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%                  SET-VALUED DIFFERENTIALS AND 
%                 THE HYBRID  MAXIMUM PRINCIPLE 
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%                       by H. J. Sussmann
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%        This paper will be presented at the 39th IEEE Conference 
%        on Decision and Control, to be held in Sydney, Australia, 
%        December 12-15, 2000,  and  will appear in the conference 
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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\\
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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@hamilton.rutgers.edu\quad\quad
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Set-valued differentials and the hybrid maximum principle$\,^{*}$}
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\begin{document}

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\twocolumn[\titlestuff]
\footnotetext[1]{Research supported in part by
NSF Grant DMS98-03411-00798 and AFOSR Grant 0923.}


\vskip4.5cm

\ssection{Introduction}
\label{sec-intro}

\noindent In recent years, it has become clear that most 
(\mbox{smooth}, nonsmooth, high-order, and hybrid)  versions
of the \mbox{maximum} principle (abbr. MP) 
for finite-dimensional, deterministic optimal
control problems without state \mbox{space} constraints can be derived in a
unified way, by using a modified version of the approach of the 
classical book
\cite{pontrybook} by Pontryagin {\it et al.}---that is, by \mbox{constructing}
``packets of needle variations,'' linearly \mbox{approximating}
(that is, ``differentiating'')
these \mbox{packets} at the base \mbox{value} of the variation parameter,
and \mbox{propagating}
the resulting linear approximations to the terminal point of
the trajectory by means of the \mbox{differentials} of the 
\mbox{reference} flow maps, in
order to construct an ``\mbox{approximating} cone'' to a suitable reachable
set.  The classical \mbox{approach} must be modified 
in three basic ways.  {\it First}, the \mbox{classical} differential 
(which is used twice, first to obtain linear approximations
to the \mbox{variations} and then to propagate these approximations
to the \mbox{terminal} point) must be replaced
by other objects, called ``\mbox{generalized} differentials'' (abbr. GDs).
Examples of GDs are: 
J.~Warga's \mbox{``derivate containers''} (cf. \cite{wargac}), 
H.~Halkin's ``screens'' (cf. \cite{halkinc}), the 
\mbox{``semidifferentials''} and ``multidifferentials'' proposed by us in
previous work (cf. \cite{sussmul}), and our more recent
``generalized \mbox{differential} quotients'' (\mbox{abbr. GQDs}) and 
``\pigds'' (\mbox{abbr.} {\pigdab}s), presented in \cite{susssmf2000}
and \cite{sussncn2000}.  {\it Second,} the time-varying vector fields
that \mbox{occur} in the classical MP must be replaced by {\it flows}.
{\it Third,} the needle variations must be replaced by
\mbox{abstract} variations.  A notion of GD will yield a
\mbox{version} of the MP 
provided it satisfies some natural properties such as the chain rule
and an appropriate ``\mbox{directional}
open mapping property'' (\mbox{abbr}. DOMP). 
In the GD \mbox{setting} the ``differentials'' of a map at a point are
sets of linear maps rather than single linear maps. It \mbox{follows}
that these differentials cannot quite be used to \mbox{propagate} cones
in a natural way, since the image of a cone under a set of linear maps
is a set of cones rather than a single cone. Hence, if we agree
to call a set of cones a ``multicone,'' the natural class of 
objects that must be used as linear approximations to sets 
is that of {\it multicones}.  In particular, the ``Pontryagin 
cone'' of the classical theory is now replaced by a 
``Pontryagin multicone.''   

In this note, we propose an axiomatic definition of the concept 
of a ``generalized differentiation theory'' (abbr. GDT) and a precise
statement of the DOMP, and we outline the definitions 
of our two most \mbox{recent} GDTs, namely, the GDQs and 
PIGDs. In addition, we give a complete statement of a ``hybrid
MP for \mbox{general} \mbox{GDTs},'', which 
now amounts to saying that ``to \mbox{every}
GDT that has the DOMP is associated
a version of the hybrid MP.''   (We find it convenient not to include
the \mbox{DOMP} among the GDT axioms, thus allowing the \mbox{possibility}
that a GDT might fail to have the \mbox{DOMP} and consequently 
fail to give rise to a \mbox{MP}. On the \mbox{other} hand, we choose
to include a form of the chain rule, 
\mbox{thereby} disqualifying concepts such as
Clarke's ``\mbox{generalized} \mbox{Jacobian}'' (cf. \cite{clarkebook}),
and Halkin's ``\mbox{shields}'' (cf. \cite{halkinc}). \mbox{Finally}, 
we limit ourselves to theories in which the GDs of 
a set-valued map $F$ from a $C^1$ manifold $M$ to a 
$C^1$ manifold $N$ at a point $(x,y)\in M\times N$ are 
{\it \mbox{nonempty} \mbox{compact} sets of linear maps} from 
$T_xM$ to $T_yN$---where $T_qQ$ denotes the tangent space
of $Q$ at $q$---thereby \mbox{excluding} theories due to Ioffe, Mordukhovich
and others, where the GDs are different kinds of \mbox{objects}. We are fully
aware that these choices 
are somewhat arbitrary, and that, when
a truly definitive version is achieved, the details of the definitions
might have to be modified.) 


\ssection{Preliminaries}\label{notprem}

\noindent By a  {\em set-valued map} (abbr. SVM) we mean a triple 
$F=(A,B,G)$ such that $A$ and $B$ are sets and $G$ is a subset of
$A\times B$. The sets $A$, $B$, $G$ are, respectively, the {\em
source}, {\em target}, and {\em graph} of $F$, and we
write $A=\so(F)$, $B=\ta(F)$, $G=\gr(F)$.  If $x$ is any object, we
write $F(x)=\{y:(x,y)\in\gr(F)\}$.  (Hence $F(x)=\emptyset$ unless
\mbox{$x\in\so(F)$}.)  The sets
$\dom(F)=\{x\in\so(F):F(x)\neq\emptyset\}$,
$\im(F)=\bigcup_{x\in\so(F)}F(x)$, are, respectively, the {\bem
domain} and {\em image} of $F$. If $F=(A,B,G)$ is an SVM, we
say that $F$ is an {\em SVM from $A$ to $B$}, and write
$F:A\tto B$.  We use $SVM(A,B)$ to denote the set of all SVMs
from $A$ to $B$.

The expression ``ppd map'' stands for ``possibly 
\mbox{partially} defined
(that is, not necessarily everywhere \mbox{defined})
ordinary (that is, single-valued) map,'' and we write
$f:A\brokarrow B$ to indicate that $f$ is a ppd map from $A$ to
$B$.  A {\it time-varying ppd map} from a set $A$ to a set $B$ is a
ppd map from $A\times\rr$ to $B$.

\medbreak
\ssection{GDTs}\label{gdts}

\noindent We use $\zz_+$ to denote the set of all nonnegative
\mbox{integers}. If $k\in\zz_+$, then $\mathcal{M}^k$,
$\mathcal{SVM(M}^k)$ will denote, \mbox{respectively}, the class of all
finite-dimensional Hausdorff manifolds of class $C^k$ without
boundary, and the class of all \mbox{SVMs} $F$ such that
$\so(F)\!\in\!\mathcal{M}^k$ and $\ta(F)\!\in\!\mathcal{M}^k$.  If
$k\ge 1$ and \mbox{$x\!\!\in\!\! M\!\!\in\!\!\mathcal{M}^k$}, then
$T_xM$, $T_x^*M$, $TM$, $T^*M$ will denote, respectively, the
\mbox{tangent} and cotangent \mbox{spaces} to $M$ at $x$, and the
tangent and cotangent bundles of $M$. Clearly, $TM$ and $T^*M$ belong
to $\mathcal{SVM(M}^{k-1})$.

If $M\in\mathcal{M}^1$, $x\in M$, and $S\subseteq M$, then
$T_x^BS$ will denote the {\em Bouligand 
tangent cone} to $S$ at $x$. By \mbox{definition}, $T_x^BS$  is the set of
all tangent vectors $v\in T_xM$ such that there exist a sequence
$\{(x_j,h_j)\}_{j\in\nn}$ of points of $S\times\rr$ having the property that
$h_j>0$ for all $j$, $h_j\downarrow 0$ as $j\to\infty$, and 
$\lim_{j\to\infty}h_j^{-1}\Big(\varphi(x_j)-\varphi(x)-h_j(v\varphi)\Big)=0$
for all functions $\varphi\in C^1(M,\rr)$.

We write $\mathcal{RLS}$,  $\mathcal{FDRLS}$, 
to denote, respectively,  the class of all linear spaces over $\rr$, and 
the class of all $X\in\mathcal{RLS}$ that are finite-dimensional.
If $X,Y\in\mathcal{RLS}$,  then 
${\rm Lin}(X,Y)$ denotes the set of all $\rr$-linear maps from
$X$ to $Y$. A subset of ${\rm Lin}(X,Y)$ will be called a {\em 
\mbox{linear} \mbox{multimap}} from $X$ to $Y$. 
A set of linear multimaps from $X$ to $Y$
will be called a {\em linear multimultimap} from $X$ to $Y$. 
We use ${\rm LinM}(X,Y)$, ${\rm LinMM}(X,Y)$,
to denote, \mbox{respectively}, 
the set of all linear  multimaps from $X$ to $Y$
and the set of \mbox{all} linear  multimultimaps from $X$ to $Y$.
If $X,Y\in\mathcal{FDRLS}$,  then 
we use ${\rm LinM}_c(X,Y)$, ${\rm LinMM}_c(X,Y)$,
\mbox{respectively}, to denote the set of all
nonempty compact linear multimaps from $X$ to $Y$, and the set of all
subsets of ${\rm LinM}_c(X,Y)$. The members of 
${\rm LinMM}_c(X,Y)$ are the {\em compact linear multimultimaps}
from $X$ to $Y$.

A {\em generalized differentiation theory} (abbr. GDT)
is a correspondence $\mathcal{D}$ that assigns to every 
$F\in\mathcal{SVM}(\mathcal{M}^1)$,
every point $(x,y)\in\so(F)\times\ta(F)$,
and every subset $S$ of the source $\so(F)$, a compact linear multimultimap
$\mathcal{D}(F;x,y;S)\in{\rm LinMM}_c(T_xM,T_yY)$, in such a way that
the following axioms are satisfied:

\ben 
\vspace{-6pt}
\item[1.] If $M,N\!\in\!\mathcal{M}^1$,  $f\!:\!M\!\mapsto\! N$ is a map 
of class $C^1$, \mbox{$x\!\in\! M$}, and 
$y\!=\!f(x)$, then $\{Df(x)\}\!\in\!\mathcal{D}(f;x,y;M)$. 

\vspace{-8pt}
\item[2.] ({\bem Monotonicity}) Whenever 
$F_1,F_2\in\mathcal{SVM}(\mathcal{M}^1)$,
$M=\so(F_1)=\so(F_2)$, $N=\ta(F_1)=\ta(F_2)$, $x\in M$, $y\in N$, 
$S\subseteq M$, $\Lambda\in\mathcal{D}(F_1;x,y;S)$, 
$\Lambda'\in{\rm LinM}_c(T_xM,T_yN)$, 
$\gr(F_1)\subseteq\gr(F_2)$ and 
$\Lambda\subseteq\Lambda'$, it follows that
$\Lambda'\in\mathcal{D}(F_2;x,y;S)$.

\vspace{-8pt}
\item[3.] {\bem (The chain rule)}  If $F_i\!\in\!\mathcal{SVM}(\mathcal{M}^1)$
and \mbox{$S_i\!\subseteq\! M_i$} for $i\!=\!1,2$,  
$M_1\!=\!\so(F_1)$, \mbox{$M_2\!=\!\ta(F_1)\!=\!\so(F_2)$}, 
\mbox{$M_3\!=\!\ta(F_2)$}, $x_i\!\in\! M_i$ for $i\!=\!1,2,3$,
$F_1(S_1)\!\subseteq\! S_2$, and
$\Lambda_2\!\in\!\mathcal{D}(F_2;x_{2},x_{3};S_2)$, then 
\mbox{$\Lambda_2\circ\Lambda_1\!\in\!\mathcal{D}(F_2\circ F_1;x_{1},x_3;S_1)$},
if either (a)~$F_1$ is a diffeomorphism of class
$C^1$ from a neighborhood $U_1$ of $x_1$ onto a neighborhood
$U_2$ of $x_2$, $F_1(x_1)=x_2$, $F_1(S_1\cap U_1)=S_2\cap U_2$, 
and \mbox{$\Lambda_1=\{DF_1(x_1)\}$}, or
(b)~$F_1$ is a single-valued map of class $C^1$ \mbox{defined} on a 
neighborhood $U$ of $x_1$, \mbox{$F_1(x_1)=x_2$},
\mbox{$\Lambda_1\in\mathcal{D}(F_1;x_{1},x_{2};S_1)$},
and \mbox{$\Lambda_1=\{DF_1(x_1)\}$}, or
(c)~$\Lambda_1\in\mathcal{D}(F_1;x_{1},x_{2};S_1)$,
$\Lambda_1\cdot T_{x_1}^BS_1\subseteq T_{x_2}^BS_2$ and,
for some coordinate chart $\kappa\!:\!U\!\to\!\rr^{n}$ on $M_2$
such that \mbox{$x_2\!\in\! U$} and $\kappa(x_2)=0$, there is
a \mbox{polyhedral} cone $C$ in $\rr^{n}$ such that 
\mbox{$\kappa(S_2\cap U)=C\cap\kappa(U)$}.

\vspace{-8pt}
\item[4.] {\bem (The product rule)}  If, for $i=1,2$,  
$F_i\in\mathcal{SVM}(\mathcal{M}^1)$, 
$x_i\!\in\! M_ i\!=\!\so(F_i)$, \mbox{$y_i\!\in\! N_ i\!=\!\ta(F_i)$}, 
$S_i\subseteq M_i$, 
and $\Lambda_i\in\mathcal{D}(F_i;x_{i},y_i;S_i)$, then 
$\Lambda_1\times\Lambda_2$ belongs to 
$\mathcal{D}(F_1\times F_2;(x_1,x_2),(y_1,y_2);S_1\times S_2)$.
\een

\ssection{The directional open mapping property}\label{domptrans}

\noindent We say that a GDT $\mathcal{D}$ has the {\bem directional 
open mapping property} if the following statement is true:.

\medbreak
\noindent{\bf (DOMP)}\ {\em 
Assume that $n,m\in\zz_+$, 
$F\!\in\!\mathcal{SVM}(\rr^n,\rr^m)$,  $v\!\in\!\rr^m$,
\mbox{$C$ is a closed} convex cone in $\rr^n$, 
$\Lambda$ \mbox{belongs} to $\mathcal{D}(F;0,0;C)$, and 
\mbox{$v\!\in\!\bigcap_{L\in\Lambda}{\rm Int}(LC)$}. Then 
there exists a closed convex cone $D$ in $\rr^n$ such that
$v\in{\rm Int}(D)$, having the property that
for every $\delta\!\in]0,\infty[$ there exists
an $\eps(\delta)\!\in]0,\infty[$ such that
\fbox{$D\!\cap\!\{y\!\in\!\rr^m\!:\!\|y\|\!\le\!\eps(\delta)\}\!\subseteq\!
F(C\!\cap\!\{x\!\in\!\rr^n\!:\!\|x\|\!\le\!\delta\})$}.}\proofend


We say that $\mathcal{D}$ has the {\bem strong directional 
open \mbox{mapping} property} if

\medbreak
\noindent{\bf (SDOMP)}\ {\em Whenever
$n,\,m,\,F,\,v,\,C,\,\Lambda$ \mbox{satisfy} the
\mbox{hypotheses} of {\bf (DOMP)}, it follows that
$D$ and the \mbox{function} $\delta\mapsto\eps(\delta)$
can be chosen so that the following
stronger conclusion holds:
\ben

\vspace{-5pt}\item[(*)] if $y\!\in\! D$ and $\|y\|\!\le\!\eps(\delta)$ then 
there \mbox{exists} a compact connected subset $Z_y$ of
\mbox{$\{x\!\in\!C\!:\!\|x\|\!\le\!\delta\}\!\times\![0,1]$} such that 
\mbox{$(0,0)\!\in\!Z_y$}, 
$(x,1)\!\in\! Z_y$ for some $x\in C$ such that $\|x\|\le\delta$,
and $ry\!\in\!F(x)$ whenever $0\!\le\!r\!\le\!1$ and 
$(x,r)\!\in\!Z_y\,.\,\,\diamondsuit$
\een
}

\noindent If, in statements (DOMP) (resp. (SDOMP)), 
the \mbox{function} \mbox{$\delta\!\mapsto\!\eps(\delta)$}
can be chosen to be linear for \mbox{small} enough $\delta$, then we say 
that $\mathcal{D}$ has the DOMP (resp. SDOMP) {\bem with linear rate}.

\ssection{Approximating multicones}\label{ap-mult}

\noindent Associated to each GDT $\mathcal{D}$ there is a notion of
``\mbox{$\mathcal{D}$-approximating} multicone to a set at a point.''  
\mbox{Recall}
that if $X\in\mathcal{RLS}$ then a {\bem convex multicone} in $X$ is a 
nonempty set of convex cones in $X$.

\begin{definition}\label{defappo}
If $\mathcal{D}$ is a GDT, $M\in\mathcal{M}^1$, 
$S\subseteq M$ and $x\in S$, a 
{\bem $\mathcal{D}$-approximating
multicone to $S$ at $x$} is a convex multicone $\mathcal{C}$ in
$T_xM$ such that there exist $X\in\mathcal{FDRLS}$, 
$F:X\tto M$, a closed convex cone $K$ in $X$, and a
$\Lambda\in\mathcal{D}(F;0,x;K)$, such that
$F(K)\subseteq S$ and $\mathcal{C}=\{LK:L\in\Lambda\}$.\proofend
\end{definition}

If $X\in\mathcal{RLS}$, then $X^\dagger$ denotes the dual space
of $X$. If $X\in\mathcal{FDRLS}$, then we identify
$X^{\dagger\dagger}$ to $X$ in the usual way.
If $\mathcal{C}$ is a convex multicone in $X$, the {\bem polar}
of $\mathcal{C}$ is the (not necessarily convex) cone 
$\mathcal{C}^\perp=\bigcup\{C^\perp:C\in
\mathcal{C}\}$, where 
$C^\perp$ denotes  the usual polar cone of $C$, that is,
$C^\perp=\{y\in X^\dagger:\langle y,x\rangle\le 0\,\,{\rm whenever}
\,\,x\in C\}$. 


\medbreak
\ssection{Generalized differential quotients (GDQs)}\label{sec-gdq}

\noindent If $X$, $Y$ are metric spaces, then 
$SVM_{comp}(X,Y)$ will denote the subset of 
$SVM(X,Y)$ whose \mbox{members} are
the set-valued maps from $X$ to $Y$
that have a \mbox{compact} graph.
We say that a sequence  $\{F_j\}_{j\in\nn}$ of 
\mbox{members} of $SVM_{comp}(X,Y)$ 
{\bem inward graph-converges} to an $F\in SVM_{comp}(X,Y)$---and
write $F_j\togr F$---if
for every open subset $\Omega$ of $X\times Y$ such that 
$\gr(F)\subseteq\Omega$ there exists a
$j_\Omega\in\nn$ such that $\gr(F_j)\subseteq\Omega$
whenever $j\ge j_\Omega$.

\begin{definition}\label{defregu} 
Assume that $X$, $Y$ are  metric spaces.  
A {\bem regular set-valued map} from $X$ to $Y$ is a set-valued map 
$F\in{}SVM(X,Y)$ such that 
\begin{itemize}
\vspace{-6pt}
\item{} for every compact subset $K$ of $X$, the restriction
$F\restr K$ of $F$ to $K$ belongs to 
$SVM_{comp}(K,Y)$  and is a limit---in the sense of inward 
graph-convergence---of a sequence of continuous
single-valued maps from $K$ to $Y$. 
\end{itemize}
\vspace{-6pt}\noindent We use $\regu{X}{Y}$ to denote the set of all regular
\mbox{set-valued}  maps from $X$ to $Y$. \proofend
\end{definition}
It is easy to see that if $F:X\mapsto Y$ is an ordinary 
(that is, single-valued and everywhere defined) map, then
$F$ belongs to $\regu{X}{Y}$ if and only if $F$ is continuous.

It is not hard to prove the following.
\begin{theorem}\label{comporeg-13}
Assume that $X$, $Y$, $Z$ are metric \mbox{spaces}. Let
\mbox{$F\in\regu{X}{Y}$}, $G\in\regu{Y}{Z}$. Then the
\mbox{composite} map
$G\circ F$ belongs to $\regu{X}{Z}$.\proofend
\end{theorem}



\vspace{-8pt}
\begin{definition}\label{defquasi} 
Let $m,n\in\zz_+$, let $F:\rr^m\tto\rr^n$ be a set-valued
map, and let $\Lambda$ be a nonempty compact subset of $\rr^{n\times m}$. 
Let $S$ be a subset of  $\rr^m$.  
We say that $\Lambda$ is a {\bem generalized differential quotient}
(abbreviated ``GDQ'') {\bem of 
$F$ at $(0,0)$ in the direction of $S$}, and write
$\Lambda\in  GDQ(F;0,0;S)$, 
if for every positive real number $\delta$ there exist $U$, $G$ such that 
\ben
\vspace{-6pt}
\item[1.] $U$  is a  compact neighborhood of $0$ in $\rr^m$ and $U\cap S$
is compact;

\vspace{-8pt}
\item[2.] $G$  is a  regular set-valued map from $U\cap S$ to the 
$\delta$-neighborhood  $\Lambda^\delta$ of $\Lambda$ in $\rr^{n\times m}$;

\vspace{-8pt}
\item[3.] $G(x)\cdot x\subseteq F(x)$
for every $x\in U\cap S$.\proofend
\een
\end{definition}

\noindent If $M$, $N$ are $C^1$ manifolds, 
\mbox{$\bar x\!\in\! M$, $\bar y\!\in\! N$, $S\!\subseteq \!M$}, 
and \mbox{$F:M\tto N$}, then we can define a linear \mbox{multimultimap}
$GDQ(F;\bar x,\bar y;S)\in{\rm LinMM}_c(T_{\bar x}M,T_{\bar y}N)$ 
by picking coordinate charts \mbox{$M\ni x\to\xi(x)\in\rr^m$}, 
\mbox{$N\ni y\to\eta(y)\in\rr^n$}---where \mbox{$m={\rm dim}\,M$,
$n={\rm dim}\,N$}---defined near $\bar x$, $\bar y$ such 
that $\xi(\bar x)=0$, $\eta(\bar y)=0$, and
declaring a subset $\Lambda$ of ${\rm Lin}(T_{\bar x}M,T_{\bar y}N)$
to \mbox{belong} to $GDQ(F;\bar x,\bar y;S)$  if 
the multimap 
$D\eta(\bar y)\circ\Lambda\circ D\xi(\bar x)^{-1}$ is in $GDQ(\eta\circ
F\circ\xi^{-1};0,0;\xi(S))$.  It turns out that, with this definition,
{\bem the set $GDQ(F;\bar x,\bar y;S)$  does not depend on the choice 
of the charts $\xi$, $\eta$.} Moreover, the following four results
can be proved.

\begin{theorem}\label{isgdt}
GDQ is a GDT.\proofend
\end{theorem}

\begin{theorem}\label{hasomp}
GDQ has the strong directional open mapping property with linear 
rate.\proofend
\end{theorem}

\begin{theorem}\label{contidif}
If $F:\rr^n\mapsto\rr^m$ is a continuous map,
$x\in\rr^n$, and $F$ is classically differentiable at $x$, then
$\{DF(x)\}\in GDQ(F;x,F(x);\rr^n)$.\proofend
\end{theorem}

\begin{theorem}\label{conticlark}
If $F:\rr^n\mapsto\rr^m$ is Lipschitz-continuous,
and $x\in\rr^n$, then the Clarke generalized Jacobian
$\partial F(x)$ belongs to $GDQ(F;x,F(x);\rr^n)$.\proofend
\end{theorem}



\noindent It is easy to exhibit maps that have  GDQs at a point $\bar x$
but are not classically 
differentiable at $\bar x$ and do not have differentials
at $\bar x$ in the sense of other theories such as Clarke's 
\mbox{generalized}
Jacobians, Warga's derivate containers, or our 
\mbox{``semidifferentials'' and ``multidifferentials''.} (A simple 
example is provided by the function $f:\rr\to\rr$ given by 
$f(x)=x\sin{1/x}$ if $x\neq 0$, and $f(0)=0$.  The set
$[-1,1]$ belongs to $GDQ(f;0,0,\rr)$, but is not a differential of $f$ at $0$ 
in the sense of any of the other theories.)

\medbreak
\ssection{Vector fields, flows, variational generators}\label{vecflovar}

\vupp
\begin{definition}
If  $M\in\mathcal{M}^1$, then
a {\em time-varying \mbox{vector} field} (abbreviated ``TVVF'') on $M$
is a ppd map \mbox{$f:M\times\rr\brokarrow TM$} such that $f(x,t)\in T_xM$
\mbox{whenever} $(x,t)\in\dom(f)$. More generally, if $E\in\mathcal{FDRLS}$,
an {\em E-augmented time-varying vector field} on $M$
is a ppd map 
$$ 
M\!\times\!\rr\!\ni\!(x,t)\!\brokarrow\!f^{@}(x,t)\!=\!
(f^0(x,t),f(x,t))\!\in\! E\!\times\! TM
$$
such that $f(x,t)\in T_xM$ whenever $(x,t)\in\dom(F)$.  

We use $\cvf(M)$, $\cvf(M;E)$, to denote, \mbox{respectively},
the set of all TVVFs on $M$ and the set of all
$E$-augmented  TVVFs on $M$. 

The abbreviation ``LAC'' will stand
for ``locally \mbox{absolutely} continuous.''
If $f\!\in\!\cvf(M)$, then a {\bem \mbox{trajectory}} 
(or {\bem integral curve}) of $f$ is a LAC
map \mbox{$\xi\!:\!I\!\mapsto\!M$}, defined on a 
\mbox{nonempty} \mbox{subinterval}
$I$ of $\rr$, such that \mbox{$(\xi(t),t)\in\dom(f)$} and
\mbox{$\dot\xi(t)=f(\xi(t),t)$} for almost all $t\in I$.
If \mbox{$f^@\!\in\!\cvf(M;E)$}, and
$f^@\!=\!(f^0,f)$, then
a {\bem \mbox{trajectory}} (or {\bem \mbox{integral} curve}) of $f^@$ is a 
trajectory of $f$, and an {\bem \mbox{augmented} \mbox{trajectory}} 
(or {\bem \mbox{augmented} integral curve}) of $f^@$ is a LAC map 
\mbox{$I\!\ni\!t\!\to\!\xi^@(t)\!=\!(\xi^0(t),\xi(t))\!\in\!E\!\times\!M$}, 
\mbox{defined} on a nonempty subinterval 
$I$ of $\rr$, such that \mbox{$(\xi(t),t)$ belongs to $\dom(F)$},
$\dot\xi(t)=f(\xi(t),t)$ and \mbox{$\dot\xi^0(t)=f^0(\xi(t),t)$}
for a.e. $t\in I$.

If $f\!\in\!\cvf(M)$, $f^@=(f^0,f)\!\in\!\cvf(M;E)$, 
we use $\traj(f)$, $\traj_c(f)$, $\traj(f^@)$, $\traj_c(f^@)$,
$\traj^@(f^@)$, $\traj_{c}^@(f^@)$,
to denote, respectively,  the set of all 
trajectories of $f$, the set of all $\xi\in\traj(f)$ whose domain
is a compact interval, the sets $\traj(f)$,  $\traj_c(f)$, 
the set of all augmented
trajectories of $f^@$, and the set of all $\xi^@\in\traj^@(f^@)$ whose domain
is a compact interval. \proofend 
\end{definition}

\begin{definition}
Assume that $M\in\mathcal{M}^1$, $E\in\mathcal{FDRLS}$,
\mbox{$f\!\in\!\cvf(M)$}, and $f^@=(f^0,f)\!\in\!\cvf(M;E)$.
If $(x,b,a)\in M\times\rr\times\rr$, we define

\noindent$\,\,\,\,\,\,\,\,\Phi^f(x,b,a)\!=\!\{\xi(b):\xi\in\traj(f):
\xi(a)=x\}\,$,

\noindent$\,\,\,\,\,\Phi^{f^@}(x,b,a)\!=\!\Phi^{f}(x,b,a)\,$,

\noindent\mbox{$\Phi^{@,f^@}(x,b,a)\!=\!\{\xi^@(b)\!:\!\xi^@\!\!
\in\!\!\traj^@(f^@),\xi^@(a)\!\!=\!\!(x,0)\}$}.

\medbreak\noindent
The set-valued maps $\Phi^f:M\times\rr\times\rr\tto M$,
\mbox{$\Phi^{@,f^@}:M\times\rr\times\rr\tto E\times M$}, are
called the {\it flow} of $f$ (or of $f^@$) and the {\it augmented
flow} of $f^@$. 

For each $(b,a)\in\rr\times\rr$, we define \mbox{set-valued maps}
$\Phi_{b,a}^f:M\tto\ M$ and $\Phi_{b,a}^{@,f^@}:M\tto E\times M$,
by \mbox{letting} $\Phi_{b,a}^f(x)=\Phi^f(x,b,a)$, $\Phi_{b,a}^{@,f^@}(x)=
\Phi^{@,f^@}f(x,b,a)$ 
for $x\in M$.  The set-valued maps $\Phi_{b,a}^f:M\tto M$, 
\mbox{$\Phi_{b,a}^{@,f^@}:M\tto M$},  are the {\it flow maps} of $f$
(or of $f^@$) and the {\it augmented flow maps} of $f^@$. \proofend
\end{definition}

\noindent If $E\!\in\!\mathcal{FDRLS}$ and 
\mbox{$A,B\!\in\! SVM(M,E\!\times\!M)$, then the}
``augmented composite'' $A\#{}B\in SVM(M,E\!\times\!M)$ is
defined by letting $(A\#{}B)(x)$ be, for each $x\!\in\! M$,
the set $\{(\alpha\!+\!\beta,z)\!:\!(\exists y)
((\alpha,y)\!\in\! B(x)\!\wedge\!(\beta,z)\!\in\! A(y)\}$,


\bfact
Let \mbox{$M\!\in\!\mathcal{M}^1$, $E\!\in\!\mathcal{FDRLS}$, 
$f\!\in\!\cvf(M)$}, $f^@\!=\!(f^0,f)\in\cvf(M;E)$.
Then the \mbox{flow maps} $\Phi_{b,a}^f$ satisfy the 
{\em flow identities} \fbox{$\Phi_{a,a}^f=\id_{M}$} and 
\fbox{$\Phi_{c,b}^f\circ\Phi_{b,a}^f=\Phi_{c,a}^f$},
if $a,b,c\in\rr$ and $a\le b\le c$. The augmented 
flow maps satisfy \fbox{$\Phi_{a,a}^{@,f^@}(x)=\{(0,x)\}$} and 
\fbox{$\Phi_{c,a}^{@,f^@}=\Phi_{c,b}^{@,f^@}\#{}\Phi_{b,a}^{@,f^@}$}.\proofend
\efact

If $k,\ell\in\zz_+$, $M\in\mathcal{M}^k$, and $0\le\ell<k$, 
we let $J^\ell_{VF}(M)$ be the space of all $\ell$-jets 
of vector fields on $M$. In particular, $J^0_{VF}(M)$ is
canonically identified with $TM$. We also let 
$J^{-1}_{VF}(M)\equaldef M$. If $\ell,\ell',k\in\zz_+\cup\{-1\}$
and $\ell\le\ell'<k$, then 
there is a natural projection 
$\Pi_{M,\ell',\ell}:J^{\ell'}_{VF}(M)\mapsto J^{\ell}_{VF}(M)$.
If $x\in M$, we write $J^\ell_{VF}(M;x)\equaldef
\Pi_{M,\ell,-1}^{-1}(x)$, so 
$J^\ell_{VF}(M;x)$  is the set of all $\ell$-jets at $x$ 
of vector fields on $M$. The sets 
$J^\ell_{VF}(M;x)$  are linear spaces, and each 
$\Pi_{M,\ell',\ell}$ maps
$J^{\ell'}_{VF}(M;x)$ linearly to $J^{\ell}_{VF}(M;x)$.
In particular, if $x\in M$ and $v\in T_xM$, the space
$\tilde J^1_{VF}(M;x,v)\equaldef\Pi_{M,1,0}^{-1}(x,v)$
is affine. Clearly, ${\rm dim}\,\tilde J^1_{VF}(M;x,v)=n^2$, 
if $n={\rm dim}\,M$.


\begin{definition}
Let \mbox{$M\!\in\!\mathcal{M}^2$}, let $I\subseteq\rr$ be a
\mbox{nontrivial} interval, and let $\xi:I\mapsto M$ be a LAC
curve. A {\bem \mbox{variational} bundle} over $\xi$ is a
measurable set-valued map $I\!\ni\! t\!\mapsto\!\mathcal{V}(t)\!
\subseteq\!\tilde J^1_{VF}(M;\xi(t),\dot\xi(t))$ having 
\mbox{compact convex} nonempty
values. We use $VB(\xi)$ to denote the set of all variational bundles
over $\xi$. If $\mathcal{V}\!\in\! VB(\xi)$, we \mbox{call} $\mathcal{V}$ 
{\em locally integrable} (abbr. LI) if, for every $C^2$ coordinate chart
$\kappa:U\to\rr^n$ on $M$, and every compact subinterval $J$ of $I$
such that $\xi(J)\subseteq U$, the absolute values of all the
components relative to $\kappa$ of all the members of
$\mathcal{V}(t)$, for $t\in J$, are bounded by integrable
functions of $t$. We use $VB_{LI}(\xi)$ to denote the set of 
all LI $\mathcal{V}\in VB(\xi)$.\proofend
\end{definition}

If $\mathcal{V}\in VB_{LI}(\xi)$, we write 
$\Gamma(\mathcal{V})$ to denote the set of all measurable selections
of $\mathcal{V}$.  Using coordinate \mbox{charts}, it easy to 
embed $\Gamma(\mathcal{V})$ as a convex weakly \mbox{compact} 
subset of a Banach space. If $\gamma\in\Gamma(\mathcal{V})$,
then $\gamma$ gives rise to a {\em connection 
$\nabla^\gamma$ along $\xi$},
in the \mbox{manner} \mbox{described} in detail in \cite{sussfree}.
(In local \mbox{coordinates}, if we write
$\partial_i\equaldef\frac{\partial\,\,\,\,\,}{\partial x_i}$,
pick for each $t$ a $C^1$ vector field
$X_t$ near $\xi(t)$ such that the $1$-jet of $X_t$ at 
$\xi(t)$ is \mbox{$\gamma(t)$, \ and write 
$\,X_t(x)=\sum_{i=1}^nX_t^i(x)\partial_i\,$, \ we can define}

\vspace{1mm}
\noindent  the ``covariant derivative'' $\nabla^\gamma v$
of a LAC vector field
$t\mapsto v(t)=\sum_{i=1}^nv^i(t)\partial_i 
\in T_{\xi(t)}M$ along $\xi$
by letting $\nabla^\gamma v(t)=\sum_{i=1}^n\Big(\dot v^i(t)
-\sum_{j=1}^n(\partial_jX_t^i)(\xi(t))v^j(t)\Big)\partial_i$.
To see that this makes intrinsic sense fix a $t$ such that
$\xi$ and $v$ are differentiable at $t$, and
pick, for $s$ near $t$, a $C^1$ vector field $V_s$ such that 
$V_s(\xi(s))=v(s)$ and the map $(s,x)\mapsto V_s(x)$ is
differentiable at $(t,\xi(t))$. Write $V_s(x)=
\sum_{i=1}^nV^i(s,x)\partial_i$. Then
$v^i(s)=V^i(s,\xi(s))$, so 
\begin{eqnarray*}
\dot v^i(t)&=&\frac{\partial V^i}{\partial s}(t,\xi(t))
+\sum_{j=1}^n\frac{\partial V^i}{\partial x_j}(t,\xi(t))\cdot
\dot\xi^j(t)\\
&=&\frac{\partial V^i}{\partial s}(t,\xi(t))
+\sum_{j=1}^n\frac{\partial V^i}{\partial x_j}(t,\xi(t))\cdot
X^j_t(\xi(t))\,.
\end{eqnarray*}
Then $\nabla^\gamma v(t)=\frac{\partial V^i}{\partial s\,\,}(t,\xi(t))
+[X_t,V_t](\xi(t))$.) We can then define a {\em linear flow}
\mbox{$L^\gamma=\{L_{t,s}^\gamma:s,t\in I,s\le t\}$}, consisting
of maps $L_{t,s}^\gamma\!:\!T_{\xi(s)}M\!\mapsto\! T_{\xi(t)}M$, 
by \mbox{letting} $L_{t,s}^\gamma(v)$ be, for 
$v\!\in\!T_{\xi(s)}M$, the ``parallel translate of
$v$ to $T_{\xi(t)}M$ by means of $\nabla^\gamma$,'' that is,
the \mbox{value} at time $t$ of the unique LAC
vector field $V$ along $\xi$ such that \mbox{$V(s)=v$} and 
$\nabla^\gamma V\equiv 0$, we can then define
a {\em \mbox{linear} \mbox{multiflow}} 
\mbox{$\Lambda^{\mathcal{V}}=\{\Lambda_{t,s}^{\mathcal{V}}:
s,t\in I,s\le t\}$}, 
\mbox{consisting} of \mbox{compact} nonempty linear multimaps
$\Lambda_{t,s}^{\mathcal{V}}$ from $T_{\xi(s)}M$ to $T_{\xi(t)}M$,
by letting $\Lambda_{t,s}^{\mathcal{V}}$ be the set of all
maps $L_{t,s}^\gamma$, for all $\gamma\in\Gamma(\mathcal{V})$.


If $E\!\in\!\mathcal{FDRLS}$, it is obvious how to define \mbox{the concept} of
{\em $E$-augmented variational bundle} along a LAC curve $\xi$ on
$M$. If $\mathcal{V}^@$ is such a bundle, and $\mathcal{V}^@$ is LI,
then each measurable selection
$\gamma^@\!=\!(\gamma^0,\gamma)\in\Gamma(\mathcal{V}^@)$ gives rise to
an {\em augmented linear flow} 
\mbox{$L^{@,\gamma^@}\!=\!\{L_{t,s}^{@,\gamma^@}\!:\!s,t\!\in\!I,
s\!\le\!t\}$}, 
consisting of linear maps 
\mbox{$L_{t,s}^{@,\gamma^@}:T_{\xi(s)}M\mapsto E\times T_{\xi(t)}M$},
and this enables us to define an augmented linear
\mbox{multiflow}
$\Lambda^{@,\mathcal{V}^@}\!=\!\{\Lambda_{t,s}^{@,\mathcal{V}^@}\!:\!s,t\!\in
\! I,s\!\le\! t\}$, 
\mbox{consisting} of \mbox{compact} nonempty linear multimaps
$\Lambda_{t,s}^{@,\mathcal{V}^@}$ from $T_{\xi(s)}M$ to
\mbox{$E\times T_{\xi(t)}M$}.

\begin{definition}
Assume that \mbox{$M\!\in\!\mathcal{M}^2$}, \mbox{$E\!\in\!\mathcal{FDRLS}$}, 
$f\in\cvf(M)$, $f^@=(f^0,f)\in\cvf(M;E)$, 
$I\subseteq\rr$ is  a \mbox{nontrivial} interval, 
and $\xi:I\mapsto M$ is an \mbox{integral}
curve of $f$. Let $\mathcal{D}$ be a GDT.
An {\bem $f$-variational} (or {\bem \mbox{$f^@$-variational}})
{\bem generator of $\mathcal{D}$-differentials
over} $\xi$ is a LI  variational
bundle $\mathcal{V}$ over $\xi$ such that 
$\Lambda_{t,s}^{\mathcal{V}}$ belongs to 
$\mathcal{D}(\Phi_{t,s}^f;\xi(s),\xi(t);M)$
for all $s,t\in I$ such that $s\le t$.
An {\bem augmented $f^@$-variational
generator of $\mathcal{D}$-differentials 
over} $\xi$ is an $E$-augmented  LI  variational
bundle $\mathcal{V}^@$ over $\xi$ such that 
$\Lambda_{t,s}^{@,\mathcal{V}^@}\in\mathcal{D}(\Phi_{t,s}^{@,f^@};
\xi(s),(\theta(s,t),\xi(t));M)$
for all $s,t\in I$ such that $s\le t$, where we let
\mbox{$\theta(s,t)=\int_s^tf^{0}(\xi(r),r)\,dr$}.\proofend
\end{definition}

\begin{example}
If $f$ is defined on an open subset of $M\times I$ containing the graph
of $\xi$, and $f(x,t)$ is measurable in $t$ for each $x$, 
and Lipschitz in $x$ for each $t$ with \mbox{Lipschitz} constants $k(t)$
such that $k$ is LI, then the 
Clarke \mbox{generalized} Jacobians $\partial f_t(\xi(t))$---where
$f_t(x)\equaldef f(x,t)$---define an
$f$-variational generator of $\mathcal{D}$-differentials
over $\xi$, if $\mathcal{D}$ is the GDT of Warga derivate containers.\proofend
\end{example}

\begin{example}
Sufficient conditions for a variational bundle $\mathcal{V}$ 
to be a an $f$-variational generator of GDQ-differentials
are given in \cite{sussncn2000}.\proofend
\end{example}

\medbreak

\ssection{Points of regular $\mathcal{D}$-differentiability}
\label{regu-dif}

\vupp
\begin{definition}
\mbox{Let $\!\mathcal{D}\!$ be $\!$a $\!$GDT. 
\mbox{$\!$Assume $\!$that \mbox{$\!M\!\in\!\mathcal{M}^1$}}}, 
$E\!\in\!\mathcal{FDRLS}$, \mbox{$f^@=(f^0,f)\in\cvf(M;E)$}.  A point 
$(x,t)\in M\times\rr$ is a {\bem point of regular 
\mbox{$\mathcal{D}$-differentiability}} of $f^@$ if  
\fbox{$\{L\}\in\mathcal{D}(\Phi^{@,f^@};(x,t,t),(0,x);M\times\rr\times\rr)$},
where $L:T_xM\times\rr\times\rr\mapsto E\times T_xM$
is the linear map given by
\fbox{$L(v,\tau,\sigma)=(0,v)+(\tau-\sigma)f^@(x,t)$}
for $v\in T_xM$, $\tau\in\rr$, $\sigma\in\rr$.\proofend
\end{definition}

\ssection{A hybrid maximum principle}
\label{sec-gdq-max}

\noindent In \cite{sussncn2000}, we stated and proved a hybrid maximum
\mbox{principle} for GDQ theory. In \cite{susssmf2000}, we stated and
proved a \mbox{general}, abstract version of the maximum principle for
\mbox{arbitrary} GDTs having the DOMP. Here we present an \mbox{intermediate}
result, which can be regarded as a \mbox{special} ``hybrid systems'' case of
the result of \cite{susssmf2000}, or as an \mbox{extension} of the theorem of
\cite{sussncn2000} to an arbitrary GDT $\mathcal{D}$.  The proof is
exactly as in \cite{sussncn2000}, except that GDQs have to replaced
throughout by $\mathcal{D}$-differentials.

{}From now on, the augmentation space $E$ considered earlier will
be $\rr$, so  all ``augmented'' objects will be $\rr$-augmented. 
(By taking $E$ to be $\rr^d$ for $d>1$ one gets, in a similar way,
a maximum principle for Pareto optimality, but we will not
discuss that here.)

For our restricted purposes, let us define a {\bem hybrid
optimal control problem} to consist of the specification
of a finite sequence $(\Sigma^1,\ldots,\Sigma^\mu)$ 
of \mbox{``ordinary control systems,''} together 
with \mbox{Lagrangians} $F^{0,1},\ldots,F^{0,\mu}$ for 
$\Sigma^1,\ldots,\Sigma^\mu$, ``switching
constraints'' $\mathcal{S}^1,\ldots,\mathcal{S}^\mu$, 
and ``switching cost functions'' $\ffi^1,\ldots,\ffi^\mu$.

Precisely, each $\Sigma^i$ is a triple 
$\Sigma^i=(Q^i,\mathcal{U}^i,F^i)$ \mbox{consisting} of a
{\bem state space} $Q^i$, a {\bem controller space}
$\mathcal{U}^i$, 
and a family $F^i=\{F_\eta^i\}_{\eta\in\mathcal{U}^i}$
such that, for each $\eta\in\mathcal{U}^i$,
$F_\eta^i$ is a \mbox{ppd} time-varying vector field on $Q^i$. Each  
$F^{0,i}$ is a family $\{F_\eta^{0,i}\}_{\eta\in\mathcal{U}^i}$
of ppd functions $F_\eta^{0,i}:Q^i\times\rr\brokarrow\rr$.
For each $i\in\{1,\ldots,\mu\}$, the switching constraint 
$\mathcal{S}^i$ is a
subset of $Q^i\times\rr\times Q^{i\tilde{+}1}\times\rr$, where 
\mbox{``$i\tilde{+}1$'' means} ``$i+1$'' if $i<\mu$, and ``$1$'' if $i=\mu$. 
The switching cost \mbox{functions} are functions
$\ffi^i:Q^i\times\rr\times Q^{i\tilde{+}1}\times\rr\to\rr$. 
A {\bem \mbox{controller}}
is a $\mu$-tuple $\pmbi{\eta}=(\eta^1,\ldots,\eta^\mu)\in
\mathcal{U}^1\times\cdots\times\mathcal{U}^\mu$. A 
{\bem \mbox{trajectory}} for a controller 
$\pmb{\eta}=(\eta^1,\ldots,\eta^\mu)$
is a \mbox{$\mu$-tuple} $\pmbi{\xi}=(\xi^1,\ldots,\xi^\mu)$
with the property that, for  each $i$, $\xi^i$ is an integral
curve of  $F_{\eta^i}^i$, defined on a compact \mbox{interval} 
$I(\xi^i)=[a(\xi^i),b(\xi^i)]$.
A {\bem trajectory-control pair} (abbr. TCP) is a pair 
$(\pmbi{\xi},\pmbi{\eta})$ 
such that $\pmbi{\eta}$ is a controller and 
$\pmbi{\xi}$ is a trajectory for $\pmbi{\eta}$.
A TCP $(\pmbi{\xi},\pmbi{\eta})$ is 
{\bem \mbox{admissible}} if, for each $i$, 
$\sigma^i(\pmbi{\xi})\in\mathcal{S}^i$, where $\hat\xi^i(t)\equaldef
(\xi^i(t),t)$ and $\sigma^i(\pmbi{\xi})\equaldef
(\hat\xi^i(b(\xi^i)),\hat\xi^{i\tilde{+}1}(a(\xi^{i\tilde{+}1})))$,
and the \mbox{functions} 
$I(\xi^i)\ni t\mapsto F^{0,i}_{\eta^i}(\hat\xi^i(t))\in\rr$ are 
a.e. defined---that is, $\hat\xi^i(t)\in\dom(F_{\eta^i}^{0,i})$ 
for a.e. $t\in I(\xi^i)$---and Lebesgue \mbox{integrable}.
The {\bem cost} of an admissible  TCP
$(\pmbi{\xi},\pmbi{\eta})$ is the number
$$
J(\pmbi{\xi},\pmbi{\eta})\equaldef
\sum_{i=1}^{\mu}\int_{a(\xi^i)}^{b(\xi^i)}
F^{0,i}_{\eta^i}(\hat\xi^i(t))\,dt+\sum_{i=1}^{\mu}\ffi^i
\Big(\sigma^i(\pmbi{\xi})\Big)\,.
$$
An {\bem optimal TCP} is an 
admissible TCP $(\pmbi{\xi},\pmbi{\eta})$ such that 
$J(\pmbi{\xi},\pmbi{\eta})\le J(\pmbi{\xi}',\pmbi{\eta}')$ 
for every admissible TCP $(\pmbi{\xi}',\pmbi{\eta}')$.

For each $i$, we define the {\bem augmented
dynamics} to be the family
$F^{@,i}=\{F_\eta^{@,i}\}_{\eta\in\mathcal{U}^i}$
of ppd maps from $Q^i\times\rr$ to 
$\rr\times TQ^i$ given by 
$F_\eta^{@,i}(q,t)=(F_\eta^{0,i}(q,t),F_\eta^i(q,t))$ for 
$(q,t)\in Q^i\times\rr$.

We write $\mathcal{Q}^i=Q^i\times\rr$, 
$\mathcal{Q}_{\#}^i=\mathcal{Q}^i\times\mathcal{Q}^{i\tilde{+}1}$,
and make the following assumptions.

\begin{itemize}
\vupp 
\item[H1.] $\mathcal{D}$ is a GDT that has the DOMP.

\vupp 
\item[H2.] $(\pmbi{\xi},\pmbi{\eta})=((\xi^1,\ldots,\xi^\mu),
(\eta^1,\ldots,\eta^\mu))$ is an
\mbox{admissible} TCP, and 
$a^i\!=\!a(\xi^i)$, $b^i\!=\!b(\xi^i)$, 
\mbox{$x^i\!=\!\xi^i(a^i)$, $y^i\!=\!\xi^i(b^i)$,} for $i=1,\ldots,\mu$. 

\vupp
\item[H3.] For $i\!=\!1,\ldots,\mu$, $\Lambda^{@,i}$
is an \mbox{augmented} \mbox{$F_{\eta^i}^{@,i}$-variational} generator 
of $\mathcal{D}$-differentials over $\xi^{i}$.

\vupp
\item[H4.] For $i=1,\ldots,\mu$, 
$\Xi_-^i=(x^i,a^i)$ and $\Xi_+^i=(y^i,b^i)$ are
points of regular $\mathcal{D}$-differentiability of
$F_{\eta^i}^{@,i}$.

\vupp
\item[H5.] For each $i\in\{1,\ldots,\mu\}$, $\mathcal{C}^i$ is a 
convex multicone in $T_{\Xi^i_+}\Q^i\times T_{\Xi^{i\tilde{+}1}_-}
\Q^{i\tilde{+}1}$ (that is, in 
$T_{(\Xi^i_+,\Xi^{i\tilde{+}1}_-)}\Q_{\#}^i$), and is 
a $\mathcal{D}$-approximating multicone for  
$\mathcal{S}^i$ at the switching point 
$P^i=\sigma^i(\pmbi{\xi})=(\Xi_+^i,\Xi_-^{i\tilde{+}1})$.

\vupp
\item[H6.] For each $i\in\{1,\ldots,\mu\}$, 
$\Omega^i$ is a subset of the \mbox{dual} space 
$\Big(T_{y^i}Q^i\times\rr\times T_{x^{i\tilde{+}1}}
Q^{i\tilde{+}1}\times\rr\Big)^\dagger$
(that is, of $\Big(T_{\Xi_+^i}\Q^i\times T_{\Xi_-^{i\tilde{+}1}}
\Q^{i\tilde{+}1}\Big)^\dagger$), 
and $\Omega^i$ belongs to $GDQ(\ffi^i;P^i,\ffi^i(P^i);
Q^i\times\rr\times Q^{i\tilde{+}1}\times\rr)$. 


\vupp
\item[H7.] Each system
$\Sigma^i$ is invariant under time-interval substitutions.
(That is, if $\eta,\zeta$ belong to $\mathcal{U}^i$,
\mbox{$\xi\in\czer{[a,b]}{Q^i}\cap\traj_c(F_\eta^i)$}, and $J$ is a 
compact subinterval of $[a,b]$ such that $(\xi(t),t)\in\dom(F_\zeta^i)$ 
for $t\in J$, then there exists a controller
$\theta\in\mathcal{U}^i$ such that 
$F_\theta^i(q,t)=F_\eta^i(q,t)$ whenever $q\in Q^i$, 
$t\in[a,b]$, $t\notin J$, and 
$F_\theta^i(q,t)=F_\zeta^i(q,t)$ whenever $q\in Q^i$, $t\in J$.)
\end{itemize}

\def\nus{\nu}
\def\xis{\xi}
\def\etas{\eta}
\def\Xis{\Xi}
\def\Is{I}
\def\qs{q}
\def\ts{t}
\def\pmbxieta{(\pmbi{\xi},\pmbi{\eta})}

\noindent We now define the notion of an
``adjoint pair'' \mbox{along} $\pmbxieta$, and what it means for such a pair 
to be ``Hamiltonian-maximizing.''  

For $i=1,\ldots,\mu$, and  $\zeta\in\mathcal{U}^i$, 
we define the {\bem \mbox{Hamiltonian}} $H^i_{\zeta}:T^*Q^i\times\rr\times\rr
\brokarrow\rr$ by letting
$$
H^i_{\zeta}(q,\lambda,t,\lambda_0)=
\lambda\cdot F_\zeta^i(q,t)-\lambda_0F_\zeta^{0,i}(q,t)\,.
$$
\begin{definition}\label{defad}
If H1-H7 hold, then an {\it adjoint pair \mbox{along}} $\pmbxieta$ 
is a pair $(\pmbi{\psi},\psi_0)$ with the property that:
\begin{itemize}
\vupp
\item $\pmbi{\psi}$ is a $\mu$-tuple $(\psi^1,\ldots,\psi^{\mu})$ 
such that each $\psi^i$ is an absolutely continuous
field of covectors along $\xi^i$; 
\vupp
\item $\psi_0\in\rr$ and $\psi_0\ge 0$; 
\vupp
\item each $\psi^i$ satisfies the {\it adjoint differential inclusion}
\end{itemize}
$$
\!\!-\dot\psi^i(t)\!\in
[-\psi_0,\psi^i(t)]\cdot\Lambda^{@,i}(t)
\,\,{\rm for\,\,a.e.}\,\,t\!\in\![a^i,b^i];
$$
\begin{itemize}
\vupp
\item for each $i\in\{1,\ldots,\mu\}$, the {\it switching condition}
$$
(-\psi^i(b^i),h_+^i,\psi^{i\tilde{+}1}(a^{i\tilde{+}1}),-h_-^{i\tilde{+}1})
\in\psi_0\Omega^i+
(\mathcal{C}^i)^{\perp}
$$
holds, where 
\begin{eqnarray*}
\quad h_+^i&=&H^i_{\eta^i}(y^i,\psi^i(b^i),b^i,\psi_0)\,,\\
\quad h_-^i&=&H^i_{\eta^i}(x^i,\psi^i(a^i),a^i,\psi_0)\,.
\qquad\qquad\qquad\diamondsuit
\end{eqnarray*}
\end{itemize}
\end{definition}
\vupp
\begin{definition}
If H1-H7 hold, and $(\pmbi{\psi},\psi_0)$ is an \mbox{adjoint} pair along
$\pmbxieta$, we say that $(\pmbi{\psi},\psi_0)$ satisfies  the 
{\it Hamiltonian maximization condition} if, for every 
\mbox{$i\in\{1,\ldots,\mu\}$}, the inequality 
$$
H^i_{\zeta}(\xi^i(t),\psi^i(t),t,\psi_0)\le
H^i_{\eta^i}(\xi^i(t),\psi^i(t),t,\psi_0)
$$
holds whenever $\zeta\in\mathcal{U}^i$, 
$t\in]a^i,b^i[$, and $(\xi^i(t),t)$ is a point of 
regular $\mathcal{D}$-differentiability
of $F_{\eta^i}^{@,i}$ and $F_\zeta^{@,i}$. \proofend
\end{definition}
\begin{definition}
If $(\pmbi{\psi},\psi_0)$ is an adjoint pair along
$\pmbxieta$, we say that $(\pmbi{\psi},\psi_0)$ satisfies  the 
{\it nontriviality condition} if either $\psi_0\neq 0$ or at least 
one of the functions $\psi^i$ is not identically zero. \proofend
\end{definition}
\begin{theorem}\label{theosmooth}
If H1-H7 hold, and the pair $\pmbxieta$ is \mbox{optimal}, 
then there exists an adjoint pair along $\pmbxieta$  that satisfies
the Hamiltonian maximization and \mbox{nontriviality} conditions.\proofend 
\end{theorem}

\begin{remark}
The switching condition of Definition \ref{defad} takes a particularly
simple form in the three most \mbox{common} situations, namely,

\bi
\vup
\item[(a)] free switching times,

\vupp
\item[(b)] fixed switching times,

\vupp
\item[(c)] equal but otherwise free switching times.  
\ei
Case (a) arises when the switching constraint $\mathcal{S}^i$
is ``\mbox{independent} of the times'' (that is, a set of the \mbox{form}
$\{(x,t,y,s)\in Q^i\times\rr\times Q^{i\tilde{+}1}\times\rr:
(x,y)\in \mathcal{S}_0^i\}$, where 
$\mathcal{S}_0^i$ is a subset of $Q^i\times Q^{i\tilde{+}1}$),
the times $b^i,a^{i\tilde{+}1}$ are \mbox{totally} free, and
the function $(x,t,y,s)\mapsto \varphi^i(x,t,y,s)$ is  independent of 
$t$ and $s$. In that case, if $\mathcal{C}_0^i$ is a 
\mbox{$\mathcal{D}$-approximating}
multicone to $\mathcal{S}_0^i$ at $(y^i,x^{i\tilde{+}1})$, 
we can take $\mathcal{C}^i$ to be the set of all cones
$\hat C$, $C\in\mathcal{C}_0^i$, where, for a cone
$C$ in $T_{y^i}Q^i\times T_{x^{i\tilde{+}1}}Q^{i\tilde{+}1}$, 
$\hat C$ is the cone 
\mbox{$\{(v,\tau,w,\sigma)\in
T_{y^i}Q^i\times\rr\times T_{x^{i\tilde{+}1}}Q^{i\tilde{+}1}\times\rr:
(v,w)\in C\}$}.  Then the polar $\hat C^\perp$ of $\hat C$ turns out to be
the set $\{(\alpha,0,\beta,0):(\alpha,\beta)\in C^\perp\}$. Also, we can take
$\Omega^i$ to be a subset of $T_{y^i}^*Q^i\times\{0\}
\times T_{x^{i\tilde{+}1}}^*Q^{i\tilde{+}1}\times\{0\}$. It then follows
from the switching condition that $h_+^i=h_-^{i\tilde{+}1}=0$, that is,
``the Hamiltonian vanishes when the switching times are free.''

Case (b) arises when the switching constraint $\mathcal{S}^i$
is a subset of 
$Q^i\times\{b^i\}\times Q^{i\tilde{+}1}\times\{a^{i\tilde{+}1}\}$.
In that case we can assume that the multicone 
$\mathcal{C}^i$ is contained in
$T_{y^i}Q^i\times\{0\}
\times T_{x^{i\tilde{+}1}}Q^{i\tilde{+}1}\times\{0\}$. It follows
that the \mbox{switching} condition imposes no restriction on
$h_+^i$ and $h_-^{i\tilde{+}1}$, so that
``the values of the Hamiltonian are free when the switching times are fixed.''


Finally, Case (c) arises when the switching cost 
\mbox{function} $(x,t,y,s)\mapsto \varphi^i(x,t,y,s)$ is  independent of 
$t$ and $s$, and the switching constraint $\mathcal{S}^i$
involves a \mbox{restriction} on the switching states only, 
plus the \mbox{requirement} that the times $b^i$ and $a^{i\tilde{+}1}$
be equal but otherwise arbitrary. Once again, we can take
$\Omega^i$ to be a subset of the product $T_{y^i}^*Q^i\times\{0\}
\times T_{x^{i\tilde{+}1}}^*Q^{i\tilde{+}1}\times\{0\}$. Moreover,
we can assume that $\mathcal{C}^i$ consists of cones
$\check C$, where, for a cone $C$ in 
$T_{y^i}Q^i\times T_{x^{i\tilde{+}1}}Q^{i\tilde{+}1}$, 
$\check C$ is the cone 
\mbox{$\{(v,r,w,r)\in
T_{y^i}Q^i\times\rr\times T_{x^{i\tilde{+}1}}Q^{i\tilde{+}1}\times\rr:
(v,w)\in C\}$}.  Clearly, the polar $\check C^\perp$ of a cone $\check C$ is 
the set \mbox{$\{(\alpha,\rho,\beta,-\rho):(\alpha,\beta)\in C^\perp,\rho\in
\rr\}$}. 
It then follows
from the switching condition that $h_+^i=h_-^{i\tilde{+}1}$, that is,
``the Hamiltonian does not jump when the switching times are equal but 
otherwise free.''

The most common situation obtains when we are in Case (c) for 
$i=1,\ldots,\mu-1$, that is, when ``the true switching times are free
but there is no clock-resetting at the switchings.'' \proofend
\end{remark}


\ssection{\Pigds} 
\label{secpigds}

\noindent GDQ theory appears to be broad enough to yield a very general
maximum principle. On the other hand, \mbox{Theorem} \ref{conticlark}
is not valid if we substitute ``Warga derivate container''
for ``Clarke generalized Jacobian,'' because  it is possible to exhibit
Lipschitz maps that have derivate containers that are not GDQs.
It is \mbox{therefore} reasonable to look for even more general
GDTs that might contain both GDQ theory and other theories
such as the derivate containers and the multidifferentials.
The following is one example of such a theory.

If $n,m\!\in\!\zz_+$, 
\mbox{$\alpha\!:\![0,1]\!\to\!\rr^n$ is a Lipschitz function},
and $h\!:\![0,1]\!\to\!\rr^{m\times n}$ is integrable,
we use $h*\alpha$ to denote the ``chronological product'' of
$h$ and $\alpha$, that is, the absolutely continuous 
function $\beta\!:\![0,1]\!\to\!\rr^m$ given
by $\beta(t)=\int_0^th(s)\cdot\dot\alpha(s)\,ds$.

Let $n\in\zz_+$, $r\in\rr$, 
and let $S$ be a subset of $\rr^n$.  We write $\aof{S(r)}$
to denote the subset of $\czer{[0,1]}{\rr^n}$ consisting of all
absolutely continuous curves \mbox{$\alpha:[0,1]\to\rr^n$} such that
(i)~$\alpha(0)=0$, and \mbox{(ii)~$\dot\alpha(t)\in S$}
 and $\|\dot\alpha(t)\|\le r$ for almost all $t\in[0,1]$.

\begin{definition}\label{defipigd}
Let $n,m\in\zz_+$, let $F:\rr^n\tto\rr^m$, 
and let $C$ be a closed convex cone in $\rr^n$.
We say that $\Lambda$ is a {\bem \pigd{} of $F$ at $(0,0)$ in the direction 
of $C$,} and write $\Lambda\in\pigdab(F,C)$, if
$\Lambda$ is a nonempty compact subset of $\rr^{m\times n}$, and
for every $\delta\in\,]\,0,\infty\,[\,$ there exists
$R\in\,]\,0,\infty\,[\,$ such that 

\ben
\vspace{-6pt}
\item[(\#)] for every $r\in\,]\,0,R]$ there exists a regular 
set-valued map
$G:\aof{C(r)}\tto\czer{[0,1]}{\rr^{m\times n}}\times\rr^m$
with the property that

\ben
\vspace{-8pt}
\item[(\#{}.a)] $h(t)\!\in\!\Lambda^\delta$ and 
$\|v\|\!\le\!\delta r$ whenever 
\mbox{$\alpha\!\in\!\aof{C(r)}$}, $(h,v)\in G(\alpha)$,  $t\in[0,1]$,

\vspace{-2pt}
\item[(\#{}.b)] $\gr(\Phi_{G})\subseteq\gr(F)$, where
$\Phi_G$ is the \mbox{set-valued} map from $\aof{C(r)}$ to $\rr^m$
such that, if \mbox{$x\in\aof{C(r)}$}, then 
$\Phi_G(x)$ is the set of all \mbox{$y\!\in\!\rr^m$} 
for which the equality $y=(h*\alpha)(1)+v$ holds for some triple
$(\alpha,h,v)\in\gr(G)$ such that $\alpha(1)=x$. \proofend
\een\een
\end{definition}

\vupp\noindent
It then turns out that every GDQ is a PIGD, and 
\mbox{every} derivate container, semidifferential, and 
\mbox{multidifferential}
is also a PIGD. Moreover, the \mbox{definition} of the PIGDs 
can easily be extended to manifolds, and one obtains 
a GDT that has the DOMP and contains all other theories
proposed so far.



%\vspace{-5mm}
\begin{thebibliography}{99}

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\vup
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\vup
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\end{document}

