\, K,
\]
that is, $\xi$ is the minimum of $S$ if and only if it is contained in the translate of the cone $K$ with vertex at $\xi$.
\[
\underset{\text{Figure 2.}}{\includegraphics[scale=0.2]{Figure2}}
\]
The basic problem for the notion of minimum is that in general does not exist since only sets with very special structure possess it. The way to overcome this difficulty is to seek instead for \emph{(Pareto) Minimals}. A point $\xi \in S$ is a (Pareto) Minimal of the set $S\sub \R^N$ with respect to the ordering $\leq_K$ when it satisfies
\[
\left.
\begin{array}{c}
\eta \in S \text{ and }\\
\eta\, \leq_K \xi
\end{array}
\right\}
\ \ \ \Longrightarrow \ \ \ \ \eta\, =\, \xi.
\] 
Minima and minimals coincide for global (linear) orderings, but in general \emph{they do not}. The geometric characterisation of minimals is 
\[
(\xi\,-\,K) \cap S\, =\, \{\xi\},
\]
that is, $\xi$ is a minimal of $S$ if and only if the reflected translated cone $\xi-K$ with vertex at $\xi$ intersects $S$ only at $\xi$. Obviously one can more generally define the minimal of a set $S$ as a point $\xi$ \emph{not necessarily contained in $S$} by considering the closure $\overline{S}$ in the place of $S$, but we will not go into that.

\[
\underset{\text{Figure 3.}}{\includegraphics[scale=0.2]{Figure3}}
\]
We will shortly see that any closed set $S$ possesses at least one minimal element with respect to an ordering generated by a proper cone.



\subsection{Dual Cones and Dual Inequalities.}  A central concept in this context is duality. Given a cone $K\sub \R^N$, we define its dual cone as
\[
K^*\, :=\, \Big\{ \eta \in \R^N\ \, \big|\, \ \eta\cdot \xi \,\geq\, 0, \ \forall\, \xi \in K   \Big\}.
\]
Geometrically, $\eta \in K^*$ if and only if $\eta$ is the inwards
pointing vector to a halfspace supporting $K$ at the origin (see
Figure 4). As usual, dual objects satisfy better properties than the
objects themselves. A cone is called \emph{self-dual} if it coincides
with its dual $K=K^*$. Simple properties of dual cones are

\begin{itemize}
\item $K^*$ is closed and convex (although $K$ might not be), \ms

\item $K'\sub K''$ implies ${K''}^*\sub {K'}^*$, \ms

\item if $\inter(K) \neq \emptyset$, then $K^*$ contains no non-trivial line,\ms

\item $K^{**}$ coincides with the closed convex hull of $K$:
\[
K^{**}\, =\, \co\,(K)
\]
\end{itemize}
The standard examples of cones given in the introduction are proper and self-dual:
\begin{itemize}
\item $[0,\infty)\, =\, \left( [0,\infty) \right)^*$, in $\R^1$, \ms

\item $\mS(N)^+ \, =\, \left( \mS(N)^+ \right)^*$,  in $\R^{N\by N}$, \ms

\item $\R^N_+ \, =\, \left( \R^N_+ \right)^*$, in $\R^N$.\ms

\end{itemize}
\[
\underset{\text{Figure 4.}}{\includegraphics[scale=0.19]{Figure4}}
\]
The only less trivial equality is the middle one, and follows by observing that $A\geq 0$ in $\mS(N)$ if and only $A:B:=A_{ij}B_{ij}\geq 0$ for all $B\in \mS(N)^+$.

The dual cone induces a partial ordering itself on $\R^N$, in general different from the ordering induced by $K$, defined as
\[
\xi \, \geq_{K^*} \eta \ \ \ \Longleftrightarrow \ \ \ \xi\,-\,\eta \, \in \, K^*.
\]
One of the main utilities of the dual objects is that they \emph{allow to characterise the ordering via a family of ordinary scalar orderings with respect to projections on the lines generated by directions in the dual cones}. Accordingly, we have
\begin{lemma}[Orderings via duality] \label{lemma1} Let $K\sub \R^N$ be a proper cone and $K^*$ its dual. Then, for any $\xi,\eta \in \R^N$, we have the equivalence
\[
\xi \, \geq_{K} \eta \ \ \ \Longleftrightarrow \ \ \ \la\cdot \xi \, \geq \, \la\cdot\eta, \ \text{ for all }\, \la  \in  K^*.
\]
\end{lemma}

\BPL \ref{lemma1}. We may assume $N\geq 2$, since the case $N=1$ is trivial. Fix $\xi,\eta \in \R^N$ and suppose first that $\xi \, \geq_{K} \eta$. By definition, this means $\xi-\eta \in K$. Hence, by definition of $K^*$, for any $\la \in K^*$ we have $\la\cdot (\xi -\eta) \geq 0$, which is what we want. 

Conversely, suppose that for any $\la \in K^*$ we have $\la\cdot (\xi -\eta) \geq 0$. For the sake of contradiction assume that $\xi-\eta \not\in K$. Since $K$ is a convex set, the projection  on $K$
\[
\text{Proj}_K\ :\ \R^N \larrow K,
\]
is uniquely defined. Since $\xi-\eta\not\in K$, we have $\text{Proj}_K(\xi-\eta) \neq \xi-\eta $ and hence we may consider the 2-dimensional plane $\Pi$ passing through the origin and the points $\xi-\eta$ and $\text{Proj}_K(\xi-\eta)$. Consider now the orthogonal matrix $O \in \text{O}(N,\R)$ which leaves the $N\!-\!2$-dimensional orthogonal complement of $\Pi$ invariant and coincides with the clockwise rotation by $\pi/2$ on $\Pi$ with respect to the orientation generated by the frame $\{\text{Proj}_K(\xi-\eta),\, \xi-\eta\}$ (See Figure 5). Then, we have
\[
\la_0\, :=\, O\, \text{Proj}_K(\xi-\eta) \  \in \, K^*,
\]
since the orthogonal complement of $\la_0$ in $\R^N$ is a hyperplane which supports $K$ at the origin. However, we have $\la_0 \cdot (\xi-\eta)<0$ because the angle between $\xi-\eta$ and $\la_0$ is greater than $\pi/2$, which contradicts our assumption. Hence,  we obtain $\xi-\eta\in K$, or equivalently $\xi \, \geq_{K} \eta$.      \qed
\[
\underset{\text{Figure 5.}}{\includegraphics[scale=0.19]{Figure5}}
\]



\subsection{The Method of Scalarisation and Existence of Minimals.} Using the duality result of Lemma \ref{lemma1} above, we now derive characterisations of minima and minimals of a set with respect to an ordering generated by a proper cone.

\bl[Scalarisation of Minima and Minimals] \label{lemma2} Let $S\sub \R^N$ be a non-empty closed set and let $K\sub \R^N$ be a proper cone with $K^*$ its dual cone. 

Then, 

\begin{enumerate}
\item (Minima) $\xi$ is the minimum of $S$ with respect to the ordering $\leq_K$ if and only if for all $\la \in K^*$, $\xi$ is the minimum of all the linear scalar functions 
\[
\eta \mapsto \la\cdot \eta\ :\ \ \R^N \larrow \R, 
\] 
over the set $S$.

\item (Minimals)
\begin{itemize}
\item If for some $\la \in \inter(K^*)$, $\xi$ is the minimum of the linear scalar function
\[
\eta \mapsto \la\cdot \eta\ :\ \ \R^N \larrow \R, 
\] 
over the set $S$, then $\xi$ is a minimal of $S$ with respect to the ordering $\leq_K$. 

\item Conversely, if in addition the set $S$ is convex ($S=\text{co}(S)$) and $\xi$ is a minimal element of $S$, then for any $\la \in K^*$, $\xi$ is the minimum of the linear scalar function
\[
\eta \mapsto \la\cdot \eta\ :\ \ \R^N \larrow \R, 
\] 
over the set $S$.

\end{itemize}
\end{enumerate}
\el

\begin{remark} In view of $(1)$ above, it follows that minima of a set in general do not exist due to \emph{obstructions} which can be rephrased as the requirement to have simultaneous minimisation of a family of linear function and the minimum being realised at the same point for all the functions. On the other hand, $(2)$ says that for every direction strictly inside the dual cone, minimising the projection along this line leads to a minimal point for the set. The converse however to this statement is true in a weaker form and convexity plays a crucial role to that.
\end{remark}

Lemma \ref{lemma2} leads immediately to the following important consequence:

\bcor[Existence of Minimals] \label{corollary1} Let $K\sub \R^N$ be a proper cone, $K^*$ its dual cone and $\leq_K$ the ordering generated by $K$. Let also $S$ be a compact non-empty set. 

\ms

Then, there exists at least one minimal element $\xi \in S$ of the set $S$ with respect to the ordering $\leq_K$. 

\ms

Moreover, for any $\la \in \inter(K^*)$, consider the supporting hyperplane of $S$ which is normal to $\la$ and such that $\la$ points inside the halfspace which contains $S$. Then, the touching point belongs to the set of minimals of $S$.
\ecor
\[
\underset{\text{Figure 6.}}{\includegraphics[scale=0.19]{Figure6}}
\]
It follows from the corollary that only the only the contact points of $S$ with its convex hull $\co(S)$ are ``attainable" candidate minimals by the scalarisation method, namely the set of points $S \cap \p (\co(S))$. 
\[
\underset{\text{Figure 7. $\xi$ is a minimal with respect to ``$\leq_{\R^2_+}$" in $\R^2$, but not attainable via scalarisation.}}{\includegraphics[scale=0.19]{Figure7}}
\]

\BPL \ref{lemma2}. We prove only the first statement of $(2)$, which is the only one we will use in the sequel. The proof of the rest claims can be found e.g. in \cite{BV}, page 54. We fix $\la \in K^*$ and suppose that $\xi$ is the minimum of the linear functional $\eta \mapsto \la \cdot \eta$ over $S$. We claim that $\{\xi\}=(\xi-K)\cap S$, which is equivalent to the statement that $\xi$ is a minimal of $S$. If this is not the case, then there is an $\eta \in S$ with $\eta \neq \xi$ such that $\eta \in (\xi-K)\cap S$. Since $\eta \in \xi-K$ and $\xi \neq \eta$, we have that $\xi-\eta \in K\set\{0\}$. Since $\la \in \inter(K^*)$, the definition of the dual cone implies that $\la\cdot(\xi-\eta)< 0$ (because $\la$ is in the interior and can not be normal to $\xi-\eta$), which is a contradition. The claim ensues.    \qed

\subsection{Viscosity Solutions of Hamilton-Jacobi-Bellman Equations.} Here we recall the definition of appropriate ``weak" solutions for fully nonlinear (1st and) 2nd order degenerate elliptic (and parabolic) PDE. The notion is standard and can be found in many sources, e.g.\ in the standard reference \cite{CIL}. However, the version below is taken from the introductory text \cite{K}. The main difference is that degenerate ellipticity is assumed as monotonicity, instead of anti-monotonicity. Let 
\[
\cF \ :\  (\Om \sub \R^n) \by \R\by \R^n \by \mS(n)\larrow \R
\]
be a continuous function and consider the PDE
\[
\cF(\cdot,u,Du,D^2u)\, =\, 0, \ \ \ u\,:\,\Om\sub \R^n\larrow\R.
\] 
We assume that $F$ satisfies
\[
X\, \leq \,Y\ \ \ \Longrightarrow \ \ \ \cF(x,r,p,X)\,\leq\,\cF(x,r,p,Y),
\]
for all $(x,r,p)\in \Om \by \R\by \R^n$, $X,Y\in \mS(n)$.

\noi \begin{definition}[Viscosity Solutions] \label{def:visc-sol} Let $u\in C^0(\Om)$, $\Om\sub \R^n$, and consider the (degenerate elliptic) PDE 
\[
\cF\big(\cdot,u,Du,D^2u\big)\, =\, 0.
\]
(a) We say that $u$ is a Viscosity Subsolution of the PDE (or a Viscosity Solution of $\cF\big(\cdot,u,Du,D^2u\big)\geq 0$) when
\[ 
\left.
\begin{array}{c}
u-\psi \leq 0 =(u-\psi)(x_0)\\
 \text{on a ball }\mB_r(x_0) \sub \Om,\\
x_0 \in \Om,\ \psi \in C^2(\R^n)
\end{array}
\right\}
 \ \ \Longrightarrow \ \ \cF\big(x_0,\psi (x_0),D\psi(x_0),D^2\psi(x_0)\big) \geq 0.
\]
(b) We say that $u$ is a Viscosity Supersolution of the PDE (or a Viscosity Solution of $\cF\big(\cdot,u,Du,D^2u\big)\leq 0$) when
\[
\left.
\begin{array}{c}
u-\phi \geq 0 =(u-\phi)(y_0)\\
 \text{on a ball }\mB_r(y_0) \sub \Om,\\
y_0 \in \Om,\ \phi \in C^2(\R^n)
\end{array}
\right\}
 \ \ \Longrightarrow \ \ \cF\big(y_0,\phi (y_0),D\phi(y_0),D^2\phi(y_0)\big) \leq 0.
\]
(c) We say that $u$ is a Viscosity Solution, when it is both a Viscosity Subsolution and a Viscosity Supersolution.
\end{definition}

\ms












\section{Deterministic Optimal Control and Hamilton-Jacobi PDE} \label{section3}

In this section we consider the first main theme of this paper, namely
the deterministic optimal control of the initial value problem
\beq \label{3.1} \left\{
\begin{array}{l}
\dot{\chi}(s)\, = \, F\big(\chi(s),\al(s) \big), \ \ \ t<s<T,\ms\\
\chi(t)\, =\, x,
\end{array}
\right.
\eeq
where $x\in \R^n$, $A\sub \R^m$ is a compact set and $F: \R^n \by A \larrow \R^n$ a continuous map. $\al$ is a measurable map is the class $\mA$ given by \eqref{1.2}. Optimal controllability is meant with respect to the vectorial cost functional
\beq \label{3.2}
C_{x,t}[\al]\ =\ g\Big( \chi\big(T,x,\al(T) \big)\Big)\ +\ \int_t^T h\Big(\chi \big(s,x,\al(s) \big),\al(s)\Big)\, \d s
\eeq
where $h,\, g$ are given maps and  $\chi$ denotes the flow map of \eqref{3.1}, which we may abbreviate to merely $\chi(s)$, suppressing the dependence in $x$ and $\al$. We assume that $F,g,h$ satisfy the assumption \eqref{1.6}. 

\ms

We begin with the definition of the appropriate vectorial minimals we will use as our vectorial extension of the value function.

\bd[Pareto Minimals of the Vectorial Cost] \label{def-PM} Let $A\sub \R^m$ be a compact set, $\mA$ the class given by \eqref{1.2} and $F,g,h$ given maps which satisfy the assumption \eqref{1.6} with $n,m,N\geq 1$. 

Consider the initial value problem \eqref{3.1} and the vectorial cost functional \eqref{3.2}. Suppose further that we are given a partial ordering $\leq_K$ on $\R^N$ generated by a proper cone $K\sub \R^N$.

We say that the map $u:\R^n\by(0,T) \larrow \R^N$ is a vectorial value map (or a Pareto Minimal) of the cost \eqref{3.2} when for any $(x,t)\in \R^n\by(0,T)$, $u(x,t)$ is a Pareto Minimal of the set 
\[
S_{x,t}\, :=\, \overline{\big\{C_{x,t}[\al]\ |\ \al \in \mA\big\}}\, \sub \R^N, 
\]
that is, when 
\[
\text{$u(x,t) \in S_{x,t}$ and for all $\eta \in S_{x,t}$ for which $\eta\leq_Ku(x,t)$, we have $\eta=u(x,t)$.}
\]
 
\ed


By using the results of the previous section, in particular Corollary \ref{corollary1} and Lemma \ref{lemma2}, we readily have the next

\begin{lemma}[Existence of Pareto Minimals of the Vectorial Cost]  \label{lemma-PM}
In the setting of Definition \ref{def-PM}, there exists at least one value maps which is a Pareto Minimals of the vectorial cost. In addition, any direction $\la \in K^*$ generates a Pareto Minimal by considering the supporting hyperplane (of the convex hull of) $S_{x,t}$ which is normal to $\la$.
\end{lemma}

\begin{remark} \label{rem-PM} We note that in general, not all the Pareto minimals of the ordering can be realised via scalarisation, but instead only those on the ``extreme points" of the convex hull (see Section \ref{section2}). 
\end{remark}

We now have the next result, regarding the vector value functions:  

\begin{proposition}[Existence and properties of the vectorial value function]
  \label{prop:exist-det}
  Given a partial ordering $\leq_K$ on $\R^N$ generated by a proper
  cone $K\sub \R^N$, let $K^*$ be its dual cone (see Section
  \ref{section2}). Then, for any $\la\in K^*$ there exists a
  vectorial \emph{value function}
  \begin{equation}
    u^\la \ :\ \R^n\by (0,T)\larrow \R^N
  \end{equation}
  which is a Pareto Minimal of the functional (\ref{3.2})
  with respect to the ordering $\leq_K$ and also satisifes
  \begin{equation}
    \label{eq:stoch-oc}
    \la \cdot u^\la(x,t)\ =\ \inf_{\al \in \mA} \big\{\la\cdot C_{x,t}[\al]\big\} , \ \ \ x\in \R^n,\, 0<t<T.
  \end{equation}
  The scalar function $\la \cdot u^\la$ is the projection of the vectorial value map $u^\la$ along the direction generated by a supporting hyperplane of $S_{x,t}$ normal to $\la$. In addition,  $\la \cdot u^\la$ is bounded and Lipschitz continuous in both variables.
\end{proposition}

\BPP \ref{prop:exist-det}. The proof is a direct consequence of Definition \ref{def-PM}, Remark \ref{rem-PM}, the scalarisation method of Section \ref{section2} and an application of the
corresponding scalar result given in \cite[Section 10.3.2]{E}.     \qed


We now state a proposition whose proof is straightforward extension of the
corresponding scalar result given in \cite[Section 10.3.2]{E} and is based on our previous analysis.

\begin{proposition}[Deterministic dynamic optimality]
  \label{prop:dynamic-opt-det}
  Let $A\sub \R^m$ be a compact set, $\mA$ the class given by
  \eqref{1.2} and $F,g,h$ are given maps which satisfy the assumption
  \eqref{1.6} with $n,m,N\geq 1$.  Then, for any $\de>0$ such that $t+\de \leq
  T$, we have
  \begin{equation}
    \la \cdot u^\la(x,t)\, =\, \inf_{\al\in A} \bigg\{ 
    \la\cdot u^\la\big(\chi(t+\de), t+\de\big) \, +\ \int_t^{t+\de} \la \cdot h(\chi(s),\al(s))\d s
    \bigg\},
  \end{equation}
  where $\chi$ is the solution of the differential equation
  (\ref{3.1}). 
\end{proposition}

The above result is a consequence of the dynamic programming principle. Roughly speaking, it states that optimal cost through the entire time interval $[t,T]$ can be achieved by running optimally in $[t,t+\de]$ and  then restarting the problem at time $t+\de$ with initial conditions
  $\chi(t+\de)$. 

\ms

We now come to the main result of this section, which is the following:

\bt[Vectorial Optimal Control, Pareto Minimals, Viscosity Solutions of HJ PDE]  \label{theorem1}

Let the conditions of Propositions \ref{prop:exist-det} and
\ref{prop:dynamic-opt-det} hold. Consider the
initial value problem \eqref{3.1} and the vectorial cost functional
\eqref{3.2}. 

Then, for any unit direction $\la \in K^*$, the projection $\la \cdot
u^\la(x,t)$ is a scalar function and a Viscosity Solution of the
initial value problem
\[
\left\{
\begin{array}{r}
v_t\ +\ H^\la(\cdot,Dv)\, =\, 0, \ \ \ \ \ \text{ in }\R^n\by (0,T),\ms\\
v\, =\, \la\cdot g,\ \text{ on }\R^n\by \{0\}.\ \ 
\end{array}
\right.
\]
Here the Hamiltonian $H^\la$ is defined by
\[
H^\la(x,p)\, := \, \min_{a\in A}\big\{F(x,a)\cdot p\, +\, \la\cdot h(x,a) \big\}.
\]

\et

\BPT \ref{theorem1}. The proof is a direct consequence of the definition of Minimals, the results of duality/scalarisation of Section \ref{section2}, Propositions \ref{prop:exist-det} and \ref{prop:dynamic-opt-det} and the standard scalar control theory (see e.g. Evans \cite{E}, pages 550-560). It is also a special case of the proof of Theorem \ref{thm:hjb} which will be proved is some detail in the subsequent section in the more general stochastic case with non-trivial noise.    \qed

\ms

\begin{remark}[Construction of feedback controls in the deterministic vector case]  \label{feedback-controls}
The family of Hamilton-Jacobi PDEs above together with their respective solutions $u^\la$ can be utilised to construct (via the method of scalarisation) for each unit direction $\la\in K^*$ a feedback control $\al^\la$ which optimally drives the dynamics of the flow generated by the  ODE system and arises as a Pareto Minimal of the vectorial cost. We refrain from giving all the techincal details, but the idea is this: at differentiability points, $\al^\la$ is defined by selecting for each $t<s<T$ the $\al(s)$ which realises the minimum:
\begin{align}
F\big(\chi^\la(s),\al^\la(s)\big) \cdot Du^\la \big(s,\al^\la(s)\big)\ \ +\ \  &\la \cdot h\big(\chi^\la (s),\al^\la (s)\big) \nonumber\\
 &= \ H^\la \Big( \chi^\la (s), Du^\la \big(\chi^\la (s),\al^\la (s)\big) \Big). \nonumber
\end{align}
In the above equality, $\chi^\la (s)$ denotes the flow map $\chi \big(s,x,\al^\la (s) \big)$.
\end{remark}


\ms 









\section{Stochastic Optimal Control and Hamilton-Jacobi-Bellman PDE} \label{section4}

In this section we extend the ideas of Section \ref{section3} to the
stochastic case. For the rudimentary facts of the theory of SDEs needed in this paper we refer to Evans \cite{E2}. We consider now the case of optimal control of the system of stochastic
differential equation
\beq
   \label{eq:sde}
  \left\{
\begin{array}{l}
    \d {\chi}(s) \,= \, F\big(\chi(s),\al(s)\big) \d s\ +\ \sigma\big(\chi(s),\al(s)\big) \d W(s), \quad t<s<T,\ms
    \\
 \ \ \   \chi(t) \,=\, x,
  \end{array}
\right.  \eeq where we use the same notation as in Section
\ref{section3}. The extra ingredients now are $W(s)$ which is an $n$
dimensional Wiener process over a stochastic basis and $\sigma:\R^n
\times A \larrow \R^n$ which is a continuous map. In the stochastic
case, the flow map $\chi$ has to be interpreted as a stochastic
process. This process is defined over a \emph{probablility space}
which is a triple $(\R^n, \cA, P)$, with $\cA$ a $\sigma$--algebra of
subsets of $\R^n$ and $P$ a probability measure over $\R^n$.

The stochastic process $\chi$ is said to solve the stochastic
differential equation (\ref{eq:sde}) if 
\begin{equation}
  \chi(s)\ =\ x \, + \int_t^s F\big(\chi(r),\al(r)\big) \d r \ + \int_t^s \sigma\big(\chi(r),\al(r)) \d W(r),
\end{equation}
interpreted as an It$\hat{\textrm{o}}$ integral, holds almost surely. 

The stochastic version of the vectorial cost functional has the same
form as the deterministic, however is given in terms of an expectation
\begin{equation}
  \label{eq:stoch-cost}
  C_{x,t}[\al] \ =\ \Exp\qb{ g\Big( \chi\big(T,x,\al(T) \big)\Big)\ +\, \int_t^T h\Big(\chi \big(s,x,\al(s) \big),\al(s)\Big)\, ds},
\end{equation}
which is defined as% with respect to $P$
\begin{equation}
  \Exp\qb{X} := \int_{\R^n} X \d P.
\end{equation}

The following two Propositions are the stochastic equivalents to
Propositions \ref{prop:exist-det} and \ref{prop:dynamic-opt-det}:

\begin{proposition}[Existence and properties of the vectorial value function]
  \label{prop:exist-sto}
  Given a partial ordering $\leq_K$ on $\R^N$ generated by a proper
  cone $K\sub \R^N$, let $K^*$ be its dual cone (see Section
  \ref{section2}). Then, for any $\la\in K^*$ there exists a
  vectorial \emph{value function}
  \[
    u^\la \ :\ \R^n\by (0,T)\larrow \R^N
  \]
  which is a Pareto Minimal of the functional (\ref{eq:stoch-cost})
  with respect to the ordering $\leq_K$ and also satisifes
  \begin{equation}
    \label{eq:stoch-oc}
    \la \cdot u^\la(x,t)\ =\ \inf_{\al \in \mA} \big\{\la\cdot C_{x,t}[\al]\big\} , \ \ \ x\in \R^n,\, 0<t<T.
  \end{equation}
  In addition, the value function is bounded and Lipschitz continuous in both variables.
\end{proposition}

\begin{proposition}[Stochastic dynamic optimality]
  \label{prop:dynamic-opt-sto}
  Let $A\sub \R^m$ be a compact set, $\mA$ the class given by
  \eqref{1.2} and $F,g,h, \sigma$ given maps which satisfy the
  assumption \eqref{1.6} with $n,m,N\geq 1$. 
  A consequence of the dynamic programming principle is that for any
  $\de > 0$ such that $t + \de \leq T$, we have
  \[
    \la \cdot u^\la(x,t) \ = \ \inf_{\al\in A} \bigg\{
    \Exp
    \qb{
      \la\cdot u^\la(\chi(t+\de), t+\de) \ + \int_t^{t+\de} \la \cdot h(\chi(s),\al(s))\d s
    }
    \bigg\},
  \]
  where $\chi$ is the solution to the system of stochastic differential equations \eqref{eq:sde}. 
\end{proposition}

The main result of this section is:

\begin{theorem}[Viscosity Solutions of HJB PDE]
  \label{thm:hjb}
  Consider the stochastic initial value problem (\ref{eq:sde})
 and the vectorial cost functional (\ref{eq:stoch-cost}).
  For each $\la \in K^*$, the projection $\la \cdot u^\la(x,t)$ is a
  scalar function and a Viscosity Solution of the initial value
  problem
  \begin{equation}
    \label{eq:HJB}
    \left\{
    \begin{array}{r}
      v_t + H^\la(\cdot,Dv,D^2 v) = 0, \qquad \ \text{  in }\R^n\by (0,T),\ms
      \\
      v = \la\cdot g, \quad \text{ on }\R^n\by \{0\},\ \ 
    \end{array}
    \right.
  \end{equation}
  where the Hamiltonian $H^\la$ is defined by
  \begin{equation}
    H^\la(x,p,Z)\, := \ \min_{a\in A}\left\{ \frac{1}{2}\sigma(x,a)\otimes \sigma(x,a) : Z \,+ \, F(x,a)\cdot p\, +\, \la\cdot h(x,a) \right\}.
  \end{equation}
\end{theorem}

\begin{proof}
  The proof of this result is just an extension of \cite[p.557 Thm
  2]{E} to the stochastic vectorial case. We include it here for
  completeness. Throughout this proof for convenience we will denote $u
  := \la\cdot u^\la$. We begin by noting that \eqref{eq:stoch-oc} implies 
  \begin{equation}
    u(x,T) \ =\ \inf_{a\in A} \big\{\Exp 
    [ \la \cdot g(\chi(T))] \big\}
    \ =\ 
    \la\cdot g(x).
  \end{equation}
  To show that $u$ is the viscosity solution of (\ref{eq:HJB}) we must
  verify the conditions given in Defintion \ref{def:visc-sol}. To that
  end, without loss of generality let $\psi\in\cont{2}(\R^n
  \times (0,T))$ and assume there exist $(x_0, t_0)$ such that 
  \begin{equation}
    \label{eq:ass2}
    u(x,t) - v(x,t) \leq u(x_0, t_0) - v(x_0,t_0),\text{ when } \norm{x - x_0} + \norm{t - t_0} \leq \epsilon.
  \end{equation}
  We now want to show that
  \begin{equation}
    \begin{split}
      v_t(x_0,t_0)
      \,+\,
      H^\la(x_0, D v, D^2 v)
      \,\geq\, 0.
    \end{split}
  \end{equation}
  Assume for the sake of contradiction that there exists an $a\in A$ such that
  \begin{equation}
    \label{eq:ass}
    \begin{split}
      \cL v(x_0, t_0) 
      &\ :=\ 
     v_t(x_0,t_0)
      \,+\,
      D v(x_0,t_0) \cdot F(x_0,a) 
      \\
      &\qquad +
      \frac{1}{2} D^2 v(x_0,t_0) : \big( \sigma(x_0,a) \otimes \sigma(x_0,a) \big) \,+\, \la \cdot h(x_0,a)
      \ <\ 0
    \end{split}
  \end{equation}
  for $\norm{x - x_0} + \norm{t - t_0} \leq \epsilon$. Let $\chi(s)$ be
  the solution of the stochastic differential equation
  \begin{equation}
    \begin{split}
      \d {\chi}(s) &\ =\  F\big(\chi(s),a \big) \d s \ +\ \sigma\big(\chi(s),a \big) \d W(s), \quad t<s<T,\ms
      \\
      \chi(t_0) &\ =\ x_0,
    \end{split}
  \end{equation}
  where $a$ is now taken to be a contant control. Now choose $\delta\in (0,\epsilon)$ such that
  \begin{equation}
    \norm{\chi(s) - x_0}\, \leq \, \epsilon, \text{ \ \ when\  } s\in [t_0,t_0 + \delta],
  \end{equation}
  then by (\ref{eq:ass}) we know
  \begin{equation}
    \cL v(\chi(s), s)\, <\, 0, \text{\ \ for } s\in  [t_0,t_0 + \delta].
  \end{equation}
  Using (\ref{eq:ass2}) we have that
  \begin{equation}
    \label{eq:stoch-proof-1}
    \begin{split}
      u\big(\chi(t_0+\de), t_0+\de\big) \,- \, u(x_0, t_0)
      &\ \leq \ v\big(\chi(t_0+\de),t_0+\de\big) \, -\, v(x_0,t_0)
      \\
      &\ \leq \ \int_{t_0}^{t_0+\de} \d v(\chi(s),s).
    \end{split}
  \end{equation}
  From Proposition \ref{prop:dynamic-opt-sto}, we have
  \begin{equation}
    \label{eq:stoch-proof-2}
    u(x_0,t_0)  \ \leq \ \Exp \qb{
      u(\chi(t_0+\de), t_0+\de) \,+ \, \int_{t_0}^{t_0+\de} \la \cdot h(\chi(s),a)\d s
    }.
  \end{equation}
  Combining (\ref{eq:stoch-proof-1}) and (\ref{eq:stoch-proof-2}) and
  dividing through by $\de$ we see  
  \begin{equation}
    \label{eq:stoch-proof-3}
    \begin{split}
    0 \
    &\ \leq
    \frac{1}{\de}
    \Exp 
    \qb{
      \int_{t_0}^{t_0+\de} \d v(\chi(s),s)
    }
    \ + \
    \frac{1}{\de}
    \Exp
    \qb{
      \int_{t_0}^{t_0+\de} \la \cdot h(\chi(s),a)\d s
    }
    \\
    &=: \
    \cE_1(\de) \ +\ \cE_2(\de).
    \end{split}
  \end{equation}
  The first term, in view of the It$\hat{\textrm{o}}$ chain rule is
  \begin{equation}
    \begin{split}
      \cE_1(\de) &\ =\ 
      \frac{1}{\de}
      \Exp 
      \qb{
        \int_{t_0}^{t_0+\de} \d \big( v(\chi(s),s) \big)
      }
      \\
      &=\ 
      \frac{1}{\de}
      \Exp 
      \bigg[
        \int_{t_0}^{t_0+\de} v_t(\chi(s),s) \d s 
        +
        D v(\chi(s),s) \cdot \d \chi(s)
        \\
        &\qquad\qquad\qquad\qquad +
        \frac{1}{2} D^2 v(\chi(s),s) : \big( \d \chi(s) \otimes \d \chi(s) \big)
      \bigg]
      \\
      &=\ 
      \frac{1}{\de}
      \Exp 
      \bigg[
        \int_{t_0}^{t_0+\de} v_t(\chi(s),s) 
        +
        D v(\chi(s),s) \cdot F(\chi(s),s) 
        \\
        &\qquad\qquad\qquad\qquad +
        \frac{1}{2} D^2 v(\chi(s),s) : \big( \sigma(\chi(s),a) \otimes \sigma(\chi(s),a) \big)
        \d s 
        \\
        &\qquad\qquad\qquad\qquad +
        D v(\chi(s),s) \cdot \sigma(\chi(s),a) \d W(s)
        \bigg].
    \end{split}
  \end{equation}
  Now using Fubini's theorem we have
  \beq 
    \label{eq:stoch-proof-4}
  \begin{array} {r} 
   \underset{\de\ri 0}\lim\,
    \cE_1(\de)
   \ =\ 
      v_t(x_0,t_0)
    \ +\ 
    D v(x_0,t_0) \cdot F(x_0,a) 
   \qquad \ms\\
    +\ \dfrac{1}{2} D^2 v(x_0,t_0) : \big( \sigma(x_0,a) \otimes \sigma(x_0,a) \big).
  \end{array}
  \eeq
  
 The second term, again by Fubini's theorem, gives
  \begin{equation}
    \label{eq:stoch-proof-5}
    \begin{split}
      \lim_{\de\ri 0}
      \cE_2(\de)
      &\ =\ 
      \lim_{\de\ri 0}
      \frac{1}{\de}
      \Exp
      \qb{
        \int_{t_0}^{t_0+\de} \la \cdot h(\chi(s),a)\d s
      }
      \\
      &\ =\ 
      \lambda \cdot h(x_0,a).
    \end{split}
  \end{equation}
  Substituting (\ref{eq:stoch-proof-4}) and (\ref{eq:stoch-proof-5})
  into (\ref{eq:stoch-proof-3}), we have
  \begin{equation}
    0 \, \leq \, \cL v(x_0, t_0),
  \end{equation}
  which contradicts (\ref{eq:ass}). Consequently, $u$ is a Viscosity
  Subsolution of (\ref{eq:HJB}). In a similar fashion one can show
  that $u$ is a Viscosity Supersolution to (\ref{eq:HJB}), hence $u$
  is a Viscosity Solution of (\ref{eq:HJB}). 
\end{proof}


\noi \textbf{Acknowledgment.} The authors are indebted to E. Zuazua and A. Porretta for their share of expertise on the status of the problem.
\ms

\bibliographystyle{amsplain}

\begin{thebibliography}{10}

\bibitem[BCD]{BCD} M. Bardi, I. Capuzzo-Dolcetta, \emph{Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations}, Modern Birkh\"auser Classics, 2008.

\bibitem[BF]{BF} R. Bellman, K. Fan, \emph{On systems of linear inequalities in Hermitian matrix
variables}, in: V. L. Klee (editor), \emph{Convexity}, vol. VII, Proceedings of the
Symposia in Pure Mathematics, Amer. Math. Soc. 1963, 1 � 11.

\bibitem[BV]{BV} S. Boyd, L.Vandenberghe, \emph{Convex Optimization}, Cambridge University Press, 7th printing, 2009.

\bibitem[BKR]{BKR} R. S. Burachik, C. Y. Kaya, M. M. Rizvi, \emph{A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets}, Journal of Optimization Theory and Applications 162 (2014), 428--446.

\bibitem[CIL]{CIL} M. G. Crandall, H. Ishii, P.L. Lions, \emph{User's Guide to Viscosity Solutions of 2nd Order Partial Differential Equations}, Bulletin of the AMS 27  (1992), 1 - 67.

\bibitem[CHY]{CHY} G.Chen, X. Huang, X. Yang, \emph{Vector Optimization: Set-valued and Variational Analysis}, Lecture Notes in Economics and Mathematical Systems, Springer, 2005.

\bibitem[DD]{DD} I. Das, J.E. Dennis, \emph{Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems}, SIAM J. Optim., 8 (2006), 631-�657.

\bibitem[D]{D} G. Debreu, \emph{Theory of Value: An Axiomatic Analysis of Economic Equilibrium}, Yale University Press, 1959.

\bibitem[E]{E} L.C. Evans, \emph{Partial differential equations}, AMS, Graduate Studies in Mathematics Vol. 19, 1998.

\bibitem[E2]{E2} L.C. Evans, \emph{An Introduction to Stochastic Differential Equations},
AMS pulications, ISBN-13: 978-1-4704-1054-4, 2013.

\bibitem[FR]{FR} W. H. Fleming, R. W. Rishel, \emph{Deterministic and Stochastic Optimal Control}, Stochastic Modelling and Applied Probability, Springer, corrected 2nd Printing, 1983. 

\bibitem[FS]{FS} W. H. Fleming, H. M. Soner, \emph{Controlled Markov Processes and Viscosity 
Solutions}, Springer, Stochastic Modelling and Applied Probability, Springer, 2nd edition, 2010.

\bibitem[GC]{GC} D. Ghosh, D. Chakraborty, \emph{A new Pareto set generating method for multi-criteria optimization problems}, Operations Research Letters
Vol. 42 (2014), 514-�521.

\bibitem[H]{H} M. I. Henig, \emph{Vector-valued dynamic programming}, SIAM J. Control and Optimization, Vol. 21, No. 3 (1983) 490 - 499.

\bibitem[I]{I} K. Isii, \emph{Inequalities of the types of Chebyshev and Cram\'er-Rao and mathematical programming}, Annals of The Institute of Statistical Mathematics 16 (1964), 277 - 293.

\bibitem[K]{K} N. Katzourakis, \emph{An introduction to Viscosity Solutions for fully nonlinear PDE with applications to Calculus of Variations in $L^\infty$}, Springer Brief monograph, BCAM series, in press.

\bibitem[Ke]{Ke} H. J. Kenneth, \emph{Stochastic Optimal Control Theory with Application in Self-Tuning Control}, Lecture Notes in Control and Information Sciences, Vol. 117, Springer, 1989.

\bibitem[L]{L} P. L. Lions, \emph{Generalized solutions of Hamilton-Jacobi equations}, Research Notes in Mathematics 69, Pitman, 1982.

\bibitem[L1]{L1} D. G. Luenberger, \emph{Quasi-convex programming}, SIAM Journal on Applied
Mathematics 16 (1968), 1090 - 1095.

\bibitem[L2]{L2} D. G. Luenberger, \emph{Optimization by Vector Space Methods}, John Wiley \& Sons, 1969.

\bibitem[L3]{L3} D. G. Luenberger, \emph{Linear and Nonlinear Programming}, Addison-Wesley,
second edition, 1984.

\bibitem[L4]{L4}  D. G. Luenberger, \emph{Microeconomic Theory}, McGraw-Hill, 1995.


\bibitem[MGGJ]{MGGJ} B. Martin, A. Goldsztejn, L. Granvilliers, C. Jermann, \emph{On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach}, Journal of Global Optimization (2014) DOI: 10.1007/s10898-014-0201-3.

\bibitem[M]{M} E. S. Maskin, \emph{The theory of implementation in Nash equilibria: a survey}, in \emph{Social Goals and Social Organization: Essays in Memory of Elisha Pazner}, 
by L. Hurwicz (Ed.), D. Schmeidler (Ed.), H. Sonnenschein (Ed.), Cambridge University Press, 1985.

\bibitem[O]{O} C. Olech, \emph{Existence Theorems for Optimal Problems with Vector-Valued Cost Function}, Transactions of the Amer. Math. Soc. Vol. 136 (1969), 159 - 180.

\bibitem[P]{P} V. Pareto, \emph{Manual of Political Economy}, A. M. Kelley Publishers, 1971.
Translated from the French edition, first published in Italian in 1906.

\bibitem[RK]{RK} N. Rastegar, E. Khorram, \emph{A combined scalarizing method for multiobjective programming problems}, European Journal of Operational Research 236 (2014), 229�237.

\bibitem[RBG]{RBG} G. Reynoso-Meza, J. Sanchis, X. Blasco, S. Garcia-Nieto, \emph{Physical programming for preference driven evolutionary multi-objective optimization}, Applied Soft Computing Vol. 24 (2014), 341�-362.

\bibitem[S]{S} M. E. Salukvadze, \emph{Vector-Valued Optimization Problems in Control Theory}, Academic Press, 1979.

\bibitem[T]{T} D. F. M. Torres, \emph{A Noether Theorem on Unimprovable Conservation Laws for Vector-Valued Optimization Problems in Control Theory}, ArXiv preprint.


\end{thebibliography}

\end{document}

%------------------------------------------------------------------------------
% End of journal.tex
%------------------------------------------------------------------------------

1409.8648.extracted_bib
[BCD]{BCD M. Bardi, I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Modern Birkh\"auser Classics, 2008.
1409.8648.extracted_bib
[BF]{BF R. Bellman, K. Fan, On systems of linear inequalities in Hermitian matrix variables, in: V. L. Klee (editor), Convexity, vol. VII, Proceedings of the Symposia in Pure Mathematics, Amer. Math. Soc. 1963, 1 � 11.
1409.8648.extracted_bib
[BV]{BV S. Boyd, L.Vandenberghe, Convex Optimization, Cambridge University Press, 7th printing, 2009.
1409.8648.extracted_bib
[BKR]{BKR R. S. Burachik, C. Y. Kaya, M. M. Rizvi, A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets, Journal of Optimization Theory and Applications 162 (2014), 428--446.
1409.8648.extracted_bib
[CIL]{CIL M. G. Crandall, H. Ishii, P.L. Lions, User's Guide to Viscosity Solutions of 2nd Order Partial Differential Equations, Bulletin of the AMS 27 (1992), 1 - 67.
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[CHY]{CHY G.Chen, X. Huang, X. Yang, Vector Optimization: Set-valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, Springer, 2005.
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[DD]{DD I. Das, J.E. Dennis, Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems, SIAM J. Optim., 8 (2006), 631-�657.
1409.8648.extracted_bib
[D]{D G. Debreu, Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Yale University Press, 1959.
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[E]{E L.C. Evans, Partial differential equations, AMS, Graduate Studies in Mathematics Vol. 19, 1998.
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[E2]{E2 L.C. Evans, An Introduction to Stochastic Differential Equations, AMS pulications, ISBN-13: 978-1-4704-1054-4, 2013.
1409.8648.extracted_bib
[FR]{FR W. H. Fleming, R. W. Rishel, Deterministic and Stochastic Optimal Control, Stochastic Modelling and Applied Probability, Springer, corrected 2nd Printing, 1983.
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[FS]{FS W. H. Fleming, H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, Stochastic Modelling and Applied Probability, Springer, 2nd edition, 2010.
1409.8648.extracted_bib
[GC]{GC D. Ghosh, D. Chakraborty, A new Pareto set generating method for multi-criteria optimization problems, Operations Research Letters Vol. 42 (2014), 514-�521.
1409.8648.extracted_bib
[H]{H M. I. Henig, Vector-valued dynamic programming, SIAM J. Control and Optimization, Vol. 21, No. 3 (1983) 490 - 499.
1409.8648.extracted_bib
[I]{I K. Isii, Inequalities of the types of Chebyshev and Cram\'er-Rao and mathematical programming, Annals of The Institute of Statistical Mathematics 16 (1964), 277 - 293.
1409.8648.extracted_bib
[K]{K N. Katzourakis, An introduction to Viscosity Solutions for fully nonlinear PDE with applications to Calculus of Variations in $L^\infty$, Springer Brief monograph, BCAM series, in press.
1409.8648.extracted_bib
[Ke]{Ke H. J. Kenneth, Stochastic Optimal Control Theory with Application in Self-Tuning Control, Lecture Notes in Control and Information Sciences, Vol. 117, Springer, 1989.
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[L]{L P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics 69, Pitman, 1982.
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[L1]{L1 D. G. Luenberger, Quasi-convex programming, SIAM Journal on Applied Mathematics 16 (1968), 1090 - 1095.
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[L2]{L2 D. G. Luenberger, Optimization by Vector Space Methods, John Wiley \& Sons, 1969.
1409.8648.extracted_bib
[L3]{L3 D. G. Luenberger, Linear and Nonlinear Programming, Addison-Wesley, second edition, 1984.
1409.8648.extracted_bib
[L4]{L4 D. G. Luenberger, Microeconomic Theory, McGraw-Hill, 1995.
1409.8648.extracted_bib
[MGGJ]{MGGJ B. Martin, A. Goldsztejn, L. Granvilliers, C. Jermann, On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach, Journal of Global Optimization (2014) DOI: 10.1007/s10898-014-0201-3.
1409.8648.extracted_bib
[M]{M E. S. Maskin, The theory of implementation in Nash equilibria: a survey, in Social Goals and Social Organization: Essays in Memory of Elisha Pazner, by L. Hurwicz (Ed.), D. Schmeidler (Ed.), H. Sonnenschein (Ed.), Cambridge University Press, 1985.
1409.8648.extracted_bib
[O]{O C. Olech, Existence Theorems for Optimal Problems with Vector-Valued Cost Function, Transactions of the Amer. Math. Soc. Vol. 136 (1969), 159 - 180.
1409.8648.extracted_bib
[P]{P V. Pareto, Manual of Political Economy, A. M. Kelley Publishers, 1971. Translated from the French edition, first published in Italian in 1906.
1409.8648.extracted_bib
[RK]{RK N. Rastegar, E. Khorram, A combined scalarizing method for multiobjective programming problems, European Journal of Operational Research 236 (2014), 229�237.
1409.8648.extracted_bib
[RBG]{RBG G. Reynoso-Meza, J. Sanchis, X. Blasco, S. Garcia-Nieto, Physical programming for preference driven evolutionary multi-objective optimization, Applied Soft Computing Vol. 24 (2014), 341�-362.
1409.8648.extracted_bib
[S]{S M. E. Salukvadze, Vector-Valued Optimization Problems in Control Theory, Academic Press, 1979.
1409.8648.extracted_bib
[T]{T D. F. M. Torres, A Noether Theorem on Unimprovable Conservation Laws for Vector-Valued Optimization Problems in Control Theory, ArXiv preprint.