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%    NEW THEORIES OF SET-VALUED DIFFERENTIALS AND NEW VERSIONS
%        OF THE MAXIMUM PRINCIPLE OF OPTIMAL CONTROL THEORY
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%                       by H. J. Sussmann
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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
Web site: http://www.math.rutgers.edu/$\widetilde{\,\,}$sussmann\\
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New theories of set-valued differentials and new versions\\
$\,\,$\\

\vspace{-10pt}
of the maximum principle of optimal control theory$\,^{*}$}
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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 The purpose of this note is to announce 
a new theory of generalized 
differentials---the ``generalized differential quotients,'' abbr. GDQs---
which has good open mapping properties,
and to use this theory to state---in Theorem \ref{theosmooth}---
a version of the maximum principle for hybrid optimal
control problems under weak regularity conditions. 
For single-valued maps, our GDQ theory essentially coincides with the one
proposed by H.~Halkin in \cite{halkinc}, but GDQ theory applies
as well to multivalued maps, thus making it possible to 
deal with non-Lipschitz vector fields, whose flow maps are in general
set-valued. 

The results presented here are much weaker than what can
actually be proved by our methods.  More general versions, involving
systems of differential inclusions, are discussed
in a detailed paper currently in preparation. 

The GDQ concept contains several other notions of generalized
differential, but does not include some important theories such as
J. Warga's \mbox{``derivate containers''} (cf. \cite{wargac}) and the
\mbox{``semidifferentials''} and ``multidifferentials'' proposed by us in
previous work (cf. \cite{sussmul}).

Recently, we have found another theory---the ``\pigds'', abbr. 
{\pigdab}s---that contains that of GDQs
as well as the other theories mentioned above. We conclude
the paper by giving, in \S\ref{secpigds}, 
a brief sketch of the definition of this concept.

\ssection{Notational preliminaries}\label{notprem}

\vupp
\noindent A {\em set-valued map} is a triple $F=(A,B,G)$ such that $A$
and $B$ are sets and $G$ is a subset of $A\times B$. If $F=(A,B,G)$ is
a set-valued map, we say that $F$ {\em is a set-valued map from $A$ to
$B$}. In that case, we refer to the sets $A$, $B$, $G$ as the {\em
source}, {\em target}, and {\em graph} of $F$, respectively, and write
$A=\so(F)$, $B=\ta(F)$, $G=\gr(F)$. 
If $x\in\so(F)$, we write $F(x)=\{y:(x,y)\in\gr(F)\}$. 
The set $\dom(F)=\{x\in\so(F):F(x)\neq\emptyset\}$ is the 
{\bem domain} of $F$. 
If $A$, $B$ are sets, we use $SVM(A,B)$ to denote 
the set of all set-valued maps from $A$ to $B$, and write 
$F:A\tto B$ to indicate that $F\in SVM(A,B)$.  

The expression ``ppd map'' stands for ``possibly partially defined
(that is, not necessarily everywhere 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 $B$.

If $k\in\nn$ and $M$ is a manifold of class $C^k$, then 
$TM$, $T^*M$  denote the tangent and cotangent bundle of $M$.
If $x\in M$, then 
$T_xM$, $T_x^*M$  denote the tangent and cotangent spaces
of $M$ at $x$. 

\ssection{Regular maps}\label{defofregmap}

\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 compact graph.
We say that a sequence  $\{F_j\}_{j\in\nn}$ of 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
set-valued  maps from $X$ to $Y$. \proofend
\end{definition}
It is easy to see that if $F:X\to Y$ is an ordinary 
(that is, single-valued and everywhere defined) map, then
$F$ belogns to $\regu{X}{Y}$ if and only if $F$ is continuous.

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

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

\vspace{-8pt}
\begin{definition}\label{defquasi} 
Let $m,n$ be nonnegative integers, 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 set 
$GDQ(F;\bar x,\bar y;S)$  of compact nonempty subsets of the space  
${\rm Lin}(T_{\bar x}M,T_{\bar y}N)$ of linear maps from
$T_{\bar x}M$ to $T_{\bar y}N$ 
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 $\bar x$, $\bar y$ and such 
that $\xi(x)=0$, $\eta(y)=0$, and
declaring a subset $\Lambda$ of ${\rm Lin}(T_{\bar x}M,T_{\bar y}N)$
to belong to $GDQ(F;\bar x,\bar y;S)$  if 
$D\eta(\bar y)\circ\Lambda\circ D\xi(\bar x)^{-1}$ 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 $\xi$, $\eta$.} In other words, {\bem the notion of a GDQ 
is invariant under $C^1$ diffeomorphisms and makes sense intrinsically
on $C^1$ manifolds.}

The following facts about GDQs can be verified.

\ben 
\vspace{-6pt}
\item[1.] If $M$, $N$ are $C^1$ manifolds, $\bar x\in M$,
$U$ is a neighborhood of $\bar x$ in $M$, 
$F:U\to N$ is a continuous map, $F$ is differentiable at
$\bar x$, $\bar y=F(\bar x)$, and $L=DF(\bar x)$, then
$\{L\}\in GDQ(F;\bar x,\bar y;M)$. 

\vspace{-8pt}
\item[2.] If $M$, $N$ are $C^1$ manifolds, $\bar x\in M$,
$U$ is a neighborhood of $\bar x$ in $M$, 
$F:U\to N$ is a Lipschitz continuous map, 
$\bar y=F(\bar x)$, and $\Lambda$ is the Clarke generalized
Jacobian of $F$ at $\bar x$, then
$\Lambda\in GDQ(F;\bar x,\bar y;M)$. 

\vspace{-8pt}
\item[3.] {\bem (The chain rule.)}  If $M_i$ is a
$C^1$ manifold and \mbox{$\bar x_i\!\in\!M_i$} for $i=1,2,3$,
$S_i\subseteq M_i$, $F_i:M_i\tto M_{i+1}$, 
and $\Lambda_i\in GDQ(F_i;\bar x_{i},\bar x_{i+1};S_i)$ for $i=1,2$,
and either $S_2\cap U$ is a retract of $U$ for some comnpact neighborhood
$U$ of $x_2$ in $M_2$ or $F_1$ is single-valued, then 
$\Lambda_2\circ\Lambda_1\in GDQ(F_2\circ F_1;\bar x_{1},\bar x_3;S_1)$.

\vspace{-8pt}
\item[4.] {\bem (The product rule.)}  If $M_1$, $M_2$, $N_1$,  $N_2$,  
are $C^1$ manifolds, and, for $i=1,2$, 
\mbox{$\bar x_i\in M_i$, $\bar y_i\in N_i$, $S_i\subseteq M_i$,} 
$F_i:M_i\tto N_{i}$, 
and $\Lambda_i\in GDQ(F_i;\bar x_{i},\bar y_i;S_i)$,
then $\Lambda_1\times\Lambda_2\in GDQ(F_1\times F_2;(\bar x_{1},\bar x_2),
(\bar y_1,\bar y_2);S_1\times S_2)$.

\vspace{-8pt}
\item[5.] {\bem (Locality.)}  If $M$, $N$, 
are $C^1$ manifolds, $\bar x\in M$, $\bar y\in N$, 
and, for $i=1,2$, $S_i\subseteq M$, $F_i:M\tto N$, 
and there exist neighborhoods $U$, $V$ of $\bar x$, $\bar y$,  in $M$, 
$N$, respectively,  such that  $U\cap S_1=U\cap S_2$ and
$(U\times V)\cap\gr(F_1)=(U\times V)\cap\gr(F_2)$, then
$GDQ(F_1;\bar x,\bar y;S_1)=GDQ(F_2;\bar x,\bar y;S_2)$.
\een

\vspace{-4pt}
\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 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.)

In addition, GDQs have the following {\bem directional 
open mapping property}.

\vupp
\begin{theorem}\label{opendirec}
Let $m,n$ be nonnegative integers, and let $C$ be a convex cone in
$\rr^m$. Let $F:\rr^m\tto\rr^n$ be a set-valued map, and let
$\Lambda\in GDQ(F;0,0;C)$.  Let $D$ be a closed convex cone in $\rr^n$
such that $D\subseteq{\rm Int}(LC)\cup\{0\}$ for every
$L\in\Lambda$. Then there exist a convex cone $\Delta$ in $\rr^n$ such
that $D\subseteq{\rm Int}(\Delta)$, and positive constants $\bar\eps$,
$\kappa$, having the property that

\ben

\vspace{-5pt}
\item[(I)] if  $y\in\Delta$ and 
$\|y\|\le\bar\eps$, then there exists an $x\in C$ such that
$\|x\|\le\kappa\|y\|$ and $y\in F(x)$. 
\een

\vspace{-5pt}\noindent Moreover, the cone $\Delta$ and the 
constants $\bar\eps$, $\kappa$ can be chosen so that the following 
stronger conclusion holds:
\ben

\vspace{-5pt}\item[(II)] if $y\in\Delta$ and $\|y\|=\eps\le\bar\eps$
then there exists a compact connected subset $Z_y$ of
$(C\cap\nball(\kappa\eps))\times[0,1]$ such that $(0,0)\in Z_y$,
$(x,1)\in Z_y$ for some $x$ belonging to $C\cap\nball(\kappa\eps)$,
and $ry\!\in\!F(x)$ whenever $0\!\le\!r\!\le\!1$ and
$(x,r)\!\in\!Z_y\,.\,\,\diamondsuit$ 
\een
\end{theorem}

Associated to  concept of a GDQ there is a notion of
``GDQ approximating multicone to a set at a point.''  
\begin{definition}\label{defappo}
If $X$ is a finite-dimensional real linear space, a {\bem convex multicone} 
in $X$ is a nonempty set of convex cones in $X$.

If $M$ is a manifold of class $C^1$, $S\subseteq M$ and $x\in S$, 
a {\bem GDQ approximating multicone to $S$ at $x$} is a convex multicone 
$\mathcal{C}$ in $T_xM$ such that there exist 
$m\in\zz_+$, $F:\rr^m\tto M$, a closed convex cone $D$ in $\rr^m$, and a
$\Lambda\in GDQ(F;0,x;D)$, such that
$F(D)\subseteq S$ and $\mathcal{C}=\{LD:L\in\Lambda\}$.\proofend
\end{definition}

If $X$ is a finite-dimensional real linear space, then $X^\perp$
denotes the dual of $X$. If $C\!\subseteq\! X$, the {\bem polar} of $C$ is
the set 
$C^\perp\!=\!\{y\!\in\! X^\perp\!:\!y(x)\!\le\!0\,
{\rm whenever}\,x\!\in\! C\}$, so $C^\perp$ is a closed convex cone in
$X^\perp$. If $\mathcal{C}$ is a convex multicone in $X$, the {\bem
polar} of $\mathcal{C}$ is the set
$\mathcal{C}^\perp={\rm Clos}\Big(\bigcup\{C^\perp:C\in \mathcal{C}\}\Big)$,
so  $\mathcal{C}^\perp$ is a (not necessarily convex) closed cone
in $X^\perp$.

\ssection{Discontinuous vector fields and their flows}

\label{secdiscvecfields} 

\noindent If $n\in\nn$,   we use $\blrn$, $\blrnr$, to denote, respectively,
the $\sigma$-algebra of Borel subsets of $\rr^n$ and the
product $\sigma$-algebra 
$\blrn\otimes{\rm Lebesgue}(\rr)$.
We let $\nrnr$ denote the set of all subsets $S$
of $\rr^n\times\rr$ such that $\Pi_n(S)$ is a Lebesgue-null subset 
of $\rr$, where $\Pi_n$ is the canonical projection
$\rr^n\times\rr\ni(x,t)\to t\in\rr$. Finally, we use 
$\blernr$ to denote the $\sigma$-algebra of subsets
of $\rr^n\times\rr$ generated by $\blxr\cup\nexr$. 
It is then clear that
$\mathcal{B}(\rr^n\times\rr)\subset\blrnr\subset\blernr$, 
and both inclusions are strict.

\begin{definition}\label{defmeastvvf}
Let $n,m\in\zz_+$, 
and let $f$ be a ppd map from $\rr^n\times\rr$ to $\rr^m$. 

\begin{itemize}
\vspace{-6pt}
\item[1.] We say that $f$ is 
{\it locally essentially Borel$\times$Lebesgue} measurable, or 
{\it locally $\blernr$-measurable}, if
$f^{-1}(U)\cap K\in\blernr$ for all open subsets $U$ of $\rr^m$
and all compact subsets $K$ of $\dom(f)$. 

\vspace{-8pt}
\item[2.] We use $\lmblernrm$ to denote the set of
all locally $\blernr$-measurable ppd maps 
{}from $\rr^n\times\rr$ to $\rr^m$.

\vspace{-8pt}
\item[3.] We call $f$ {\it locally integrably bounded} (LIB) 
if for every compact subset $K$ of $\dom(f)$ there exists an 
integrable funtion $\rr\ni t\to\ffi(t)\in[0,+\infty]$ such that
$||f(x,t)||\le\ffi(t)$ for all $(x,t)\in K$. \proofend
\end{itemize}
\end{definition}

If $n,m\in\zz_+$, $f:\rr^n\times\rr\brokarrow\rr^m$, $I$ is a compact
interval, and $S\subseteq\dom(f)$, we write $\Xi_S(f,I)$ to denote the
set of all curves $\xi\!\in\!\czer{I}{\rr^n}$ such that 
$(\xi(t),t)\!\in\! S$ for all $t\in I$.  
We write $\Xi(f,I)$ for $\Xi_{\dom(f)}(f,I)$, 

If $\psi:\rr\to\rr_+\cup\{+\infty\}$ is a function,
we use $\Xi_{S}^{\psi}(f,I)$ to denote the set of
all curves $\xi\in\Xi_S(f,I)$ such that

\vupp
\begin{equation}\label{limo}
\limsup_{t\downarrow\bar t}\frac{||\xi(t)-\xi(\bar t)||}{t-\bar t}
\le\psi(\bar t)
\end{equation}

\vspace{-6pt}\noindent for almost every  $\bar t\in I$.


\vupp
\bfact\label{factomeascont}
If $n,m\in\zz_+$, $f\in\lmblernrm$, $I$ is a compact interval, 
and $\xi\in\Xi(f,I)$, then $f\compo\xi$ is
measurable. \proofend
\efact
If $n,m\in\zz_+$, $f\in\lmblernrm$, $I$ is a compact interval, 
and $f$ is locally integrably bounded, then we can define a map 
$\mathcal{T}_{f,I}:I\times I\times\Xi(f,I)\to\rr^m$ 
by letting

\vspace{-10pt}
\begin{equation}\label{defoftf}
\mathcal{T}_{f,I}(a,t,\xi)=\int_a^tf(\xi(s),s)\,ds
\end{equation}

\vspace{-6pt}\noindent if $a\in I$, $t\in I$, $\xi\in\Xi(f,I)$.

\vupp
\begin{definition}\label{asin-d}
If $n,m\in\zz_+$ and $f:\rr^n\times\rr\brokarrow\rr^m$, we
call $f$ {\bem locally integrally continuous} if 

\begin{itemize}
\vspace{-6pt}
\item[1.] $f\in\lmblernrm$,

\vspace{-8pt}
\item[2.] For every compact subset $K$ of $\dom(f)$ and every 
compact interval $I$ there exists an 
integrable funtion $\rr\ni t\to\psi(t)\in[0,+\infty]$ such that
$||f(x,t)||\le\psi(t)$ for all $(x,t)\in K$
and the restriction to $\Xi_{K}^{\psi}(f,I)$ of the map 
$\mathcal{T}_{f,I}$ is continuous. \proofend
\end{itemize}
\end{definition}

\vuppp
\noindent By taking coordinate charts, it is easy to see that the 
concept of a ``locally integrally continuous time-varying ppd section
\mbox{$f\!:\!M\!\times\!\rr\!\brokarrow\!E$}''
is well defined if $M$ is a manifold and $E$ is a vector bundle over $M$. 

\begin{definition}
Let $M$ be a manifold of class $C^1$.

\begin{itemize}
\vspace{-7pt}
\item[1.] A {\it time-varying vector field} (abbreviated ``TVVF'') on $M$
is a ppd map $f:M\times\rr\brokarrow TM$ such that $f(x,t)\in T_xM$
whenever $(x,t)\in\dom(f)$. 

\vspace{-7pt}
\item[2.] We use $\cvf(M)$ to denote the set of all TVVFs on 
$M$. 

\vspace{-7pt}
\item[3.] If $f\!\in\!\cvf(M)$, a {\bem trajectory} 
(or {\bem integral curve}) of $f$ is a locally absolutely
continuous map $\xi\!:\!I\!\to\! M$, defined on a nonempty subinterval 
$I$ of $\rr$, such that the conditions $(\xi(t),t)\in\dom(f)$ and
$\dot\xi(t)=f(\xi(t),t)$ are satisfied  for a.e. $t\in I$.

\vspace{-7pt}
\item[4.] We use $\traj(f)$ to denote the set of all trajectories of $f$, 
and $\traj_c(f)$ to denote the set of all $\xi\in\traj(f)$ whose domain
is a compact interval.  

\vspace{-7pt}
\item[5.] For $(x,b,a)\in M\times\rr\times\rr$, we define
\begin{equation}
\Phi^f(x,b,a)=\{\,\xi(b):\xi\in\traj(f):\xi(a)=x\,\}\,.
\end{equation}
The map $\Phi^f:M\times\rr\times\rr\tto M$ is the {\it flow} of $f$. 

\vspace{-7pt}
\item[6.] For each $(b,a)\in\rr\times\rr$, we define
$\Phi_{b,a}^f:M\tto M$ by letting $\Phi_{b,a}^f(x)=\Phi^f(x,b,a)$
for $x\in M$. The maps $\Phi_{b,a}^f:M\tto M$ are
the {\it flow maps} of $f$. \proofend
\end{itemize}
\end{definition}

\vuppp
\bfact
Let $M$ be a manifold of class $C^1$, and assume $f\in\cvf(M)$.
Then the flow maps $\Phi_{b,a}^f$ satisfy the 
identities $\Phi_{a,a}^f\!=\!\id_{M}$ and 
\mbox{$\Phi_{c,b}^f\circ\Phi_{b,a}^f\!=\!\Phi_{c,a}^f$},
if \mbox{$a,b,c\!\in\!\rr$} and $a\le b\le c$. \proofend
\efact

\ssection{Approximate limits} %{and points of approximate continuity}
\label{sec-pessc}

\noindent If $n,m\in\zz_+$, 
$f$ is a ppd map from $\rr^n\times\rr$ to $\rr^m$, 
$\beta>0$, $\bar x\in\rr^n$, and ${\bf v}$ is a nonempty subset of $\rr^m$,  
we define  
$$
\sigma_{f,\bar x,{\bf v},\beta}(t)=
\sup\{\,{\rm dist}(f(x,t),{\bf v}):x\in\rr^n\,,\,\|x-\bar x\|\le\beta\,\}\,,
$$
so $\sigma_{f,\bar x,{\bf v},\beta}$ is a function on $\rr$ with
values in $[0,\infty]$.  (We take the value of the right-hand side 
to be zero if the set of those $x\in\rr^n$ such
that $\|x-\bar x\|\le\beta$ and $f(x,t)$ is defined is empty.)

If $I$ is a subinterval of $\rr$, $\bar x\in\rr^n$, and $\beta>0$, we define
\begin{equation}\label{massenet-thesets}
S_{\bar x,I,\beta}=\{(x,t):x\in\rr^n,\|x-\bar x\|\le\beta,t\in I\}\,.
\end{equation}
\begin{fact}
Assume that $n,m\!\in\!\zz_+$, $f$ belongs to
$\lmblernrm$,
$x\!\in\!\rr^n$, ${\bf v}\!\subseteq\!\rr^m$, and 
${\bf v}\neq\emptyset$.  Let $I$ be a nonempty
subinterval of  $\rr$, and assume that 
$\bar\beta>0$ and $S_{x,I,\bar\beta}\subseteq\dom(f)$.
Then $\sigma_{f,\bar x,{\bf v},\beta}$ is
measurable on $I$ whenever $0<\beta\le\bar\beta$.\proofend
\end{fact}


We let $\mathcal{C}(1)$ be the set of all cones in $\rr$. Then
$\mathcal{C}(1)$ has exactly four members, and
``$C\in\mathcal{C}(1)$'' is an alternative way of saying that $C$ is
one of the four sets $\{0\}$, $[0,+\infty\,[\,$, $\,]-\infty,0]$,
$\rr$.  If $C\in\mathcal{C}(1)$, $\bar t\in\rr$, and $h>0$, we use
$\bar t+C(h)$, $\bar t+C$, to denote the sets $\{\bar t+r:r\in
C,|r|\le h\}$, $\{\bar t+r:r\in C\}$, respectively.

\begin{definition}\label{approxlim} 
Assume that $n,m\!\in\!\zz_+$, $f$ belongs to
$\lmblernrm$, ${\bf v}\subseteq\rr^m$, ${\bf v}\neq\emptyset$, 
$(\bar x,\bar t)\in\rr^n\times\rr$.  Let $C\in\mathcal{C}(1)$.

\ben
\vspace{-7pt}
\item[1.] We say that ${\bf v}$ is an {\bem approximate  limit set of $f$ at 
$(\bar x,\bar t)$ along $C$}, and write
$$
{\bf v}\in\Applim_{x\to\bar x\,,\,t\to\bar t,t-\bar t\in C}f(x,t)\,,
$$
if $(\bar x,\bar t)\in{\rm Int}_{\rr^n\times(\bar t+C)}(\dom(f))$
and
$$
\lim_{_{\,\,(h,\beta)\to (0,0)\,,\,\,h>0,\,\,\beta\ge 0}}\,\,\,\frac{1}{h}
\int_{\bar t+C(h)}\sigma_{f,\bar x,{\bf v},\beta}(t)\,dt=0\,\,.
$$


\vspace{-7pt}
\item[2.] We say that a vector $v$ is an {\bem approximate limit value of $f$ at 
$(\bar x,\bar t)$  along $C$}, and write
$$
v=\applim_{x\to\bar x\,,\,t\to\bar t,t-\bar t\in C}f(x,t)\,,
$$
if $\{v\}\in\Applim_{x\to\bar x\,,\,t\to\bar t,t-\bar t\in C}f(x,t)$.\proofend
\een
\end{definition}
We say that $(\bar x,\bar t)$ is a {\bem point of approximate continuity
along $C$} if 
$f(\bar x,\bar t)=\applim_{x\to\bar x,t\to\bar t,t-\bar t\in C}f(x,t)$.

We use the expressions ``approximate right limit,''
``approximate left limit,'' ``approximate  limit,''
``point of approximate right continuity,'' 
``point of approximate left continuity,'' 
``point of approximate continuity,'' respectively, 
as alternative names of ``approximate limit along $[0,+\infty\,[\,$,''
``approximate limit along $\,]-\infty,0]$,'' 
``approximate limit along $\rr$,''
``point of approximate continuity along $[0,+\infty\,[\,$,''
``point of approximate continuity along $\,]-\infty,0]$,'' and
``point of approximate continuity along $\rr$.'' 
By taking coordinate charts, it is easy to see that {\bem all 
these concepts are well defined if 
$M$ is a manifold, $E$ is a vector bundle over $M$,  
and $f$ is a time-varying ppd section}
\mbox{$f\!:\!M\!\times\!\rr\!\brokarrow\!E$}.

The following lemma gives a useful sufficient condition for a point
$(x,t)$ to be a point of approximate continuity of a time-varying
vector field.  To state the lemma, we first introduce the obvious
one-sided analogues of the usual notions of a Lebesgue point and a
point density.

\begin{itemize}
\vspace{-7pt}
\item{} if $I\subseteq\rr$ is an interval, and $C\in\mathcal{C}(1)$,
then a {\bem $C$-Lebesgue point} of a locally integrable function
\mbox{$\ffi:I\to\rr\cup\{+\infty\}$} is a point $\bar t\in I$ such that
$|\ffi(\bar t)|<\infty$, $\bar t+C(\bar h)\subseteq I$ for some 
positive number $\bar h$, and 

\vuppppp\vup
$$\lim_{h\to 0+}\frac{1}{h}
\int_{\bar t+C(h)}\Big|\ffi(t)-\ffi(\bar t)\Big|\,dt=0\,.
$$

\vspace{-7pt}
\item{} If $E\subseteq\rr$ is a measurable set, a 
{\bem point of $C$-density} of $E$ is a point $\bar t\in E$ such that 
$$
\lim_{h\to 0+}h^{-1}{\rm meas}\Big((\bar t+C(h))\backslash E\Big)=0\,.
$$
\end{itemize}
\begin{lemma}\label{defregupoint}
Assume that $n,m\!\in\!\zz_+$, $C\!\in\!\mathcal{C}(1)$, $C\!\neq\!\{0\}$, 
$(\bar x,\bar t)\in\rr^n\times\rr$, and $f\in\lmblernrm$.  Assume, moreover, 
that $(\bar x,\bar t)$ belongs to 
${\rm Int}_{\rr^n\times(\bar t+C)}(\dom(f))$,
and there exist positive numbers  $\bar h,\bar\beta$,  
an integrable function  $\ffi:\bar t+C(\bar h)\to\rr\cup\{+\infty\}$,
and a measurable subset $E$ of $\bar t+C(\bar h)$, such that 

\begin{itemize}
\vspace{-7pt}
\item[(a)] the set $S=S_{\bar x,\bar t+C(\bar h),\bar\beta}$
is contained in $\dom(f)$,  

\vspace{-7pt}
\item[(b)] $||f(x,t)||\le\ffi(t)$ for all $(x,t)\in S$,

\vspace{-7pt}
\item[(c)] $\bar t$ is a $C$-Lebesgue point of $\ffi$,

\vspace{-7pt}
\item[(d)] $\bar t$ is a point of $C$-density of $E$,

\vspace{-7pt}
\item[(e)] $\lim_{x\to\bar x,t\to\bar t,t\in E\cap(\bar t+C)}f(x,t)=
f(\bar x,\bar t)$. 

\end{itemize}

\vspace{-7pt}\noindent Then $(\bar x,\bar t)$ is a point of approximate 
continuity of $f$.\proofend
\end{lemma}

\ssection{Variational generators}

\noindent If $\xi:[a,b]\to\rr^n$ is a continuous curve, and $\alpha>0$, we use
$\tubn(\xi,\alpha)$ to denote the ``$\alpha$-tube about $\xi$ in $\rr^n$,''
that is, the set 
$$
\tubn(\xi,\alpha)\,\,\equaldef\,\,
\{(x,t):x\in\rr^n,a\le t\le b,\|x-\xi(t)\|\le\alpha\}\,.
$$ 
\begin{definition}\label{miscella}
Let $n,m\in\zz_+$, and let $f$ 
be a ppd map from $\rr^n\times\rr$ to $\rr^m$.
Let $a,b,\xi$ be such that $a,b\in\rr$, $a\le b$, and 
$\xi\in\czer{[a,b]}{\rr^n}$. 
A  {\bem variational generator for $f$ about $\xi$} is a
measurable set-valued map $\Lambda:[a,b]\tto\rr^{m\times n}$ 
with compact convex nonempty values such that
there exist $k_\Lambda$, $\bar\alpha$, ${\bf k}$ having the 
following three properties: 

\ben
\item[1.] $k_\Lambda:[a,b]\to[0,+\infty]$ is integrable and such that 
$\sup\Big\{\|L\|:L\in\Lambda(t)\Big\}\le k_\Lambda(t)$
for all $t\in[a,b]$;

\vspace{-7pt}
\item[2.] $\bar\alpha>0$ and $\tubn(\xi,\bar\alpha)\subseteq\dom(f)$;

\vspace{-6pt}
\item[3.]  ${\bf k}=\{k^\alpha\}_{0<\alpha\le\bar\alpha}$ is a family
of Lebesgue-integrable functions $k^{\alpha}:[a,b]\to[0,+\infty]$, for
$0\!<\!\alpha\!\le\!\bar\alpha$, such that
$\lim_{\alpha\downarrow 0}\int_a^bk^\alpha(t)dt\!=\!0$ and
\een

\vspace{-11pt}
$$
\sup\!\!
\Big\{\!\!\min\!\Big\{\!\|\Delta^{f}_{\xi}(x,t,L)\|\!:\!
L\!\in\!\Lambda(t)\!\Big\}\!:\!\|x\!-\!\xi(t)\|\!\le\!\alpha\!\!\Big\}
\!\!\le\!\alpha k^\alpha(t)
$$

\ben
\vspace{-8pt}
\item[] for all $t\in[a,b]$ and all $\alpha\in\,]\,0,\bar\alpha]$,  where 
\een

\vspace{-11pt}
$$
\Delta^{f}_{\xi}(x,t,L)\,\equaldef\,f(x,t)-f(\xi(t),t)-L\cdot(x-\xi(t))\,.
$$
We use $VG(f,\xi)$ to denote the set of all variational generators 
of $f$ about $\xi$.  \proofend
\end{definition}

If $n$, $m$, $f$, $a$, $b$, $\xi$, $\Lambda$ are as in Definition 
\ref{miscella}, we use 
$\Gamma(\Lambda)$ to denote the set of all measurable single-valued
selections of $\Lambda$. Then
$\Gamma(\Lambda)$ is a nonempty convex weakly compact subset
of $L^1([a,b];\rr^{m\times n})$. 

We now specialize to the case when $m=n$.
If $L$ belongs to $L^1([a,b];\rr^{n\times n})$,
we let $M_L$ be the fundamental matrix solution of 
the linear time-varying equation 
$\dot M=L(t)\cdot M$.  That is, 
$M_L$ is a continuous map from $[a,b]\times[a,b]$ to 
$\rr^{n\times n}$ that satisfies 
\begin{equation}\label{forgronml}
M_L(t,s)=\id_{\rr^n}+\int_s^tL(r)\cdot M_L(r,s)\,dr\,.
\end{equation}
\begin{fact}\label{fact-ml-is-cont}
If $\mathcal{B}\subseteq L^1([a,b],\rr^{n\times n})$ and 
$\mathcal{B}$ is bounded,
then the map 
$\mathcal{B}_{weak}\!\ni\! L\!\to\! M_L\!\in\!\czer{[a,b]\!\times\![a,b]}
{\rr^{n\times n}}$ is continuous. \proofend
\end{fact}
If $n$, $m$, $f$, $a$, $b$, $\xi$, $\Lambda$ are as in Definition 
\ref{miscella}, and $m=n$, we define 
$$
\mathcal{M}(\Lambda)\,\equaldef\,\{\,M_L:L\in\Gamma(\Lambda)\,\}
\subseteq \czer{[a,b]\times[a,b]}{\rr^{n\times n}}\,.
$$
\begin{fact}\label{fact-ml-gugugu}
$\mathcal{M}(\Lambda)$ is nonempty and compact.
In particular, if $t,s\in[a,b]$ then the set 
\begin{equation}
\mathcal{M}_{t,s}(\Lambda)\,\equaldef\,\{\,M_L(t,s):L\in\Gamma(\Lambda)\,\}
\end{equation}
is a nonempty compact subset of $\rr^{n\times n}$. \proofend
\end{fact}

\begin{remark}\label{rem-bundle}
The concept of a variational generator also makes intrinsic
sense on manifolds. The important new point is that 
now $\Lambda$ has to  be taken to be a section along 
$\xi$ of an appropriate bundle. We make this precise in the only case
that will be used here, namely, when $f$
is an ``augmented ppd vector field'' on a manifold $M$,
that is, a time-varying ppd section of the bundle
$TM\times\rr$.
In that case, we let $E_M$ be the vector bundle over $M$ whose fiber
$E_M(x)$ at each $x\in M$ is the product 
$J_x^1(VF(M))\times T_x^*M$, where 
$J_x^1(VF(M))$ is the space of $1$-jets at $x$ of smooth
vector fields on $M$. Then, if 
$a,b,\xi$ are such that $a,b\in\rr$, $a\le b$, and 
$\xi\in\czer{[a,b]}{M}$, 
a  {\bem variational generator for $f$ about $\xi$} is a
measurable set-valued map $\Lambda:[a,b]\tto E_{M}$ 
such that 

\begin{itemize}
\vuppp
\item{} $\Lambda(t)$ is, for a.e. $t$,
a compact convex nonempty subset of $E_{M}(\xi(t))$
such that $\pi_{1,0}(v)=f_1(\xi(t),t)$ for every $v\in\Lambda(t)$,
where $\pi_{1,0}$ is the canonical projection from 
$J^1(VF(M))$ to $J^0(VF(M))$ and $f_1$ is the $TM$ component of $f$;

\vupp
\item{} locally,
relatively to suitable coordinate charts, 
there exist $k_\Lambda$, $\bar\alpha$, ${\bf k}$ having the 
three properties of Definition \ref{miscella}.
\end{itemize}

\vupp\noindent With this definition of variational generator
Theorem \ref{theodifvecfield-etimes} remains true 
on manifolds. 

A more detailed discussion of the invariant definition 
of variational generators is given in Sussmann \cite{sussfree}.
\proofend
\end{remark}



\ssection{Differentiation of flows}

\noindent We let $\mathcal{C}(2)$ be the set of all cones in $\rr^2$
that are products $C_+\times C_-$, where 
$C_+\in\mathcal{C}(1)$ and $C_-\in\mathcal{C}(1)$. Then 
$\mathcal{C}(2)$ has exactly sixteen members. 


If $X_-$, $X_+$ are finite-dimensional real linear spaces,
${\bf M}$ is a set of linear maps from $X_-$ to $X_+$,
and ${\bf v}_+$, ${\bf v}_-$, are nonempty subsets of $X_-,X_+$, 
we define a set 
$[{\bf M};{\bf v}_+,{\bf v}_-]$ of linear 
maps from $X_-\times\rr^2$ to $X_+$ 
by letting $[{\bf M};{\bf v}_+,{\bf v}_-]$ be the set 
of all maps $[M;v_+,v_-]$, for all $M\in{\bf M}$, $v_+\in{\bf v}_+$, 
$v_-\in{\bf v}_-$, where
$[M;v_+,v_-]$ is the linear map from $X_-\times\rr^2$ to $X_+$
given by 
$$
[M;v_+,v_-](y,\beta,\alpha)\,\equaldef\,
M\cdot y+\beta\cdot v_+-\alpha\cdot M\cdot v_-\,.
$$

It is then clear that {\bem if ${\bf M}$, ${\bf v}_+$, ${\bf v}_-$ are
compact, then $[{\bf M};{\bf v}_+,{\bf v}_-]$ is compact.}

The following result is the general theorem on GDQ differentiation
with respect to the state and the endpoints, along a trajectory
$\xi\in\traj_c(f)$ of the flow $\Phi^f$ generated by a time-varying
vector field $f$.  The basic requirements are local integral continuity,
the existence of a variational generator,
and the existence of approximate limits of $f$ at
the endpoints of $\xi$.

\begin{theorem}\label{theodifvecfield-etimes}
Assume that $Q$ is a manifold of class $C^1$,
$f$ is a time-varying pdd vector field on $Q$, 
$a,b\in\rr$, $a\le b$, $C\in\mathcal{C}(2)$, 
$C_+,C_-\in\mathcal{C}(1)$, 
$C=C_{+}\times C_-\in\mathcal{C}(2)$, and 
$C^{b,a}=(b,a)+C=(b+C_{+})\times(a+C_-)$.

Assume, in addition, that

\ben
\vspace{-8pt}
\item[1.] $f$ is locally integrally continuous;

\vspace{-8pt}
\item[2.] $\Lambda$  is a variational generator for $f$
along $\xi$,

\vspace{-8pt}
\item[3.] ${\bf v}_+$, ${\bf v}_-$ are nonempty compact convex subsets of 
$T_{\xi(a)}Q$, $T_{\xi(b)}Q$,  such that 
\begin{eqnarray*}
{\bf v}_+&\in&\Applim_{x\to\xi(b)\,,\,t\to b,t-b\in C_+}f(x,t)\,,\\
{\bf v}_-&\in&\Applim_{x\to\xi(a)\,,\,t\to a,t-a\in C_-}f(x,t)\,.
\end{eqnarray*}
\een

\vuppp\noindent Then the set 
$[\mathcal{M}_{b,a}(\Lambda);{\bf v}_+,{\bf v}_-]$ belongs to 
$GDQ\Big(\Phi^{f};(\xi(a),b,a),\xi(b);Q\times C^{b,a}\Big)$.\proofend
\end{theorem} 


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


\noindent Theorem \ref{theodifvecfield-etimes}, together
with the directional open mapping theorem \ref{opendirec},
imply a version of the maximum principle that contains and
improves upon several previous smooth and nonsmooth versions,
for vector field systems as well as for
differential inclusions and  systems of 
differential inclusions. Moreover, one can also allow ``jump
maps,'' and obtain a ``hybrid'' version. We state this 
more general version directly but, for simplicity, we only discuss
the vector field case.   

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}'' $L^1,\ldots,L^\mu$ for 
$\Sigma^1,\ldots,\Sigma^\mu$, ``switching
constraints'' $\mathcal{S}^1,\ldots,\mathcal{S}^\mu$, 
``switching cost functions'' $\ffi^1,\ldots,\ffi^\mu$,
and ``time sets''  $T_-^1,T_+^1,T_-^2,T_+^2,\ldots,T_-^\mu,T_+^\mu$.


Precisely, each $\Sigma^i$ is a triple 
$\Sigma^i=(Q^i,\mathcal{U}^i,F^i)$ consisting of a
{\bem state space} $Q^i$, a {\bem controller space}
$\mathcal{U}^i$, and a {\bem controlled dynamics}, that is, 
a parametrized
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  
$L^i$ is a family $\{L_\eta^i\}_{\eta\in\mathcal{U}^i}$
of ppd functions $L_\eta^i:Q^i\times\rr\brokarrow\rr$.
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 functions are functions
$\ffi^i:Q^i\times\rr\times Q^{i\tilde{+}1}\times\rr\to\rr$. 
The time sets  $T_-^i,T_+^i$ are subsets of $\rr$.
A {\bem controller}
is a $\mu$-tuple $\pmbi{\eta}=(\eta^1,\ldots,\eta^\mu)\in
\mathcal{U}^1\times\cdots\times\mathcal{U}^\mu$. A 
{\bem trajectory} for a controller 
$\pmb{\eta}=(\eta^1,\ldots,\eta^\mu)$
is a $\mu$-tuple $\pmbi{\xi}=(\xi^1,\ldots,\xi^\mu)$
with the property that, for  each $i$, $\xi^i$ is an absolutely 
continuous  curve 
in $Q^i$, defined on a compact interval $I(\xi^i)=[a_-(\xi^i),a_+(\xi^i)]$,
that satisfies the conditions $\hat\xi^i(t)\in\dom(F_{\eta^i}^i)$ 
and $\dot\xi^i(t)=F_{\eta^i}^i(\hat\xi^i(t))$ 
for a.e. $t\in I(\xi^i)$, and is such that 
$\sigma^i(\pmbi{\xi})\in\mathcal{S}^i$, where $\hat\xi^i(t)\equaldef
(\xi^i(t),t)$ and $\sigma^i(\pmbi{\xi})\equaldef
(\hat\xi^i(a_+(\xi^i)),\hat\xi^{i\tilde{+}1}(a_-(\xi^{i\tilde{+}1})))$.
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 admissible} if, for each $i$, 
$a_-(\xi^i)\in T_-^i$, $a_+(\xi^i)\in T_+^i$, 
and the functions 
$I(\xi^i)\ni t\to L^i_{\eta^i}(\hat\xi^i(t))\in\rr$ are 
a.e. defined---that is, $\hat\xi^i(t)\in\dom(L_{\eta^i}^i)$ 
for a.e. $t\in I(\xi^i)$---and Lebesgue integrable.
The {\bem cost} of an admissible  TCP
$(\pmbi{\xi},\pmbi{\eta})$ is the number
$$
J(\pmbi{\xi},\pmbi{\eta})\equaldef
\sum_{i=1}^{\mu}\int_{a_-(\xi^i)}^{a_+(\xi^i)}
L^i_{\eta^i}(\hat\xi^i(t))\,dt+\sum_{i=1}^{\mu}\ffi
\Big(\sigma^i(\pmbi{\xi})\Big)\,.
$$
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 $L$-augmented
dynamics} to be the family
$\tilde F^i=\{\tilde F_\eta^i\}_{\eta\in\mathcal{U}^i}$
of ppd maps from $Q^i\times\rr$ to 
$TQ^i\times\rr$ given by 

\vupppp
$$
\tilde F_\eta^i(q,t)=\left[\begin{array}{c}
F_\eta^i(q,t)\\L_\eta^i(q,t)\end{array}\right]\quad{\rm for}\quad
(q,t)\in Q^i\times\rr\,.
$$
Now assume that

\begin{itemize}
\vupp 
\item[H1.] $(\pmbi{\xi},\pmbi{\eta})=((\xi^1,\ldots,\xi^\mu),
(\eta^1,\ldots,\eta^\mu))$ is an
admissible TCP, and 
$a_-^i\!=\!a(\xi^i)$, $a_+^i\!=\!a_+(\xi^i)$, 
\mbox{$x_-^i\!=\!\xi^i(a_-^i)$, $x_+^i\!=\!\xi^i(a_+^i)$,} for
$i=1,\ldots,\mu$. 

\vupp
\item[H2.] $\tilde\Lambda^1,\ldots,\tilde\Lambda^\mu$
are variational generators for $\tilde F_{\eta^1}^1,\ldots, 
\tilde F_{\eta^\mu}^\mu$ along $\xi^1,\ldots,\xi^\mu$.

\vupp
\item[H3.] $C_\sigma^i$ are, for $i=1,\ldots,\mu$, $\sigma\in\{+,-\}$, 
cones in $\rr$ such that 
$a_\sigma^i\in{\rm Int}_{a_\sigma^i+C_\sigma^i}T_\sigma^i$ and
$\Xi_\sigma^i\equaldef(x_\sigma^i,a_\sigma^i)$ 
is a point of approximate continuity of
$\tilde F_{\eta^i}^i$ along $C_\sigma^i$.

\vupp
\item[H4.] 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 $T_{(\Xi^i_+,\Xi^{i\tilde{+}1}_-)}\Q_{\#}^i$)
where, for $\sigma\in\{+,-\}$,
\begin{eqnarray*}
R_{\sigma}^i&\equaldef&\left\{\begin{array}{ccl}
\{0\}&{\rm if}&C_{\sigma}^i=\{0\}\,,\\
\rr&{\rm if}&C_{\sigma}^i\neq\{0\}\,,\end{array}\right.\\
\mathcal{Q}_{\sigma}^i&\equaldef&Q^i\times(a_{\sigma}^i+ R_{\sigma}^i)\,.
%\mathcal{Q}_{\#}^i&=&\mathcal{Q}_+^i\times\mathcal{Q}_-^{i\tilde{+}1}\,.
\end{eqnarray*}

\vupp
\item[H5.] $\mathcal{C}^i$ is a GDQ approximating multicone for  
the restricted switching set 
$$
\mathcal{S}_{rest}^i\,\equaldef\,\mathcal{S}^i\cap
(\mathcal{Q}_+^i\times\mathcal{Q}_-^{i\tilde{+}1})\,.
$$ 
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 dual space 
$\Big(T_{x_+^i}Q^i\times R_+^i\times T_{x^{i\tilde{+}1}}
Q^{i\tilde{+}1}\times R_-^{i\tilde{+}1}\Big)^\dagger$
(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 $GDQ(\ffi^i;P^i,\ffi^i(P^i);
\Q_+^i\times\Q_-^{i\tilde{+}1})$. 


\vupp
\item[H7.] For each $i\in\{1,\ldots,\mu\}$ and each $\eta\in
\mathcal{U}^i$, the time-varying map $\tilde F_\eta^i$ 
is locally integrally continuous.

\vupp
\item[H8.] Each control system $\Sigma^i$ is invariant under
time-interval substitutions.  (That is, if
$\eta,\zeta\in\mathcal{U}^i$,
$\xi\in\czer{[a,b]}{Q^i}\cap\traj_c(F_\eta^i)$, 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})}

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

For $i=1,\ldots,\mu$, and  $\zeta\in\mathcal{U}^i$, 
we define the {\bem Hamiltonian} $H^i_{\zeta}:T^*Q^i\times\rr\times\rr
\brokarrow\rr$ by letting
$$
H^i_{\zeta}(q,\lambda,t,\lambda_0)=
\lambda\cdot F_\zeta^i(q,t)-\lambda_0L_\zeta^i(q,t)\,.
$$
\begin{definition}\label{defad}
If H1-H8 hold, then a  {\it multiplier along} $\pmbxieta$ 
is a triple $(\pmbi{\psi},\psi_0,\pmb{\kappa})$ 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 a field of covectors along $\xi^i$ 
(that is, $\psi^i$ is a map from the interval
$[a_-^i,a_+^i]$ to $T^*Q^i$, 
such that $\psi^i(t)$ belongs to
$T_{\xi^i(t)}^*Q^i$ for every $t\in[a_-^i,a_+^i]$);
\vupp
\item $\pmbi{\kappa}$ is a $2\mu$-tuple 
$(\kappa_-^1,\kappa_+^1,\ldots,\kappa_-^{\mu},\kappa_+^{\mu})$ 
such that \mbox{$\kappa_\sigma^i\!\in\!R_\sigma^i$}
whenever $i\in\{1,\ldots,\mu\}$ and $\sigma\in\{+,-\}$;
\vupp
\item each $\psi^i$ is absolutely continuous and satisfies the 
{\it adjoint differential inclusion}
\end{itemize}

\vuppppp
$$
\!\!-\dot\psi^i(t)\!\in
[\psi^i(t),-\psi_0]\cdot\tilde\Lambda(t)
\,\,{\rm for\,\,a.e.}\,\,t\!\in\![a_-^i,a_+^i];
$$

\begin{itemize}
\vuppp
\item $\psi_0\in\rr$ and $\psi_0\ge 0$; 

\vupp
\item for each $i\!\in\!\{1,\ldots,\mu\}$, $\sigma\!\in\!\{+,-\}$,
the {\it switching conditions}
$$
\!\!\!\!\!\!\!\!\!\!\!\!
(-\psi^i(a_+^i),-\kappa_+^i\!+\!h_+^i,\psi^{i\tilde{+}1}(a_-^{i\tilde{+}1}),
\!-\!\kappa_-^i\!-\!h_-^{i\tilde{+}1})\!\in
\!\psi_0\Omega^i\!+\!(\mathcal{C}^i)^{\perp},
$$
$$
\qquad\qquad\qquad\qquad\qquad\qquad\quad\kappa_\sigma^i\in (C_\sigma^i)^\perp
$$
hold, where 
$$
h_\sigma^i=H^i_{\eta^i}(x_\sigma^i,\psi^i(a_\sigma^i),a_\sigma^i,
\psi_0)\,.
$$
\end{itemize}
\end{definition}
\vupp
\begin{definition}
If H1-H8 hold, and $(\pmbi{\psi},\psi_0,\pmb{\kappa})$ is a multiplier  
along $\pmbxieta$, we say that $(\pmbi{\psi},\psi_0,\pmb{\kappa})$ 
is {\it Hamiltonian-maximizing} 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 $\zeta\in\mathcal{U}^i$, 
$t\in]a_-^i,a_+^i[$, and $(\xi^i(t),t)$ is a point of approximate
continuity of $\tilde F_{\eta^i}^i$ and $\tilde F_\zeta^i$. \proofend
\end{definition}

\vupppp
\begin{definition} 
If $(\pmbi{\psi},\psi_0,\pmb{\kappa})$ is a multiplier along
$\pmbxieta$, we say that $(\pmbi{\psi},\psi_0,\pmb{\kappa})$ is
{\it nontrivial} if it is not true that
$\psi_0=\kappa_-^1=\kappa_+^1=\ldots=\kappa_-^\mu=\kappa_+^\mu=0$
and all the functions $\psi^i$ are identically zero. \proofend
\end{definition}
\begin{theorem}\label{theosmooth}
If H1-H8 hold, and the pair $\pmbxieta$ is optimal, 
then there exists a nontrivial Hamiltonian-maximizing
multiplier along $\pmbxieta$.\proofend 
\end{theorem}

\vupp\noindent By taking $\mu=1$, Theorem \ref{theosmooth} can be shown to 
include the classical ``non-hybrid'' smooth
and nonmsooth versions of the maximum principle given, for example, in
Pontryagin {\it et al.}  \cite{pontrybook}, Berkovitz
\cite{berkovitz}, Clarke \cite{clarkebook,clarkeminimal}.
In that case, the switching condition of 
Definition \ref{defad} becomes the transversality condition.
When the augmented  vector fields 
$\tilde F_{\eta^i}^i$ are of class $C^1$, one can take 
\begin{equation}\label{sop}
\tilde\Lambda^i(t)=\Biggl\{\frac{\partial\tilde F_{\eta^i}^i}
{\partial x\,\,}(\xi^i(t),t)\Biggr\}\,,
\end{equation}
and the adjoint differential inclusion becomes the classical
adjoint equation. On the other hand, if the function
$x\to\tilde F_{\eta^i}^i(x,t)$ is differentiable at $\xi^i(t)$ for 
almost all $t$, then one can still take $\tilde\Lambda^i$ to be given
by (\ref{sop}), and $\tilde\Lambda^i$ is a variational generator,
provided that the differentiability of
$x\to\tilde F_{\eta^i}^i(x,t)$ at $\xi^i(t)$ has an obvious 
integral uniformity property with respect to $t$.  So 
Theorem \ref{theosmooth} is in fact stronger than the classical
versions, even in the setting of single-valued differentials. 

In addition, when the $\tilde F_{\eta^i}^i$ are Lipschitz
continuous on some tube about the reference trajectory,
with an integrable Lipschitz constant, then one can take
%always 
%\begin{equation}\label{sop-ba}
%\tilde\Lambda^i(t)=\partial\tilde F_{\eta^i,t}^i(\xi^i(t))\,,
%\end{equation}
$\tilde\Lambda^i(t)$ to be $\partial\tilde F_{\eta^i,t}^i(\xi^i(t))$, 
where $\tilde F_{\eta^i,t}^i$ is the map 
$x\to\tilde F_{\eta^i}^i(x,t)$, and ``$\partial$'' stands
for ``Clarke generalized Jacobian.''  Moreover,  in the Lipschitz case
one can often take the $\tilde\Lambda^i$ to be smaller than the  
Clarke generalized Jacobian (for example, equal to the classical differential, 
when it exists), so even in the Lipschitz case 
Theorem \ref{theosmooth} often yields a stronger conclusion than 
the usual nonsmooth results.  

Theorem \ref{theosmooth} also applies to problems where 
the vector fields are only continuous with respect to the state
(in which case the flow maps are set-valued) and to 
problems with discontinuous vector fields. An important 
class of such problems arises from {\bem differential inclusion
systems}. As long as the inclusions under consideration
are almost lower semicontinuous, then there exist sufficiently
many integrally continuous selections to make our theorem 
applicable.
All these applications will be discussed in a subsequent paper.

\ssection{\Pigds}
\label{secpigds}

\noindent 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 continous 
function $\beta\!:\![0,1]\!\to\!\rr^m$ given
by 
$$
\beta(t)=\int_0^th(s)\cdot\dot\alpha(s)\,ds\,.
$$
Let $n\in\zz_+$, and let $S$ be a subset of $\rr^n$.  We write $\aof{S}$
to denote the subset of $\czer{[0,1]}{\rr^n}$ consisting of all
absolutely continuous curves $\alpha:[0,1]\to\rr^n$ such that
$\alpha(0)=0$ and $\dot\alpha(t)\in S$ for almost all $t\in[0,1]\,$.

If $C$ is a convex cone in $\rr^n$, and $r>0$, we write
$C(r)=\{v\in C:\|v\|\le r\}$.

\vup
\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 positive real number $\delta$ there exists a number
$R\in\,]\,0,\infty\,[\,$ with the property that 

\ben
\vspace{-6pt}
\item[(\#)] for every $r\in\,]\,0,R]$ there exists a map
\mbox{$G\!\in\!\regu{\aof{C(r)}}{\czer{[0,1]}{\rr^{m\times n}}\times\rr^m}$}
such 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 set-valued map from $\aof{C(r)}$ to $\rr^m$
such that, if $x$ belongs to $\aof{C(r)}$, then 
$\Phi_G(x)$ is the set of all $y\!\in\!\rr^m$ 
for which $y\!=\!(h*\alpha)(1)\!+\!v$ for some triple
$(\alpha,h,v)\!\in\!\gr(G)$ such that $\alpha(1)\!=\!x$. \proofend
\een\een
\end{definition}

\vuppp\noindent
It then turns out that every GDQ is a PIGD, and 
every derivate container, semidifferential, and multidifferential
is also a PIGD. Moreover, PIGDs are intirinsicall defined of manifolds,
and satisfy all the desirable 
properties such as the chain rule (for polyhedral cones)
and the directional open mapping theorem.  The resulting version
of the maximum principle is thus even more general than the one involving
GDQs. 



\vspace{-3mm}
\begin{thebibliography}{99}

\vupp
\bibitem{berkovitz}{Berkovitz,~L.~D., {\it Optimal Control Theory.}
Springer-Verlag, New York, 1974.}

\vupp
\bibitem{clarkeminimal}{Clarke,~F.~H., \textit{The Maximum Principle 
under minimal hypotheses.}   SIAM J. Control Optim. \textbf{14},
1976, pp.~1078--1091.}

\vupp
\bibitem{clarkebook}{Clarke,~F.~H., \textit{Optimization and Nonsmooth
Analysis.} Wiley Interscience, New York, 1983.}

\vupp
\bibitem{halkinc}{Halkin,~H., \textit{Necessary conditions for optimal
control problems with differentiable or nondifferentiable data.} In
\textit{Mathematical Control Theory}, \underline{Lect. Notes in Math.}
\textbf{680}, Springer-Verlag, Berlin, 1978, pp.~77--118.}

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

\vupp
\bibitem{sussfree}{Sussmann, H. J., 
\textit{An introduction to the coordinate-free maximum principle.}
In \textit{Geometry of Feedback and Optimal Control}, 
B.~Jakubczyk and W.~Respondek Eds., M.~Dekker, Inc., New York,
1997, pp.~463-557.}

\vupp
\bibitem{sussmul}{Sussmann,~H.~J., \textit{Multidifferential calculus: chain
rule, open mapping and transversal intersection theorems.} In \textit{Optimal
Control: Theory, Algorithms, and Applications}, W.~W.~ Hager and
P.~M.~Pardalos Eds., Kluwer, 1998, pp.~436--487.}

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

\vupp
\bibitem{wargac}{Warga,~J., \textit{Optimization and controllability without
differentiability  assumptions.}  SIAM J. Control and Optimization
\textbf{21}, 1983, pp.~837--855.} 

\end{thebibliography}

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