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\def\gua{H.~J.~SUSSMANN}
\def\guo{THE MAXIMUM PRINCIPLE}

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\begin{center}
{\Huge{\bf TRANSVERSALITY,\\
$\,$\\
\vspace{-5mm}
SET SEPARATION,\\
$\,$\\
\vspace{-5mm}
AND THE PROOF OF\\
$\,$\\
\vspace{-4mm}
THE MAXIMUM PRINCIPLE}}\\
$\,\,$\\
$\,\,$\\
{\Large{\it H\'ector J. Sussmann} --- January 3, 2001}
\end{center}

\vspace{1cm}
\medbreak
\begin{center}
{\huge\bf CONTENTS}
\end{center}

\medbreak

\thispagestyle{empty}

\medbreak

{\large 
\noindent 1.\ THE TRANSVERSALITY CONDITIONS\hfill 2

\noindent \ 1.1\ Cones \hfill 2

\noindent \ 1.2\ Boltyanskii tangent cones to a set at a point \hfill 2

\noindent \ 1.3\ The maximum principle with transversality conditions for fixed
time\\
\null$\,\qquad$  interval problems \hfill 3

\noindent \ 1.4\ The maximum principle with transversality conditions for 
variable\\
\null$\,\qquad$  time interval problems \hfill 4

\noindent \ 1.5\ An example \hfill 5

\noindent \ 1.6\ A remark on more general endpoint conditions \hfill 7

\medbreak

\medbreak
\noindent 2.\ THE SEPARATION THEOREM \hfill 7

\noindent \ 2.1\ The maximum principle 
as a set separation condition for fixed time\\
\null$\,\qquad$ interval problems \hfill 7

\noindent \ 2.2\ The maximum principle 
as a set separation condition for variable\\
\null$\,\qquad$  time interval problems \hfill 9

\medbreak
\noindent 3.\ THE PROOF OF THE MAXIMUM PRINCIPLE\hfill 10

\noindent \ 3.1\ The variational equation \hfill 11

\noindent \ 3.2\ Packets of needle variations \hfill 11

\noindent \ 3.3\ The endpoint map \hfill 12

\noindent \ 3.4\ The main technical lemma \hfill 12


\newpage

\section{THE  TRANSVERSALITY CONDITIONS}

\noindent We now move one step further, and present two even more
general versions of the Pontryagin maximum principle. 
The new versions are still rather simple as far as technical
hypotheses go, but contain a new ingredient, namely, 
the ``transversality condition.'' This shows
up as an additional necessary condition for optimality for a problem
in which the terminal constraint, instead of being of the form
``$\xi(b)=\hat x$'' considered so far, is of the more general form
$$
\xi(b)\in S\,,
$$
where $S$ is some given subset of $Q$.

\ssubsection{Cones}

\noindent If $X$ is a real linear space, a {\em cone} in $X$ is a 
subset $C$ of $X$ such that 

\medbreak
\begin{itemize}
\item[1.] $C\neq\emptyset$,
\item[2.] $C$ is a union of rays (that is, whenever 
$r\in\rr$, $r\ge 0$, and $c\in C$, it follows that $r\cdot c\in C$).
\end{itemize}

\medbreak\noindent If $C$ is a cone in $X$ then necessarily $0\in C$.
In general, a cone need not be convex. A cone $C$ is convex if and
only if it is closed under addition, that is, $c_1+c_2\in C$ whenever 
$c_1,c_2\in C$. 

If $X$ is endowed with a topology, then we can talk about {\em closed
cones}. We will only be interested in cones in {\em finite-dimensional}
real linear spaces $X$. In that case, $X$ has a canonical topology,
so the concept of a closed cone is well defined.

The (algebraic) {\em polar} of a cone $C$ in a real linear space $X$ 
is the set 
$$
C^\perp\equaldef\{w\in X^{\dagger}: \langle w,c\rangle\le 0\,\,
{\rm for\,\,all}\,\,c\in C\}\,,
$$
where $X^\dagger$ denotes the (algebraic) dual of $X$, that is, the set of 
all linear functionals $w:X\mapsto\rr$.   (When $X$ is a topological 
linear space one wants to consider the {\em topological dual}, consisting 
of the continuous linear functionals, and the corresponding 
{\em topological polar} of a cone. But for us, since we are 
only interested in the finite-dimensional case, algebraic duals and polars
are sufficient.)

 
It is easy to see that {\em the polar of any cone $C$ in $X$ is 
a convex cone}. Moreover, if $X$ is finite-dimensional and $C$ is
a cone in $X$, then $C^\perp$ is closed in $X^\dagger$, and 
$C^{\perp\perp}$----which is a subset of $X^{\dagger\dagger}$,
a space which is canonically identified with $X$---is the closed
convex hull of $C$. In particular, $C^{\perp\perp}=C$ if and only
if $C$ is closed and convex.


\ssubsection{Boltyanskii tangent cones to a set at a point}

Let $S$ be a subset of a finite-dimensional real linear space
$X$, and let $\hat x$ be a point of $S$. A {\bem Boltyanskii
tangent cone} (or ``approximating cone'') to $S$ at $\hat x$ 
is a closed convex cone $C$ in $X$ such that 
there exist a neighborhood $U$ of $0$ in $X$ and a continuous 
map $\varphi:U\cap C\mapsto S$ having  the property that
$$
\varphi(v)=\hat x+v+o(\|v\|)\qquad{\rm as}\qquad v\to 0,\,\,v\in C\,.
$$

\ssubsection{The maximum principle with transversality conditions for fixed
time interval problems}

\noindent We assume that the following conditions are satisfied:

\medbreak\begin{itemize}
\item[C1.] $n$, $m$ are nonnegative integers;

\item[C2.] $Q$ is an open subset of $\rr^n$;

\item[C3.] $U$ is a closed subset of $\rr^m$;

\item[C4.] $a$, $b$ are real numbers such that $a\le b$;

\item[C5.] $Q\!\times\! U\!\times\![a,b]\!\ni\!(x,u,t)\!\mapsto\! f(x,u,t)=
(f^1(x,u,t),\ldots,f^n(x,u,t))\!\in\!\rr^n$ and 
$Q\times U\times[a,b]\ni(x,u,t)\mapsto L(x,u,t)\in\rr$ are continuous maps; 

\item[C6.] for each $(u,t)\in U\times[a,b]$ the maps 
$$
Q\ni x\mapsto f(x,u,t)=\Big(f^1(x,u,t),\ldots,f^n(x,u,t)\Big)\in\rr^n
$$
and 
$$
Q\ni x\mapsto L(x,u,t)\in\rr
$$
are continuously differentiable, 
and their partial derivatives with respect to the $x$ coordinates
are continuous functions of $(x,u,t)$;

\item[C7.] $\bar x$ is a given point of $Q$, and $S$ is a subset of $Q$;


\item[C8.] $TCP_{[a,b]}(Q,U,f)$ (the set of all ``trajectory-control pairs
defined on $[a,b]$ for the data $Q,U,f$'') is the set of all pairs
$(\xi,\eta)$ such that:

\begin{itemize}
\item[a.] $[a,b]\ni t\mapsto\eta(t)\in U$ is a measurable bounded map,

\item[b.] $[a,b]\ni t\mapsto\xi(t)\in Q$ is an absolutely continuous map,

\item[c.] $\dot\xi(t)=f(\xi(t),\eta(t),t)$ for almost every $t\in[a,b]$;.
\end{itemize}

\item[C9.] $TCP_{[a,b]}(Q,U,f)\ni(\xi,\eta)\mapsto J(\xi,\eta)\in\rr$ is the 
functional given by 
$$
J(\xi,\eta)\equaldef\int_a^bL(\xi(t),\eta(t),t)\,dt\,;
$$

\item[C10.] $\gamma_*=(\xi_*,\eta_*)$ (the ``reference TCP'') is
such that 

\begin{itemize}
\item[a.] $\gamma_*\in TCP_{[a,b]}(Q,U,f)$,

\item[b.] $\xi_*(a)=\bar x$ and $\xi_*(b)\in S$,

\item[c.] $J(\xi_*,\eta_*)\le J(\xi,\eta)$ for all 
$(\xi,\eta)\in TCP_{[a,b]}(Q,U,f)$ such that 
$\xi(a)=\bar x$ and $\xi(b)\in S$,

\end{itemize}

\item[C11.] $C$ is a Boltyanskii tangent cone to $S$ at the point 
$\hat x=\xi_*(b)$.
\end{itemize}

\begin{theorem}\label{theo-timenovar}
Assume that the data $n$, $m$, $Q$, $U$, $a$, $b$, $f$, 
$L$, $\bar x$, $S$, satisfy conditions C1-C7, $TCP_{[a,b]}(Q,U,f)$ 
and $J$ are defined by C8-C9, $\gamma_*=(\xi_*,\eta_*)$ satisfies C10,
$\hat x$ is defined by C11 and $C$ satisfies C11. 
Define the Hamiltonian $H$ to be the function
$$
Q\times U\times\rr^n\times\rr\times[a,b]\ni(x,u,p,p_0,t)
\mapsto H(x,u,p,p_0,t)\in\rr
$$ 
given by
$$
H(x,u,p,p_0,t)\equaldef\langle p,f(x,u,t)\rangle-p_0 L(x,u,t)\,.
$$
Then there exists a pair $(\pi,\pi_0)$ such that
\begin{itemize}
\item[E1.] $[a,b]\ni t\mapsto\pi(t)\in\rr^n$ is an absolutely continuous
map;
\item[E2.] $\pi_0\in\rr$ and $\pi_0\ge 0$;
\item[E3.] $(\pi(t),\pi_0)\neq(0,0)$ for every $t\in[a,b]$;
\item[E4.] the ``adjoint equation'' holds, that is, 
$$
\dot\pi(t)=-\frac{\partial H}{\partial x}(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)
$$
for almost every $t\in[a,b]$;
\item[E5.] the ``Hamiltonian maximization condition'' holds, that is,
$$
H(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)=
\max\Big\{H(\xi_*(t),u,\pi(t),\pi_0,t):u\in U\Big\}
$$
for almost every $t\in[a,b]$,
\item[E6.] the ``transversality condition'' holds, that is, 
$$
-\pi(b)\in C^\perp\,.
$$
\end{itemize} 
\end{theorem}

\ssubsection{The maximum principle with transversality conditions 
for variable time interval problems}

\noindent We assume that the following conditions are satisfied:

\medbreak\begin{itemize}
\item[C1.] $n$, $m$ are nonnegative integers;

\item[C2.] $Q$ is an open subset of $\rr^n$;

\item[C3.] $U$ is a closed subset of $\rr^m$;

\item[C4.] the maps $Q\times U\ni(x,u)\mapsto f(x,u)=
(f^1(x,u),\ldots,f^n(x,u))\in\rr^n$ and 
$Q\times U\ni(x,u)\mapsto L(x,u)\in\rr$ are continuous; 

\item[C5.] for each $u\in U$ the maps 
$$
Q\ni x\mapsto f(x,u)=\Big(f^1(x,u),\ldots,f^n(x,u)\Big)\in\rr^n
$$
$$
\!{\rm and}\qquad\qquad\qquad Q\ni x\mapsto L(x,u)\in\rr\qquad
\qquad\qquad\qquad\qquad
$$
are continuously differentiable, 
and their partial derivatives with respect to the $x$ coordinates
are continuous functions of $(x,u)$;

\item[C6.] $\bar x$ is a given point and $S$ is a given subset of $Q$;

\item[C7.] $TCP(Q,U,f)$ (the set of all ``trajectory-control pairs
for the data $Q,U,f$'') is the set given by 
$$
TCP(Q,U,f)\equaldef\bigcup
\Big\{TCP_{[a,b]}(Q,U,f):a,b\in\rr,a\le b\Big\}\,,
$$
where, for $a,b\in\rr$ such that $a\le b$, 
$TCP_{[a,b]}(Q,U,f)$ is the set of all pairs
$(\xi,\eta)$ such that:

\begin{itemize}
\item[a.] $[a,b]\ni t\mapsto\eta(t)\in U$ is a measurable bounded map;

\item[b.] $[a,b]\ni t\mapsto\xi(t)\in Q$ is an absolutely continuous map;

\item[c.] $\dot\xi(t)=f(\xi(t),\eta(t))$ for almost every $t\in[a,b]$.
\end{itemize}

\item[C8.] $TCP(Q,U,f)\ni(\xi,\eta)\mapsto J(\xi,\eta)\in\rr$ is the 
functional given by 
$$
J(\xi,\eta)\!\equaldef\!\int_a^bL(\xi(t),\eta(t))\,dt\,\,\,{\rm for}\,\,
(\xi,\eta)\!\in\! TCP_{[a,b]}(Q,U,f),\,a,b\!\in\!\rr,\,a\le b.
$$

\item[C9.] $\gamma_*=(\xi_*,\eta_*)$ (the ``reference TCP'') and
$a_*$, $b_*$ are such that 

\begin{itemize}
\item[a.] $a_*\in\rr$, $b_*\in\rr$, and $\gamma_*\in TCP_{[a_*,b_*]}(Q,U,f)$;

\item[b.] $\xi_*(a_*)=\bar x$ and $\xi_*(b_*)\in S$;

\item[c.] $J(\xi_*,\eta_*)\le J(\xi,\eta)$ for all $a,b\in\rr$ such that
$a\le b$ and all $(\xi,\eta)\in TCP_{[a,b]}(Q,U,f)$ such that 
$\xi(a)=\bar x$ and $\xi(b)\in S$.
\end{itemize}
\item[C10.] $C$ is a Boltyanskii tangent cone to $S$ at the point 
$\hat x=\xi_*(b_*)$.
\end{itemize}

\begin{theorem}\label{theo-timevar}
Assume that the data $n$, $m$, $Q$, $U$, $f$, 
$L$, $\bar x$, $S$ satisfy conditions C1-C6, $TCP(Q,U,f)$ 
and $J$ are defined by C7-C8, and $a_*$, $b_*$,  
$\gamma_*=(\xi_*,\eta_*)$, $\hat x$, $C$, satisfy C9.
Define the Hamiltonian $H$ to be the function
$$
Q\times U\times\rr^n\times\rr\ni(x,u,p,p_0)
\mapsto H(x,u,p,p_0)\in\rr
$$ 
given by
$$
H(x,u,p,p_0)\equaldef\langle p,f(x,u)\rangle-p_0 L(x,u)\,.
$$
Then there exists a pair $(\pi,\pi_0)$ such that
\begin{itemize}
\item[E1.] $[a_*,b_*]\ni t\mapsto\pi(t)\in\rr^n$ is an absolutely continuous
map;
\item[E2.] $\pi_0\in\rr$ and $\pi_0\ge 0$;
\item[E3.] $(\pi(t),\pi_0)\neq(0,0)$ for every $t\in[a_*,b_*]$;
\item[E4.] the ``adjoint equation'' holds, that is, 
$$
\dot\pi(t)=-\frac{\partial H}{\partial x}(\xi_*(t),\eta_*(t),\pi(t),\pi_0)
$$
for almost every $t\in[a_*,b_*]$;
\item[E5.] the ``Hamiltonian maximization condition with zero value'' 
holds, that is,
$$
0=H(\xi_*(t),\eta_*(t),\pi(t),\pi_0)=
\max\Big\{H(\xi_*(t),u,\pi(t),\pi_0):u\in U\Big\}
$$
for almost every $t\in[a,b]$,
\item[E6.] the ``transversality condition'' holds, that is, 
$$ 
-\pi(b_*)\in C^\perp\,.
$$
\end{itemize} 
\end{theorem}

\ssubsection{An example}

We consider the ``one-dimensional landing problem'' in which it is 
desired to find, for a given initial point $(\alpha,\beta)\in\rr^2$,
a trajectory-control pair $(\xi,\eta)$ of the system 
$$
\dot x=y\,,\qquad\dot y=u\,,\qquad -1\le u\le 1\,,
$$
such that $\xi$ goes from $(\alpha,\beta)$ to some 
point in the $y$ axis in minimum time.
This is exactly the same as the ``soft landing problem'' discussed
before, except that now we are not asking the landing to be soft,
that is, our terminal condition is that the terminal position
$x$ be zero, but we are not imposing any condition on the terminal
velocity.

The configuration space $Q$ for our problem is $\rr^2$, the control set
$U$ is the compact interval $[-1,1]$, the dynamical law $f$ is given
by 
$$
f(x,y,u)=\left[\begin{array}{c}y\\u\end{array}\right]\,,
$$
and the Lagrangian is identically equal to $1$. The terminal
constraint is $\xi(b)\in S$, where 
$$
S=\{(x,y)\in\rr^2:x=0\}\,.
$$

The Hamiltonian $H$ is given by 
$$
H(x,y,u,p_x,p_y,p_0)=p_xy+p_yu-p_0\,,
$$
where we are using $p_x$, $p_y$ to denote the two components
of the momentum variable $p$. 

Assume that $(\xi_*,\eta_*)$ is a solution of our minimum time problem,
and that $(\xi_*,\eta_*)\in TCP_{[a_*,b_*]}(Q,U,f)$.

Then Theorem \ref{theo-timevar} tells us that there exists a pair 
$(\pi,\pi_0)$ satisfying all the conditions of the conclusion.
Write $\pi(t)=(\pi_x(t),\pi_y(t))$. The adjoint equation
then implies 
\begin{eqnarray*}
\dot\pi_x(t)&=&0\,,\\
\dot\pi_y(t)&=&-\pi_x(t)\,.
\end{eqnarray*}
Therefore the function $\pi_x$ is constant. 
Let $A\in\rr$ be such that
$\pi_x(t)=A$ for all $t\in[a_*,b_*]$. Then there must exist a constant
$B$ such that
$$
\pi_y(t)=B-At\quad{\rm for}\quad t\in[a_*,b_*]\,.
$$
Now, if $A$ and $B$ were both equal to zero, the function 
$\pi_y$ would vanish identically and then the Hamiltonian 
maximization condition would say that the function
$$
[-1,1]\ni u\mapsto 0
$$
is maximized by taking $u=\eta_*(t)$, a fact that would give us no 
information whatsoever about $\eta_*$. 

Fortunately, the conditions of Theorem \ref{theo-timevar} 
imply that {\bem $A$ and $B$ cannot both vanish.} To see this, observe 
that if $A=B=0$ then it follows that $\pi_x(t)\equiv\pi_y(t)\equiv 0$. 
But then the nontriviality condition tells us that $\pi_0\neq 0$. 
So the value $H(\xi_*(t),\eta_*(t),\pi(t),\pi_0)$ would be equal to 
$-\pi_0$, which is not equal to zero. This contradicts the fact that,
for our time-varying problem, the Hamiltonian is supposed to vanish.

It is clear that the set $S$ itself is a Boltyanskii tangent
cone to $S$ at the terminal point $\xi_*(b_*)$. Therefore the transversality 
condition says that $\pi(b_*)\in S^\perp$, i.e., that
$$
\pi_y(b_*)=0\,.
$$
So $\rr\ni t\mapsto\pi_y(t)=B-At$ is a linear function
which is not identically zero (because $A$ and $B$ do not both vanish)
but vanishes at the endpoint $b_*$ of the interval $[a_*,b_*]$.
Therefore {\bem $\pi_y(t)$ never vanishes on $[a_*,b_*[$}.
It follows that {\bem the optimal control $\eta_*$ is either
constantly equal to $1$ or constantly equal to $-1$.}
We have thus proved that {\bem all optimal
controls are bang-bang and constant}.


\ssubsection{A remark on more general endpoint conditions}
Suppose we wanted to consider endpoint conditions of the 
form 
$$
(\xi(a),\xi(b))\in S\,,
$$
where $S$ is a subset of $Q\times Q$.  (For example, 
we could require that $\xi(a)\in S_1$ and 
$\xi(b)\in S_2$, in which case we would take $S=S_1\times S_2$.
Or we could require ``periodic endpoint conditions''
$\xi(a)=\xi(b)$, in which case we would take $S$ to be the diagonal
of $Q$, that is, the set $\{(x,x):x\in Q\}$.) 

These problems can easily be reduced to the ones considered in \S 1.3 and
\S 1.4, by means of the following trick: work with a new system 
in $Q\times Q$, with state evolution equations 
$\dot x=f(x,u,t)$, $\dot x'=0$.  (This has the effect of keeping  track 
of the initial state so that it is still there as part of the state
at any later time.) One needs an aditional transformation
because now we do not want to have to initialize the state at a
prespecified point $(\bar x,\bar x')$ of $Q\times Q$. 
For this reason, we add an extra piece of dynamics 
prior to time $a_*$, by means of the dynamical law
$\dot x=v$, $\dot x'=v$,  on the time interval
$[a_*-1,a_*]$, where $v$ is a new control with values in $\rr^n$.
This has the effect of allowing us to fix a point
$(\bar x,\bar x')$  of $Q\times Q$ to 
be the initial condition at time $a_*-1$, while letting the state
at time $a_*$ be of the form $(x,x)$ with $x$ arbitrary. 

\section{THE  SEPARATION THEOREM}

\noindent We now proceed to ``geometrize'' the problem by completely
eliminating the optimization aspect, and working instead with 
a question about {\bem separation of two sets.}

Precisely, let us say that two subsets $S_1$, $S_2$ 
of a set $A$ are {\bem separated at a point $\hat x\in S_1\cap S_2$}
if 
$$
S_1\cap S_2=\{\hat x\}\,,
$$
that is, if $S_1\cap S_2$ contains no points other than $\hat x$. 

If $A$ is a topological space, we say that 
$S_1$ and $S_2$ are {\bem locally separated at $\hat x$}
if there exists a neighborhood $U$ of $\hat x$ such that
$$
U\cap S_1\cap S_2=\{\hat x\}\,.
$$

\ssubsection{The maximum principle 
as a set separation condition for fixed time interval problems}

\noindent We assume that the following conditions are satisfied:

\medbreak\begin{itemize}
\item[C1.] $n$, $m$ are nonnegative integers;

\item[C2.] $Q$ is an open subset of $\rr^n$;

\item[C3.] $U$ is a closed subset of $\rr^m$;

\item[C4.] $a$, $b$ are real numbers such that $a\le b$;

\item[C5.] $Q\!\times \!U\!\times\![a,b]\!\ni\!(x,u,t)\!\mapsto \!f(x,u,t)=
(f^1(x,u,t),\ldots,f^n(x,u,t))\!\in\!\rr^n$ is a continuous map; 

\item[C6.] for each $(u,t)\in U\times[a,b]$ the map
$$
Q\ni x\mapsto f(x,u,t)=\Big(f^1(x,u,t),\ldots,f^n(x,u,t)\Big)\in\rr^n
$$
is continuously differentiable, 
and its partial derivatives with respect to the $x$ coordinates
are continuous functions of $(x,u,t)$;

\item[C7.] $\bar x$ is a given point of $Q$, and $S$ is a subset of $Q$;


\item[C8.] $TCP_{[a,b]}(Q,U,f)$ (the set of all ``trajectory-control pairs
defined on $[a,b]$ for the data $Q,U,f$'') is the set of all pairs
$(\xi,\eta)$ such that:

\begin{itemize}
\item[a.] $[a,b]\ni t\mapsto\eta(t)\in U$ is a measurable bounded map,

\item[b.] $[a,b]\ni t\mapsto\xi(t)\in Q$ is an absolutely continuous map,

\item[c.] $\dot\xi(t)=f(\xi(t),\eta(t),t)$ for almost every $t\in[a,b]$;.
\end{itemize}

\item[C9.] $\mathcal{R}_{[a,b]}(Q,U,f;x)$  (the ``reachable set
from $x$ for the data $Q,U,f$ over the interval $[a,b]$'') is defined by
\begin{eqnarray*}
&\,&\mathcal{R}_{[a,b]}(Q,U,f;x)\equaldef\\
&\,&\,\,\Big\{\,y:(\exists(\xi,\eta)\in
TCP_{[a,b]}(Q,U,f))(\xi(a)=x\wedge \xi(b)=y)\,\Big\}\,.
\end{eqnarray*}

\item[C10.] $\gamma_*=(\xi_*,\eta_*)$ (the ``reference TCP'') is
such that 

\begin{itemize}
\item[a.] $\gamma_*\in TCP_{[a,b]}(Q,U,f)$,

\item[b.] $\xi_*(a)=\bar x$ and $\xi_*(b)\in S$,

\item[c.] $\mathcal{R}_{[a,b]}(Q,U,f;\bar x)$  and 
$S$ are blocally separated at 
the point $\hat x=\xi_*(b)$;
\end{itemize}

\item[C11.] $C$ is a Boltyanskii tangent cone to $S$ at $\hat x$.
\end{itemize}

\begin{theorem}\label{theo-timenovarsep}
Assume that the data $n$, $m$, $Q$, $U$, $a$, $b$, $f$, 
$\bar x$, $S$, satisfy conditions C1-C7, $TCP_{[a,b]}(Q,U,f)$ 
and $\mathcal{R}_{[a,b]}(Q,U,f;x)$ 
are defined by C8-C9, $\gamma_*=(\xi_*,\eta_*)$ satisfies C10,
$\hat x$ is defined by C10, and $C$ satisfies C11. 

Assume, moreover, that $C$ is not a linear subspace of $\rr^n$.

Define the Hamiltonian $H$ to be the function
$$
Q\times U\times\rr^n\times[a,b]\ni(x,u,p,t)
\mapsto H(x,u,p,t)\in\rr
$$ 
given by
$$
H(x,u,p,t)\equaldef\langle p,f(x,u,t)\rangle\,.
$$
Then there exists a $\pi$ such that
\begin{itemize}
\item[E1.] $[a,b]\ni t\mapsto\pi(t)\in\rr^n$ is an absolutely continuous
map;
\item[E2.] $\pi(t)\neq 0$ for every $t\in[a,b]$;
\item[E3.] the ``adjoint equation'' holds, that is, 
$$
\dot\pi(t)=-\frac{\partial H}{\partial x}(\xi_*(t),\eta_*(t),\pi(t),t)
$$
for almost every $t\in[a,b]$;
\item[E4.] the ``Hamiltonian maximization condition'' holds, that is,
$$
H(\xi_*(t),\eta_*(t),\pi(t),t)=
\max\Big\{H(\xi_*(t),u,\pi(t),t):u\in U\Big\}
$$
for almost every $t\in[a,b]$,
\item[E5] the ``transversality condition'' holds, that is, 
$$
-\pi(b)\in C^\perp\,.
$$
\end{itemize} 
\end{theorem}

\ssubsection{The maximum principle 
as a set separation condition for variable time interval problems}

\noindent We assume that the following conditions are satisfied:

\medbreak\begin{itemize}
\item[C1.] $n$, $m$ are nonnegative integers;

\item[C2.] $Q$ is an open subset of $\rr^n$;

\item[C3.] $U$ is a closed subset of $\rr^m$;

\item[C4.] the map $Q\times U\ni(x,u)\mapsto f(x,u)=
(f^1(x,u),\ldots,f^n(x,u))\in\rr^n$ is continuous; 

\item[C5.] for each $u\in U$ the map
$$
Q\ni x\mapsto f(x,u)=\Big(f^1(x,u),\ldots,f^n(x,u)\Big)\in\rr^n
$$
is continuously differentiable, 
and its partial derivatives with respect to the $x$ coordinates
are continuous functions of $(x,u)$;

\item[C6.] $\bar x$ is a given point and $S$ is a given subset of $Q$;

\item[C7.] $TCP(Q,U,f)$ (the set of all ``trajectory-control pairs
for the data $Q,U,f$'') is the set given by 
$$
TCP(Q,U,f)\equaldef\bigcup
\Big\{TCP_{[a,b]}(Q,U,f):a,b\in\rr,a\le b\Big\}\,,
$$
where, for $a,b\in\rr$ such that $a\le b$, 
$TCP_{[a,b]}(Q,U,f)$ is the set of all pairs
$(\xi,\eta)$ such that:

\begin{itemize}
\item[a.] $[a,b]\ni t\mapsto\eta(t)\in U$ is a measurable bounded map;

\item[b.] $[a,b]\ni t\mapsto\xi(t)\in Q$ is an absolutely continuous map;

\item[c.] $\dot\xi(t)=f(\xi(t),\eta(t))$ for almost every $t\in[a,b]$.
\end{itemize}

\item[C8.] $\mathcal{R}(Q,U,f;x)$  (the ``reachable set
from $x$ for the data $Q,U,f$'') is defined by
$$
\mathcal{R}(Q,U,f;x)\equaldef
\bigcup_{-\infty<a\le b<+\infty}\mathcal{R}_{[a,b]}(Q,U,f;x)
$$
where
\begin{eqnarray*}
&\,&\mathcal{R}_{[a,b]}(Q,U,f;x)\equaldef\\
&\,&\,\,\Big\{\,y:(\exists(\xi,\eta)\in
TCP_{[a,b]}(Q,U,f))(\xi(a)=x\wedge \xi(b)=y)\,\Big\}\,.
\end{eqnarray*}

\item[C9.] $\gamma_*=(\xi_*,\eta_*)$ (the ``reference TCP'') and
$a_*$, $b_*$ are such that 

\begin{itemize}
\item[a.] $a_*\in\rr$, $b_*\in\rr$, and $\gamma_*\in TCP_{[a_*,b_*]}(Q,U,f)$;

\item[b.] $\xi_*(a_*)=\bar x$ and $\xi_*(b_*)\in S$;

\item[c.] $\mathcal{R}(Q,U,f;\bar x)$
and $S$ are locally separated at  the point
$\hat x=\xi_*(b_*)$.
\end{itemize}
\item[C10.] $C$ is a Boltyanskii tangent cone to $S$ at $\hat x$.
\end{itemize}

\begin{theorem}\label{theo-timevarsep}
Assume that the data $n$, $m$, $Q$, $U$, $f$, 
$\bar x$, $S$ satisfy conditions C1-C6, $TCP(Q,U,f)$ 
and $\mathcal{R}(Q,U,f;\bar x)$ are defined by C7-C8, and $a_*$, $b_*$,  
$\gamma_*=(\xi_*,\eta_*)$, $\hat x$, $C$, satisfy C9 and C10.

Assume, moreover, that $C$ is not a linear subspace of $\rr^n$. 

Define the Hamiltonian $H$ to be the function
$$
Q\times U\times\rr^n\ni(x,u,p)
\mapsto H(x,u,p)\in\rr
$$ 
given by
$$
H(x,u,p)\equaldef\langle p,f(x,u)\rangle\,.
$$
Then there exists a $\pi$ such that
\begin{itemize}
\item[E1.] $[a_*,b_*]\ni t\mapsto\pi(t)\in\rr^n$ is an absolutely continuous
map;
\item[E2.] $\pi(t)\neq0$ for every $t\in[a_*,b_*]$;
\item[E3.] the ``adjoint equation'' holds, that is, 
$$
\dot\pi(t)=-\frac{\partial H}{\partial x}(\xi_*(t),\eta_*(t),\pi(t))
$$
for almost every $t\in[a_*,b_*]$;
\item[E4.] the ``Hamiltonian maximization condition with zero value'' 
holds, that is,
$$
0=H(\xi_*(t),\eta_*(t),\pi(t))=
\max\Big\{H(\xi_*(t),u,\pi(t)):u\in U\Big\}
$$
for almost every $t\in[a,b]$,
\item[E5.] the ``transversality condition'' holds, that is, 
$$ 
-\pi(b_*)\in C^\perp\,.
$$
\end{itemize} 
\end{theorem}

\section{THE  PROOF OF THE MAXIMUM PRINCIPLE}

\noindent We now present a proof of Theorem \ref{theo-timenovarsep}.
The basic strategy is to make ``needle 
variations,'' combine these variations into ``packets of needle variations,''
``propagate'' the effects of these variations to the terminal point of 
the reference trajectory, and construct an ``approximating cone'' to the 
reachable set. Then a topological argument will be used to derive, from the 
separation assumption for the reachable set and the set, the existence of 
a hyperplane separating the corresponding tangent cones. By propagating 
backwards via the adjoint equation
the linear functional that defines this hyperplane, we will get the desired 
momentum vector $\pi$. 

In order to avoid technical complications, we will first do the proof 
under the following additional simplifying assumption:

\medbreak
\begin{itemize}
\item[(SA)] the reference control $\eta_*$ is continuous.
\end{itemize}

\medbreak\noindent Assumption (SA) is not needed, but the proof without 
it is a little bit more technical, so we will first assume (SA), and
then explain how to get rid of this condition.

So from now on {\bem we assume that the data 
$n$, $m$, $Q$, $U$, $a$, $b$, $f$, 
$\bar x$, $S$, satisfy conditions C1-C7 of \S 2.1, $TCP_{[a,b]}(Q,U,f)$ 
and $\mathcal{R}_{[a,b]}(Q,U,f;x)$ 
are defined by C8-C9 of \S 2.1, $\gamma_*=(\xi_*,\eta_*)$ satisfies C10 of
\S2.1, $\hat x$ is defined by C10 of \S 2.1, $C$ satisfies C11 of \S 2.1,
and (SA) holds}. 




\ssubsection{The variational equation}
Define
$$
A(t)=\frac{\partial f}{\partial x}(\xi_*(t),\eta_*(t),t)\,.
$$
Then, under our hypotheses---including the additional condition
(SA)---the map
$[a,b]\ni t\mapsto A(t)\in\rr^{n\times n}$ is continuous.



The linear time-varying ordinary differential equation 
\begin{equation}\label{vareqqq}
\dot v(t)=A(t)\cdot v(t)
\end{equation}
is the {\bem variational equation along the reference TCP 
$(\xi_*,\eta_*)$}. 

We use $M(t,s)$ to denote the fundamental matrix solution of (\ref{vareqqq}).
That is, for each $s\in[a,b]$ the map 
$$
[a,b]\ni t\mapsto M(t,s)\in\rr^{n\times n}
$$
is the solution of the matrix-valued initial value problem
\begin{equation}\label{vareqqqmat}
\left\{\begin{array}{l}
\dot X(t)=A(t)\cdot X(t)\,,\\
X(s)=\pmb{1}_n\,,\end{array}\right.
\end{equation}
where $\pmb{1}_n$ is the identity matrix of $\rr^n$.

It then follows that the map 
$$
[a,b]\times[a,b]\ni(t,s)\mapsto M(t,s)\in\rr^{n\times n}
$$
is continuous and satisfies the integral equation
$$
M(t,s)=\pmb{1}_n+\int_s^t A(r)\cdot M(r,s)\,dr\quad{\rm for}\quad
t,s\in[a,b]\,.
$$

\ssubsection{Packets of needle variations}

Fix 

\begin{itemize}
\item[NV1.] a positive integer $k$

\item[NV2.] a $k$-tuple  $\vec{\tau}=(\tau_1,\ldots,\tau_k)$ (the ``variation
times'') such that
$$
a\le\tau_1<\tau_2<\cdots<\tau_k<b\,,
$$
\item[NV3.] a $k$-tuple  $\vec{u}=(u_1,\ldots,u_k)$ (the ``variation
controls'') of members of the control set $U$.
\end{itemize}
Use $MB([a,b],U)$ to denote the set of all controls defined
on the interval $[a,b]$. (That is, the members of 
$MB([a,b],U)$ are the bounded measurable maps $\eta:[a,b]\mapsto U$.) 

For a positive real number $\alpha$, define
$$
\rr^k_+(\alpha)\equaldef\{(x_1,\ldots,x_k)\in\rr^k:
0\le x_j\le\alpha\,\,{\rm for}\,\,j=1,\ldots,k\}\,.
$$
Let
$$
\bar\eps=\min(\tau_2-\tau_1,\tau_3-\tau_2,\ldots,\tau_k-\tau_{k-1},b-\tau_k)\,.
$$

We then define the {\bem packet of needle variations of the 
reference control $\eta_*$ associated to $\vec{\tau},\vec{u}$} to be 
the map 
$$
\rr^k_+(\bar\eps)\ni\vec{\eps}=(\eps_1,\ldots,\eps_k)
\mapsto\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}\in MB([a,b],U)
$$
that associates to every 
$\vec{\eps}=(\eps_1,\ldots,\eps_k)\in\rr^k_+(\bar\eps)$ a control 
$\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}$ defined as follows:
$$
\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)=\left\{
\begin{array}{lcl}
\eta_*(t)&{\rm if}&t\in[a,b]\backslash
\bigcup_{j=1}^k[\tau_j,\tau_j+\eps_j]\,,\\
u_j&{\rm if}&t\in[\tau_j,\tau_j+\eps_j],\,j\in\{1,\ldots,k\}\,.\end{array}
\right.
$$

\ssubsection{The endpoint map}
Given $k,\vec{\tau},\vec{u}$ as above, we let 
$\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}$ be, for 
$\vec{\eps}\in\rr_+^k(\bar\eps)$, the unique  solution
$t\mapsto\xi(t)$ of the equation
$$ 
\dot\xi(t)=f(\xi(t),\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(t),t)
$$
such that $\xi(a)=\bar x$.

Uniqueness follows because the map 
$$
Q\times[a,b]\ni(x,t)\mapsto f(x,\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(t),
t)
$$
is Lipschitz with respect to $x$ with a bounded
Lipschitz constant as long as $(x,t)$ belongs to a compact subset 
of $Q\times[a,b]$.

In principle, $\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}$ need not exist
on the whole interval $[a,b]$. The {\bem endpoint map}
$$
\vec{\eps}\mapsto\mathcal{E}^{\vec{\tau},\vec{u}}(\vec{\eps})
$$
{\bem associated to $\vec{\tau},\vec{u}$} is the map 
that assigns to each $\vec{\eps}\in\rr_+^k(\bar\eps)$ 
the terminal point
\begin{equation}\label{defendpo}
\mathcal{E}^{\vec{\tau},\vec{u}}(\vec{\eps})\equaldef
\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(b)
\end{equation}
of the trajectory $\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}$. Precisely,
the {\bem domain} $\mathcal{D}^{\vec{\tau},\vec{u}}$ of
$\mathcal{E}^{\vec{\tau},\vec{u}}$ is the set of those
$\vec{\eps}\in\rr_+^k(\bar\eps)$ such that 
$\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}$ exists on the whole interval
$[a,b]$ and, for 
$\vec{\eps}\in\mathcal{D}^{\vec{\tau},\vec{u}}$, 
$\mathcal{E}^{\vec{\tau},\vec{u}}(\vec{\eps})$ is defined by 
(\ref{defendpo}). 

\ssubsection{The main technical lemma}

\begin{lemma}
Given $k,\vec{\tau},\vec{u}$ as above, write 
$$
x_j=\xi_*(\tau_j)\,,\quad u_{*,j}=\eta_*(\tau_j)\,.
$$
Then there exists an 
$\tilde\eps$ such that:
\begin{itemize}
\item[1.] $0<\tilde\eps\le\bar\eps$,
\item[2.] $\rr_+^k(\tilde\eps)\subseteq\mathcal{D}^{\vec{\tau},\vec{u}}$,
\item[3.] the endpoint map $\mathcal{E}^{\vec{\tau},\vec{u}}$ is continuous
on $\rr_+^k(\tilde\eps)$,
\item[4.] $\mathcal{E}^{\vec{\tau},\vec{u}}$ is differentiable at 
$0\in\rr^k$, and the differential  $D\mathcal{E}^{\vec{\tau},\vec{u}}(0)$
of $\mathcal{E}^{\vec{\tau},\vec{u}}$
at $0$ is the linear map
$$
\rr^k\ni\vec{\alpha}=(\alpha_1,\ldots,\alpha_k)
\mapsto \Lambda(\vec{\alpha})\in\rr^n
$$
given by
$$
\Lambda(\vec{\alpha})\equaldef
\sum_{j=1}^k\alpha_j\cdot M(b,\tau_j)\cdot\Big(f(x_j,u_j,\tau_j)
-f(x_j,u_{*,j},\tau_j)\Big)\,.
$$
\end{itemize}
\end{lemma}

\medbreak\noindent Conclusion 4 says that 
$$
\mathcal{E}^{\vec{\tau},\vec{u}}(\vec{\eps})=
\xi_*(b)+\sum_{j=1}^k\eps_j\cdot M(b,\tau_j)\cdot\Big(f(x_j,u_j,\tau_j)
-f(x_j,u_{*,j},\tau_j)\Big)+o(\|\vec{\eps}\|)
$$
as $\vec{\eps}\to 0$ via values in $\rr_+^k$.

\begin{proof}
We fix a positive number $\delta$ such that the 
set 
$$
\tilde K=\{(x,t)\in\rr^n\times[a,b]:\|x-\xi_*(t)\|\le\delta\}
$$
is entirely contained in $Q\times[a,b]$. 

We let $U_0$ be the union of the sets 
$\{\eta_*(t):a\le t\le b\}$ and $\{u_1,\ldots,u_k\}$. Then $U_0$ 
is compact. We define $K=\{(x,u,t):(x,t)\in\tilde K\wedge u\in U_0\}$, 
so $K$ is a compact subset of $Q\times U\times[a,b]$. 

Since $f$ and $\frac{\partial f}{\partial x}$ are continuous on 
$Q\times U\times[a,b]$, we can pick a constant $\kappa$ such that
$$
\|f(x,u,t)\|+\Big\|\frac{\partial f}{\partial x}(x,u,t)\Big\|\le\kappa
\quad{\rm whenever}\quad(x,u,t)\in Q\times U\times[a,b]\,.
$$
For $\vec{\eps}\in\rr_+^k(\bar\eps)$, let $\beta(\vec{\eps})$ be the 
maximum of all real numbers $\beta$ such that 
(i)~$a\le\beta\le b$, (ii)~$\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}$ exists
on $[a,\beta]$, (iii)~$(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t),t)\in
\tilde K$ whenever $t\in[a,\beta]$. (It is easy to prove that such a maximum 
exists.) 

Fix $\vec{\eps}=(\eps_1,\ldots,\eps_k)\in\rr_+^k(\bar\eps)$. If 
$t\in[a,\beta(\vec{\eps})]$, then 
\begin{eqnarray*}
\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)&=&
\int_{a}^t\Big(f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),s)-
f(\xi_*(s),\eta_*(s),s)\Big)\,ds\\
&=&\int_{a}^t\Big(f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*(s),s)-f(\xi_*(s),\eta_*(s),s)\Big)\,ds\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\int_{[a,t]\cap E(\vec{\eps})}
\Big(f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),s)
-f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),\eta_*(s),s)\Big)\,ds\,,
\end{eqnarray*}
where 
$$
E(\vec{\eps})=\bigcup_{j=1}^k[\tau_j,\tau_j+\eps_j]\,.
$$
Then
\begin{eqnarray*}
\|\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)\|&\le&
\int_{a}^t\Big\|f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*(s),s)-f(\xi_*(s),\eta_*(s),s)\Big\|\,ds\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\int_{[a,t]\cap E(\vec{\eps})}
\Big\|f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),s)
-f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),\eta_*(s),s)\Big\|\,ds\\
&\le&\kappa\int_{a}^t\Big\|\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s)-
\xi_*(s)\Big\|\,ds+2\kappa\|\vec{\eps}\|\,,
\end{eqnarray*}
where 
$$
\|\vec{\eps}\|=\eps_1+\ldots+\eps_k\,.
$$
Gronwall's inequality then implies
\begin{equation}\label{condia}
\|\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)\|\le
2\kappa\|\vec{\eps}\|e^{\kappa(b-a)}\,.
\end{equation}
Choose $\tilde\eps$ such that 
\begin{equation}\label{condi1}
0<\tilde\eps\le\bar\eps
\end{equation}
and 
$$
4k\kappa\tilde\eps e^{\kappa(b-a)}\le\delta\,.
$$
Then 
$$
\vec{\eps}\in\rr_+^k(\tilde\eps)\Longrightarrow
\Biggl(\|\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)\|\le\frac{\delta}{2}
\quad{\rm whenever}\quad t\in[a,\beta(\vec{\eps})]\Biggr)\,.
$$
It then follows that 
$$
\vec{\eps}\in\rr_+^k(\tilde\eps)\Longrightarrow
\beta(\vec{\eps})=b\,,
$$
so that 
\begin{equation}\label{condi2}
\rr_+^k(\tilde\eps)\subseteq\mathcal{D}^{\vec{\tau},\vec{u}}
\end{equation}
and
$$
\Biggl(\vec{\eps}\in\rr_+^k(\tilde\eps)\wedge t\in[a,b]\Biggr)
\Longrightarrow\Biggl(
\|\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)\|\le
2\kappa\|\vec{\eps}\|e^{\kappa(b-a)}\le\frac{\delta}{2}\Biggr)\,.
$$
A similar Gronwall inequality argument establishes that 
\begin{eqnarray*}
&\,&\Biggl(\vec{\eps}\in\rr_+^k(\tilde\eps)\wedge 
\vec{\eps}'\in\rr_+^k(\tilde\eps)\wedge 
t\in[a,b]\Biggr)\Longrightarrow\\
&\,&\quad\Biggl(
\|\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-
\xi_*^{\vec{\tau},\vec{u},\vec{\eps}'}(t)\|\le
2\kappa\Big(\sum_{j=1}^k|\eps_j-\eps'_j|\Big)
e^{\kappa(b-a)}\le\frac{\delta}{2}\Biggr)\,.
\end{eqnarray*}
It follows, in particular, that 
$$
\Big(\vec{\eps}\in\rr_+^k(\tilde\eps)\wedge 
\vec{\eps}'\!\in\rr_+^k(\tilde\eps)\Big)\Rightarrow
\|\mathcal{E}^{\vec{\tau},\vec{u}}(\vec{\eps})-
\mathcal{E}^{\vec{\tau},\vec{u}}(\vec{\eps}')\|\le
2\kappa\Big(\sum_{j=1}^k|\eps_j-\eps'_j|\Big)e^{\kappa(b-a)}\,,
$$
so the endpoint map $\mathcal{E}^{\vec{\tau},\vec{u}}$ is continuous
on $\rr_+^k(\tilde\eps)$.

Finally, we have to prove that $\mathcal{E}^{\vec{\tau},\vec{u}}$ is 
differentiable at $0$ and that the differential is given by the 
formula of our statement.  For this purpose, we go back to the formula
\begin{eqnarray*}
\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)&=&
\int_{a}^t\Big(f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*(s),s)-f(\xi_*(s),\eta_*(s),s)\Big)\,ds\\
&\,&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\int_{[a,t]\cap E(\vec{\eps})}
\Big(f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),s)
-f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),\eta_*(s),s)\Big)\,ds\,,
\end{eqnarray*}
that we rewrite as
\begin{equation}\label{rewri}
\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)=
\int_{a}^t\Delta_1(s)\,ds
+\int_{[a,t]\cap E(\vec{\eps})}\Delta_2(s)\,ds\,.
\end{equation}
We observe that 

\begin{itemize}
\item[1.] the difference 
$$
\Delta_1(s)\equaldef f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*(s),s)-f(\xi_*(s),\eta_*(s),s)
$$
is ``approximately equal'' to 
$\frac{\partial f}{\partial x}(\xi_*(s),\eta_*(s),s)\cdot
(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s)-\xi_*(s))$,
i.e., to $A(s)\cdot(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s)-\xi_*(s))$,
whenever $t\in[a,b]$.
\item[2.] the difference
$$
\Delta_2(s)\equaldef f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),
\eta_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),s)-
f(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s),\eta_*(s),s)
$$  is 
``approximately equal'' to 
$$
w_j\equaldef f(x_j,u_j,\tau_j)-f(x_j,u_{*,j},\tau_j)
$$
whenever $s\in[\tau_j,\tau_j+\eps_j]$,
\medbreak
\end{itemize}

\noindent To make this precise, we let $\theta:[a,b]\to\rr^n$
be the function such that  
$$
\theta(t)=\left\{\begin{array}{lcl}
0&{\rm if}&t\notin E(\vec{\eps})\,,\\
f(x_j,u_j,\tau_j)-f(x_j,u_{*,j},\tau_j)&{\rm if}&t\in[\tau_j,\tau_j+\eps_j]\,,
\end{array}\right.
$$
and we let $\mu:[a,b]\to\rr^n$ be the solution of
$$
\dot\mu(t)=A(t)\cdot\mu(t)+\theta(t)\,,\qquad\mu(a)=0\,.
$$
Then 
\begin{equation}\label{rewri2}
\mu(t)=\int_{a}^t(A(s)\cdot\mu(s)+\theta(s))\,ds\,.
\end{equation}
If we subtract (\ref{rewri2}) from (\ref{rewri}), we get
\begin{eqnarray*}
&\,&\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)-\mu(t)\\
&\,&=\int_{a}^t\Big(\Delta_1(s)-A(s)\cdot\mu(s)\Big)\,ds
+\int_{[a,t]\cap E(\vec{\eps})}
\Big(\Delta_2(s)-\theta(s)\Big)\,ds\\
&\,&=\int_{a}^t\Big(
A(s)\cdot(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s)-\xi_*(s)-\mu(s)\Big)\,ds+R(t)\,,
\end{eqnarray*}
where the remainder $R(t)$ is given by
$$
R(t)\equaldef\int_{a}^t\Big(\Delta_1(s)-A(s)\cdot
(\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(s)-\xi_*(s))\Big)\,ds
\!+\!\int_{[a,t]\cap E(\vec{\eps})}
\Big(\Delta_2(s)-\theta(s)\Big)\,ds\,.
$$
Then Gronwall's inequality yields
\begin{equation}\label{rewri20}
\|\xi_*^{\vec{\tau},\vec{u},\vec{\eps}}(t)-\xi_*(t)-\mu(t)\|
\le e^{\kappa(b-a)}\sup\{\|R(s)\|:a\le s\le b\}\,.
\end{equation}
The functions $\theta$ and $\mu$ depend on $\vec{\eps}$. The variations
of constants formula yields
$$
\mu^{\vec{\eps}}(b)=\int_{a}^{b}M(b,s)\cdot\theta^{\vec{\eps}}
(s)\,ds\,,
$$
where we have now made the $\vec{\eps}$-dependence explicit. 
So 
\begin{eqnarray}
\mu^{\vec{\eps}}(b)&=&\sum_{j=1}^k
\int_{\tau_j}^{\tau_j+\eps_j}M(b,s)\cdot w_j\,ds\label{rewri30}\\
&=&\sum_{j=1}^k\eps_j\cdot M(b,\tau_j)\cdot w_j\nonumber\\
&\,&\,\,
+\sum_{j=1}^k\int_{\tau_j}^{\tau_j+\eps_j}\Big(M(b,s)-M(b,\tau_j)\Big)
\cdot w_j\,ds\nonumber\\
&=&\sum_{j=1}^k\eps_j\cdot M(b,\tau_j)\cdot w_j +o(\|\vec{\eps}\|)
\nonumber\,.
\end{eqnarray}

If we apply (\ref{rewri20}) with $t=b$, and use (\ref{rewri30}), we get
\begin{equation}\label{rewri50}
\|\mathcal{E}^{\vec{\tau},\vec{u}}(\vec{\eps})-\xi_*(b)
-\sum_{j=1}^k\eps_j\cdot M(b,\tau_j)\cdot w_j\|
\le e^{\kappa(b-a)}\omega(\vec{\eps})+o(\|\vec{\eps}\|)\,,
\end{equation}
where
$$
\omega(\vec{\eps})\equaldef\sup\{\|R^{\vec{\eps}}(s)\|:a\le s\le b\}\,.
$$
So our desired conclusion will follow if we prove that 
$$
\omega(\vec{\eps})=o(\|\vec{\eps}\|)\quad{\rm as}\quad\eps\to 0\,.
$$
The proof of this is a fairly routine and somewhat tedious calculation,
that will be done later.

\end{proof}
}\end{document}








