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\begin{document}
\bsl
\begin{center}\begin{minipage}{16.4cm}
\baselineskip=40pt
\stapageno


{\Large\it These transparencies constitute  a combined version of three
\mbox{different} talks, and include a lot of supplementary 
material that was not actually presented in the lectures. }

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}
\stapageno
\begin{center}
{\Large {\bf GENERALIZED}}\\

\vspace{-16mm}
{\Large {\bf DIFFERENTIATION}}\\

\vspace{-16mm}
{\Large {\bf THEORIES}}\\

\vspace{-16mm}
{\Large {\bf THAT SATISFY}}\\

\vspace{-16mm}
{\Large {\bf THE CHAIN RULE}}\\

\vspace{-16mm}
{\Large{\bf AND HAVE AN OPEN}}\\

\vspace{-16mm}
{\Large{\bf MAPPING PROPERTY }}\\

\vspace{-10mm}
{\large H\'ector J. Sussmann}\\

\vspace{-2mm}
{Rutgers University --- Piscataway, NJ}\\

\medbreak
%\vspace{-3mm}
{\small and (Spring 2003)}\\

%\vspace{-3mm}
Institut Mittag-Leffler --- Djursholm, Sweden

\vspace{4mm}
{{\bf Mathematical Control Theory Conference}}\\

{Baton Rouge, Louisiana, April 10-13, 2003}\\

\end{center}

{\small {\it  (these transparencies are available at}\\
{\it http://www.math.rutgers.edu/$\widetilde{\,\,}$sussmann/slides.html)}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}
\stapageno
\begin{center}
{\Huge${\cal SET-VALUED}$}\\

\vspace{-7mm}
{\Huge${\cal DIFFERENTIALS}$}\\

\vspace{7mm}
{\large{\it H\'ector J. Sussmann}}\\

%\vspace{-2mm}
{\it Rutgers University}\\

\vspace{-2mm}
{\it Piscataway, New Jersey, USA}\\

\vspace{5mm}
Institut Mittag-Leffler\\

\vspace{-2mm}
Djursholm, Sweden\\

\vspace{-2mm}
May 20, 2003

\end{center}

{\small {\it  (these transparencies are available at}\\
{\it http://www.math.rutgers.edu/$\widetilde{\,\,}$sussmann/slides.html)}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}
\stapageno
\begin{center}

{\Large${\cal DIFFERENTIALS\,\,\,OF}$}\\

\vspace{-10mm}
{\Large${\cal NON\!-\!DIFFERENTIABLE}$}\\

\vspace{-10mm}
{\Large${\cal MAPS}$}\\

%\vspace{7mm}
{\large{\it H\'ector J. Sussmann}}\\

%\vspace{-2mm}
{\it Rutgers University}\\

\vspace{-2mm}
{\it Piscataway, New Jersey, USA}\\

\vspace{5mm}
Brazilian Colloquium of Mathematics\\

\vspace{-2mm}
IMPA\\

\vspace{-2mm}
Rio de Janeiro\\

\vspace{-2mm}
July 27-August 1, 2003

\end{center}

{\small {\it  (these transparencies are available at}\\
{\it http://www.math.rutgers.edu/$\widetilde{\,\,}$sussmann/slides.html)}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}
\stapageno
\begin{center}

{\Huge DO NOT WORRRY!}\\

%\vspace{-10mm}
{\Huge THE TALK}\\

{\Huge WILL BE}\\

%\vspace{-10mm}
{\Huge IN {\bf ENGLISH}}

%\vspace{7mm}

\end{center}

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}
\stapage

\begin{center}
{\Large{\bf Outline of the talk}}
\end{center}

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

6. Regular maps.

7.$\,\,\,$Generalized  \ \ \ differential \ \ \ \mbox{quotients}

\ \ \ (GDQs).

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}
\stapage

%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}

{\large\bf 1. Motivation: the Pontryagin \\
\phantom{aaa}Maximum Principle and\\
\phantom{aaa}generalized derivatives.}

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

6. Regular maps.

7.$\,\,\,$Generalized  \ \ \ differential \ \ \ \mbox{quotients}

\ \ \ (GDQs).

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$

\vspace{9mm}
Everybody knows that if we want \mbox{nonsmooth} functions 
such as $|x|$ to have a ``derivative'' at every point, then these derivatives 
will have to be {\bf sets of numbers} (for functions
$f:\rr\mapsto\rr$) or {\bf sets of linear maps} (for functions
$f:\rr^n\mapsto\rr^m$).

For example:

\vspace{-5mm}
\begin{center}\fbox{\fbox{
\begin{minipage}{15cm}
If $f:\rr\mapsto\rr$ is given by $f(x]=|x|$, then 
$f'(0)$ must be the interval $[-1,1]$.
\end{minipage}}}\end{center}

\vspace{6mm}
\begin{center}\begin{minipage}{15.8cm}
\baselineskip=20pt
(Even if you don't agree with this, you should accept that
$f'(0)$ contains at least $1$ and $-1$, because the graph 
has two slopes at $0$. Why do we actually take the convex hull?
There are many reasons, and this will \mbox{become} clearer later.
However, the only \mbox{purpose} of this slide is to persuade you
that $f'(0)$ {\bf must be a set}, so if you believe that
$f'(0)=\{-1,1\}$, I have won.)
\end{minipage}\end{center}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\
$\quad$\\
\begin{center}{\bf OTHER EXAMPLES}\end{center}

\vspace{2mm}
\begin{center}\fbox{\fbox{
\begin{minipage}{15cm}
If $f(x)=|x|+|x|^{3/2}\sin(1/x)$, then 
$f'(0)$ must be the interval $[-1,1]$.
\end{minipage}}}\end{center}

\vspace{4mm}
\begin{center}\fbox{\fbox{
\begin{minipage}{15cm}
If $f(x)=2x+|x|+|x|^{3/2}\sin(1/x)$, then 
$f'(0)$ must be the interval $[1,3]$.
\end{minipage}}}\end{center}

\vspace{4mm}
{\bf Homework problem:} If $f(x)=x\sin(1/x)$ for $x\neq 0$, $f(0)=0$, 
find $f'(0)$

{\bf Solution:} Change the definition of ``\mbox{derivative},''   so that
$$
f'(x)\equaldef\lim_{\delta\downarrow 0}
\Biggl\{\frac{f(x+h)-f(x)}{h}:0<|h|\le\delta\Biggr\}\,.
$$
Then $f'(0)=[-1,1]$.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
QUESTION: what are {\bf real \mbox{mathematical} definitions}
that would make the above \mbox{solutions} correct?

\vspace{4mm}
REMARK: This situation is similar to that encountered by the founders
of the \mbox{Calculus} three centuries ago
(and also by our Calc I students today). They knew
that the derivative of $x^2$ is $2x$, that the derivative of $\sin\,x$
is $\cos\,x$, that the chain rule and the Leibniz product rule hold,
etc. But they did not have a good working \mbox{definition}
of ``derivative''  that could be used to prove general theorems.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.2cm}\stapage
$\quad$\\
$\quad$\\
BUT THERE IS A BIG DIFFERENCE. \mbox{Today}, we know so much about 
\mbox{mathematical}
rigor, that finding a rigorous \mbox{mathematical} definition
that would make the above \mbox{solutions} correct is a trivial thing to do.

The problem is, we are so good at \mbox{concocting}
\mbox{rigorous} definitions, that 
dozens of different definitions of ``generalized derivative'' 
exist. all of which agree on simple examples, but yield different results 
in more complicated cases. 

So the really interesting question is {\Large\bf how do we choose
the right \mbox{definition}?} Or, who knows, 
maybe we do not have to choose,
and can work with all of them, as long as they satisfy some desirable 
properties. 

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\
$\quad$\\
$\quad$\\
{\bf WHAT WE NEED IS \fbox{MOTIVATION!!}.} 

\vspace{2mm}
We have to know what these ``generalized derivatives'' are supposed to be 
good for. 

\vspace{2mm}
In this talk, the purpose is 

\vspace{-4mm}
\begin{center}$\!\!\!$\fbox{\fbox{\fbox{
\begin{minipage}{16cm}\baselineskip=46pt
{\huge\bf To \mbox{differentiate} flows of \mbox{nonsmooth} vector fields
so as to be able to prove general versions of the Pontryagin Maximum
\mbox{Principle}.}
\end{minipage}}}}\end{center}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
%\vspace{-2mm}
\begin{center}
{\large\bf A VERY NON-SMOOTH, HYBRID,
TECHNICALLY HYBRID, MAXIMUM PRINCIPLE}
\end{center}

{\bf Principle:} A statement not yet precise enough to be a true theorem,
but ready to become a theorem (in many ways) by 
\mbox{making} the terms precise.


{\bf Hybrid:} Including discrete decisions, e.g., jumps.


{\bf Technically hybrid:} Such that the \mbox{technical} conditions of
different versions of the MP may be satisfied in different regions of
the state space (or at different times), and our reference trajectory
visits several of these regions.


\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
\begin{center}{\large\bf Time-varying vector fields (TVVFs)
and their flows}\end{center}

\vspace{5mm}
A {\bf TVVF} on $M$ is a (possibly partially \mbox{defined}) map
$$
M\times\rr\ni(x,t)\mapsto f(x,t)\in TM
$$ 
such that $f(x,t)\in T_xM$ whenever \mbox{$(x,t)\in{\rm Dom}(f)$}.

\vspace{5mm}
A {\bf trajectory} of a TVVF $f$ on $M$ is a 
\mbox{locally} absolutely continuous 
curve $\xi:I\mapsto M$, \mbox{defined} on a nonempty subinterval 
$I$ of $\rr$,  such that 
$$
\fbox{$(\xi(t),t)\in{\rm Dom}(f)\,\,\,{\rm and}\,\,\,\dot\xi(t)=f(\xi(t),t)$}
$$
for almost all $t\in I$. 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
\begin{center}{\large\bf Flows}\end{center}

\vspace{5mm}
If $t,s\in\rr$, $x\in M$,  we define
$$
\fbox{$
\left.\begin{array}{l}\Phi_{t,s}^f(x)\equaldef\Psi^f(x,t,s)\equaldef\\
\Big\{\xi(t):\xi\in{\rm Traj}(f),\xi(s)=x\Big\}\end{array}\right.$}\,.
$$

\vspace{5mm}
Then $\Psi^f$ is the {\bf flow} of $f$, and the 
$\Phi_{t,s}^f$ are the {\bf flow maps} of $f$.

\vspace{5mm}
Naturally, these maps are in  principle set-valued, and some values
may be empty, \mbox{because} we are not assuming existence or uniqueness
of trajectories. 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
$\quad$\\

\vspace{-5mm}
\begin{center}{\large\bf Differentials of flows}\end{center}

\vspace{3mm}
Suppose $f$ is a {\bf TVVF} on $M$, and 
\mbox{$\xi_*:[a,b]\mapsto M$} is a trajectory. Let 
\mbox{$x_*=\xi_*(a)$}, 
$y_*=\xi_*(b)$. Then the flow map $\Phi_{b,a}^f$ sends 
$x_*$ to $y_*$, so we may ask {\bf what is the derivative
of $\Phi_{b,a}^f$ at $(x_*,y_*)$?}

\vspace{2mm}
If $(x,t)\mapsto f(x,t)$ is of class $C^1$ with \mbox{respect} to
$x$, measurable with respect to $y$, and locally 
bounded---together with its \mbox{$x$-derivative}---by an integrable 
\mbox{function} of $t$, we can compute the 
{\bf \mbox{classical} \mbox{differential}} $D\Phi_{b,a}^f(x_*)$ 
by solving the {\bf \mbox{variational} \mbox{equation}}
$$
\dot V(t)=\frac{\partial f}{\partial x}\Biggl(\xi_*(t),t\Biggr)\cdot V(t)
$$
with initial condition $V(a)={\rm identity}$. 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
$\quad$\\


\vspace{-7mm}
\begin{center}{\large\bf Lebesgue points}\end{center}

\vspace{4mm}
Suppose $f$ is a {\bf TVVF} on $M$, and $(x,t)\in M$. We say that 
$(x,t)$ is a {\bf Lebesgue point} of $f$
if $f$ is defined near $(x,t)$ and
the limit of 

\vspace{-7mm}
$$
\frac{1}{h}\int_{t-h}^{t+h}\sup\Biggl\{\|f(y,s)-f(x,t)\|:\|y-x\|\le\delta\Biggr\}ds\,.
$$

\vspace{5mm}
as $\delta\downarrow 0$ and $h\downarrow 0$ is equal to $0$.

\vspace{-2mm}
\begin{center}{\large\bf The differentiation theorems
for flows in the $C^1$ case}\end{center}

\vspace{5mm}
{\bf Theorem I:} $D\Phi_{b,a}^f(x_*)=V(b)$. 

\vspace{5mm}
{\bf Theorem II:}  
{\em If $(x_*,a)$ and  $(y_*,b)$ are Lebesgue points of $f$, then 
\fbox{$D\Psi^f(x_*,b,a)\!=\!f(y_*,b)dt\!+\!V(b)\Biggl((dx\!-\!f(x_*,a)ds\Biggr)$}.}

\vspace{4mm}

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
\begin{center}{\large\bf Differentials of Lipschitz flows}\end{center}

\vspace{6mm}
Now suppose that the TVVF $(x,t)\mapsto f(x,t)$ is Lipschitz with respect to
$x$, measurable with respect to $y$, and with a Lipschitz 
constant locally bounded by an integrable function of $t$. We can 
compute a {\bf Warga derivate container} $\Lambda_{b,a}^f(x_*)$ 
by solving the {\em variational differential inclusion}
$$
\dot V(t)\in\partial_xf\Biggl(\xi_*(t),t\Biggr)\cdot V(t)
$$
with initial condition $V(a)={\rm identity}$. (Here 
$\partial_xf$ stands for ``Clarke generalized Jacobian of $f$
with respect to $x$.'')

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
\begin{center}{\large\bf The Warga differentiation theorem for
Lipschitz flows}\end{center}

\vspace{6mm}
{\bf Theorem I$_{lip}$:} {\em The set $\Lambda_{b,a}^f(x_*)$ 
of all $V(b)$s is a WDC of $\Phi_{b,a}^f$ at $x_*$.}

\vspace{10mm} It would be nice of one could prove:

\vspace{6mm}
{\bf Theorem II$_{lip}$:}  {\em If $(x_*,a)$ and  $(y_*,b)$) are Lebesgue 
points of $f$, then the set $\hat\Lambda_{b,a}^f(x_*)$ of all
maps $f(y_*,b)dt+V(b)\Biggl((dx-f(x_*,a)ds\Biggr)$
is  a WDC of $\Psi^f$ at $(x_*,b,a)$.}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
$\quad$\\
$\quad$\\

However:

\vspace{-10mm}
{\Large\bf Theorem II$_{lip}$ is not true in general.}


\vspace{6mm}
The reason is very simple: if we only \mbox{assume} $f$ to be bounded
by an integrable \mbox{function} of $t$, then the time-derivative
of $\Psi^f$ need not be bounded, so 
$\Psi^f$ need not be \mbox{Lipschitz}, so 
$\Psi^f$ need not have a�\ Warga derivate \mbox{container}, \mbox{because}
only Lipschitz maps have Warga derivate containers.

\vspace{-6mm}
{\Large\bf Question: can we make \mbox{Theorem} II$_{lip}$ true?}

\vspace{3mm}
We will answer this later.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage

\stapageno

\stapageno

\stapageno

\vspace{8mm}
\stapageno

\begin{center}{\large\bf A nonsmooth Max. Principle}\end{center}

The Maximum Principle will be a {\em necessary condition} for the 
{\bf separation property}
$$
{\rm (SEP)}\qquad\quad
\fbox{${\cal R}_{[a,b]}^{\Sigma,{\cal U}}(\xi_*(a))\cap S=\{\xi_*(b)\}$}\,,
\qquad\quad
$$
where ${\cal R}_{[a,b]}^{\Sigma}(x)$ is the $\Sigma$-reachable set
from $x$ over the interval $[a,b]$ using controls in ${\cal U}$,
and �$S$ is some given set.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage


\begin{center}{\large\bf Data}\end{center}

\begin{itemize}
\item[(D1)] a system
$$
\Sigma:\quad
\fbox{$\dot x=f(x,u,t)\,,\,\,x\in M\,,\,\, u\in U\,,\,\,t\in\rr$}
\,;
$$
\item[(D2)] a ``reference trajectory-control pair'' $(\xi_*,\eta_*)$, 
defined on an interval $[a,b]$;
\item[(D3)] a subset $S$ of $M$ s. that $y_*\equaldef\xi_*(b)\in S$; 
\item[(D4)] a closed convex cone $C$ in the tangent space $T_{y_*}M$;
\item[(D5)] a set ${\cal U}$ of functions \mbox{$\eta:[a,b]\mapsto U$}
(the ``class of admissible \mbox{controls}'').
\item[(D6)] a subset $G$ (the ``good set'') of $[a,b]$ such that 
$a\in G$ and $b\in G$.
\item[(D7)] a family $\Lambda=\{\Lambda_{t,s}\}_{s,t\in[a,b],s\le t}$
(the ``derivative of the reference flow'') such that 
each $\Lambda_{t,s}$ is a set of linear maps from
$T_{\xi_*(s)}M$ to $T_{\xi_*(t)}M$.
\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
\begin{center}{\large\bf Hypotheses}\end{center}

\vspace{4mm}
\begin{itemize}
\item[(H1)] ${\cal U}$ contains every function 
\mbox{$\eta:[a,b]\mapsto U$} 
\mbox{obtained} from $\eta_*$ by modifying $\eta_*$ on finitely
many disjoint subintervals $I_1,\ldots,I_m$ of $[a,b]$ 
by substituting a \mbox{constant} control $u_i$ for
$\eta_*(t)$ on $I_i$.
\item[(H2)] The flow $\Psi^{f_*}$ of the \mbox{reference} vector field
$f_*$ (defined by $f_*(x,t)\equaldef f(x,\eta_*(t),t)$) 
has a \fbox{\fbox{differential}}
$$
\hat\Lambda_{t,s}=\dot\xi_*(t)dt+
\Lambda_{t,s}\Biggl(dx-\dot\xi_*(s)ds\Biggr)
$$
at $\Biggl((\xi_*(s),t,s),\xi_*(t)\Biggr)$ 
whenever $s,t\in G$ and $s\le t$.
\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage

$\quad$\\
$\quad$\\

\vspace{-3mm}
\begin{itemize}
\item[(H3)] The sets �$\Lambda_{t,s}$ are compact and nonempty�
and satisfy the {\em flow identities}
$$
\Lambda_{t,t}=\Biggl\{{\rm id}_{_{T_{\xi_*(t)}M}}\Biggr\}\,,
$$
$$
\Lambda_{t,s}\circ\Lambda_{s,r}=\Lambda_{t,r}\,\,
{\rm if}\,\,r,s,t\in G\,,\,\,r\le s\le t\,.
$$
\item[(H4)] $C$ is a  \fbox{\fbox{tangent cone}} to $S$ at 
$y_*$. 
\item[(H5)] (SEP) holds.
\end{itemize}
\end{minipage}\end{center}

\vspace{-3mm}
\begin{center}-----------------\end{center}

\vspace{-3mm}
\begin{center}\begin{minipage}{16cm}
The \fbox{\fbox{double-boxed terms}} are not yet \mbox{precisely} defined,
and this---and {\it only this}---is why the principle we are stating is not yet
a true theorem.   

\vspace{3mm}
There are exactly {\Large\bf two} double-boxed terms:

\vspace{3mm}
\begin{center}\fbox{\fbox{differential}} \ \ and
\ \ \fbox{\fbox{tangent cone}}\end{center}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage

\vspace{-3mm}
\begin{center}{\large\bf Conclusion}\end{center}


\vspace{7mm}
There exist a {\bf compatible selection}
$$
L=\{L_{t,s}\}_{s,t\in G,s\le t}
$$
of $\Lambda$, and covectors
$\pi(t)\in T_{\xi_*(t)}^*M$, for \mbox{$t\in G$}, such that

\vspace{5mm}
\begin{itemize}\item[]\begin{itemize}
\item[(C1)] {\it (The adjoint equation in integrated form.)}
\fbox{$\pi(s)=L_{t,s}^\dagger(\pi(t))$} whenever 
$s\le t$, $s,t\in G$,  
\end{itemize}\end{itemize}

\vspace{9mm}
\begin{center}\begin{minipage}{15cm}
\small\baselineskip=25pt 
NOTE: A {\bf compatible selection} of $\Lambda$ is a family 
$L=\{L_{t,s}\}_{s,t\in G,s\le t}$ such that 
$L_{t,s}\in\Lambda_{t,s}$ 
whenever $s\le t$ and $s,t\in G$, 
and $L_{t,s}L_{s,r}=L_{t,r}$ whenever $r,s,t\in G$ and
$r\le s\le t$. 
\end{minipage}\end{center}

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage	
\begin{itemize}\item[]\begin{itemize}
\item[(C2)] {\it (The Hamiltonian maximization \mbox{condition}.)}
$$
\!\!\!\!\!\!\!\!\!\!\!\!
\fbox{$\pi(t)\cdot f(\xi_*(t),\eta_*(t),t)\ge\pi(t)\cdot f(\xi_*(t),u,t)$}
$$ 
whenever $t\in G$, $u\in U$, and $t$ is such that 
$\Psi^{f(\cdot,u,\cdot)}$ is \fbox{\fbox{differentiable}}
at $\Biggl((\xi_*(t),t,t),\xi_*(t)\Biggr)$ with \fbox{\fbox{differential}}
equal to $dx+f(\xi_*(t),u,t)(dt-ds)$.

\vspace{5mm}
\item[(C3)] {\it (The nontriviality condition.)}
$\pi(b)\neq 0$.

\vspace{5mm}
\item[(C4)] {\it (The transversality condition.)}
\mbox{$\pi(b)\cdot c\ge 0$} whenever $c\in C$.
\end{itemize}\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}\stapage	
$\quad$\\
$\quad$\\
EXAMPLE: The {\it reflected brachistochrone}.

\vspace{2mm}
PROBLEM: Using trajectories of
$$
\dot x=u\sqrt{|y|}\,,\,\,\,\dot y=v\sqrt{|y|}\,,\,\,\,u^2+v^2=1\,,
$$
steer $A\in\rr^2$ to $B\in\rr^2$ in minimum time.

\vspace{2mm}
The {\it existence} of an optimal trajectory for any given $A$, $B$
is trivial (Ascoli-Arzel\`a).

\vspace{2mm}As long as $y>0$ or $y<0$ the optimal \mbox{trajectories} are
{\it cycloids}. (This follows from any classical version of the MP.) 

\vspace{2mm}With a small {\it ad hoc} argument, one can \mbox{exclude}
arcs that intersect the $x$ axis in more than point. 

\vspace{2mm}What is left is: arcs of the form
``cycloid from $A$ to $C$ followed by 
cycloid from $C$ to $B$,'' with $C$ belonging to the $x$ axis.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}\stapage	

\stapageno

\stapageno

\stapageno

\stapageno

\stapageno

\vspace{6mm}
\stapageno
How does one find $C$?

The dynamics is not Lipschitz. So \mbox{neither} the classical versions of
the MP nor the \mbox{Lipschitz} ones (Clarke, Vinter, Ioffe, 
\mbox{\L{}ojasiewicz} Jr.)
help.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}\stapage
$\quad$\\
Formally, it is clear that {\bf the ``adjoint \mbox{vector},'' if it exists,
will have to blow up as we cross the $x$ axis.} 

{\it Proof:} \ the Hamiltonian is 
$$
H=(p_1u+p_2v)\sqrt{|y|}+p_0\,.
$$ 
Along our curve, $H\equiv 0$. So $p_1$ and $p_2$
\mbox{cannot} both vanish, because if they did then $p_0$ would be $0$
as well, \mbox{contradicting} \mbox{nontriviality}. Then 
$$
u=\frac{p_1}{\sqrt{p_1^2+p_2^2}}\qquad{\rm and}\qquad
v=\frac{p_2}{\sqrt{p_1^2+p_2^2}}\,,
$$
so the maximized Hamiltonian is 
$$
{\cal H}=\sqrt{p_1^2+p_2^2}\sqrt{|y|}-p_0\,.
$$
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}\stapage
$\quad$\\$\quad$\\
Since ${\cal H}\equiv 0$ along our trajectory, and we know that 
$(p_1,p_2)\neq(0,0)$, and $y$ is not $\equiv 0$, \mbox{we\ conclude}
that
$$
\sqrt{p_1^2+p_2^2}=\frac{p_0}{\sqrt{|y|}}\,,
$$
so $\sqrt{p_1^2+p_2^2}\to+\infty$ as $y\to 0$. 

\vspace{5mm}
So for this problem {\bf the ``adjoint \mbox{vector}'' 
cannot satisfy any classical or \mbox{nonsmooth} ``adjoint \mbox{equation}'' 
whose solutions are 
\mbox{absolutely} continuous by \mbox{definition}.}  

\vspace{5mm}
If there is an ``adjoint \mbox{equation},'' it will have to be allowed
to have solutions that are {\it \mbox{discontinuous}} and in fact can even be\\

\vspace{-27mm}
{\Huge\bf unbounded!!!!!!}.

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}\stapage
With our version of the maximum \mbox{principle} there is no problem, because
one can \mbox{easily} embed our candidate piecewise cycloid in a {\it flow}
of piecewise cycloids, which will be smooth except when you cross the
$x$ axis. The flow will then be (classically) \mbox{differentiable} 
\mbox{except} at the one time $\tau$ when the reference \mbox{trajectory} crosses
the $x$ axis. If we form the set $G$ by just excluding $\tau$,
we can apply our MP. 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}\stapage
\stapageno

\stapageno

\stapageno

\stapageno

\stapageno

\stapageno

The final result is:  

\vspace{8mm}
\begin{center}\fbox{\begin{minipage}{15cm}
{\Large\em Let $R_+$, $R_-$ be the radii of the rolling circles
that \mbox{generate} the upper and lower cycloids. Then $R_1=R_2$.}
\end{minipage}}\end{center}


\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}\stapage
%\vspace{-7mm}
%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}
$\quad$\\$\quad$\\
1. Motivation: the Pontryagin \mbox{Maximum}
\phantom{aaa}Principle and generalized derivatives.

\vspace{-12mm}
{\large\bf 2. $\!\!\!\!\!\!\!\!\!\!\!\!$Categories, \ \ \ functors,
\mbox{\phantom{aaa}multifunctors}.}

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

6. Regular maps.

7.$\,\,\,$Generalized  \ \ \ differential \ \ \ \mbox{quotients}

\ \ \ (GDQs).

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
A {\bf \ category} \ is \ a \ mathematical \ entity \ ${\cal C}$ 
\mbox{consisting} of the following three ingredients: 

\vspace{2mm}
\begin{itemize}
\item[1.] A class ${\cal O}({\cal C})$ of {\bf objects},

\vspace{2mm}
\item[2.] for each pair $(A,B)$ of objects, a set 
${\rm Hom}_{\cal C}(A,B)$ of {\bf morphisms} from $A$ to $B$,

\vspace{2mm}
\item[3.]  a {\bf composition operation} \ $\circ_{_{\cal C}}\,$,  \ which 
assigns to each pair $\mu$, $\nu$ such that $\mu\in{\rm Hom}_{\cal C}(A,B)$ 
and $\nu\in{\rm Hom}_{\cal C}(B,C)$ for some triple $(A,B,C)$ 
of objects of ${\cal C}$ a morphism $\nu\circ_{_{\cal C}}\mu$ (or just 
$\nu\circ\mu$) belonging to ${\rm Hom}_{\cal C}(A,C)$, 
\end{itemize}

\vspace{2mm}
and such that {\bf composition is associative} and {\bf every object $A$
admits an identity morphism} ${\rm id}_A\in{\rm Hom}_{\cal C}(A,A)$.
\end{minipage}\end{center}
\esl
\bsl 
\begin{center}\begin{minipage}{15.5cm}
\stapage
$\quad$\\$\quad$\\
NOTATION: \fbox{$\mu:A\mapsto_{\cal C}B$} (or even  
\fbox{$\mu:A\mapsto B$}, if it
is clear what ${\cal C}$ is) is an alternative notation for 
$\mu\in{\rm Hom}_{\cal C}(A,B)\,$.

\bigbreak {\bf Composition is associative} means: if 

\vspace{-14mm} 
$$\mu:A\mapsto B\,,\quad\nu:B\mapsto C\,,
\quad{\rm and}\quad\rho:C\mapsto D
$$ 
then \fbox{$\rho\circ(\nu\circ\mu)=(\rho\circ\nu)\circ\mu$}.

\bigbreak An {\bf identity morphism}  of an object $A$ is a morphism
$\iota:A\mapsto A$ such that 
$$
\fbox{$\iota\circ\mu=\mu\,\,{\rm if}\,\,B\,\,{\rm is\,\,in}\,\,
{\cal O}({\cal C})\,\,{\rm and}\,\,\mu:B\mapsto A,$}
$$
and 
$$
\fbox{$\mu\circ\iota=\mu\,\,{\rm if}\,\,B\,\,{\rm is\,\,in}\,\,{\cal O}({\cal C})\,\,{\rm and}\,\,\mu:A\mapsto B,$}
$$
\medbreak 
FACT: The identity morphism $\iota_A$ of $A$ is unique.

\end{minipage}\end{center}
\esl
\bsl 
\begin{center}\begin{minipage}{15.5cm}
\stapage

\vspace{9mm} 
We also require (to simplify a number of things) the following:

\vspace{9mm} 
\begin{center}\begin{minipage}{10cm}
{\em If $\mu:A\mapsto B$ and $\mu:A'\mapsto B'$ then 
$A=A'$ and $B=B'$.}
\end{minipage}\end{center}

\vspace{14mm} 
In the definition of ``category'' one could do 
\mbox{without} \ $\,$the \ $\,$objects, \ \ \ $\,$and \ just \ take \ a 

category \ to \ consist \ of \ a \ class \ \ $\,$${\cal M}$$\,$ \ \ of

``morphisms'' \ $\,$$\,$$\,$together \ $\,$with \ $\,$a \ $\,$partially 

defined binary operation $\circ$ on ${\cal M}$ called ``composition.''

\vspace{6mm} 
The {\bf objects} \ would then be defined as the {\em
\mbox{idempotent} morphisms}, i.e., the morphisms $\mu$ such that
$\mu\circ\mu=\mu$.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage

\vspace{10mm}
One could even go further and do away with the morphisms as well, and
just \mbox{define} the \mbox{category} to be the \mbox{composition}
\mbox{operation}. 

\vspace{6mm}
In other words, a category would be a \mbox{partially}
\mbox{defined} binary operation that 
\mbox{satisfies} certain axioms.  

\vspace{6mm}	
Since a binary \mbox{operation} is a \mbox{collection} of triples, the 
``morphisms'' \ could then be \mbox{defined} as the entities that
occur as entries of one of these triples, and the ``objects'' as the
idempotent morphisms.
\end{minipage}\end{center}
\esl
\bsl
%\begin{minipage}{16cm}
%\baselineskip=23pt
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{center}{\Large\bf FUNCTORS}\end{center}

\vspace{8mm}
If ${\cal C}_1,{\cal C}_2$  are 
categories, a {\bf functor} from  ${\cal C}_1$ to ${\cal C}_2$
is a correspondence ${\cal F}$ that assigns to \mbox{every}
\mbox{object} $A$ of ${\cal C}_1$ an object 	${\cal F}(A)$ of ${\cal C}_2$,
and to each morphism $\mu:A\mapsto_{_{{\cal C}_1}}B$ \mbox{between}
\mbox{two} \mbox{objects} 
\ \ $A,\,B$ \ of \ ${\cal C}_1$ \ a \ \mbox{morphism}
\mbox{${\cal F}(\mu):{\cal F}(A)\mapsto_{_{{\cal C}_2}}{\cal F}(B)$}
\ \ between \ \ the \mbox{corresponding} objects 
${\cal F}(A),{\cal F}(B)$ of ${\cal C}_2$, in such a way that

\vspace{8mm}
\begin{itemize}
\item[a.] \fbox{${\cal F}(\iota_A)=\iota_{{\cal F}(A)}$} whenever $A$ is in 
${\cal O}({\cal C}_1)$,

\vspace{8mm}
\item[b.] \fbox{${\cal F}(\nu\circ\mu)={\cal F}(\nu)\circ{\cal F}(\mu)$}
whenever \mbox{$\mu\!:\!A\!\mapsto_{_{{\cal C}_1}}\!\!\!B$} 
and $\nu:B\mapsto_{_{{\cal C}_1}}C$.
\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\
\begin{center}
\fbox{\fbox{\fbox{\fbox{
\begin{minipage}{15cm}
{\huge{\em 
Since we are doing set-valued \mbox{analysis}, 
why not introduce set-valued functors?}}
\end{minipage}}}}}
\end{center}

\vspace{3mm}
A {\bf \ multifunctor} \ from  \ ${\cal C}_1$ \ to \ ${\cal C}_2$
\ is \ a 
\mbox{correspondence} \ \ ${\cal F}$ \ which is \mbox{exactly} like a functor 
\mbox{except} that if $A,\,B$ are  \mbox{objects} of 
${\cal C}_1$ and $\mu\in{\rm Hom}_{{\cal C}_1}(A,B)$ 
then \mbox{${\cal F}(\mu)$} is a {\bf subset} 
rather than a member 
of $ {\rm Hom}_{{\cal C}_2}({\cal F}(A),{\cal F}(B))$, and 

\vspace{1.5mm}
a. \fbox{$\iota_{_{{\cal F}(A)}}\in{\cal F}(\iota_{_A})$} whenever $A$ is in 
${\cal O}({\cal C}_1)$,

\vspace{2.5mm}  
b. \fbox{${\cal F}(\nu)\circ{\cal F}(\mu)\subseteq
{\cal F}(\nu\circ\mu)$} whenever $A,B,C$ 
\phantom{aaa}are \ in \ ${\cal O}({\cal C}_1)$,
$\mu:A\mapsto_{_{{\cal C}_1}}B$, and 
\phantom{aaa}\mbox{$\nu:B\mapsto_{_{{\cal C}_1}}C$.}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\vspace{8mm}
NOTE: If we adopt the point of view that a category consists of the
\mbox{specification} of a class of morphisms (no  objects), then a 
multifunctor 
from  ${\cal C}_1$ to ${\cal C}_2$ is just a
\mbox{correspondence} �$\mu\mapsto{\cal F}(\mu)$ that \mbox{assigns} to each
morphism $\mu$ of  ${\cal C}_1$ a set of \mbox{morphisms} ${\cal F}(\mu)$ of
${\cal C}_2$ and satisfies certain \mbox{properties}. The \mbox{correspondence}
$A\mapsto{\cal F}(A)$ \mbox{determined} by ${\cal F}$ on the objects
can then be   \mbox{recovered}, \mbox{because} the objects of ${\cal C}_i$
are the \mbox{idempotents} of ${\cal C}_i$, and if $\mu$ is an 
idemptotent of ${\cal C}_1$, then ${\cal F}(\mu)$ contains one and only one 
idemptotent of ${\cal C}_2$.

\vspace{4mm}
A {\bf functor} is just a multifunctor ${\cal F}$ such that 
${\rm card}({\cal F}(\mu))=1$ for every morphism $\mu$.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\
\begin{center}
EXAMPLES OF CATEGORIES
\end{center}

\vspace{4mm}	
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal SVM}\,$}}. {\em (Sets \ and \ set-valued \ maps).} 
\mbox{Objects}: sets. \mbox{Morphisms}: set-valued 
maps. {\small (\mbox{Precisely}: if $A$, $B$ are sets, the members of 
${\rm Hom}_{\cal SVM}(A,B)$ are the triples $\mu=(A,B,G)$
such that $G\subseteq A\times B$. Then $A$, $B$, $G$ are, respectively, 
the {\em source}, {\em target} and {\em graph} of $\mu$.
{\bf \mbox{NOTATION:}} \fbox{$F:A\tto B$} means 
$F\in{\rm Hom}_{\cal SVM}(A,B)$, and
\fbox{${\rm So}\,F$, ${\rm Ta}\,F$, ${\rm Gr}\,F$, ${\rm Do}\,F$, 
${\rm Im}\,F$}, are, respectively,�
the source, target, graph, domain, and image of $F$.)}
\end{minipage}

\vspace{4mm}	
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal OM}\,$.}}  {\em (Sets and ordinary maps.)} 
\mbox{Objects}: \ sets. \ \ \mbox{Morphisms:} \mbox{everywhere}
defined single-valued maps.
{\small \mbox{(Precisely:} if $A$, $B$ are sets, then the \mbox{members} of 
${\rm Hom}_{\cal OM}(A,B)$ are the $\mu\in{\rm Hom}_{\cal SVM}(A,B)$
such that \mbox{${\rm Do}(\mu)=A$} and the map $\mu$ is single-valued.
{\bf \ NOTATION:} \fbox{$F:A\mapsto B$} means 
\mbox{$F\in{\rm Hom}_{\cal OM}(A,B)$}.)}
\end{minipage}

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal FDRLS}\,$}}. {\em (Finite-dimensional real linear spaces
and linear maps.)} 
Objects: finite-dimensional real \mbox{linear} 
spaces. Morphisms: \mbox{linear} maps. 
{\small \mbox{(Precisely:} if $A$, $B$ are 
FD real linear spaces, then the members of 
${\rm Hom}_{\cal FDRLS}(A,B)$ are the $\mu\in{\rm Hom}_{\cal OM}(A,B)$
that are linear.)}
\end{minipage}

\vspace{8mm}
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal MC}^k{\cal C}^\ell\,\,$}}, 
if $k,\ell\in\nn\cup\{\infty\}\cup\{\omega\}$ and $k\ge\ell$. 
{\em (Finite-dimensional manifolds of class 
$C^k$ and maps of class $C^\ell$.)} 
\mbox{Objects}: finite-dimensional \mbox{manifolds} of class $C^k$.
The \mbox{morphisms} from $M$ to $M'$ are the maps 
\mbox{$F:M\mapsto M'$} of class $C^\ell$.{\small 
{\bf \ NOTATION:} \fbox{$C^\ell(M,M')$} is the set 
${\rm Hom}_{{\cal MC}^k{\cal C}^\ell}(M,M')$.}
\end{minipage}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal PMC}^k{\cal C}^\ell\,$}}. 
{\em (Pointed finite-dimensional \mbox{manifolds} of class $C^k$
and base-point \mbox{preserving} maps of class $C^\ell$.)} 
\mbox{Objects}: pairs $(M,p)$ such that $M$ is a 
finite-dimensional manifold of class $C^k$ and \mbox{$p\in M$.}
The morphisms from $(M,p)$ to $(M',p')$ are the maps $F\in C^\ell(M, M')$ 
such that $F(p)=p'$.{\small 
{\bf \ NOTATION:} \fbox{$C^\ell((M,p),(M'p'))$}  is the set 
${\rm Hom}_{{\cal PMC}^k{\cal C}^\ell}((M,p),(M',p'))$.}
\end{minipage}

\vspace{6mm}	
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal PMC}^1{\cal C}^0{\cal D}^1\,$}}. 
{\em (Pointed finite-dimensional \mbox{manifolds} of class $C^1$
and base-point \mbox{preserving}
continuous maps that are \mbox{differentiable} at the base point.)} 
\mbox{Objects}: same as for ${\cal PMC}^1\,$.
The morphisms from $(M,p)$ to $(M',p')$ are the \mbox{continuous} 
maps $F:M\mapsto M'$  such that $F(p)=p'$ and 
$F$ is differentiable at $p$.{\small 
{\bf \ NOTATION:} \fbox{$C^0D^1((M,p),(M'p'))$}  is the set 
${\rm Hom}_{{\cal PMC}^1{\cal C}^0{\cal D}^1}((M,p),(M',p'))$.}
\end{minipage}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.8cm}\stapage
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal PMC}^1{\cal LIP}\,$}}. 
{\em (Pointed finite-dimensional \mbox{manifolds} of class $C^1$
and base-point \mbox{preserving} Lipschitz-continuous maps.)} 
Objects: same as for ${\cal PMC}^1\,$.
The morphisms from $(M,p)$ to $(M',p')$ are the Lipschitz-continuous 
maps $F:M\mapsto M'$  such that \mbox{$F(p)=p'$}.{\small 
{\bf \ NOTATION:} \fbox{$C^{Lip}((M,p),(M'p'))$}  is the set 
${\rm Hom}_{{\cal PMC}^1{\cal LIP}}((M,p),(M',p'))$.}
\end{minipage}

\vspace{5mm}	
\begin{minipage}{15.2cm}
\baselineskip=23pt
\fbox{\fbox{${\cal PMC}^1{\cal SVM}$}}.
{\em (Pointed finite-dimensional \mbox{manifolds} of class $C^1$
and set-valued maps.)} Objects: same as for ${\cal PMC}^1\,$.
The morphisms from $(M,p)$ to $(M',p')$ are the set-valued maps
from $M$ to $M'$.{\small (Precisely, the 5-tuples $(M,p,M',p',G)$, where 
$G$ is a subset of $M\times M'$.)} 
\end{minipage}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\
$\quad$\\
1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

\vspace{-20mm}
{\Large\bf 3. $\!$Statement \ \ \ of \ \ \ the 

\vspace{-16mm}
\mbox{\ \ \ extension} problem.}

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Regular maps.

6.$\,\,\,$Generalized  \ \ \ differential \ \ \ \mbox{quotients}

\ \ \ (GDQs).

7. Warga derivate containers (WDCs).

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
{\bf THE ORDINARY DIFFERENTIAL IS A FUNCTOR FROM ${\cal PMC}^1{\cal C}^1\,$
TO  ${\cal FDRLS}\,$.}

If we call this functor ${\cal D}_{C^1}$, then 

\begin{itemize}
\item[$\bullet$] ${\cal D}_{C^1}(M,p)$	is the usual tangent space $T_pM$ of 
\phantom{aa}$M$ at $p$, 

\item[$\bullet$] if 
$\mu:(M,p)\mapsto_{_{\!\!\!\!{\cal PMC}^1{\cal C}^1}}(M',p')$, then 
${\cal D}_{C^1}(\mu)$ is the ordinary differential 
of $\mu$ at $p$, which, as is well known, is a linear map
from $T_pM$  to $T_{p'}M'$, i.e., a member of the space
${\rm Hom}_{{\cal FDRLS}}({\cal D}_{C^1}(M,p),{\cal D}_{C^1}(M',p'))$ of 
all ${\cal FDRLS}$-morphisms from ${\cal D}_{C^1}(M,p)$ 
to ${\cal D}_{C^1}(M',p')$.
\end{itemize}

\vspace{4mm}
{\bf THE CHAIN RULE IS  PRECISELY THE STATEMENT THAT 
${\cal D}_{C^1}$ IS A FUNCTOR.}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
\begin{center}
{\bf THREE VERY IMPORTANT\\ 
PROPERTIES OF ${\cal D}_{C^1}$}
\end{center}

\begin{center}
\fbox{\fbox{
\begin{minipage}{15.2cm}\begin{center}
{\bf THE PRODUCT RULE:}
\end{center}
\medbreak
THE DIFFERENTIAL OF A \mbox{PRODUCT} IS THE 
PRODUCT OF THE \mbox{DIFFERENTIALS}

If, for $i=1,2$, $\mu_i\in C^1((M_i,p_i),(M'_i,p'_i))$, then 
$$
{\cal D}_{C^1}(\mu_1\times\mu_2)=
{\cal D}_{C^1}(\mu_1)\times{\cal D}_{C^1}(\mu_2)\,.
$$
\end{minipage}}}\end{center}

\begin{center}
\fbox{\fbox{
\begin{minipage}{15.2cm}
\begin{center}
{\bf LOCALITY}
\end{center}

If $\mu,\nu\in C^1((M,p),(M',p'))$ and $\mu\equiv\nu$
on some neighborhood of $p$, then 
$$
D\mu(p)=D\nu(p)\,.
$$
\end{minipage}}}\end{center}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage

\begin{center}
\fbox{\fbox{
\begin{minipage}{15.2cm}
\begin{center}
$\qquad$\\
{\bf THE OPEN MAPPING\\
THEOREM:}
\end{center}

\bigbreak
If $\mu\in C^1((M,p),(M',p'))$ and ${\cal D}_{C^1}(\mu)$
is \mbox{surjective}  (i.e., $D\mu(p)$ is a surjective 
\mbox{linear} map from $T_pM$ to $T_{p'}M'$), then
$\mu$ maps neighborhoods of $p$ in $M$ to 
neighborhoods of $p'$ in $M'$.

\phantom{aaaaaa}
\end{minipage}}}\end{center}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\baselineskip=24pt
$\quad$\\$\quad$\\
Our goal here is to discuss ways to \mbox{extend} ${\cal D}_{C^1}$ 
to \ local \ multifunctors \ defined \ on 
\mbox{categories} larger than 
${\cal PMC}^1{\cal C}^1$ in such a way that the product rule and an open 
\mbox{mapping} theorem are still satisfied. 

\vspace{5mm}	
The {\bf first step} is to make plenty of room for such extensions by 
allowing the targets of our \mbox{multifunctors} to take values
in a \mbox{category} larger than ${\cal FDRLS}$. 

\vspace{7mm}
\begin{minipage}{15.4cm}\baselineskip=13pt
{\small NOTE: ``larger'' means ``more morphisms.'' 
This could happen because the larger category contains 
more objects or, even when the class of objects is unchanged, 
if the are more morphisms. {\bf Example:} ${\cal PMC}^1{\cal SVM}$
is ``larger'' than ${\cal PMC}^1{\cal C}^1$}.
\end{minipage}

\vspace{8mm}	
We will consider multifunctors with values in 
$\overline{{\cal FDRLS}}$, the category whose objects are the same as 
those of ${\cal FDRLS}$, but whose \mbox{morphisms} are {\bf 
nonempty compact sets of linear maps} rather than single linear maps.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\vspace{6mm}	
In our words, our ``generalized \mbox{differentials}'' will be 
nonempty compact sets of \mbox{linear} maps. Furthermore, 
{\bf they will not be unique},
and the ``chain rule''  
\fbox{\fbox{${\cal F}(\nu)\circ{\cal F}(\mu)\subseteq
{\cal F}(\nu\circ\mu)$}} will say that 

\vspace{4mm}	
\begin{center}
\fbox{\fbox{
\begin{minipage}{13cm}
The \mbox{composite} of two differentials is a differential of the
composite map
\end{minipage}}}
\end{center}

\vspace{7mm}	but will
{\bf not} say that every differential of the \mbox{composite} arises
as a composite of \mbox{differentials}. 

\vspace{7mm}
\begin{minipage}{15.4cm}\baselineskip=13pt
{\small {\bf Example:} If $f:\rr\mapsto\rr$ is given 
by \mbox{$f(x)=2x+|x|$}, and $g(x)=\frac{1}{3}(2x-|x|)$, so that  
$g=f^{-1}$ (\mbox{compositional} inverse),
then in some theories if $\Lambda\in f'(0)$ and
$\Xi\in g'(0)$ then $[1,3]\subseteq\Lambda$ and 
$[\frac{1}{3},1]\subseteq\Xi$, but 
$\{1\}$ is a derivative of $g\circ f$ at $0$, since
$(g\circ f)(x)\equiv x$.}
\end{minipage}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
DEFINITION: \ \ A \ \ multifunctor \ \ ${\cal D}$ \ from 
${\cal PMC}^1{\cal SVM}$
to $\overline{{\cal FDRLS}}$ is {\bf local} if, whenever 

\vspace{4mm}
a. $F_i:(M,p)\mapsto_{_{{{\cal PMC}^1{\cal SVM}}}}(M',p')$ for $i=1,2$,

\vspace{4mm}
b. $({\rm Gr}\,F_1)\cap U=({\rm Gr}\,F_2)\cap U$ \ \ for \ \ some 
\phantom{aaa}neighborhood $U$ of $(p,p')$ in $M\times M'$, 

\vspace{4mm}
it follows that ${\cal D}(F_1)={\cal D}(F_2)$. 

\vspace{6mm}
DEFINITION: We say that a multifunctor ${\cal D}$ from 
\ ${\cal PMC}^1{\cal SVM}$ \ to \ $\overline{{\cal FDRLS}}$ 
{\bf \ satisfies \ the 
\mbox{product} rule} if, whenever 

\vspace{-13mm}
$$
F_i:(M_i,p_i)\mapsto_{�{{{\cal PMC}^1{\cal SVM}}}}(M_i',p_i')
\,\,{\rm for}\,\,i=1,2\,,
$$ 

\vspace{2mm}
it follows that 
${\cal D}(F_1)\times{\cal D}(F_2)\subseteq{\cal D}(F_1\times F_2)$ .

\vspace{7mm}
\begin{minipage}{16cm}
\baselineskip=12pt 
{\small REMARK: If you don't want your Calculus I
students to get strange ideas, you can talk about the ``Cartesian
\mbox{product} rule,'' rather than the ``\mbox{product} rule.''}
\end{minipage}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\$\quad$\\
\fbox{\fbox{\fbox{\fbox{
\begin{minipage}{15.2cm}
\begin{center}\begin{minipage}{15cm}
\baselineskip=42pt
\null
\vspace{-23mm}
{\Large

\begin{center}DEFINITION \end{center}

\vspace{-15mm}
A {\bf Non-directional $\quad$
\mbox{Generalized} \mbox{Differentiation} 
Theory} (NDGDT) is a \mbox{local} multifunctor ${\cal D}$ from 
${\cal PMC}^1{\cal SVM}$ to $\overline{{\cal FDRLS}}$ 
that satisfies the \mbox{product} rule and is an extension
of ${\cal D}_{C^1}$.} 
\vspace{7mm}
\end{minipage}\end{center}
\end{minipage}}}}}

\vspace{6mm}
\begin{center}\begin{minipage}{13cm}
\small\baselineskip=14pt
NOTE: ``${\cal D}$ is an extension of ${\cal D}_{C^1}$'' 
means: ``if $F:(M,p)\mapsto_{{\cal PMC}^1{\cal C}^1}(M',p')$ 
then $\{DF(p)\}\in{\cal D}(F)$.''
\end{minipage}\end{center}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\$\quad$\\
DEFINITION: If 
$\quad\Lambda:X\mapsto_{_{_{\overline{{\cal FDRLS}}}}}Y\,,\quad$
we call $\Lambda$ is {\bf nonsingular} if every member of 
$\Lambda$ is \mbox{surjective} as a linear map from $X$ to $Y$

	
DEFINITION: If \ ${\cal D}$ \ is \ a \ non-directional \mbox{generalized}
differentiation theory, we say that {\bf ${\cal D}$ has the open
mapping property (OMP)} if, whenever 

%\vspace{-7mm}
a. $F:(M,p)\mapsto_{�{{{\cal PMC}^1{\cal SVM}}}}(M',p')$

%\vspace{-9mm}
b. $\Lambda\in{\cal D}(F)$,

%\vspace{-9mm}
and 

%\vspace{-9mm}
c. $\Lambda$ is nonsingular, 

%\vspace{-7mm}
it follows that $F$ maps neighborhoods of $p$ to neighborhoods of 
$p'$ (that is, \mbox{whenever} $U\subseteq M$ and $p\in{\rm Int}\,U$,
it follows that \mbox{$p'\in{\rm Int}\,F(U)$)}.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}
\stapage
$\quad$\\$\quad$\\
%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

\vspace{-12mm}
{\large\bf 
4. $\!\!\!\!\!\!\!\!\!\!\!\!\!\!$A trivial example of an \mbox{$\quad$extension}: the 
classical 
\mbox{$\quad$differential} at one point of 
\mbox{$\quad$a continuous} map.}

5. Warga derivate containers (WDCs).

6. Regular maps.

7.$\,\,\,$Generalized  \ \ \ differential \ \ \ \mbox{quotients}

\ \ \ (GDQs).

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}
\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}
\baselineskip=24pt\stapage
$\quad$\\$\quad$\\
One trivial extension of ${\cal D}_{C^1}$ that has all the \mbox{desired}
properties is obtained by using \mbox{instead} of ${\cal PMC}^1{\cal C}^1$ 
the category ${\cal PMC}^1{\cal C}^0{\cal D}^1$
with the same objects as ${\cal PMC}^1{\cal C}^1$ 
but whose morphisms are the \mbox{continuous} 
base-point preserving maps that are differentiable at the base point.

The {\bf open mapping theorem} says exactly the same as before (that is,
if $F(p)=p'$, 
$F$  is \mbox{continuous} near $p$, and $DF(p)$ exists and is surjective, then 
$F$  maps neighborhoods of $p$ to neighborhoods of $p'$). 

But the proof
involves a {\bf topological \mbox{argument}} in an essential way, because 
the result \mbox{depends} on (and, in fact, is 
\mbox{elementarily} equivalent to)
the Brouwer fixed point theorem. 

We will not pursue the study of this \mbox{particular} extension, because we
will move on right away to a much larger extension.  
\end{minipage}\end{center}
\esl 

\bsl
\begin{center}\begin{minipage}{15.5cm}
\baselineskip=25pt\stapage
$\quad$\\$\quad$\\
THEOREM: Let $Y$ be a real normed linear space. Write  
$B_Y=\{y\in Y:\|y\|\le 1\}$. Then the following
five conditions are \mbox{equivalent:}

\vspace{3mm}
\begin{itemize}
\item[a.] Every $F\!\in\!C^0(B_Y,B_Y)$ has
a fixed point.
%\vspace{-8mm}
\item[b.] If $X$ is any normed space, $\Omega$ is open in
$X$, $F\!\in\!C^0(\Omega,Y)$, \mbox{$x\in\Omega$}, 
$F$ is Fr\'echet-differentiable at $x$, and $DF(x)$ 
is a \mbox{surjective} linear map from $X$ to $Y$, then 
$F$ is open at $x$ (i.e., $F$ maps \mbox{neighborhoods} 
of $x$ to \mbox{neighborhoods} of $F(x)$).
\item[c.] If $F\!\in\!C^0(Y,Y)$, $F(0)=0$, $F$ 
is Fr\'echet-differentiable at $0$, 
and $DF(0)={\rm id}_Y$, then 
$F$ is open at $0$.
%\vspace{-8mm}
\item[d.] There does not exist a continuous 
\mbox{retraction} from $B_Y$ to the unit sphere 
$\{y\in Y:\|y\|=1\}$.
%\vspace{-8mm}
\item[e.] $\,Y$ is finite-dimensional.
\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

\vspace{-14mm}
{\large\bf 5. $\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$Warga derivate containers
\phantom{aai}(WDCs).}

6. Regular maps.

7.$\,\,\,$Generalized  \ \ \ differential \ \ \ \mbox{quotients}

\ \ \ (GDQs).

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
\begin{center}
{\bf WARGA DERIVATE CONTAINERS:}
\end{center}

If $F:\rr^m\mapsto\rr^n$ maps $0$ to $0$, and $\Lambda$ is a nonempty compact 
(not necessarily convex) subset of $\rr^{n\times m}$, we say that 
$\Lambda$ is a {\bf Warga derivate container} of 
$F$ at $(0,0)$, and write
$$
\Lambda\in WDC(F;0,0)\,,
$$
if for every $\delta>0$ there exist 

(a)~a neighborhood  $U_\delta$ of $0$ in $\rr^m$ 

and 

(b)~a sequence $\{F_j\}_{j=1}^\infty$ of maps of class $C^1$ 
\phantom{iaaa}from $U_\delta$ to $\rr^n$ 

such that 

(i)~$DF_j(x)\in\Lambda^\delta$ whenever $x\in U_\delta$
and $j\in\nn$ 

and 

(ii)~$F_j\to F$ uniformly on $U_\delta$ as $j\to\infty$.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
Clearly, a necessary condition for $F$ to have a WDC 
at $(0,0)$ is that $F$ be Lipschitz-continuous near $0$.

If this condition is satisfied, then 

(a)~$\partial F(0)\in WDC(F;0,0)$, 

(b)~$\partial F(0)$ is the smallest of all convex WDCs.

\begin{center}
-----------------------------------
\end{center}

The {\bf chain rule} and {\bf product rule} are trivial. 

One can prove an {\bf open mapping theorem} as well. 

The WDC theory \mbox{extends} immediately to pointed
manifolds of class $C^1$, 
and \ is \ an \ \mbox{extension} \ of \ ${\cal D}_{C^1}$ \ to 
\ a \ local \mbox{multifunctor} \ from 
\mbox{${\cal PMC}^1{\cal LIP}\,$} \ to  \ $\overline{{\cal FDRLS}}$ 
that satisfies 
the \mbox{product} rule and has the open \mbox{mapping} property.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
$\quad$\\$\quad$\\$\quad$\\
\fbox{\fbox{\fbox{\begin{minipage}{16cm}
\baselineskip=34pt
{\Large
{\bf WDC is a non-directional generalized differentiation theory 
that satisfies the OMP,
contains the Clarke generalized \mbox{Jacobians}
(and in particular the classical differentials of maps 
of class $C^1$), and assigns a nonempty set of \mbox{differentials} 
to every Lipschitz-continuous map, but does not 
contain the differentials at one point of continuous maps.
}}
\end{minipage}}}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}
\stapage
$\quad$\\$\quad$\\
%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

\vspace{-13mm}
{\large\bf 6. Regular maps.}

7.$\,\,\,$Generalized  \ \ \ differential \ \ \ \mbox{quotients}

\ \ \ (GDQs).

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}
\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\

\vspace{-6mm}
\begin{center}{\Large\bf REGULAR MAPS}\end{center}

DEFINITION. \ Let $X$, $Y$  be metric spaces. 
A set-valued map $F:X\tto Y$ is {\em regular} if for every compact subset 
$K$ of $X$ 

%\vspace{-10mm}
\bi
\item[(1)] ${\rm Gr}\,(F\restr K)$ is compact;

%\vspace{-12mm}
\item[(2)] there exists a sequence $\{f_j\}_{j=1}^{\infty}$
of single-valued \mbox{continuous} maps \mbox{$f_j:K\mapsto Y$} that 
converges to $F\restr K$ in the following sense: 
\ei

\vspace{4mm}
\fbox{\fbox{
\begin{minipage}{15cm}
$$
\lim_{j\to\infty}\Delta\Biggl(\,
{\rm Gr}\,(f_j),{\rm Gr}\,(F\restr K)\,\Biggr)=0\,,
$$
where $\Delta(A,B)=\sup\Big\{\,{\rm dist}(q,B):q\in A\,\Big\}$.
\end{minipage}
}}

\vspace{4mm}
(That is: ``given any neighborhood $U$ of the graph of $F\restr K$, the graphs 
of the $f_j$ are \mbox{eventually} inside $U$.'') {\small NOTE: 
``$\restr$'' means ``\mbox{restriction}.''}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
REGULARITY is a ``good'' property, which serves as a reasonable 
set-valued substitute for continuity. For \mbox{example}, it is 
\mbox{invariant} 
under composition (on either side) with \mbox{continuous} maps. (Actually, 
{\bf a composite of regular maps is regular}.) 

\vspace{14mm}


\begin{center}
{\bf THE FIXED POINT THEOREM}
\end{center}


\vspace{4mm}
Let $K$ be a nonempty compact convex \mbox{subset} of a normed space
$X$. Let $F:K\tto K$ be a regular set-valued map. Then 
$F$ has a fixed point (i.e., $(\exists x\in K)(x\in F(x))$).
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{center}
{\bf THE KAKUTANI-KY FAN THEOREM}
\end{center}

%\vspace{-4mm}
Let $S$ be a subset of a normed space $X$. Let $F:S\tto X$ be a
set-valued map with nonempty convex values
such that ${\rm Gr}(F\restr K)$ is 
compact whenever $\subseteq S$ and $K$ is compact. 
Then $F$ is regular. 

In particular, {\em if $S$ is nonempty compact \mbox{convex} and
$F(S)\subseteq S$, then $F$ has a fixed
point.} 

\vspace{4mm}
\begin{minipage}{15.4cm}
\baselineskip=17pt
\small
This last statement is {\bf Ky Fan's theorem}. The finite-dimensional case is 
{\bf Kakutani's theorem}. When $F$ is single-valued we get the 
{\bf Schauder fixed point \mbox{theorem}}, whose finite-dimensional case 
is the {\bf Brouwer fixed point theorem}.
\end{minipage}

$$
\left.\begin{array}{ccc}
{\rm Brouwer}&\longrightarrow&{\rm Schauder}\\
\downarrow&\,&\downarrow\\
{\rm Kakutani}&\longrightarrow&{\rm Ky Fan}\end{array}\right.
$$
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}
\stapage

%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}
$\quad$\\$\quad$\\

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

6. Regular maps.

\vspace{-13mm}
{\large\bf \mbox{7.$\,\,$Generalized}  differential \mbox{$\quad$quotients}
(GDQs).}

8. The GDQ and WDC theories are not 

$\,$\ \ \ \mbox{comparable}.

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}
\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{center}
{\bf GENERALIZED DIFFERENTIAL QUOTIENTS}
\end{center}

Let $m,n\in\zz_+$, let $F:\rr^m\tto\rr^n$ be a set-valued
map, and let $\Lambda$ be a nonempty \mbox{compact} \mbox{subset}
of $\rr^{n\times m}$. 
We say that $\Lambda$ is a {\bf generalized \mbox{differential} quotient}
(\mbox{abbreviated} ``GDQ'') {\bf of 
$F$ at $(0,0)$}, and write
$\Lambda\in  GDQ(F;0,0)$, 
if for every positive real number $\delta$ there exist $U$, $G$ such that 
\begin{itemize}

\vspace{6pt}
\item[1.] $U$  is a  compact neighborhood of $0$ in $\rr^m\,\,$;

\vspace{3pt}
\item[2.] $G$  is a  regular set-valued map from $U$ to the 
$\delta$-neighborhood  $\Lambda^\delta$ of $\Lambda$ in $\rr^{n\times m}\,\,$;

\vspace{3pt}
\item[3.] $G(x)\cdot x\subseteq F(x)$
for every $x\in U$.
\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
The {\bf chain rule} for GDQs is trivial:

\vspace{4mm}
Suppose that $F:\rr^m\tto\rr^n$ and \mbox{$F':\rr^n\tto\rr^p$},
are set-valued maps,  and assume that \mbox{$\Lambda\in  GDQ(F;0,0)$}, 
$\Lambda'\in  GDQ(F';0,0)$.

\vspace{4mm}
Let $\hat F=F'\circ F$, $\hat\Lambda=\Lambda'\circ\Lambda$. Given
$\delta\,$:

\vspace{6mm}
$\bullet$ find $\delta_{_0}$ such that
$(\Lambda')^{\delta_0}\circ\Lambda^{\delta_0}\subseteq
\Lambda^\delta$;

\vspace{6mm}
$\bullet$ find  $U'$, $G':U'\tto(\Lambda')^{\delta_0}$ such that
\mbox{$G'(y).y\subseteq F'(y)$} whenever $y\in U'$;

\vspace{6mm}
$\bullet$ find $U$, $G:U\tto\Lambda^{\delta_0}$ s. that 
$G(x)x\subseteq F(x)$ whenever
$x\in U$; 

\vspace{6mm}
$\bullet$ by shrinking $U$, if necessary, assume that $Lx\in U'$
whenever $x\in U$, $L\in\Lambda^{\delta_0}$.  

\vspace{6mm}
$\bullet$ \mbox{Let $\hat G(x)=\{L'L:L\in G(x), L'\in G'(Lx)\}$}.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
Then if $z\in\hat G(x).x$ we
have $z=L'Lx$ for some $L\in G(x)$ and some \mbox{$L'\in G'(Lx)$}.  

\vspace{6mm}
If $y=Lx$, then $y\in G(x)x$, so \mbox{$y\in F(x)$}.  

\vspace{6mm}
Also, $y\in U'$, and
$L'\in G'(y)$, so \mbox{$z=L'y\in G'(y)y\subseteq F'(y)$}, so $z\in F'(y)$,
and then $z\in\hat F(x)$. 

\vspace{6mm}
Therefore \mbox{$\hat G(x).x\subseteq\hat F(x)$}
whenever $x\in U$.

\vspace{6mm}
To conclude the proof one has to show that $\hat G$ is regular. 

\vspace{6mm}
But that's easy! All we have to do is express $\hat G$ as a
composition of regular maps.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$$
\hat G=\Phi\circ\Psi\circ\Theta\circ\Gamma
$$
where 
\begin{eqnarray*}
\Gamma(x)&=&G(x)\times\{x\}\,, \\
\Theta(L,x)&=&(Lx,L)\,,\\
\Psi(y,L)&=&G'(y)\times\{L\}\,,\\
\Phi(L',L)&=&L'L\,.
\end{eqnarray*}

\fbox{\fbox{\begin{minipage}{16cm}
$$
\left.\begin{array}{ccccccccc}
\,&\Gamma&\,&\Theta&\,&\Psi&\,&\Phi&\,\\
x&\mapsto&(L,x)&\mapsto&(Lx,L)&\mapsto &L',L)&\mapsto&L'L\,.
\end{array}\right.
$$
$\qquad$
\end{minipage}
}}

\vspace{6mm}
Hence \fbox{\fbox{$\hat\Lambda\in GDQ(\hat F;0,0)$}}, as desired.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\
\begin{center}
{\bf The Clarke Generalized Jacobian of a Lipschitz map is a GDQ:}
\end{center}

%\vspace{-7mm}
PROOF: Suppose $F:\rr^m\mapsto\rr^n$ maps $0$ to $0$ and is 
Lipschitz-continuous near $0$.   Let $\Lambda=\partial F(0)$.
Fix $\delta>0$. Then $\Lambda^\delta$ is 
\mbox{compact} convex. 
Let $F_j$ be the usual \mbox{regularizations} of $F$ \mbox{(obtained}
by \mbox{convolving} with an \mbox{approximate} identity), \mbox{modified}
so that $F_j(0)=0$. Then for $x$ in some \mbox{compact} convex 
\mbox{neighborhood} 
$U$ of $0$, all the $DF_j(x)$, for $x\in U$ and $j$ large, are in 
$\Lambda^\delta$, and we can write

%\vspace{-18mm}
$$
F_j(x)=\Biggl(\int_0^1DF_j(tx)\,dt\Biggr)x\,.
$$

%\vspace{-10mm}
So $F_j(x)\in\Lambda^\delta.x$ if $j$ is large, and then 
$F(x)$ \mbox{belongs} to $\Lambda^\delta.x$ because 
$\Lambda^\delta.x$ is compact. 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
Given $x$, define

%\vspace{-18mm}
$$
G^\delta(x)=\{L\in\Lambda^\delta:Lx=F(x)\}\,.
$$

%\vspace{-7mm}
Then $G^\delta(x)$ is compact, convex, nonempty.  
Furthermore, $G^\delta$ is  clearly upper \mbox{semicontinuous},
since the graph of $G^\delta$ is obviously compact.
Therefore $G^\delta$ is regular. Since 
$G^\delta$ takes \mbox{values} in $\Lambda^\delta$, and $G^\delta(x)x=F(x)$
for all $x$, we are done. 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage	
\begin{center}
{\bf If $F(0)=0$, and $F$ is differentiable at $0$ and continuous near 
$0$ then $\{DF(0)\}\in GDQ(F;0,0)$.}
\end{center}

This is is easy, and goes back to Botsko and Gosser.

\begin{center}
-----------------------------------
\end{center}

COROLLARY (with two completely \mbox{different} trivial proofs) : The
singleton of the ordinary differential of a map of class $C^1$ is a
GDQ.  
\end{minipage}\end{center}
\esl 
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage	
\begin{center}
{\bf GDQs ARE WELL DEFINED\\
ON MANIFOLDS:}
\end{center}

\vspace{4mm}
Indeed, let us suppose that $(M,p)$ and $(M',p')$ are pointed manifolds of 
class $C^1$, and that \fbox{$F:(M,p)\to_{{\cal PMC}^1{\cal SVM}}(M',p')$}.

\vspace{4mm}
Pick charts $\Phi$, $\Phi'$, mapping \mbox{neighborhoods}
$U$, $U'$ of $p$ in $M$ and of  $p'$ in $M'$
\mbox{diffeomorphically} 
onto open balls $B$, $B'$ of radius $r$, $r'$ centered at $0$
in \mbox{Euclidean} spaces of dimensions $n$, $n'$, and 
\mbox{sending} $p$, $p'$ to the origin. Then 
$\Phi'\circ F\circ\Phi^{-1}$ is a set-valued map 
from $\rr^n$ to $\rr^{n'}$. 

\end{minipage}\end{center}
\esl 
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage	

If $\Lambda$ \mbox{belongs} to $GDQ(\Phi'\circ F\circ\Phi^{-1};0,0)$,
then clearly 
\fbox{$D\Phi'(p')^{-1}\circ \Lambda\circ D\Phi(p)$} is a nonempty compact 
set of linear maps from $T_pM$ to $T_{p'}M'$. Furthermore, 
{\bf the chain rule implies that this set does not depend on the choice of 
charts $\Phi$, $\Phi'$.}

\vspace{4mm}
So we can define

\vspace{4mm}
\fbox{\fbox{\begin{minipage}{14cm}
\begin{eqnarray*}
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!GDQ(F;p,p')\,=\\
&\,&\,\Big\{\,
D\Phi'(p')^{-1}\circ\Lambda\circ D\Phi(p):\\
&\,&\,\,\Lambda\in GDQ(\Phi'\circ F\circ\Phi^{-1};0,0)\,\Big\}\,.
\end{eqnarray*}
\end{minipage}}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16cm}\stapage	
$\quad$\\$\quad$\\
The {\bf GDQ open mapping theorem} is trivial:

\vspace{4mm}
Suppose that $F:\rr^m\tto\rr^n$ is a set-valued map, $\Lambda\in  GDQ(F;0,0)$, 
and every $L\in\Lambda$ is onto. 

\vspace{4mm}
For a surjective $L\in\rr^{n\times m}$, let 
\fbox{$L^{\#}\,=L^{\dagger}\Big(LL^{\dagger}\Big)^{-1}$}, 
so $L^{\#}$ is the \mbox{{\em Moore-Penrose pseudoinverse} of $L$.}

\vspace{7mm}
$\bullet$ Fix a $\delta>0$ such that every $L\in\Lambda^\delta$ is onto.

\vspace{7mm}
$\bullet$ Find $U$, and a regular $G:U\tto\Lambda^{\delta}$, such that 
\mbox{$G(x)x\subseteq F(x)$} whenever $x\in U$.
Assume  $U$ is the closed $r$-ball centered at $0$.

\vspace{7mm}
$\bullet$ For  $x\in U$, let $G^{\#}(x)=\{L^{\#}:L\in G(x)\}$.
Then $G^{\#}$ is a regular map from $U$ to $\rr^{m\times n}$. 

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage	
$\quad$\\$\quad$\\
$\bullet$ Let $C$ be such that $\|M\|\le C$ \mbox{whenever}
$x\in U$ and $M\in G^{\#}(x)$.

\vspace{7mm}
$\bullet$ Given $y$ such that $C\|y\|\le r$, let $\mu_y$ be the set-valued map
$U\ni x\mapsto G^{\#}(x)y$. Then $\mu_y$ maps $U$ to $U$ and is regular.
So $\mu_y$ has a fixed point $x$. Then $x\in G^{\#}(x)y$, so 
\mbox{$x=My$} for some \mbox{$M\in G^{\#}(x)$}. Then $M=L^{\#}$ for some
\mbox{$L\in G(x)$}. So $x=L^{\#}y$, and then \mbox{$Lx=LL^{\#}y=y$}, so
$y\in G(x)x\subseteq F(x)$. {\bf END}

\vspace{7mm}
\begin{center}
------------------------------
\end{center}

\vspace{7mm}
{\bf Locality} is trivially true.

\vspace{7mm}
\begin{center}
------------------------------
\end{center}

\vspace{7mm}
The {\bf GDQ product rule} is trivial as well.



\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage	
$\quad$\\$\quad$\\
Hence

\vspace{17mm}

\fbox{\fbox{\fbox{\begin{minipage}{15cm}
\baselineskip=43pt

{\Large
%$\quad$\\
{\bf GDQ is a non-directional generalized differentiation theory 
that satisfies the open \mbox{mapping} property
and contains the classical differentials at one point of continuous maps
as well as the Clarke generalized \mbox{Jacobians}.
}}
\end{minipage}}}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.4cm}
\stapage
\begin{center}{\large\bf GDQs of Lipschitz flows}\end{center}

Suppose that $M\times\rr\ni(x,t)\mapsto f(x,t)$ is 
\mbox{Lipschitz} with respect to
$x$, measurable with respect to $y$, and with a Lipschitz 
constant \mbox{locally} bounded by an integrable function of $t$.
Let $\xi_*:[a,b]\mapsto M$ be a trajectory of $f$.
Solve the {\em variational differential inclusion}

\vspace{-12mm} 
$$
\dot V(t)\in\partial_xf\Biggl(\xi_*(t),t\Biggr)\cdot V(t)
$$
with initial condition $V(a)={\rm identity}$. 

{\bf Theorem I$_{lip}$:} {\em The set $\Lambda_{b,a}^f(x_*)$ 
of all $V(b)$s is a GDQ of $\Phi_{b,a}^f$ at $(x_*,y_*)$.}

\vspace{4mm} 

{\bf Theorem II$_{lip}$:}  {\em If $(x_*,a)$ and  $(y_*,b)$) are Lebesgue 
points of $f$, then the set $\hat\Lambda_{b,a}^f(x_*)$ of all
maps $f(y_*,b)dt+V(b)\Biggl((dx-f(x_*,a)ds\Biggr)$
is  a GDQ of $\Psi^f$ at $((x_*,b,a),y_*)$.}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}
\stapage

%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}
$\quad$\\$\quad$\\

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

6. Regular maps.

\mbox{7.$\,$Generalized}  differential \mbox{quotients}
\phantom{aai}(GDQs).

\vspace{-13mm}
{\large\bf \mbox{8.$\,$The GDQ} and WDC theories 
\phantom{aai}are not \mbox{comparable}.}

9. Path-integral \ \ generalized \ \ differentials 
\phantom{aai}\mbox{(PIGDs)}.

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{center}
{\Large{\bf An example of a GDQ\\
that is not a WDC}}
\end{center}
Let $f:\rr\mapsto\rr$ be given by 
$$
f(x)=\left\{\begin{array}{lll}
x\sin{\frac{1}{x}}&{\rm if}&x\neq 0\\
0&{\rm if}&x=0\end{array}\right.\,.
$$
\def\into{[-1,1]}
Then $f(x)=G(x)x$, where 
$$
G(x)=\left\{\begin{array}{lll}
\Biggl\{\sin{\frac{1}{x}}\Biggr\}&{\rm if}&x\neq 0\\
\into&{\rm if}&x=0\end{array}\right.\,.
$$
Furthermore, $G$ is regular, because it is usc and has compact convex 
nonempty values.  So 
$$
\into\in GDQ(f,0,0)\,.
$$
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{center}
{\Large{\bf An example of a WDC\\
which is not a GDQ}}
\end{center}
Let $f:\rr\mapsto\cc$ be given by 
$$
f(x)=\int_0^xe^{i/t}dt\,.
$$
Then $f$ is Lipschitz, and $f$ is in fact of class $C^1$ on
$\rr\backslash\{0\}$. 

\vspace{4mm}
The circle $\ss^1=\{z\in\cc:\|z\|=1\}$
is a WDC of $f$ at $(0,0)$, but is not a GDQ. (\mbox{Reason:}
$f(x)=O(x^2)$ as $x\to 0$. Therefore if 
$G(x)x=f(x)$ then $G(x)=\frac{f(x)}{x}$ for $x\neq 0$, and then
$G(x)=O(|x|)$ as $x\to 0$.)

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16cm}\stapage
$\quad$\\$\quad$\\
We have seen examples  of (i)~ a WDC which is not a GDQ
and (ii)~ a GDQ which is not a WDC.  Therefore 

\vspace{4mm}
\fbox{\fbox{\begin{minipage}{16cm}
{\Large{\bf The theories WDC and GDQ
are not comparable.}}\end{minipage}}}

\vspace{4mm}	
\begin{minipage}{15.5cm}
\baselineskip=20pt
Several extensions of WDC have been \mbox{proposed} (e.g., Halkin's
``screens,'' \ as well \ as HJS's ``semidifferentials'' 
\ and \ ``\mbox{multidifferentials}'') which are not 
\mbox{comparable} with GDQ either.
(Halkin also proposed a theory of ''fans'' which goes in the GDQ
direction but is slightly more restrictive.)  

Here we propose a theory that contains all the above. We call the 
differentials of this theory
{\large\bf path-integral generalized differentials}
\end{minipage}
\end{minipage}\end{center}
\esl 
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

6. Regular maps.


\mbox{7.$\,$Generalized}  differential \mbox{quotients}
\phantom{aai}(GDQs).

8. The GDQ and WDC theories  are not \mbox{comparable}.


\vspace{-13mm}
{\large\bf \mbox{9.$\,$Path-integral} generalized 
\phantom{aa}\mbox{differentials} \mbox{(PIGDs)}.}

$\!$$\!$$\!$10. Directional generalized differentials.
\end{minipage}\end{center}
\esl
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
{\bf BASIC IDEA:}    Let us write $\bb_n(r)$ to denote the closed ball
$\bb_n(r)\equaldef\{\,x\in\rr^n:\|x\|\le r\,\}$.

\vspace{6mm}
Use ${\cal A}_n(r)$ to denote the set of all Lipschitz paths
$\alpha:[0,1]\mapsto\rr^n$ such that $\alpha(0)=0$ and
$\|\dot\alpha(t)\|\le r$ for almost all $t\in[0,1]$.

\vspace{6mm}
We want to think of ${\cal A}_n(r)$ as a ``blowup'' of $\bb_n(r)$. The map
${\cal P}_{n,r}:{\cal A}_n(r)\mapsto\bb_n(r)$
such that ${\cal P}_{n,r}(\alpha)=\alpha(1)$ 
for $\alpha\in{\cal A}_n(r)$ is onto, and we can think of maps
from $\bb_n(r)$ to some space as constructed starting from maps 
\mbox{defined} on ${\cal A}_n(r)$. 

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\mbox{Actually},

\vspace{6mm}

\begin{center}\fbox{\begin{minipage}{14cm}
{\em  the inverse
${\cal P}_{n,r}^{-1}:\bb_n(r)\tto{\cal A}_n(r)$
is a regular set-valued map.}
\end{minipage}}\end{center}

\vspace{6mm} PROOF: the graph of 
${\cal P}_{n,r}^{-1}$ is the set 
$$
\{(x,\alpha):x\in\bb_n(r)\wedge\alpha\in{\cal A}_n(r)\wedge\alpha(1)=x\}\,,
$$
which is clearly compact. Furthermore, each set 
${\cal P}_{n,r}^{-1}(x)$ is convex and nonempty.
\end{minipage}\end{center}
\esl 
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
We will consider set-valued maps
\mbox{$F:\rr^n\mapsto\rr^m$} that contain maps 
\mbox{$f_r:\bb_n(r)\mapsto\rr^m$} of the 
form $\Phi_r\circ{\cal P}_{n,r}^{-1}$, where 
$\Phi_r:{\cal A}_n(r)\mapsto\rr^m$ is of the form

%\vspace{4mm}
\begin{equation}\label{eqphir}
\Phi_r(\alpha)=\int_0^1h_{\alpha}(t)\dot\alpha(t)dt+v_{\alpha}\,,
\end{equation}

\vspace{6mm}
and $v_{\alpha}$ is a ``small error.''

\vspace{1mm}
The map $\alpha\mapsto (h_{\alpha},v_{\alpha})$ will take values in
the product $C^0([0,1],\rr^{m\times n})\times\rr^n$, and will in fact
be allowed to be set-valued, but will be \mbox{required} to be 
\mbox{regular}.

\end{minipage}\end{center}
\esl 
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\
This suffices to imply that {\bf the set-valued map 
$\Phi_r$ is regular too}. 

\vspace{3mm}
PROOF: it suffices to see that the map 
$(h,\alpha)\mapsto\int_0^1h(t)\dot\alpha(t)dt$ is \mbox{continuous}
on $\Sigma_r\equaldef C^0([0,1],\rr^{m\times n})\times{\cal A}_n(r)$.
If $(h_j,\alpha_j)$ converge to $(h,\alpha)$ in $\Sigma_r$, then 
$$
\int_0^1h_j\dot\alpha_j-\int_0^1h\dot\alpha=
\int_0^1(h_j-h)\dot\alpha_j+\int_0^1h(\dot\alpha_j-\dot\alpha)\,.
$$
Clearly, $\int_0^1(h_j-h)\dot\alpha_j\to 0$.  Given $\eps>0$,
find a $g\in C^1$ such that $\|h-g\|_{sup}\le\eps$. 
Then, if we let $\beta_j=\alpha_j-\alpha $, we have
$$
\int_0^1\!\!g(\dot\alpha_j\!-\!\dot\alpha)\!=\!
g(1)\beta_j(1)\!-\!g(0)\beta_j(0)\!-\!\int_0^1\dot g(t)\beta_jdt,
$$
which goes to zero, because $\beta_j\to 0$ \mbox{uniformly}. And, finally,
\mbox{$\|\int_0^1(h-g)(\dot\alpha_j-\dot\alpha)\|\le 2\eps r$}.
\end{minipage}\end{center}
\esl

\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
$\quad$\\$\quad$\\

Finally, if we want $\Lambda$ to be a ``differential'' of 
$F$, we will require that the $h_\alpha$'s that occur in 
(\ref{eqphir}) be ``$\Lambda$-valued modulo a small error'' (i.e.,
$\Lambda^\delta$-valued with arbitrarily small $\delta$), 
and that the error $v_{\alpha}$ be $o(r)$, i.e., bounded by
$\delta r$ with \mbox{arbitrarily} small $\delta$.

\vspace{5mm}
SUMMARIZING: We will look at set-valued maps 
\mbox{$F:\rr^n\tto\rr^m$} whose graphs contain graphs of maps 
\mbox{$f_r:\bb_n(r)\tto\rr^m$} of the 
form $\Phi_r\circ{\cal P}_{n,r}^{-1}$, where 
$\Phi_r:{\cal A}_n(r)\tto\rr^m$ is of the form

\vspace{-6mm}
\begin{equation}\label{eqphir-set}
\!\!\!\!\!\!\!\!\Phi_r(\alpha)\!=\!\Biggl\{v\!+\!
\int_0^1h(t)\dot\alpha(t)dt\!:\!(h,v)\!\in\!G_r(\alpha)\Biggr\},
\end{equation}

\vspace{2mm}
where $G_r$ is a regular set-valued map from
${\cal A}_n(r)$ to $C^0([0,1],\Lambda^\delta)\times\bb_m(\delta r)$.
\end{minipage}\end{center}
\esl 
\bsl 
\begin{center}\begin{minipage}{16.4cm}\stapage

\begin{center}
{\Large{\bf PIGDs}}
\end{center}

DEFINITION: 
Let $n,m\in\zz_+$, $F:\rr^n\tto\rr^m$, and let 
$\Lambda$ be a nonempty compact subset of $\rr^{m\times n}$.
\ We \ say \ that \ $\Lambda$ is a {\bf \ path-integral generalized
differential} of $F$ at $(0,0)$,
and write \fbox{$\Lambda\in\pigdab(F;0,0)$}, if
for every $\delta>0$ there exists 
$R\in\,]\,0,\infty\,[\,$ such that 
for every \mbox{$r\in\,]\,0,R]$} there exists a regular 
set-valued map

\vspace{-9mm}
$$
G:{\cal A}_n(r)\tto C^0([0,1],\rr^{m\times n})\times\rr^m
$$
such that

\ben
%\vspace{-6mm}
\item[(1)] $h(t)\in\Lambda^\delta$ and $\|v\|\le\delta r$ 
whenever
\mbox{$\alpha\in{\cal A}_n(r)$}, $(h,v)\in G(\alpha)$,  $t\in[0,1]$,

%\vspace{-6mm}
\item[(2)] \ \ \ $v+\int_0^1h(t)\dot\alpha(t)dt\in F(\alpha(1))$
\ \ whenever\\
\phantom{aai}\mbox{$\alpha\in{\cal A}_n(r)$}, $(h,v)\in G(\alpha)$.
\een
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
The {\bf chain  rule for PIGDs} is trivial:
%\vspace{-5mm}
\mbox{Suppose} that \mbox{$F:\rr^m\tto\rr^n$, $F':\rr^n\tto\rr^p$},
are set-valued maps,  and that 
\mbox{$\Lambda\in PIGD(F;0,0)$ and $\Lambda'\in  PIGD(F';0,0)$}.

%\vspace{-5mm}
Let $\hat F=F'\circ F$, $\hat\Lambda=\Lambda'\circ\Lambda$. Given
$\delta\,$:


%\vspace{-8mm}
$\bullet$ define 
\begin{eqnarray*}
C&=&\sup\{\|L\|:L\in\Lambda^{\delta}\}\,,\\
C'&=&\sup\{\|L\|:L\in(\Lambda')^{\delta}\}\,.
\end{eqnarray*}

%\vspace{-8mm}
$\bullet$ find $\delta_{_0}$ such that 
\begin{eqnarray*}
0<\delta_{_0}&\le&\delta\,,\\
\delta_0(C+\delta_0)+C'\delta_0&\le&\delta\,,\\
(\Lambda')^{\delta_0}\circ\Lambda^{\delta_0}&\subseteq&\Lambda^\delta\,;
\end{eqnarray*}

%\vspace{-8mm}
$\bullet$ find  $R'$ and for each $r'\in]0,R']$ a regular 
\phantom{aa}map \mbox{$G'_{r'}:{\cal A}_n(r')\tto C^0([0,1],(\Lambda')
^{\delta_0}\times\rr^p$} \phantom{aa}such that 
$$
w+\int_0^1k\dot\beta\in F'(y)\quad{\rm and}\quad\|w\|\le\delta_0 r'
$$
\phantom{aa}\mbox{whenever} 
$\beta$ belongs to ${\cal A}_n(r)$, $y=\beta(1)$, 
\phantom{aa}and \mbox{$(k,w)\in G'_{r'}(\beta)$} , 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
$\bullet$ find $R$ and for each $r\in]0,R]$ a regular map
\phantom{aa}$G_r:{\cal A}_m(r)\tto C^0([0,1],(\Lambda)^{\delta_0}
\times\rr^n$ such that 
$$
v+\int_0^1h(t)\dot\alpha(t)dt\in F(x)\quad{\rm and}\quad\|v\|\le\delta_0 r
$$ 
\phantom{aa}\mbox{whenever} $\alpha\!\in\!{\cal A}_m(r)$, $x\!=\!\alpha(1)$, 
$(h,v)\!\in\!G(\alpha)$, 

%\vspace{-8mm}
$\bullet$ for $r\in]0,R]$, $\alpha\in{\cal A}_m(r)$, $(h,v)\in G_r(\alpha)$, 
\phantom{aa}\mbox{define} 
$$
\beta_{r,\alpha,h,v}(t)=\int_0^t\Biggl(v+h(s)\dot\alpha(s)\Biggr)ds\,.
$$
\phantom{aa}Then $\beta_{r,\alpha,h,v}\in{\cal A}_m(r')$ if 
$r'=(C+\delta_0)r$. 

%\vspace{-8mm}
$\bullet$ Let $\hat G_r(\alpha)$ be the set of all pairs
$$
\Biggl(\,kh,w+\Biggl(\int_0^1k(t)dt\Biggr)v\,\Biggr)\,,
$$
\phantom{aa}for all pairs $(h,v)$ belonging to $G_r(\alpha)$ and 
\phantom{aa}all \mbox{$(k,w)\in G'_{(C+\delta_0)r}(\beta_{r,\alpha,h,v})$}. 
(This makes 
\phantom{aa}sense if $(C+\delta_0)r\le R'$.) 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
Then if 
$$
z=u+\int_0^1\ell(t)\dot\alpha(t)dt
$$ 
\phantom{aa}for some $(\ell,u)\in\hat G(\alpha)$, we 
can write 
$$
u=w+\Big(\int_0^1k(t)dt\Big)v\,,\,\,\,\ell(t)=k(t)h(t)\,,
$$ 
and get 
%\vspace{-16mm}
\begin{eqnarray*}
z&=&w+(\int_0^1k(t)dt)v+\int_0^1k(t)h(t)\dot\alpha(t)dt\\
&=&w+\int_0^1\Big(k(t)(v+h(t)\dot\alpha(t)\Big)dt\\
&=&w+\int_0^1k(t)\dot\beta_{r,\alpha,h,v}(t)dt\,.
\end{eqnarray*}
If $y=\beta_{\alpha,h,v}(1)$, then $y\in F(x)$. 
Then $z\in F'(y)$. So $z\in\hat F(x)$.   
Furthermore, 
$$
\|w\|\le\delta_0(C+\delta_0)r+C'\delta_0r\le\delta r\,.
$$
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
Hence the maps $\hat G_{r}$, constructed for 
$r\in]0,\hat R]$, where 
$$\hat R=\min\Biggl(R,\frac{R'}{C+\delta_0}\Biggr)\,,
$$ 
satisfy all the desired properties, provided that
every $\hat G_{r}$ is regular. 

%\vspace{-6mm}
To conclude the proof one has to show that the maps $\hat G_r$ are regular. 
But that's easy! All we have to do is express $\hat G$ as a
composition of regular maps, as we did for the GDQ case.

%\vspace{-6mm}
Hence \fbox{\fbox{$\hat\Lambda\in PIGD(\hat F;0,0)$}}, as desired.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
\begin{center}
{\bf PIGDs ARE WELL DEFINED\\
ON MANIFOLDS:}
\end{center}

%\vspace{-4mm}
Indeed, let us suppose that $(M,p)$ and $(M',p')$ are pointed manifolds of 
class $C^1$, and that \fbox{$F:(M,p)\to_{{\cal PMC}^1{\cal SVM}}(M',p')$}.
Pick charts $\Phi$, $\Phi'$, mapping neighborhoods 
$U$, $U'$ of $p$ in $M$ and of  $p'$ in $M'$ diffeomorphically 
onto open balls $B$, $B'$ of radius $r$, $r'$ centered at $0$
in \mbox{Euclidean} spaces of dimensions $n$, $n'$, and 
\mbox{sending} $p$, $p'$ to the origin. Then 
$\Phi'\circ F\circ\Phi^{-1}$ is a set-valued map 
from $\rr^n$ to $\rr^{n'}$. 

%\vspace{-4mm}
If $\Lambda$ \mbox{belongs} to $PIGD(\Phi'\circ F\circ\Phi^{-1};0,0)$,
then clearly 
\fbox{$D\Phi'(p')^{-1}\circ \Lambda\circ D\Phi(p)$} is a nonempty compact 
set of linear maps from $T_pM$ to $T_{p'}M'$. Furthermore, 
{\bf the chain rule implies that this set does not depend on the choice of 
charts $\Phi$, $\Phi'$.}

\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage

\stapageno

\stapageno

\stapageno

\stapageno

\stapageno

So we can define

\vspace{4mm}
\fbox{\fbox{\begin{minipage}{16cm}
\begin{eqnarray*}
PIGD(F;p,p')\,&=&\,\Big\{\,
D\Phi'(p')^{-1}\circ\Lambda\circ D\Phi(p):\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\Lambda\in GDQ(\Phi'\circ F\circ\Phi^{-1};0,0)\,\Big\}\,.
\end{eqnarray*}
\end{minipage}}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
The {\bf open mapping theorem for PIGDs} is trivial:

%\vspace{-7mm}
Suppose that $F:\rr^m\tto\rr^n$ is a set-valued map, $\Lambda\in PIGD(F;0,0)$, 
and every $L\in\Lambda$ is onto. 

%\vspace{-7mm}
For an onto  $L\in\rr^{m\times n}$, let 
\fbox{$L^{\#}\,=L^{\dagger}\Big(LL^{\dagger}\Big)^{-1}$}, 
so $L^{\#}$ is the \mbox{{\em Moore-Penrose pseudoinverse} of $L$.}

%\vspace{-9mm}
$\bullet$ Fix a $\bar\delta>0$ such that every $L\in\Lambda^{\bar\delta}$ 
is onto.

%\vspace{-9mm}
$\bullet$ Let $C=\max\{\|L^{\#}\|:L\in\Lambda^{\bar\delta}\}$.

%\vspace{-9mm} 
$\bullet$ Pick $\delta$ such that $0<\delta<\min\Big(\frac{1}{C},\bar\delta\Big)$. 
%Let $C'=C(1-\delta)$.

%\vspace{-9mm} 
$\bullet$ Find $r$, and a regular 

\vspace{-9mm} 
$$
G:{\cal A}_n(r)\tto C^0([0,1],\rr^{m\times n})\times\rr^m
$$ 
\phantom{aai}such that  
$v+\int_0^1h(t)\dot\alpha(t)dt\in F(\alpha(1))$ and 
\phantom{aai}$\|v\|\le\delta r$ whenever
\mbox{$\alpha\in{\cal A}_n(r)$}, $(h,v)\in G(\alpha)$, 
\phantom{aai}and $h(t)\in\Lambda^\delta$ whenever $t\in[0,1]$,
\mbox{$\alpha\in{\cal A}_n(r)$}, 
\phantom{aai}and $(h,v)\in G(\alpha)$.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage

$\bullet$ Given $y\in\rr^m$ such that $C\|y\|\le (1-C\delta)r$, 
\phantom{aai}we want to find an $x\in\bb_n(r)$ such that  
\phantom{aai}\mbox{$y\in F(x)$}.  For this purpose, it suffices to 
\phantom{aai}find $\alpha\in{\cal A}_n(r)$, $(h,v)\in G(\alpha)$ such that 
$$
v+\int_0^1h(t)\dot\alpha(t)dt=y\,. 
$$

So we want to solve
\fbox{$\int_0^1h(t)\dot\alpha(t)dt=y-v$},
for which it suffices to find $\alpha,h,v$ with the property that 
\fbox{$\beta_{\alpha,h,v,y}=\alpha$}
where 
\begin{equation}\label{eqbeta}
\beta_{\alpha,h,v,y}(t)=\int_0^th(s)^{\#}(y-v)ds\,.
\end{equation}
(Indeed, (\ref{eqbeta}) implies 
$$
\dot\beta_{\alpha,h,v,y}(t)=h(t)^{\#}(y-v)\qquad {\rm a.e.},
$$
from which it follows that $h(t)\dot\beta_{\alpha,h,v,y}(t)=y-v$ (because 
$h(t)h(t)^{\#}={\rm id}_{\rr^n}$), and then we get 
$h(t)\dot\alpha(t)=y-v$, so \ $\int_0^1h(t)\dot\alpha(t)\,dt=y-v$, \ as 
desired.) 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
Let $\mu_y$ be the set-valued map that sends a path
$\alpha\in{\cal A}_n(r)$ to the set

\vspace{-4mm}
$$
\Biggl\{\,\beta_{\alpha,h,v,y}:(h,v)\in G(\alpha)\,\Biggr\}\subseteq
C^0([0,1],\rr^n)\,.
$$

Then $\mu_y$ maps ${\cal A}_n(r)$ to ${\cal A}_n(r)$ and is regular.
So $\mu_y$ has a fixed point $\alpha$, and we are done.


\vspace{6mm}
\begin{minipage}{16.5cm}
\baselineskip=19pt
{\small (NOTE: Part of this 
argument is based on an old idea that Waszewski
used in his work on the domain of an impicitly defined function.)}
\end{minipage}\vspace{-9mm}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
\begin{center}
{\bf Warga Derivate Containers of Lipschitz maps are PIGDs:}
\end{center}

%\vspace{-7mm}
Indeed, suppose $F:\rr^m\mapsto\rr^n$ maps $0$ to $0$ and the map
is Lipschitz-continuous near $0$.   Let \mbox{$\Lambda\in WDC(F;0,0)$}.
Fix $\delta>0$. Then $\Lambda^\delta$ is compact. 

%\vspace{-7mm}
Let $F_j$ be maps of class $C^1$ that are defined on a ball
$\bb_m(R)=\{x\in\rr^m:\|x\|\le R\}$ and are such that 
\mbox{$F_j(0)=0$}, $DF_j(x)\in\Lambda^\delta$
for all  \mbox{$x\in\bb_m(R)$} and all $j$, and 
$F_j\to F$ uniformly on $\bb_m(R)$. 

%\vspace{-7mm}
Given a number $r\in]0,R]$, pick a $j\in\nn$ such that
\mbox{$\|F_j(x)-F(x)\|\le\delta r$} for all $x\in\bb_m(r)$, and
define 

%\vspace{-17mm}
\begin{eqnarray*}
h_\alpha(t)&=&DF_j(\alpha(t))\,,\\ 
v_\alpha&=&F(\alpha(1))-F_j(\alpha(1))\,,
\end{eqnarray*}

%\vspace{-7mm}
for $\alpha\in{\cal A}_n(r)$. 
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
Then


%\vspace{-17mm}
\begin{eqnarray*}
F(\alpha(1))&=&v_\alpha+F_j(\alpha(1))\\
&=&v_\alpha+\int_0^1\frac{d\phantom{t}}{dt}\Biggl(F_j(\alpha(t))\Biggr)dt \\
&=&v_\alpha+\int_0^1DF_j(\alpha(t))\dot\alpha(t)dt\\
&=&v_\alpha+\int_0^1h_\alpha(t)\dot\alpha(t)dt\,.
\end{eqnarray*}
\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\bf END}	
\end{minipage}\end{center}
\esl

\bsl 
\begin{center}\begin{minipage}{16.4cm}\stapage
\begin{center}
{\large\bf GDQs are PIGDs:}
\end{center}

\vspace{4mm}
{\it Proof}. \ Suppose that $F:\rr^m\tto\rr^n$, and \mbox{$\Lambda$}
belongs to $GDQ(F;0,0)$. Fix $\delta>0$. Let 
\mbox{$G:\bb_n(r)\tto\Lambda^\delta$} be a regular set-valued map
such that the inclusion
\mbox{$G(x)x\subseteq F(x)$} is true \mbox{whenever} $x\in\bb_n(r)$. Define
$$
{\cal G}(\alpha)=\Big\{(\hat L,0):L\in G(\alpha(1))\Big\}\,,
$$
where, if $L\in\rr^{n\times m}$,  $\hat L$ is the constant
matrix-valued function $[0,1]\ni t\mapsto L\in\rr^{n\times m}$.

%\vspace{-7mm}
Then, if $(h,v)\in{\cal G}(\alpha)$, and we let $x=\alpha(1)$, we have
$h(t)\equiv L$ for some $L\in G(x)$, and then
$v+\int_0^1h(t)\dot\alpha(t)dt=Lx$, so 
$$
v+\int_0^1h(t)\dot\alpha(t)dt\in G(x)x\subseteq F(x)\,.
$$
\end{minipage}\end{center}
\esl
\bsl 
\begin{center}\begin{minipage}{16.4cm}\stapage
Furthermore, ${\cal G}(\alpha)$ is regular, because 
$$
{\cal G}(\alpha)=\Theta\circ G\circ{\cal P}_{m,r}
$$
where $\Theta$ is the map
$$
\rr^{n\times m}\ni L\mapsto(\hat L,0)\in
C^0([0,1],\rr^{n\times m})\times\rr^n\,.
$$
\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\bf END}	
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage


The {\bf \ PIGD \ product \ rule} \ and \ the {\bf \ locality 
\mbox{property}} are trivial as well. \ \ Hence

\vspace{8mm}
\fbox{\fbox{\fbox{\fbox{\begin{minipage}{15cm}

\begin{center}
\begin{minipage}{14.1cm}


\baselineskip=30pt

{\Large\bf

$\quad$\\

\vspace{-30mm}

\mbox{PIGD is a non-directional} \mbox{generalized differentiation}
\mbox{theory \ that \ satisfies the}
\mbox{open \ $\,$mapping \ $\,$property} 
and \mbox{\mbox{$\,$contains} $\,\,\,$GDQ $\,\,$and}
\vspace{-6mm}
\begin{minipage}{14.2cm}
\baselineskip=23pt		
WDC. {\small In addition, PIGD contains all the other 
theories (e.g., semidifferentials and
\mbox{multidifferentials}) mentioned but not 
\mbox{discussed} in detail in this talk.}\\

$\qquad$
\end{minipage}
}\end{minipage}
\end{center}

\end{minipage}}}}}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{16.4cm}\stapage
\begin{minipage}{16cm}
\baselineskip=48pt

{\Large
{\bf This concludes the \mbox{EASY} part of the talk,
in which all the categories that were sources of our 
\mbox{extensions}
had as objects those of ${\cal PMC}^1{\cal C}^1$. (That is, we 
\mbox{extended} ${\cal D}_{C^1}$ to larger classes of maps, but 
\mbox{without} enlarging the class of \mbox{objects}.) }}
\end{minipage}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
%\begin{center}
%{\Large{\bf Outline of the talk}}
%\end{center}

1. Motivation: the Pontryagin Maximum 

$\,$\ \ \ Principle and generalized derivatives.

2. Categories, functors, multifunctors. 

3. Statement of the extension problem.

4. A trivial example of an extension: the 

$\,$\ \ \ classical differential at one point of a

$\,$\ \ \ \mbox{continuous} map.

5. Warga derivate containers (WDCs).

6. Regular maps.

\mbox{7. Generalized}  differential \mbox{quotients}
\phantom{aai}(GDQs).

\mbox{8. The} GDQ and WDC theories  are not 
\phantom{aai}\mbox{comparable}..

\mbox{9. Path-integral} generalized \mbox{differentials} 
\phantom{aai}\mbox{(PIGDs)}.

\vspace{-13mm}
{\large\bf \mbox{$\!$$\!\!\!$$\!$10. Directional} generalized 
\phantom{aai}\mbox{differentials}.}
\end{minipage}\end{center}
\esl
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
It would be desirable to do similar things for {\bf \mbox{directional} 
derivatives}, i.e., derivatives along subsets. 

\vspace{4mm}
That is, one should consider \mbox{categories} whose 
objects \ are \ triples \ $(M,S,p)$ \mbox{consisting} \ of \ a 
\mbox{manifold} $M$ of class $C^1$, a subset
$S$ of $M$ such that $p\in S$, and a point $p$ of $M$, and such that the
morphisms from $(M,S,p)$ to $(M',S',p')$
are set-valued maps from $S$ to $S'$. 

\vspace{4mm}	
When we try to do that, all kinds of \mbox{technical} issues arise that need
to be \mbox{addressed}. 

\vspace{4mm}	
Some are trivial, some seem truly hard.
\end{minipage}\end{center}
\esl

\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
Among the trivial issues are:

\vspace{6mm}	
\begin{itemize}
\item[1.] Shall we take {\bf all} subsets $S$ or only some ``nice'' sets?


\item[2.] What are the ``tangent objects'' (i.e., the ``object'' parts of our 
functors) of the triples $(M,S,p)$. Should they still be the spaces 
$T_pM$, or should they be ``\mbox{tangent} cones'' in some sense to the sets
$S$? 

\item[3.] And if we choose the latter possibility, what kind of 
tangent cone? 
\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{itemize}
\item[4.] If we are going to work with set-valued maps from $S$ to 
$S'$, and try to define their \mbox{``differentials''}
at $(p,p')$, then it is clear
why we need $S$, $S'$, $p$ and $p'$. But why do we need 
$M$ and $M'$?   Shouldn't  we just \mbox{formulate}
the \mbox{theory} in terms of ``pointed sets,'' i.e. pairs $(S,p)$,
 \ without \ bringing \ in \ the \ ``\mbox{ambient} 
\mbox{manifolds}'' $M$ at all? But if we do this
we would still need some extra structure on $S$ that enables us to talk about 
``derivatives at $p$'' of at least {\it some} \ maps 
defined on $S$. How shall such a 
structure be defined intrinsically, \mbox{without} \mbox{embedding}
$S$ in a manifold $M$?
\end{itemize}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
These questions are trivial, not in the sense that they have an obvious 
unique answer, but in the sense that they have lots of 
\mbox{different} equally 
reasonable answers, and one can list them all
and feel free to choose one.

But all these answers seem to come with a price. For example, if one 
insists on having too many sets $S$, then proving the chain rule 
becomes harder. 
\end{minipage}\end{center}
\esl
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
\vspace{-5mm}
\vbox{
\begin{center}
{\Large{\bf LOCAL DIFFCONES}}
\end{center}

\vspace{5mm}

DEFINITION: A {\bf local diffcone of class $C^k$} is a triple $(M,S,p)$ such 
that

%\vspace{-6mm}
a. $M$ is a manifold of class $C^k$,  

%\vspace{-11mm}
b. $S$ is a subset of $M$, 

%\vspace{-11mm} 
c. $p\in S$,  

%\vspace{-11mm}
d. there \mbox{exist} $\Phi$, $U$, $r$, $C$ such that 

%\vspace{-5mm}
\mbox{\phantom{aaa}d1}. $U$ is a neighborhood of $p$ in $M$,

%\vspace{-11mm}
\mbox{\phantom{aaa}d2}. $r>0$, 

%\vspace{-11mm}
\mbox{\phantom{aaa}d3}. $\Phi$ is a diffeomorphism of class $C^k$ 
\mbox{\phantom{aaaaaaa}from}
$U$ to the closed ball $\bb_n(r)$,

%\vspace{-11mm}
\mbox{\phantom{aaa}d4}. $\Phi(p)=0$, 

%\vspace{-11mm}
\mbox{\phantom{aaa}d5}. $C$ is a closed convex cone in $\rr^n$, 

%\vspace{-11mm}
\mbox{\phantom{aaa}d6}. $\Phi(S\cap U)=\bb_n(r)\cap C$.
}
\end{minipage}\end{center}
\esl
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
TRIVIAL REMARK: If $(M,S,p)$ is a \mbox{local} diffcone of class $C^1$, then all
existing \mbox{notions} of ``\mbox{tangent} cone of $S$ at $p$'' 
(e.g., \mbox{Bouligand}, Clarke) agree, and yield a closed convex 
cone in $T_pM$ that \mbox{certainly} deserves to be called 
{\bf the} tangent cone of $(M,S,p)$.

\vspace{6mm}
In particular: if we want to \mbox{define} ``\mbox{directional}
generalized differentiation \mbox{theories}'' to be multifunctors 
on the category ${\cal LDCC}^1$ of local diffcones of class 
$C^1$, then it is clear how to define the ``object-to-object''
part of the multifunctor: to each object $(M,S,p)$ 
of ${\cal LDCC}^1$ we assign the tangent cone to $S$m at $p$.
\end{minipage}\end{center}
\esl

\bsl 
\begin{center}\begin{minipage}{16.2cm}\stapage
\baselineskip=28pt
DISGUSTINGLY PEDANTIC AND \mbox{TOTALLY} USELESS REMARK, FOR BOURBAKI FANS ONLY
{\it (all others please skip this)}: One can define the \mbox{notion} of 
\mbox{\ ``diffcone''}  \ intrinsically,
\ without \mbox{referring} to the ambient manifold 
$M$ at all.  (Idea: \ a {\it \ diffcone of class
$C^k$} \ is a pointed \mbox{topological} space $(S,p)$ endowed with a sheaf
${\cal S}$ of rings of real-valued functions such there exist 

(a)~a neighborhood $U$ of $p$ in $S$, 

(b)~a neighborhood $V$ of $0$ in $\rr^n$ for some $n$, 

and 

(c)~a closed convex cone $C$ in $\rr^n$, 

such that the \ ringed \ spaces \ $(U,{\cal S}\restr U)$ and
$(C,C^k(C)\restr V)$ are isomorphic by means of a map $\Phi:U\mapsto
V$ such that $\Phi(p)=0$, where $C^k(C)$ is the sheaf of restrictions
to $C$ of real-valued functions of class $C^k$ defined on open subsets
of $\rr^n$.)  
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\begin{center}
{\Large{\bf PATH-INTEGRAL GENERALIZED
DIFFERENTIALS ON LOCAL DIFFCONES}}
\end{center}

\vspace{4mm}
{\bf BASIC IDEA:}    

Let $C$ be a closed convex cone in $\rr^n$. Recall that
$\bb_n(r)\equaldef\{\,x\in\rr^n:\|x\|\le r\,\}$. Set
$$
C(r)\equaldef C\cap\bb_n(r)\,.
$$
Use ${\cal A}(C(r))$ to denote the set of all 
\mbox{Lipschitz} paths
$\alpha:[0,1]\mapsto\rr^n$ such that $\alpha(0)=0$ and
$\dot\alpha(t)\in C(r)$ (i.e., $\dot\alpha(t)\in C$ and 
$\|\dot\alpha(t)\|\le r$) for almost all $t\in[0,1]$.

%\vspace{-6mm}
We want to think of ${\cal A}(C(r))$ as a ``blowup'' of $C(r)$. 

The map ${\cal P}_{C,r}$ such that 
\fbox{${\cal P}_{C,r}(\alpha)=\alpha(1)$}
for $\alpha\in{\cal A}(C(r))$ is onto, and we can think of maps
from $C(r)$ to some space as constructed starting from maps 
defined on ${\cal A}(C(r))$. 
\end{minipage}\end{center}
\esl 
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
%\vspace{-6mm}
\mbox{Actually}, the inverse
$$
{\cal P}_{C,r}^{-1}:C(r)\tto{\cal A}(C(r))
$$
is a regular set-valued map. (Proof: the graph of 
${\cal P}_{C,r}^{-1}$ is the set 
$$
\{(x,\alpha):x\in C(r)\wedge\alpha\in{\cal A}(C(r))\wedge\alpha(1)=x\}\,,
$$
which is clearly compact. Furthermore, each set 
${\cal P}_{C,r}^{-1}(x)$ is convex and nonempty.)
\end{minipage}\end{center}
\esl 

\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
We will consider set-valued maps
$F$ that contain maps \mbox{$f_r:C(r)\mapsto\rr^m$} of the 
form $\Phi_r\circ{\cal P}_{C,r}^{-1}$, where 
$\Phi_r:{\cal A}(C(r))\mapsto\rr^m$ is of the form

\vspace{-7mm}
\begin{equation}\label{eqphir-ux}
\Phi_r(\alpha)=\int_0^1h_{\alpha}(t)\dot\alpha(t)dt+v_{\alpha}\,,
\end{equation}

\vspace{-3mm}
and $v_{\alpha}$ is a ``small error.''

\vspace{1mm}
The map $\alpha\mapsto (h_{\alpha},v_{\alpha})$ will take values in
the product $C^0([0,1],\rr^{m\times n})\times\rr^n$, and will in fact
be allowed to be set-valued, but will be \mbox{required} to be 
\mbox{regular}.

\vspace{1mm}
This suffices to imply that the set-valued map 
$\Phi_r$ is regular too. 

\vspace{2mm}
Finally, if we want $\Lambda$ to be a ``differential'' of 
$F$, we will require that the $h_\alpha$'s that occur in 
(\ref{eqphir}) be ``$\Lambda$-valued modulo a small error'' (i.e.,
$\Lambda^\delta$-valued with arbitrarily small $\delta$), 
and that the error $v_{\alpha}$ be $o(r)$, i.e., bounded by
$\delta r$ with \mbox{arbitrarily} small $\delta$.
\end{minipage}\end{center}
\esl 
\bsl 
\begin{center}\begin{minipage}{15.5cm}\stapage
Let $n,m\in\zz_+$, let $C$, $C'$ be closed convex cones
in $\rr^n$, $\rr^m$, respectively, let $F:C\tto C'$, and let 
$\Lambda$ be a nonempty compact subset of ${\rm Lin}(C,C')$.
We say that $\Lambda$ is a {\bf path-integral generalized
differential} of $F$ at $(0,0)$,
and write $\Lambda\in\pigdab(F;0,0)$, if
for every $\delta>0$ there exists 
$R\in\,]\,0,\infty\,[\,$ such that 
for every \mbox{$r\in\,]\,0,R]$} there exists a regular 
set-valued map

\vspace{-8mm}
$$
G:{\cal A}(C(r))\tto C^0([0,1],\rr^{m\times n})\times\rr^m
$$
such that

\ben
%\vspace{-6mm}
\item[(1)] $h(t)\in\Lambda^\delta$ and $\|v\|\le\delta r$ 
whenever
\mbox{$\alpha\in{\cal A}(C(r))$}, $(h,v)\in G(\alpha)$,  $t\in[0,1]$,

%\vspace{-6mm}
\item[(2)] \ \ \ $v+\int_0^1h(t)\dot\alpha(t)dt\in F(\alpha(1))$
\ \ whenever\\
\phantom{aai}\mbox{$\alpha\in{\cal A}_n(r)$}, $(h,v)\in G(\alpha)$.
\een
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
If one could prove the chain rule, then one could define 
PIGDs on local diffcones of class $C^1$. 

It then turns out that the locality property and the product rule are 
trivial.

Furthermore, \ the {\bf open mapping \mbox{theorem}}, 
\mbox{properly} reformulated, can also be proved.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage


THEOREM O: {\em \ Let \ $(M,S,p)$, \ $(M',S',p')$, \ be \mbox{diffcones} of 
\ class \ $C^1$, \ let \ $F:S\tto S'$, \ and \ let 
$\Lambda\in PIGD(F,p,p')$.  Let $C$, $C'$ be the
tangent cones \ of \ $(M,S,p)$, \ $(M',S',p')$, \mbox{respectively}.  
\mbox{Assume} that
$v$ is an interior point of $C'$ (as a closed convex cone) and $D$ is
a closed \mbox{convex} cone such that $D\subseteq C'$ and 
\mbox{$v\in{\rm Int}_{C'}D$}. \mbox{Assume} that $D\subseteq LC$ for 
every $L\in\Lambda$. Let
\mbox{$\xi:[0,\bar\eps]\mapsto S'$} be, for some positive $\bar\eps$, a curve
of class $C^1$ such that $\xi(0)=p'$ and $\dot\xi(0)=v$. Then 
for \mbox{every} neighborhood $N$ of $p$ in $S$ the image $F(N)$ 
\mbox{contains} \ a \mbox{\ neighborhood} \ of \ the \ arc 
\mbox{$\{\gamma(t):0<t<\eps\}$} in $S'$ for some $\eps\in]0,\bar\eps[$.}
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
Hence, if one can prove the chain rule, one gets a good theory.

So far, I have not been able to prove the chain rule in full generality.
All I know is the following weaker results: 

%\vspace{-6mm}
I. \ The chain rule is true for {\bf polyhedral 
\mbox{\phantom{aaa}cones}}, but the proof  that I have is very 
\mbox{\phantom{aaa}long and technical}. 

%\vspace{-11mm}
II. The following disjunction is true

%\vspace{-9mm}
\mbox{\phantom{aaa}either}

%\vspace{-11mm}
\mbox{\phantom{aaa}(a)~the} chain rule does {\bf not} hold for 
\mbox{\phantom{aaaaaa}general closed} convex cones, 

%\vspace{-11mm}
\mbox{\phantom{aaa}or}

%\vspace{-11mm}
\mbox{\phantom{aaa}(b)~the} chain rule for general closed 
\mbox{\phantom{aaaaaa}convex cones} is true but very hard 
\mbox{\phantom{aaaaaa}to prove},
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
\vspace{16mm}

or

\vspace{30mm}

(c)~I am very stupid.
\end{minipage}\end{center}
\esl
\bsl
\begin{center}\begin{minipage}{15.5cm}\stapage
The proof of the chain rule for polyhedral cones is very long and
technical, and does not seem to generalize to arbitrary closed convex
cones.

\vspace{5mm}
In any case, one can of course define the \mbox{notion} of 
``polyhedral local diffcone'', and the theory works perfectly 
for local \mbox{polyhedral} diffcones. 

\vspace{9mm}

CONCLUDING QUESTIONS:

A. Can one have a chain rule for PIGDS 
\phantom{aaa}on arbitrary diffcones? (Answer: {\it I 
\mbox{\phantom{aaa}don't know}.)}

B. Does it matter? (Answer: {\it for the 
\mbox{\phantom{aaa}Maximum} Principle without constraints 
\mbox{\phantom{aaa}it certainly} does not, \mbox{because} the only 
\mbox{\phantom{aaa}cones} that are used are orthants, but 
\mbox{\phantom{aaa}for} a maximum principle with state space 
\mbox{\phantom{aaa}constraints} it probably will \mbox{matter} a lot.
\mbox{\phantom{aaa}But} this is just a guess}.)
\end{minipage}\end{center}
\esl
\end{document}
