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%                  A LOCAL SECOND- AND THIRD-ORDER
%                        MAXIMUM PRINCIPLE
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%                       by H. J. Sussmann
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%        Control Conference, to be held in Anchorage, Alaska, 
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%               Date of this version: February 12, 2002
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\vspace{3cm}

{\Large  

\bi
\item[]
This is the back of {\huge Page \arabic{pagono}}
\item[] of the paper
\ei

\begin{center}
A Maximum Principle for\\
hybrid optimal control problems\\

$\quad$\\

by\\

$\quad$\\ 
H\'ector J. Sussmann\\

\end{center}
}

\newpage 

\null

\vspace{3.3cm}


{\Large 

Paper ID:\\

$\,\,\,${\Huge CDC99-INV0007} \\

Session Title:\\

$\,\,\,$ Optimal Control II\\

Session Code: {\Huge TuM01-4} \\

Paper Category: REGULAR 

\newpage
}}

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H\'ector J.\ Sussmann\footnotemark[1]\\
Department of Mathematics\\
Rutgers, the State University of New Jersey\\
Hill Center---Busch Campus\\
110 Frelinghuysen Road\\
Piscataway, NJ 08854-8019, USA\\
{\small\tt sussmann@hamilton.rutgers.edu}
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Benedetto Piccoli\\
SISSA\\
Via Beirut 2-4\\
34014 Trieste\\
ITALY\\
{\small\tt piccoli@sissa.it}
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H\'ector J. Sussmann\footnotemark[1]\\
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Department of Mathematics; Rutgers, the State University of New Jersey\\
Hill Center---Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ
08854-8019, USA\\ 
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E-mail: sussmann@math.rutgers.edu\quad\quad
Web site: http://www.math.rutgers.edu/$\widetilde{\,\,}$sussmann\\
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{\bf A Local Second- and Third-order Maximum Principle}
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\begin{document}

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\twocolumn[\titlestuff]
\footnotetext[1]{Supported in part by
NSF Grant DMS98-03411-00798.}


\vskip4.5cm

\ssection{Introduction}

\noindent In this note we announce preliminary results obtained
by applying the methodology of 
Sussmann \cite{sussram, sussshyb, susssmf2000, sussncn2000,
susssyd2000} to the problem studied by
Ledzewicz and Sch\"attler (cf. \cite{ls,ls2}), of deriving high-order 
necessssary \mbox{conditions} for a minimum in optimal control theory,
extending the \mbox{classical} Pontryagin Maximum Principle (abbr. MP)
of \cite{pontrybook}. 
The work of \cite{ls,ls2} uses high-order \mbox{generalizations}
of the theorems of Lyusternik and Avakov. We \mbox{pursue}
a different approach,
constructing needle variations and, at the crucial 
point where a ``topological argument'' is needed, applying
the Brouwer fixed point \mbox{theorem}. The result is a
high-order version of the MP that \mbox{contains} and extends the 
results of  \cite{ls,ls2}. In \mbox{particular}, our 
\mbox{version} makes it clear that 
new multipliers \mbox{occur} for the first time for the third-order
principle.  It \mbox{turns} out that what is called a ``second-order
MP'' in \cite{ls,ls2} is naturally a ``third-order MP''
in our \mbox{setting}. Our third-order MP contains new
\mbox{multipliers} exactly as the result of \cite{ls2}  does, but in 
\mbox{addition}
it \mbox{also} contains the classical multipliers, to which the new multipliers
are coupled in a precise way, described in
Theorem \ref{theocontrol-third}.

We present an abstract necessary condition for set separation (Theorem
\ref{maintheor-bbb}), a necessary condition for a minimum 
in an abstract setting (Theorem \ref{theoopti-uuu}), and, finally, 
the optimal control result (Theorem \ref{theocontrol-third}). We also
present an example showing that our results apply in cases where those
of \cite{ls,ls2} do not.

For lack of space, we omit the proofs, we limit our discussion to
problems with fixed endpoints, and we only consider the second and
third-order cases.

\ssection{Preliminary definitions}

\noindent We assume in this section 
that $\cx$, $\cy$ are normed spaces, $\cs\subseteq\cx$,
$x_*\in\cs$, and $\cf$ is a map from $\cs$ to $\cy$.

\subssection{High-order differentiability.}
Let $k\in\nn$. We say that {\em $\cf$ is of class $D^k$ at $x_*$
in the direction of $\cs$} if 
\begin{itemize}
\item[1.] $\cf$ is continuous on $N\cap\cs$ for some neighborhood
$N$ of $x_*$ in $\cx$,

\vup\vup
\item[2.] there exists a continuous polynomial map 
\mbox{$\cp:\cx\mapsto\cy$} of \mbox{degree}
$\le k$ without constant term such that
$$
\!\!\!\!
\lim_{x\to x_*,x\in\cs,x\neq x_*}\frac{\cf(x)-\cf(x_*)-\cp(x-x_*)}
{\|x-x_*\|^k}=0\,.\,\,\quad\diamondsuit
$$
\end{itemize}


\subssection{Variations and variational \mbox{directions}.}
A {\em \mbox{variation} of $x_*$ in $\cs$} is a \mbox{continuous} curve
\mbox{$[0,\bar\eps_\eta]\!\ni\!\eps\!\mapsto\!\eta(t)\!\in\!\cs$},
defined on the \mbox{interval} $[0,\bar\eps_\eta]$ for some 
positive $\bar\eps_\eta$, such that $\eta(0)=x_*$.
We use $Var(x_*,\cs)$ to \mbox{denote} the set of all variations
of $x_*$ in $\cs$. 

Let $k\in\nn$. A {\em $k$th-order variational direction of 
$\cs$ at $x_*$} is a $k$-tuple $(v_1,\ldots,v_k)\in\cx^k$
for which there exists a variation 
\mbox{$\eta\!\in\!Var(x_*,\cs)$} such that
$$
\eta(\eps)\!=\!x_*\!+\!\eps v_1\!+\!\ldots+\eps^kv_k+o(\eps^k)
$$ 
as $\eps\downarrow 0$. We use $var_k(x_*,\cs)$ to denote
the set of all $k$th-order variational directions
of $\cs$ at $x_*$. 

A {\em limiting first-order variational direction of $\cs$ at $x_*$} is a 
\mbox{$v\in\cx$} which is the limit of a sequence $\{v_j\}_{j\in\nn}$
of first-order variational directions of $\cs$ at $x_*$. We use 
$\overline{var}_1(x_*,\cs)$ to denote 
the set of all limiting first-order variational directions of 
$\cs$ at $x_*$. 

If $v_*\in\cx$, then $var_{2,1}(x_*,v_*,\cs)$,
$var_{3,1}(x_*,v_*,\cs)$ will denote, respectively, 
the set of all $w\in\cx$ such that $(v_*,w)\in var_2(x_*,\cs)$, and 
the set of all pairs \mbox{$(w,z)\in\cx\times\cx$} such that 
$(v_*,w,z)\in var_3(x_*,\cs)$. 
If $(v_*,w_*)\in\cx\times\cx$, we use $var_{3,2}(x_*,v_*,w_*,\cs)$ to denote
the set of all $z\in\cx$ such that 
$(v_*,w_*,z)\in var_3(x_*,\cs)$. 

\subssection{Abundant subsets.}  We say that
a subset $\cs_0$ of $\cs$ is {\em $(\cs,\cf)$-abundant}
if there exists a sequence $\{\sigma_j\}_{j\in\nn}$ of
maps $\sigma_j:\cs\mapsto\cs_0$ such that 

\vup\vupi
\begin{itemize} 
\item[1.] the map $\cf\circ\sigma_j$ is continuous for each $j$,

\vupi\vup
\item[2.] $\cf\circ\sigma_j$ converges to $\cf$ 
uniformly on compact subsets of $\cs$ as $j\to\infty$.
\end{itemize}

\subssection{Feasible sets and 2-regularity.}
\noindent Following \cite{ls}, we define  the notion of a 
``second-order feasible set,'' and of ``2-regularity,'' as follows.

\begin{definition}\label{fbi}
{\rm The {\sl second-order feasible set of $\cs$ in $\cx$ at $x_*$ in 
the direction of 
$v_*$} is the set $FS_\cx^{(2)}(\cs;x_*,v_*)$ of all $w\in\cx$ for
which there exist a neighborhood $W$ of $w$ in $\cx$
and an $\bar\eps\in\rr$ such that $\bar\eps\!>\!0$ and
$x_*\!+\!\eps v_*\!+\!\eps^2W\!\subseteq\!\cs$ for all
$\eps\in[0,\bar\eps]$.}\proofend
\end{definition}

%\input proofs-alaska-d.tex

\begin{definition}\label{regudef}
{\rm Assume that $\cf$ is of class $D^2$
at $x_*$ in the direction of $\cs$.  Let $P_1$, $P_2$ be, respectively,
a \mbox{continuous} linear map from $\cx$ to $\cy$ and a 
\mbox{continuous} bilinear symmetric map
from $\cx^2$ to $\cy$, such that 
\mbox{$\cf(x_*\!+\!v)\!=\!\cf(x_*)\!+\!P_1(v)\!+\!\frac{1}{2}P_2(v,v)
\!+\!o(\|v\|^{2})$ as $v\!\to\!0$},
\mbox{$x_*\!+\!v\!\in\!\cs$}. Let $\pi$ be the canonical projection
from $\cy$ to $\cy/P_1(\cx)$. We say that {\em $\cf$ is 
2-regular at $(x_*,v_*,\cx,\cs)$} if
(a)~\mbox{$FS_{\cx}^{(2)}(\cs;x_*,v_*)\neq\emptyset$},
(b)~the linear  map
\mbox{$\cx\!\ni\!u\!\mapsto\!\Big(P_1\cdot u,\pi_v
\big(P_2(v_*,u)\big)\Big)\!
\in\!P_1(\cx)\!\times\!\big(\cy/P_1(\cx)\big)$ is}
\mbox{onto}, and (c)~$P_2(v_*,v_*)\in P_1(\cx)$.}\proofend
\end{definition}



\ssection{The basic assumptions}\label{secnotations}

\noindent We let $\nu$ be a positive integer, which in the results to 
be stated below will be equal to $1$, $2$, or $3$. If $k$ is a
positive integer, and $v$ is any object, then we use $v^{<k>}$ 
to denote the $k$-tuple consisting of $v$ repeated $k$ times.

For the separation theorem, we will assume that

\noindent$\,\,$\begin{minipage}{8.3cm}\begin{itemize}
\item[(A1)] {\em $\cua$ (the ``ambient source space'')
and $Y$ (the ``target space'') are normed \mbox{spaces}, 
$Y$ is finite-dimensional,
$\cu$ is a convex subset of $\cua$, $u_*\in\cu$,  
\mbox{$\cf:\cu\mapsto Y$} is a map,
$\omega\in Y$ and $\omega\neq 0$.}
\end{itemize}\end{minipage}
\noindent$\,\,$\begin{minipage}{8.3cm}\begin{itemize}
\item[(A2)] {\em $\ca$ is an $(\cu,\cf)$-abundant 
subset of $\cu$ such that \mbox{$u_*\in\ca$}.}
\end{itemize}\end{minipage}
\noindent$\,\,$\begin{minipage}{8.3cm}\begin{itemize}
\item[(A3)] {\em $\cf$ is of class $D^\nu$ at $u_*$,}
\end{itemize}\end{minipage}

\noindent$\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(A4)] {\em $\{P_k\}_{k=1}^\nu$ is a family such that,
(a)~for each $k$, $P_k$ is a continuous $k$-linear 
symmetric map from $(\cua)^k$ to $Y$, and (b)~as $v\to 0$, $u_*+v\in\cu$,
we have}

\vup\vup\vup\vup\vup\vup
$$
\!\!\!\!
\cf(u_*+v)=\cf(u_*)+\sum_{k=1}^\nu\frac{1}{k!}P_k(v^{<k>})+o(\|v\|^{\nu})\,.
$$ 
\end{itemize}
\end{minipage}
\noindent We define

\vup\vup\vup\vup\vup\vup
\begin{eqnarray}
y_*&\equaldef&\cf(u_*)\,,\label{ystar}\\
\Omega&\equaldef&\{y_*+t\omega:t\in\rr,t\ge 0\}\,,\label{laline}\\
\Omega_0&\equaldef&\{t\omega:t\ge 0\}\,,\label{lambdao}
\end{eqnarray}

\vspace{0.3mm}
\noindent so that $y_*\!\in\!\cf(\ca)\cap\Omega$ and
$\Omega\!=\!y_*\!+\!\Omega_0\!=\!\{y_*\!+\!z\!:\!z\!\in\!\Omega_0\}$.

\vspace{0.3mm}
We will look for necessary conditions for the following {\em
separation property}:

\medbreak\vup\noindent
(SEP) $\qquad\qquad\cf(\ca)\cap\Omega=\{y_*\}$\,. 

\medbreak\vup\noindent
We write 

\vup\vup\vup\vup\vup
\begin{equation}\label{firstovar}
\cv=var_1(u_*,\cu)\,.
\end{equation}
If $\nu>1$, we will consider, in addition to $\cv$, a finite set $\hat\cv$  of
``independent first-order variational directions with higher-order
effects.'' This means, to begin with, that

\medbreak\vup\noindent
(A5) {\em $\hat\cv$ is a finite subset of $\cv$.}

\medbreak\vup
We will need additional assumptions on $\hat\cv$. The first one is the
``independence condition,''  whose purpose
is to enable us to add variations 
$$
\eps\mapsto\eta_v(\eps)=u_*+\tilde\eta_v(\eps)=
u_*+\eps v+o(\eps)
$$
of $u_*$ in $\cu$ corresponding to different members $v$ of $\hat\cv$,
and still obtain a variation of $u_*$ in $\cu$. We will 
\mbox{guarantee} this
by requiring that the values $\tilde\eta_v(\eps)$ belong to sets 
$\cu_v^0$ such that $u_*+\sum_{\ell}\cu_v^0\subseteq\cu$, 
from which it will follow that 
$u_*+\sum_{v}\tilde\eta_v(\eps)\in\cu$. 
The sets $\cu_v^0$ will actually play a role in the statement of
our main separation \mbox{theorems}. Therefore, rather than merely assume that
they exist, we will actually regard them as additional data
for our separation problem. If we write 
$\cu_v=\cu_v^0+u_*$, the ``\mbox{independence} condition'' becomes:

\medbreak\noindent$\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(A6)] {\em $\{\cu_v\}_{v\in\hat\cv}$ is a family of 
convex subsets of $\cua$ such that
\begin{itemize}
\item[1.] $u_*\in\cu_v$ for every $v\in\hat\cv$, 

\vup\vup
\item[2.] $u_*+\sum_{v\in\hat\cv}\Big(\cu_v-u_*)\subseteq\cu$,

\vup\vup
\item[3.] $v\in var_1(u_*,\cu_v)$ for each $v\in\hat\cv$. 
\end{itemize}}\end{itemize}
\end{minipage}

\medbreak\noindent In addition to (A5) and (A6), we will 
impose---if \mbox{$\nu>1$}---the 
``cancellation and  cross-term conditions''

\noindent$\,\,\,\,$\begin{minipage}{8.1cm}
\begin{itemize}
\item[(A7)] {\em the vector $P_1\cdot v$ belongs to $\Omega_0$
whenever $v\in\hat\cv$,}

\vup\vup
\item[(A8)] {\em whenever $v,v'\in\hat\cv$ are such that
$v\neq v'$, the vector $P_2(v,v')$  belongs to $\Omega_0$.}
\end{itemize}
\end{minipage}

\medbreak\noindent When $\nu=3$, we will need the following conditions:

\medbreak\noindent$\,\,\,\,\,\quad$\begin{minipage}{7.8cm}
\begin{itemize}
\item[(A$_3$9.a)] {\em $\hat\cv^2$ and $\hat\cv^3$ are subsets
of $\cv$, $\hat\cv^2\cup\hat\cv^3=\cv$ and 
$\hat\cv^2\cap\hat\cv^3=\emptyset$;}

\vup\vup
\item[(A$_3$9.b)] {\em whenever $v,v',v''\in\hat\cv^3$ and
$v\neq v'\neq v''\neq v$, it follows that 
$P_3(v,v',v'')\in\Omega_0$;}

\vup\vup
\item[(A$_3$9.c)] {\em whenever $v,v'\in\hat\cv^3$,
$v\neq v'$, $w$ belongs to $var_{2,1}(u_*,v,\cu_{v})$, and
$\frac{1}{2}P_2(v,v)+P_1\cdot w\in\Omega_0$, it follows that 
$\frac{1}{2}P_3(v,v,v')+P_2(w,v')\in\Omega_0$.}
\end{itemize}
\end{minipage}

\medbreak\noindent
If $\nu=3$, then for each $v\in\hat\cv^3$ and each subset $S$ of
$var_{3,1}(u_*,v,\cu_v)$ we let $Y_{1,v,S}$ be the linear span of 
$\omega$ and all the vectors  $\frac{1}{2}P_2(v,v)+P_1\cdot w$,
for all $(w,z)\in S$, and define
$Y_{1,v}\equaldef Y_{1,v,var_{3,1}(u_*,v,\cu_v)}$. 
%Let $\pi^v$ denote 
%the canonical projection from $Y$ to the quotient space
%$Y/Y_1^v$. 



\medbreak For the abstract minimization theorem,
we will \mbox{assume} that 
\medbreak\noindent$\,\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(M1)] {\em $\cua$ and $X$ are normed \mbox{spaces},
$X$ is finite-dimensional, $\cu$ is a convex subset of $\cua$,
$u_*\in\cu$,  \mbox{$\ce:\cu\mapsto X$} is a map, and
\mbox{$\ci:\cu\mapsto \rr$} is a function,}
\end{itemize}
\end{minipage}

\noindent$\,\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(M2)] {\em $\ca$ is an $(\cu,\cf)$-abundant 
subset of $\cu$ such that $u_*\in\ca$, where $\cf$ is the
map}
\begin{equation}\label{defioff}
\cu\ni u\mapsto\cf(u)\equaldef(\ce(u),\ci(u))\in Y\equaldef X\times\rr\,,
\end{equation}
\end{itemize}
\end{minipage}

\noindent$\,\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(M3)] {\em $\ce$ and $\ci$ are of class $D^\nu$ at $u_*$,}
\end{itemize}
\end{minipage}

\noindent$\,\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(M4)] {\em for each $k\in\{1,\ldots,\nu\}$, 
$P_k$ and  $p_k$ are \mbox{continuous} $k$-linear symmetric maps
from $(\cua)^k$ to $X$, $\rr$, \mbox{respectively}, 
and the formulas}
\end{itemize}\end{minipage}
$$
\ce(u_*+v)=\ce(u_*)+\sum_{k=1}^\nu\frac{1}{k!}P_k(v^{<k>})
+o(\|v\|^{\nu})\,,
$$ 
$$
\ci(u_*+v)=\ci(u_*)+\sum_{k=1}^\nu\frac{1}{k!}p_k(v^{<k>})
+o(\|v\|^{\nu})
$$ 

{\em $\qquad$hold as $v\to 0$, $u_*+v\in\cu$.}


\medbreak\noindent If $\nu=2$ or $\nu=3$, we will also assume that
conditions (A5) and (A6) hold, as well as the following
{\em second-oder 
cancellation and cross-term conditions}:

\medbreak\noindent$\,\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(M7)] {\em $P_1\cdot v=0$ and $p_1\cdot v\le 0$ 
whenever $v\in\hat\cv$,}

\vup\vup
\item[(M8)] {\em $P_2(v_1,v_2)=0$  and $p_2(v_1,v_2)\le 0$  
whenever $v_1,v_2\in\hat\cv$ are such that
$v_1\neq v_2$,}
\end{itemize}\end{minipage}

\medbreak\noindent If $\nu=3$, we will assume the following {\em third-order
\mbox{cancellation} and cross-term conditions\/}:
\medbreak\noindent$\quad\,\,\,\,\,$\begin{minipage}{7.8cm}
\begin{itemize}
\item[(M$_3$9.a)] {\em  $\hat\cv^2\cap\hat\cv^3=\emptyset$, 
and $\hat\cv=\hat\cv^2\cup\hat\cv^3$,}

\vup\vup
\item[(M$_3$9.b)] {\em Whenever $v,v',v''\in\hat\cv^3$ and
\mbox{$v\neq v'\neq v''\neq v$}, it follows that 
$P_3(v,v',v'')=0$ and $p_3(v,v',v'')\le 0$.}

\vup\vup
\item[(M$_3$9.c)] {\em Whenever $v,v'\!\in\!\hat\cv^3$,
$v\!\neq\! v'$, and $w$ is such that $w\in var_{2,1}(u_*,v,\cu_{v})$,
\mbox{$\frac{1}{2}P_2(v,v)+P_1\cdot w=0$} and
$\frac{1}{2}p_2(v,v)+p_1\cdot w\le 0$, 
it follows that 
$\frac{1}{2}P_3(v,v,v')+P_2(w,v')=0$ and also that
$\frac{1}{2}p_3(v,v,v')+p_2(w,v')=0$.}
\end{itemize}\end{minipage}

\medbreak\noindent We let

\vup\vup\vup\vup\vup
\begin{equation}\label{zstar-uuu}
x_*=\ce(u_*)\,.
\end{equation}
We will seek necessary conditions for the {\em minimization condition}:

\vspace{0.5mm}
\noindent$\,\,\,\,\,\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(MIN)] {\em $\ci(u_*)\le\ci(u)$ for all $u\in\ca$ such that
$\ce(u)=x_*\,$.}
\end{itemize}
\end{minipage}



\medbreak For the optimal control result, we will \label{seccic-ontr}
\mbox{consider} a \mbox{control} system

\vup\vup\vupi
\begin{equation}\label{cosys}
\dot x=f(x,u,t)\,,\,\,\,x\in\rr^n\,,\,\,\,u\in U\,,a\le t\le b\,,
\end{equation}

\vspace{0.6mm}
\noindent and a Lagrangian 
\begin{equation}\label{cosyslag}
\rr^n\times U\times[a,b]\ni(x,u,t)\mapsto L(x,u,t)\in\rr\,.
\end{equation}
We let $F(x,u,t)\equaldef (f(x,u,t),L(x,u,t))$, and
\mbox{assume}:

\medbreak\noindent$\,\,$\begin{minipage}{8.3cm}\begin{itemize}
\item[(C1)] {\em $m\in\nn$ and $U$ is a closed convex subset of $\rr^m$,}
\end{itemize}\end{minipage}

\noindent$\,\,$\begin{minipage}{8.3cm}\begin{itemize}
\item[(C2)] {\em the partial map
\mbox{$\rr^n\!\times\!U\!\ni\!(x,u)\!\mapsto\! F(x,u,t)\!\in\!\rr^{n+1}$}
is of class $C^\nu$ for every $t\in[a,b]$,}
\end{itemize}\end{minipage}

\noindent$\,\,$\begin{minipage}{8.3cm}\begin{itemize}
\item[(C3)] {\em the partial  map 
\mbox{$[a,b]\!\ni\!t\!\mapsto\!F(x,u,t)\!\in\!\rr^{n+1}$}
is \mbox{measurable} for every $(x,u)\in\rr^n\times U$,} 
\end{itemize}\end{minipage}

\noindent$\,\,$\begin{minipage}{8.3cm}\begin{itemize}
\item[(C4)] {\em the maps of condition {\rm (C2)} are bounded, 
\mbox{together} with 
all their partial derivatives with respect to $(x,u)$
of order $\le\nu$, on every compact subset of the product
$\rr^n\times U\times[a,b]$.}
\end{itemize}\end{minipage}

We let $\cua=L^\infty([a,b],\rr^m)$, and let $\cu$ be the set
of those controls $u(\cdot)\in\cua$ that are $U$-valued.

A {\em trajectory-control pair} is a pair $(x(\cdot),u(\cdot))$ 
\mbox{consisting} of an absolutely continuous curve 
$x(\cdot):[a,b]\mapsto\rr^n$ and a control 
$u(\cdot):[a,b]\mapsto U$ with the property  that 
\mbox{$\dot x(t)=f(x(t),u(t),t)$} for almost all $t\in[a,b]$.

We fix an initial state $\bar x$ and a terminal state $\hat x$,
and let $TCP(\bar x,\hat x)$ denote the set of all trajectory-control
pairs $(x(\cdot),u(\cdot))$ such that $x(a)=\bar x$ and $x(b)=\hat x$.

We assume that we are given a pair $(x_*(\cdot),u_*(\cdot))$
(the ``reference trajectory-control pair'') such that 

\medbreak\noindent$\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(C5)] {\em $(x_*(\cdot),u_*(\cdot))\in TCP(\bar x,\hat x)$,
and $(x_*(\cdot),u_*(\cdot))$ \mbox{minimizes} the functional}
$$
TCP(\bar x,\hat x)\ni(x(\cdot),u(\cdot))\mapsto
J(x(\cdot),u(\cdot))\in\rr\,,
$$
where $J(x(\cdot),u(\cdot))\equaldef\int_a^bL(x(t),u(t),t)dt$.
\end{itemize}
\end{minipage}

\medbreak\noindent We write $f_x$, $f_u$, $L_x$, $L_u$ 
for $\frac{\partial f}{\partial x}$,
$\frac{\partial f}{\partial u}$, $\frac{\partial L}{\partial x}$,
$\frac{\partial f}{\partial u}$, and similarly for higher order
derivatives. Also, we write 
$$
f_x^*(t)\equaldef f_x(x_*(t),u_*(t),t)\,,\quad
f_u^*(t)\equaldef f_u(x_*(t),u_*(t),t)\,,
$$
$$
L_x^*(t)\equaldef L_x(x_*(t),u_*(t),t)\,,\quad
L_u^*(t)\equaldef f_u(x_*(t),u_*(t),t)\,,
$$
and use a similar notation for higher order derivatives, so that,
for example, $f_{xxu}^*(t)$ is, for each time $t\in[a,b]$,
an $\rr^n$-valued trilinear form 
on $\rr^n\times\rr^n\times\rr^m$  and, if $\varphi_1$, $\varphi_2$, 
$\theta$ are bounded measurable functions on $[a,b]$ with values 
in $\rr^n$, $\rr^n$, $\rr^m$, respectively, then 
$f_{xxu}^*(\varphi_1,\varphi_2,\theta)$ denotes the function
$[a,b]\ni t\mapsto f_{xxu}^*(t)(\varphi_1(t),\varphi_2(t),\theta(t))$.



\medbreak\noindent  We also assume that 

\medbreak\noindent$\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(C6)] {\em $\ch$ is a finite set of pairs $(h,v)$ such that
$v\in\cua$, $h$ is an absolutely continuous map from $[a,b]$ to $\rr^n$,
$h(a)=h(b)=0$, 
$$
\!\!\!\!\!\!\dot h(t)=\frac{\partial f}{\partial x}
(x_*(t),u_*(t),t)\cdot h(t)+\frac{\partial f}{\partial u}
(x_*(t),u_*(t),t)\cdot v(t)
$$
for a.e. $t\in[a,b]$.
and}
\begin{equation}\label{free}
\int_a^b\Big(L_x^*(t)\cdot h(t)+L_u^*(t)\cdot v(t)\Big)\,dt\le 0\,.
\end{equation}
\end{itemize}\end{minipage}

\medbreak Furthermore, we assume that

\medbreak\noindent$\,\,$\begin{minipage}{8.3cm}
\begin{itemize}
\item[(C7)] {\em $\{I_{h,v}\}_{(h,v)\in\ch}$ is a family
of pairwise disjoint subintervals of $[a,b]$ such that} 
\begin{equation}\label{disjov-ug}
\!\!\!\!\!\!\!\!\!\!\!\!
\{t\in[a,b]:v(t)\neq 0\}\subseteq I_{h,v}\,\,
{for}\,\,all\,\,(h,v)\in\ch.
\end{equation}
\end{itemize}\end{minipage}

\medbreak We remark that conditions (C6) and (\ref{disjov-ug})
imply that

\vup\vup
\begin{equation}\label{disjoh-ug}
\{t\in[a,b]:h(t)\neq 0\}\subseteq I_{h,v}\,\,
{\rm for\,\,all}\,\,(h,v)\in\ch.
\end{equation}


\medbreak We define the {\em input-to-trajectory map} 
\mbox{$\ct:\cu\mapsto C^0([a,b],\rr^n)$}
by first defining $\ct$ as a map from $\tilde\cu$ to
$C^0([a,b],\rr^n)$, where $\tilde\cu$ is the set of all controls
$u\in\cu$ such that the initial value problem
$\dot x=f(x,u(t),t)$, $x(a)=\bar x$, has a solution (which, of
course, is unique) defined on $[a,b]$.  We then extend 
$\ct$ to all of $\cu$ by defining $\ct(u)$ to be the curve $x_*$
if  $u\in\cu\backslash\tilde\cu$. Naturally, with this
definition $\ct$ need not be continuous on $\cu$, but it is 
continuous (and, in fact, $\nu$ times continuously Fr\'echet differentiable)
on $\tilde\cu$, which is a relative neighborhood of $u_*$ in 
$\cu$, and this more than suffices to make our abstract 
theorems applicable. 

We then define the {\em terminal point map} $\ce$
by letting $\ce(u)=\ct(u)(b)$ for $u\in\cu$.

For each $(h,v)\in\ch$, we define $\cu_{v}^a$ to be the 
space $L^\infty(I_{h,v},\rr^m)$, so $\cu_{v}^a$ can be identified
with a subspace of $\cu^a$ in an obvious way. Also, 
we let $\cu_v$ be the subset of $\cu$ consisting
of all functions $u\in\cu$ such that 
$u(t)=u_*(t)$ for $t\notin I_{h,v}$. Finally, we let
$\cu_v^0=\{u-u_*:u\in\cu_v\}$, so 
$\cu_v^0$ is a convex subset of $\cu_v^a$ and $\cu_v=\cu_v^0+u_*$.

We define the {\em restricted terminal map} $\ce_v^0:\cu_v^0\mapsto\rr^n$
to be the map given by 
$$
\ce_v^0(u)=\ce(u+u_*)\quad{\rm for}\quad u\in\cu_v\,.
$$

%\input trajectory-map-alaska.tex

\vbox{
\ssection{The separation theorem}
\noindent We define $C_1$ to be the smallest 
closed convex cone in $Y$ 
that contains all the vectors $P_1\cdot v$, $v\in\cv$.
}

If $\nu>1$, we let $C_2$ be the smallest closed convex cone 
in $Y$ that contains $C_1(\cv)$ as well as all the vectors 
$\frac{1}{2}P_2(v,v)+P_1\cdot w$, for  all $(v,w)$ such that
$v\in\hat\cv$ and $(v,w)\in var_2(u_*,\cu_v)$.

If $\nu=3$, we define
\begin{eqnarray}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\tilde P(v,w)&\equaldef&\frac{1}{2}P_2(v,v)+P_1\cdot w\,,\label{deftildep}\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\hat P(v,w,z)&\equaldef&\frac{1}{6}P_3(v,v,v)\!+\!P_2(v,w)\!+
\!P_1\cdot z\,.\label{defhatp}
\end{eqnarray}
and let $C_3$ be the smallest closed
convex cone in $Y$ that contains $C_2$ as well as
all the vectors $y$ of the \mbox{form}
\mbox{$y=\hat P(v,w,z)$}
for  some $(v,w,z)$ such that $v\in\hat\cv^3$,
$(v,w,z)\in var_{3}(u_*,\cu_v)$, and
$\tilde P(v,w)\in\Omega_0$.

\medbreak\noindent
For $\nu=1,2,3$, the {\em $\nu$-th-order surjectivity condition} says:

\begin{itemize}\item[]\begin{itemize}
\item[(SUR-$\nu$)] {\em $\omega$ is an interior point of the cone 
$C_\nu$.}
\end{itemize}\end{itemize}
The separation theorem can now be stated.
\begin{theorem}\label{maintheor-bbb}
Assume that $\nu\in\{1,2,3\}$, and the data $\cua$, $Y$, $\cu$, $u_*$, 
$\cf$, $\omega$, $\ca$, 
$P_1,\ldots,P_\nu$ satisfy Assumptions (A1), (A2), (A3), (A4).
Let $y_*$, $\Omega$, $\cv$, $\Omega_0$ be defined by (\ref{ystar}),
(\ref{laline}), (\ref{firstovar}), (\ref{lambdao}).  
If $\nu=2$ or $\nu=3$, assume in addition that 
$\hat\cv$, $\{\cu_v\}_{v\in\hat\cv}$ are such that 
(A5), (A6), (A7) and (A8) hold. If $\nu=3$, assume that (A$_3$9.a), 
(A$_3$9.b) and (A$_3$9.c) are satisfied. Then the $\nu$th-order 
surjectivity condition 
(SUR-$\nu$) implies that the separation property (SEP) does not 
hold. \proofend
\end{theorem}

Theorem \ref{maintheor-bbb} implies the following
``multiplier'' result:
\begin{theorem}\label{maincoro-bbb}
Assume that the hypotheses of Theorem \ref{maintheor-bbb} are satisfied
and (SEP) holds. Then
there exists a nonzero covector $\bar{\bar\mulb}\in Y^*$, such that 
\begin{itemize}
\item[(1)] $\,\langle\bar{\bar\mulb},\omega\rangle\le 0\,$,

\vup\vup
\item[(2)] $\,\langle\bar{\bar\mulb},P_1\cdot v\rangle\ge 0\,$ 
for all $v\in\overline{var}_1(u_*,\cu)\,$,

\vup\vup
\item[(3)] if $\,\nu>1$ then 
$\,\langle\bar{\bar\mulb},\tilde P(v,w)\rangle\ge 0\,$
whenever $v\in\hat\cv$ and $w\in var_{2,1}(u_*,v,\cu_v)\,$,

\vup\vup
\item[(4)] if $\nu=3$ then for every $v\in\hat\cv^3$ 
\begin{itemize}
\item[(4.{\rm \&})] 
$\langle\bar{\bar\mulb},\hat P(v,w,z)\rangle\ge 0$ whenever $w,z$ are such
that $(w,z)\in var_{3,1}(u_*,\cu_v)$ and
$\tilde P(v,w)\in\Omega_0\,$.
\end{itemize}\end{itemize}
In addition, if $\nu>1$, then
Condition {\rm (3)} implies, in particular, that
\begin{itemize}\item[]\begin{itemize}
\item[(3.\#)] $\,\langle\bar{\bar\mulb},P_1\cdot w\rangle
\ge 0$ whenever \mbox{$v\in\hat\cv$} and  
\mbox{$(v,w)\in var_{2}(u_*,\cu_v)\,$}.
\end{itemize}\end{itemize}
Furthermore, if $\nu=3$, then
\begin{itemize}
\item{} For each $v\in\hat\cv^3$, Condition {\rm (4.\&)}
is equivalent to the following multiplier statement:
\begin{itemize}
\item[(4.\#)] for every finite subset
$S$ of $var_{3,1}(u_*,v,\cu_v)$ there exists a covector
$\bar{\bar\mula}^{v,S}\in Y_{1,v,S}^*$ such that 
$\,\,\langle\bar{\bar\mula}^{v,S},\omega\rangle\le 0\,$,
and the inequality
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\Big\langle\bar{\bar\mulb},\hat P(v,w,z)\Big\rangle
+\Big\langle\bar{\bar\mula}^{v,S},
\tilde P(v,w)\Big\rangle\ge 0\label{bigineq-gurgur}
\end{eqnarray}
holds for all $(w,z)\in S$.
\end{itemize}
\item{} Condition {\rm (4.\&)}---or its equivalent form 
{\rm (4.\#)}---implies the following statement:
\begin{itemize}
\item[(4.\%)] for every $v\!\in\!\hat\cv^3$ there exist a covector
\mbox{$\bar{\bar\mula}^{v}\!\in\! Y_{1,v}^*$} and a nonnegative number
$\theta^{v}$  such that 
\begin{itemize}
\item[(4.\%.a)] the inequalities

\vup\vup
\begin{eqnarray}\label{bigineq-gnu-wewe}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\theta^{v}\langle\bar{\bar\mulb},P_1\cdot z\rangle&\ge&0\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\theta^{v}\Big\langle\bar{\bar\mulb},\hat P(v,w,z)\Big\rangle
+\Big\langle\bar{\bar\mula}^{v},
\tilde P(v,w)\Big\rangle&\ge& 0\label{bigineq-gnu}
\end{eqnarray}
hold for all $(w,z)\in var_{3,1}(u_*,v,\cu_v)$,
\item[(4.\%.b)] $\,\,\Big\langle\bar{\bar\mula}^{v},\omega\rangle\le 0\,$, 
\item[(4.\%.c)] either $\,\,\theta^{v}>0$ or $\bar{\bar\muld}^{v}
\in Y_{1,v}^*\backslash\{0\}$.
\end{itemize}\end{itemize}\end{itemize}
\end{theorem}

%\input proofs-alaska-a.tex
%\input proofs-alaska-h.tex

\ssection{Necessary conditions for a minimum}

\noindent For each $v\in\hat\cv^3$ and each subset $S$ of
$var_{3,1}(u_*,v,\cu_v)$, we let $X_{1,v,S}$ be the linear \mbox{span} of 
all the vectors  $\tilde P(v,w)$, for all $(w,z)\in S$, and write\
$X_{1,v}\equaldef X_{1,v,var_{3,1}(u_*,v,\cu_v)}$.

We write
\begin{eqnarray}
\tilde p(v,w)&\equaldef&p_1\cdot w+\frac{1}{2}p_2(v,v)
\,,\label{deftildepsmall}\\
\hat p(v,w,z)&\equaldef&\frac{1}{6}p_3(v,v,v)+p_2(v,w)+p_1\cdot z
\,,\label{defhatpsmall}
\end{eqnarray}


\begin{theorem}\label{theoopti-uuu}
Assume that $\nu\in\{1,2,3\}$, and 
the data $\cua$, $X$, $\cu$, $u_*\in\cu$, $\ce$, $\ci$,
$\ca$, $\cf$, $P_1,\ldots,P_\nu$, $p_1,\ldots,p_\nu$,
satisfy Assumptions (M1) to (M4) of \S\ref{secnotations}. If $\nu>1$, 
\mbox{assume} that $\hat\cv$ and $\{\cu_v\}_{v\in\hat\cv}$, 
are such that (A5) and (A6) hold, as well as the second-oder 
cancellation and cross-term conditions (M7) and (M8).
If $\nu=3$, assume that the third-order
cancellation and cross-term conditions (M$_3$9.a), (M$_3$9.b),
(M$_3$9.c) are satisfied. 

Let $x_*=\ce(u_*)$, and assume that (MIN) holds.
Then there exist a covector $\bar\mulb\in X^*$ and a $\mulb_0\in\rr$
such that
\begin{itemize}
\item[(1)]  $\mulb_0\ge 0$ and $\|\bar\mulb\|+\mulb_0>0$,

\vup\vup
\item[(2)] $\langle\bar\mulb,P_1\cdot v\rangle+\mulb_0p_1\cdot v\ge 0$
for all $v\in\overline{var}_1(u_*,\cu)$,

\vup\vup
\item[(3)] if $\nu>1$ then 
$\langle\bar\mulb,\tilde P(v,w)\rangle+\mulb_0\tilde p(v,w)\ge 0$ 
\mbox{whenever} $v\in\hat\cv$ and
$w\in var_{2,1}(u_*,v,\cu_v)$, 

\vup\vup
\item[(4)] if $\nu=3$ then for every $v\in\hat\cv^3$, 
\begin{itemize}
\item[(4.{\rm \&})] 
\mbox{$\big\langle\bar\mulb,\hat P(v,w,z)\big\rangle\!+\!\mulb_0\hat p(v,w,z)\!
\ge\!0$} whenever
$(w,z)\!\in\!var_{3,1}(u_*,\cu_v)$,
\mbox{$\tilde P(v,w)\!=\!0$} and $\tilde p(v,w)\le 0$.
\end{itemize}\end{itemize}
In addition, if $\nu\!>\!1$, then {\rm (3)} implies, 
\mbox{in particular, that}
\begin{itemize}\item[]\begin{itemize}
\item[(3.\#)] $\,\langle\bar\mulb,P_1\cdot w\rangle
+\mulb_0\,p_1\cdot w\ge 0$ whenever $v\in\hat\cv$ and
$(v,w)\in var_{2}(u_*,\cu_v)\,$.
\end{itemize}\end{itemize}
Furthermore, if $\nu=3$, then
\begin{itemize}
\item{} For each $v\in\hat\cv^3$, Condition {\rm (4.\&)} 
is equivalent to the following multiplier statement:
\begin{itemize}
\item[(4.\#)] for every finite subset
$S$ of $var_{3,1}(u_*,v,\cu_v)$ there exists a covector
$\bar\mula^{v,S}\in X_{1,v,S}^*$ and a nonnegative real number
$\mula_0^{v,S}$  such that 
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\Big\langle\bar\mulb,\hat P(v,w,z)\Big\rangle+
\Big\langle\bar\mula^{v,S},\tilde P(v,w)\Big\rangle\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\mulb_0\Big(\hat p(v,w,z)\Big)
+\mula_0^{v,S}\Big(\tilde p(v,w)\Big)\ge 0\label{bigineq}
\end{eqnarray}
for all $(w,z)\in S$.
\end{itemize}
\item{} \mbox{{\rm (4.\&)}---or its equivalent form 
{\rm (4.\#)}---implies:}
\begin{itemize}
\item[(4.\%)] for every $v\in\hat\cv^3$ there exist a covector
\mbox{$\bar\mula^{v}\in X_{1,v}^*$} and nonnegative numbers
$\mula_0^{v}$, $\theta^{v}$  such that 
\begin{itemize}
\item[(4.\%.a)] the inequalities
\begin{equation}\label{onzc-b}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\theta^v\Big\langle\bar\mulb,P_1\cdot z\Big\rangle\ge 0,
\end{equation}
\begin{equation}\label{bigineq-uuu}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\theta^{v}\Big\langle\bar\mulb,\hat P(v,w,z)\Big\rangle\!+\!
\Big\langle\bar\mula^{v},\tilde P(v,w)\Big\rangle
\!+\!\mula_0^{v}\Big(\tilde p(v,w)\Big)\!\ge\! 0
\end{equation}
hold for all $(w,z)\in var_{3,1}(u_*,v,\cu_v)$,
\item[(4.\%.b)] either $\theta^{v}\!\neq\!0$, or $\mula_0^{v}\!\neq\!0$, or 
\mbox{$\bar\mula^{v}\!\in\!X_{1,v}^*\backslash\{0\}$},
\item[(4.\%.c)] $\theta^v\mulb_0=0$.
\end{itemize}\end{itemize}\end{itemize}
\end{theorem}

%\input proofs-alaska-e.tex
%\input proofs-alaska-j.tex

\ssection{A high-order maximum principle}
\noindent We now study the same optimal control problem as in
\S\ref{seccic-ontr}, and state the 
necessary condition for a minimum that follows by applying
Theorem \ref{theoopti-uuu}.

Following \cite{ls2},  we write 
\begin{eqnarray}
\!\!\!\!\!\!\!R_2[f][h,v]\!\!\!\!&=&\!\!\!\!\frac{1}{2}f_{xx}^*(h,h)\!
+\!f_{xu}^*(h,v)\!+\!\frac{1}{2}f_{uu}^*(v,v),\label{defio1}\\
\!\!\!\!\!\!\!R_2[L][h,v]\!\!\!\!&=&\!\!\!\!\frac{1}{2}L_{xx}^*(h,h)
\!+\!L_{xu}^*(h,v)\!+\!\frac{1}{2}L_{uu}^*(v,v),\label{defio2}\\
\!\!\!\!\!\!\!R_3[f](h,v)\!\!\!\!&=&\!\!\!\!\frac{1}{6}f_{xxx}^*(h,h,h)+
\frac{1}{2}f_{xxu}^*(h,h,v)\nonumber\\
&\,&\!\!\!\!\!\!\!+\frac{1}{2}f_{xuu}^*(h,v,v)+
\frac{1}{6}f_{uuu}^*(v,v,v)\,,\label{defio3-ug}\\
\!\!\!\!\!\!\!R_3[L](h,v)\!\!\!\!&=&\!\!\!\!\frac{1}{6}L_{xxx}^*(h,h,h)+
\frac{1}{2}L_{xxu}^*(h,h,v)\nonumber\\
&\,&\!\!\!\!\!\!\!+\frac{1}{2}L_{xuu}^*(h,v,v)+
\frac{1}{6}L_{uuu}^*(v,v,v),\label{defio3-ug-l}
\end{eqnarray}
and let
\begin{eqnarray}
(\nabla^1f(h,v))_x\cdot g&=&f_{xx}^*(g,h)+f_{xu}^*(g,v)\,,\label{novak-e}\\
(\nabla^1L(h,v))_x\cdot g&=&L_{xx}^*(g,h)+L_{xu}^*(g,v)\,,\label{novak-f}\\
(\nabla^1f(h,v))_u\cdot w&=&f_{xu}^*(h,w)+f_{uu}^*(v,w)\,,\label{novak-g}\\
(\nabla^1L(h,v))_u\cdot w&=&L_{xu}^*(h,w)+L_{uu}^*(v,w)\,,\label{novak-h}
\end{eqnarray}
if $g\in L^\infty([a,b],\rr^n)$,  $w\in L^\infty([a,b],\rr^m)$.



 
%\input some-preliminary-alaska.tex

\begin{theorem}\label{theocontrol-third}
Assume that $\nu\!\in\!\{1,2,3\}$, and 
\mbox{conditions} (C1) to (C7) of 
\S\ref{seccic-ontr} hold.
Then there exist an absolutely continuous 
field of covectors $[a,b]\ni t\mapsto\lambda(t)\in\rr_n$ 
(where $\rr_n$ denotes the space of real
$n$-dimensional  row vectors) and a nonnegative $\lambda_0\in\rr$ such that
\begin{itemize}
\item[1.] $\lambda_0+\|\lambda(b)\|>0\,$;

\vup\vup
\item[2.] the adjoint equation
$\dot\lambda(t)=-\lambda(t)\cdot f_x^*(t)-\lambda_0 L_x^*(t)$
is satisfied for almost all $t\in[a,b]\,$;

\vup\vup
\item[3.] for almost all $t\in[a,b]$, the inequality 
\begin{equation}\label{mman-a}
\Big(\lambda_0L_u^*(t)+\lambda(t)\cdot f_u^*(t)\Big)\cdot(u-u_*(t))\ge 0
\end{equation}
holds for all $u\in U\,$;

\vup\vup
\item[4.] if $\nu>1$, then for every $(h,v)\in\ch$, the inequalities
\begin{equation}\label{mman-b}
\int_a^b(\lambda_0L_u^*+
\lambda\cdot f_u^*)\cdot w\,dt\ge 0\,,
\end{equation}
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int_a^b\Big((\lambda_0L_u^*+\lambda\cdot f_u^*)\cdot w
+\lambda_0 R_2[L](h,v)\nonumber\\
&\,&+\lambda(t)\cdot R_2[f](h,v)\Big)
\,dt\ge 0\label{mman-c}
\end{eqnarray}
hold for every $w\in var_{2,1}(u_*,v,\cu)$ such that
\mbox{$\{t:w(t)\neq 0\}\subseteq I_{h,v}\,$};

\vup\vup
\item[5.] if $\nu\!=\!3$ then  for every $(h,v)\!\in\!\ch$ the inequality
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\int_a^b(\lambda_0L_u^*+\lambda\cdot f_u^*)\cdot z\,dt\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\int_a^b\lambda_0\,\Big((\nabla^1L(h,v))_x\cdot
{\bf y}^{v,w}+(\nabla^1L(h,v))_u\cdot w\Big)\,dt\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\int_a^b\lambda\cdot \Big((\nabla^1f(h,v))_x\cdot
{\bf y}^{v,w}+(\nabla^1f(h,v))_u\cdot w\Big)\,dt\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\int_a^b\Big(\lambda_0\,R_3[L](h,v)
+\lambda\cdot R_3[f](h,v)\Big)\,dt\,\ge\,0\label{vibrat}
\end{eqnarray}
holds for every $(w,z)\in var_{3,1}(u_*,v,\cu)$ such that
$\{t:w(t)\neq 0\}\subseteq I_{h,v}$,
$\{t:z(t)\neq 0\}\subseteq I_{h,v}$, and
${\bf y}^{v,w}(b)=0$, where ${\bf y}^{v,w}$is 
the solution of the initial value problem
$$
\left\{\begin{array}{rcl}
\dot{\bf y}^{v,w}&=&f_x^*\cdot{\bf y}^{v,w}+f_u^*\cdot w
+R_2[f](h,v)\,,\\
{\bf y}^{v,w}(a)&=&0\,\,;\end{array}\right.
$$

\vup\vup
\item[6.] if $\nu\!=\!3$ then for every member $(h,v)$ of the set 
$\ch$ there \mbox{exist} \mbox{absolutely} continuous
functions \mbox{$[a,b]\ni t\mapsto\psi^{h,v}(t)\in\rr^n$},
$[a,b]\ni t\mapsto\rho^{h,v}(t)\in\rr^n$, and 
nonnegative numbers
$\psi_0^{h,v}$,
$\theta^{h,v}$  such that 
\begin{itemize}
\item[6.i.] $\dot\rho^{h,v}(t)=-\rho^{h,v}(t)\cdot f_x^*(t)$
a.e. on $[a,b]$\,,
\item[6.ii.] the extended adjoint equation 
\begin{equation}\label{mman-adj-ext}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\dot\psi^{h,v}=-\psi^{h,v}\cdot f_x^*-
\psi_0^{h,v}\,L_x^*-\rho^{h,v}\cdot(\nabla^1f(h,v))_x
\end{equation}
holds for almost all $t\in[a,b]$,
\item[6.iii.] $\rho^{h,v}(t)=\theta^{h,v}\lambda(t)$ for all $t\in[a,b]$,
\item[6.iv.] the inequality
\begin{eqnarray}\label{bigineq-uuu-grgrgr}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int_a^b\Big(\psi^{h,v}\!\cdot\! f_u^*\cdot w
\!+\!\psi_0^{h,v}\,L_u^*\cdot w\!+\! 
\rho^{h,v}\!\cdot\!f_{xu}^*(h,w)\!\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\rho^{h,v}\cdot f_{uu}^*(v,w)
\!+\!\rho^{h,v}\!\cdot\!f_u^*\cdot z\Big)dt\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\int_a^b\Big(\psi^{h,v}\cdot R_2[f](h,v)
+\psi_0^{h,v}\,R_2[L](h,v)\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\rho^{h,v}\cdot R_3[f](h,v)\Big)\,dt\ge 0
\end{eqnarray}
holds for all $(w,z)\in var_{3,1}(u_*,v,\cu)$  such that
$w\equiv 0$ and $z\equiv 0$ outside $I_{h,v}$,
\item[6.v.] either $\,\,\theta^{h,v}\neq 0$, or 
$\,\,\mula_0^{h,v}\neq 0$, or 
the \mbox{restriction} of $\mula^{\bar v}$ to $X_{1,v}$ 
does not vanish \mbox{identically}, where $X_{1,v}$ is the linear span
of the vectors $\frac{1}{2}\ce''(u_*)(v,v)+\ce'(u_*)(w)$ 
for all pairs $(w,z)\in var_{3,1}(u_*,v,\cu)$ such that
$w\equiv 0$ and $z\equiv 0$ outside $I_{h,v}$,
\item[6.vi.] $\,\,\theta^{h,v}\lambda_0=0$,
\end{itemize}
\item[7.] for every member $(h,v)$ of $\ch$ such that the restricted 
terminal point map 
$\ce_v^0$ is 2-regular at $(0,v,\cu_v^a,\cu_v^0)$, 
the \mbox{multipliers} $\psi^{h,v}$, $\rho^{h,v}$, 
$\psi_0^{h,v}$, $\theta^{h,v}$, satisfy:
\begin{itemize}
\item[7.i.] $\rho^{h,v}(t)\cdot f_u^*(t)\equiv 0$ for almost all
$t\in I_{h,v}$, 
\item[7.ii.] the inequality
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int_a^b\Big(\psi^{h,v}\!\cdot\! f_u^*\cdot w
\!+\!\psi_0^{h,v}\,L_u^*\cdot w\!+\!\rho^{h,v}\!\cdot\!
f_{xu}^*(h,w)\!\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\rho^{h,v}\cdot f_{uu}^*(v,w)\Big)dt
+\int_a^b\Big(\psi^{h,v}\cdot R_2[f](h,v)\nonumber\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\psi_0^{h,v}\,R_2[L](h,v)+\rho^{h,v}\cdot R_3[f](h,v)\Big)\,dt\ge 0
\label{bigineq-uuu-grgrgraa}
\end{eqnarray}
is true for all $w\in var_{2,1}(u_*,v,\cu)$  such that
$w\equiv 0$ outside $I_{h,v}$,
\item[7.iii.] either $\psi_0^{h,v}\neq 0$, or the functional
\begin{eqnarray*}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
L^\infty(I_{h,v},\rr^m)\ni
w\mapsto\int_{I_{h,v}}\Big(\psi^{h,v}\!\cdot\! f_u^*\cdot w
\!+\!\psi_0^{h,v}\,L_u^*\cdot w\!\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\!\rho^{h,v}\!\cdot\!
f_{xu}^*(h,w)\!+\!\rho^{h,v}\cdot f_{uu}^*(v,w)\Big)dt\in\rr
\end{eqnarray*}
does not vanish identically on $L^\infty(I_{h,v},\rr^m)$.
\end{itemize}\end{itemize}
\end{theorem}


%\input proofs-alaska-b.tex


%\input proofs-alaska-f.tex


\ssection{A simple example}

We consider the control system in $\rr^2$ given by
$$
\left\{\begin{array}{rcl}
\dot x_1&=&u_1\phantom{u_2}+\varphi_1(u_1,u_2,x_1,x_2)\,,\\
\dot x_2&=&u_1u_2+\varphi_2(u_1,u_2,x_1,x_2)\,,\end{array}\right.
$$
where $\varphi_1,\varphi_2$ are real-valued functions on $\rr^4$ 
of class $C^3$ and such that $\varphi_1$ and $\varphi_2$ vanish at
$(0,0,0,0)$ together with all their partial derivatives of order $\le 3$. 

We choose a fixed time interval, namely, $[0,1]$. 
The controls are all the bounded measurable functions
$[0,1]\ni t\mapsto (u_1(t),u_2(t))\in\rr^2$. 
Therefore the space 
$U$ of control values is $\rr^2$, and
the space $\mathcal{U}$ of controls is $L_{\infty}([0,1],\rr^2)$. 

We let $\Xi$ be the set of all trajectory-control pairs 
$(x(\cdot),u(\cdot))$, so the members of $\Xi$ are the pairs
$(x(\cdot),u(\cdot))$ such that $u(\cdot)\in\mathcal{U}$ and 
$x(\cdot)$ is an absolutely continuous map $[0,1]\ni t\mapsto
x(t)=(x_1(t),x_2(t))$ such that 
\begin{eqnarray*}
\dot x_1(t)&=&u_1(t)\phantom{u_2(t)}+\varphi_1(u_1(t),u_2(t),x_1(t),x_2(t))\\
\dot x_2(t)&=&u_1(t)u_2(t)+\varphi_2(u_1(t),u_2(t),x_1(t),
x_2(t))
\end{eqnarray*}
for almost every $t\in[0,1]$.

We define a cost functional $J:\Xi\mapsto\rr$ by letting
$$
J(x(\cdot),u(\cdot))=\int_0^1x_1(t)^3\,dt\qquad
{\rm for}\quad (x(\cdot),u(\cdot))\in\Xi\,.
$$
Our optimal control problem is that of minimizing the 
cost among all trajectory-control pairs 
$(x(\cdot),u(\cdot))\in\Xi_0$, where
$\Xi_0$ is the set of all $(x(\cdot),u(\cdot))\in\Xi$ 
such that $x_1(0)=x_2(0)=x_1(1)=x_2(1)=0$.

We take our reference trajectory-control pair to be 
$\xi_*=(x_*,u_*)$, where $x_*(t)\equiv(0,0)$ and $u_*(t)\equiv(0,0)$.
It is clear that $(x_*,u_*)$ is not optimal, since, for 
\mbox{example}, 
it is easy to connect $(0,0)$ to $(0,0)$ by 
\mbox{means} of a trajectory 
$[0,1]\ni t\mapsto(x_1(t),x_2(t))$ such that $x_1(t)<0$ 
and $u_2(t)=0$ whenever 
$0<t<1$, and such a trajectory will have a strictly negative cost. 
(That the trajectory 
exists is completely obvious if, say, $\varphi_1\equiv\varphi_2\equiv 0$, and 
is quite easy to prove in the general case).

Our goal is to see how well our necessary \mbox{conditions}
perform in ruling out the optimality of $\xi_*$. The 
condition of Theorem \ref{theocontrol-third} says that for $\xi_*$ to
be optimal there have to exist absolutely continuous functions
$\lambda_1:[0,1]\mapsto\rr$ and $\lambda_2:[0,1]\mapsto\rr$, and a
\mbox{nonnegative} $\lambda_0$ such that a nontriviality
condition and certain inequalities hold, and the adjoint equation is
satisfied.

It is clear that $f_x^*\equiv 0$ and $L_x^*\equiv 0$. Therefore
the adjoint equation says that 
$\lambda(t)\equiv{\rm constant}\equiv (\lambda_1,\lambda_2)$,
where $\lambda_1$ and $\lambda_2$ are real numbers.
Clearly, 
\begin{equation}\label{mman-c-guau-hhh}
f_u^*\equiv\left[\begin{array}{ll} 1&0\\0&0\end{array}\right]\qquad
{\rm and}\qquad L_u^*\equiv\left[\begin{array}{l} 0\\0\end{array}\right]\,.
\end{equation}
Therefore (\ref{mman-a}) implies that 
$\lambda_1u_1\ge 0$ for all pairs $(u_1,u_2)\in\rr\times[0,\infty[$.
It follows that $\lambda_1=0$. 

If $h=(h_1,h_2)$ and $v=(v_1,v_2)$ we have 
\begin{equation}\label{mman-c-guau-ggg}
R_2[f](h,v)=
\left[\begin{array}{l} 0\\v_1v_2\end{array}\right]\,,\quad
R_2[L](h,v)=0\,.
\end{equation}

To satisfy Hypothesis (C6) we have to make sure that every pair
$(h,v)=((h_1,h_2),(v_1,v_2))\in\ch$ is such that
\begin{equation}\label{aredet}
h_1(t)=\int_0^tv_1(s)\,ds\,,\quad h_2(t)\equiv 0\,,\quad h_1(1)=0\,.
\end{equation}
This means
that we can choose $v_1$ and $v_2$ to be arbitrary, provided only that
$\int_0^1v_1(t)\,dt=0$, and then $h_1$ and $h_2$ are determined
by (\ref{aredet}).

The second-order inequality (\ref{mman-c}) says that
\begin{eqnarray}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\int_0^1\Big((\lambda_0L_u^*+\lambda\cdot f_u^*)\cdot w+\lambda_0 R_2[L]
(h,v)\nonumber\\
&\,&+\lambda(t)\cdot R_2[f](h,v)\Big)\,dt\ge 0\,.\label{mman-c-guau}
\end{eqnarray}
Then (\ref{mman-c-guau-hhh}) and
(\ref{mman-c-guau-ggg}) imply, since $\lambda_1=0$, that 
\begin{equation}\label{mman-c-guau-xxx}
\lambda_2\int_0^1 v_1(t)v_2(t)\,dt\ge 0\,.
\end{equation}
We can choose $\ch$ to consist of two pairs $\,\,(h^1,v^1)\,$, 
$\,\,(h^2,v^2)\,$, where \mbox{$v^1=(v_1^1,v_2^1)$}, $v^2=(v_1^2,v_2^2)$, 
$v_1^1$, $v_1^2$ are functions supported in the intervals
\mbox{$I_{h^1,v^1}=[0,\frac{1}{2}]$}, $I_{h^2,v^2}=
[\frac{1}{2},1]$,  not identically zero, and such  that 
$$
\int_0^1v_1^1(t)\,dt=\int_0^1v_1^2(t)\,dt=0\,,\quad
v_2^1\equiv v_1^1\,,\,\,{\rm and}\,\,_2^2\equiv -v_1^2\,.
$$
Then 
(\ref{mman-c-guau-xxx}) says that

\vup\vup\vup
\begin{equation}\label{mman-c-guau-xxx-ai}
\lambda_2\int_0^1 (v_1^1(t))^2\,dt\ge 0\,\,\,{\rm and}
\,\,\,\lambda_2\int_0^1 (v_2^1(t))^2\,dt\le 0\,.
\end{equation}

\vup\noindent Therefore $\lambda_2=0$. 
Then the third-order inequality (\ref{vibrat}) 
reduces to the assertion that
$\lambda_0\int_0^1\,R_3[L](h^i,v^i)\,dt\ge 0$
for all $i\in\{1,2\}$ such that there exists a 
$w=(w_1,w_2)\in L_\infty([0,1],\rr^2)$ having the property that 
$\int_0^1w_1(t)dt\!=\!0$, $\int_0^1w_2(t)dt\!=\!-\int_0^1v_1^i(t)v_2^i(t)dt$,
and $\{t\!:\!w(t)\!\neq\! 0\}\!\subseteq\! I_{h^i,v^i}$.

It is clear that $w$ does exist for $i=1,2$, since we can take
$w_1(t)\equiv 0$, $w_2(t)=v_1^i(t)v_2^i(t)$. Moreover,
$R_3[L](h^i,v^i)=(h_1^i)^3$. So the inequality 

\vup\vup\vup
\begin{equation}\label{mman-c-guau-yyy}
\lambda_0\int_0^1h_1^i(t)^3\,dt\ge 0
\end{equation}

\vup\vup\noindent must be true for $i=1,2$. 
Clearly, we can choose $(h^1,v^1)$ such that $h_1^1(t)\le 0$ for 
all $t$. (Take, e.g.,  $v_1^1(t)=-\sin{4\pi\,t}$ for
$0\le t\le\frac{1}{2}$.) Then $\int_0^1h_1^i(t)^3\,dt< 0$, so
(\ref{mman-c-guau-yyy}) implies that $\lambda_0\le 0$. Since
$\lambda_0\ge 0$, we conclude that $\lambda_0=0$,
contradicting the nontriviality condition, and proving that
$\xi_*$ is not optimal. \proofend

\begin{remark}
{\rm For comparison, we point out that 
\mbox{Theorem} 6.1 of 
Ledzewicz-Sch\"attler \cite{ls2}, as stated, is {\em not} applicable
to our example, since the theorem \mbox{requires} that the first derivative of
the cost functional not be identically zero (\mbox{Hypothesis}
(iii) in Page 833 of \cite{ls2}). }\proofend
\end{remark}

\vup\vup
\begin{thebibliography}{A}

\vup\vup
\bibitem{ls} Ledzewicz, U., and H. Sch\"attler, ``An extended maximum 
principle.'' {\it Nonlineanr Analysis, Theory, Methods and Applications}
{\bf 29}, 1997, pp. 159-183.

\vup\vup
\bibitem{ls2} Ledzewicz, U., and H. Sch\"attler, ``A high-order
\mbox{generalized} local maximum principle.'' {\it SIAM J. Control Optim.}
{\bf 38}, 2000, pp. 823-854.

\vup\vup
\bibitem{pontrybook}{Pontryagin,~L.S., V.G.~Boltyanskii, 
\mbox{R.~V.} \mbox{Gamkrelidze}
and E.F.~Mischenko, \textit{The \mbox{Mathematical} Theory of Optimal 
Processes.} Wiley, New York, 1962.} 

\vup\vup
\bibitem{sussram}{Rampazzo, F., and H.J.~Sussmann, 
``Set-valued \mbox{differentials} and a nonsmooth version of 
Chow's Theorem.''  In \textit{Proc. 40th IEEE Conf. Decision
and Control, Orlando, Fla., December 20001}, IEEE
\mbox{publications}, \mbox{New York}, 2001.}


\vup\vup
\bibitem{sussshyb}{Sussmann,~H.~J., ``A maximum principle
for \mbox{hybrid} optimal control problems.'' In \textit{Proc. 38th IEEE
\mbox{Conf}. Decision and Control, Phoenix, AZ, Dec. 1999}.  IEEE
publications, New York, 1999, \mbox{pp.~425--430}.} 

\vup\vup
\bibitem{susssmf2000}{Sussmann,~H.~J., ``R\'esultats r\'ecents
sur les courbes optimales.''  In \textit{15$\,^e$ Journ\'ee Annuelle
de la Soci\'et\'e Math\'emathique de France (SMF)}, Publications de
la SMF, Paris, 2000, pp.~1-52.}

\vup\vup
\bibitem{sussncn2000}{Sussmann,~H.~J., ``New theories of set-valued
\mbox{differentials} 
and new versions of the maximum \mbox{principle} of optimal control 
theory.'' In \textit{Nonlinear Control in the year 2000},
A.~Isidori, F.~Lamnabhi-Lagarrigue and  W.~Respondek Eds.,
Springer-Verlag, London, 2000, pp.~487-526.}

\vup\vup
\bibitem{susssyd2000}{Sussmann,~H.~J., ``Set-valued differentials and 
the hybrid maximum principle.''  In \textit{Proc. 39th IEEE Conf. 
Decision and Control, \mbox{Sydney}, \mbox{Australia}, \mbox{December} 
12-15, 2000}, IEEE \mbox{publications}, \mbox{New York}, 2000.}
\end{thebibliography}

\end{document}

