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\begin{document}
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\noindent{\bf NOTATIONS AND DEFINITIONS:} 
\begin{itemize}
\item[$\bullet$] $\rr^n$ is the space of all real {\em column}
vectors $x=\left(\begin{array}{c}x^1\\x^2\\\vdots\\x^n\end{array}\right)$.
\item[$\bullet$] $\rr_n$ is the space of all real {\em row}
vectors $p=(p_1,p_2,\ldots,p_n)$. 
\item[$\bullet$] $\rr^{m\times n}$ is the space of all real {\em matrices}
with $m$ columns and $n$ rows.
\item[$\bullet$] Given a metric space $U$, a {\em $U$-control} is a
function $[a,b]\ni t\mapsto\eta(t)\in U$ whose domain is a compact interval.
\item[$\bullet$] A $U$-control $[a,b]\ni t\mapsto\eta(t)\in U$ is 
{\em piecewise continuous} if there exists a finite subset $F$ of the open 
interval $\,]a,b[\,$ such that $\eta$ is continuous at all points $t$ of 
$[a,b]\backslash F$ and the left and right limits of $\eta(t)$ as $t$ 
approaches $\tau$ exist for all $\tau\in F$. Notice that
\begin{center}
\fbox{\begin{minipage}{9cm} 
{\em The closure of the graph ${\rm Gr}(\eta)$ of a piecewise continuous
$U$-control $\eta$ is a compact subset of $U\times\rr$.}
\end{minipage}}.
\end{center}
We use $C^{0,pc}([a,b],U)$ to denote the set of all piecewise
continuous $U$-controls.
\item[$\bullet$] A {\em trajectory} for a piecewise continuous control
$\eta:[a,b]\mapsto U$ is a continuous map $\xi:[a,b]\mapsto\Omega$ such that
\begin{equation}\label{dyneq}
\dot\xi(t)=f(\xi(t),\eta(t),t)
\end{equation}
for all but finitely many points $t\in[a,b]$.
Equivalently, a continuous map $\xi:[a,b]\mapsto\Omega$ is a trajectory for 
$\eta$ if and only if
$$
\xi(t)-\xi(s)=\int_s^tf(\xi(r),\eta(r),r)\,dr\qquad
{\rm for\,\,\,all}\,\,\, s,t\in[a,b]\,.
$$
\item[$\bullet$] A {\em trajectory-control pair} for 
$(\Omega,U,\mathcal{U},f)$
is a pair $(\xi,\eta)$ such that $\eta\in\mathcal{U}$ and 
$\xi$ is a trajectory for $\eta$.
\item[$\bullet$] $TCP(\Omega,U,\mathcal{U},f)$ is the set of all
trajectory-control pairs for the data 
$(\Omega,U,\mathcal{U},f)$. 
\item[$\bullet$] A {\em cone} in a real linear space $X$ is a nonempty subset 
$C$ of $X$ such that $(\forall c\in C)(\forall r\in\rr)(r\ge 0\Rightarrow
rc\in C)$. 
\item[$\bullet$] If $X$ is a normed real linear space (or, more generally, 
if $X$ is a real topological linear space), and $C$ is a cone
in $X$, the {\em polar} of $C$ is the set
$$
C^\dagger\equaldef\{p\in X^\dagger:p(c)\le 0\,\,{\rm for\,\,all}\,\,
c\in C\}\,,
$$
where $X^\dagger$ is the dual of $X$, i.e., the space of all
continuous linear functionals $p:X\mapsto\rr$.
\item[$\bullet$] For each function $h:\Omega\times\rr_n\mapsto\rr$ 
of class $C^1$, $\vec{h}$ is the {\em Hamilton vector field} corresponding 
to $h$, defined by 
$$
\vec{h}=\sum_{k=1}^n\Biggl(
\frac{\partial h}{\partial p_k}\frac{\partial\,\,}{\partial x^k}
-\frac{\partial h}{\partial x^k}\frac{\partial\,\,}{\partial p_k}\Biggr)\,.
$$
\end{itemize}

\newpage
\noindent{\bf ASSUMPTIONS FOR THEOREM OC-FT:} 
\begin{itemize}
\item[(A1)] $\Omega$ is an open subset of $\rr^n$.
\item[(A2)] $U$ is a metric space.
\item[(A3)] $a,b\in\rr$ and $a<b$.
\item[(A4)] $\Omega\times U\times[a,b]\ni(x,u,t)\mapsto f(x,u,t)\in\rr^n$ and 
$\Omega\times U\times[a,b]\ni(x,u,t)\mapsto L(x,u,t)\in\rr$ are continuous maps
such that 
\begin{itemize}
\item[(A4.a)] the maps $\Omega\ni x\mapsto f(x,u,t)\in\rr^n$ and 
$\Omega\ni x\mapsto L(x,u,t)\in\rr$ are of class $C^1$ for each fixed
$(u,t)$,
\item[(A4.b)] the maps $\Omega\times U\times[a,b]\ni(x,u,t)\mapsto 
\frac{\partial f}{\partial x}(x,u,t)\in\rr^{n\times n}$ and 
$\Omega\times U\times[a,b]\ni(x,u,t)\mapsto\frac{\partial L}{\partial x}(x,u,t)
\in\in\rr_n$ are continuous.
\end{itemize}
\item[(A5)] $\mathcal{U}=C^{0,pc}([a,b],U)$.
\item[(A6)] $S$ is a subset of $\Omega$.	
\item[(A7)] $(\xi_*,\eta_*)\in TCP(\Omega,U,\mathcal{U},f)$
and $\xi_*(b)\in S$. 
\item[(A8)] $C$ is a tangent cone of $S$ at $\xi_*(b)$. 
\item[(A9)] $J(\xi_*,\eta_*)\le J(\xi,\eta)$ for all 
$(\xi,\eta)\in TCP(\Omega,U,\mathcal{U},f)$ such that $\xi(b)\in S$,
where
$$
J(\xi,\eta)\equaldef\int_a^bL(\xi(t),\eta(t),t)\,dt\,.
$$
\end{itemize}

\medbreak
\noindent{\bf THEOREM OC-FT:}  {\bf\em (Optimal control theorem, fixed
time.)} {\em Assume that (A1,2,3,4,5,6,7,8,9) hold. 
Define the {\bf\em Hamiltonian} 
$$
\Omega\times\rr_n\times U\times\rr\times[a,b]\ni(x,p,u,p_0,t)\mapsto 
H(x,p,u,p_0,t)\in\rr
$$
by letting 
$$
H(x,p,u,p_0,t)\equaldef p\cdot f(x,u,t)-p_0(x,u,t)\,.
$$
Then there exists  a pair $(\pi,\pi_0)$ 
such that 
\begin{itemize}
\item[(\#)] $\pi:[a,b]\mapsto\rr_n$ is a continuous map 
and $\pi_0\in\rr$;
\item[(AE)] (the "adjoint equation") 
\begin{equation}\label{adjeq-ocft}
\dot
\pi(t)=-\pi(t)\cdot\frac{\partial f}{\partial x}(\xi_*(t),\eta_*(t),t)
+\pi_0\frac{\partial L}{\partial x}(\xi_*(t),\eta_*(t),t)
\end{equation}
at all points of continuity $t$ of $\eta_*$;
\item[(HM)] (the "Hamiltonian maximization condition")
$$
H(\xi_*(t),\pi(t),\eta_*(t),\pi_0,t)=
\max\Big\{H(\xi_*(t),\pi(t),u,\pi_0,t):u\in U\}
$$ 
at all points of continuity $t$ of $\eta_*$; 
\item[(TC)] (the "transversality condition") 
$$
\pi(b)\in -C^\dagger\,;
$$ 
\item[(NT)] (the "nontriviality condition") 
$(\pi(t),\pi_0)\neq(0,0)$ for some $t$ (which implies
that $(\pi(t),\pi_0)\neq(0,0)$ for all $t$);
\item[(NN)] (the "nonnegativtiy condition") $\pi_0\ge 0$.
\end{itemize}}

\medbreak
\noindent{\bf REMARK:}  The adjoint equation (\ref{adjeq-ocft})
can also be written as
\begin{equation}\label{ae-ham}
\dot\pi(t)=-\frac{\partial H}{\partial x}(\xi_*(t),\pi(t),\eta_*(t),\pi_0,t)\,.
\end{equation}
The dynamical equation (\ref{dyneq}) is equivalent to
\begin{equation}\label{dyneq-ham}
\dot\xi_*(t)=\frac{\partial H}{\partial p}(\xi_*(t),\pi(t),\eta_*(t),
\pi_0,t)\,.
\end{equation}
The pair of equations (\ref{dyneq-ham}), (\ref{ae-ham}) is equivalent
to the single equation
\begin{equation}\label{full-ham}
\dot\Theta(t)=\vec{H}_{\eta_*(t),\pi_0,t}(\Theta(t))\,,
\end{equation}
evolving in $\Omega\times\rr_n$, i.e., $(x,p)$ space
(the cotangent bundle of $\Omega$), where 
$$
\Theta(t)=(\xi_*(t),\pi(t))\,,
$$
and 
$$
H_{u,p_0,t}(x,p)\equaldef H(x,p,u,p_0,t)\,.
$$


\bigbreak
\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
$\qquad$
\begin{center}
{\bf THE INTRINSIC FORM OF THE ADJOINT EQUATION:}
\end{center}
{\large\em The adjoint equation (\ref{adjeq-ocft}}) 
is equivalent to the following statement:
\begin{center}\begin{minipage}{10.5cm}
{\large\em For every vector field $X$ of class $C^1$ on 
$\Omega$, the equality
$$
\frac{d\,\,}{dt}\Big(\pi(t)\cdot X(\xi_*(t))\Big)
=\pi(t)\cdot[f_{\eta_*(t),t},X](\xi_*(t))+\pi_0 X
L_{\eta_*(t),t}(\xi_*(t))
$$
holds, where
$$
f_{u,t}(x)\equaldef f(x,u,t)\quad{\rm and}\quad
L_{u,t}(x)\equaldef L(x,u,t)\,.
$$}
\end{minipage}\end{center}
\end{minipage}}}\end{center}

\vspace{1cm}
\noindent{\bf ASSUMPTIONS FOR THEOREM OC-VT:} 
\begin{itemize}
\item[(\tA 1)] $\Omega$ is an open subset of $\rr^n$.
\item[(\tA 2)] $U$ is a metric space.
\item[(\tA 3)] $T_*\in\rr$ and $T_*>0$.
\item[(\tA 4)] $\Omega\times U\ni(x,u)\mapsto f(x,u)\in\rr^n$ and 
$\Omega\times U\ni(x,u)\mapsto L(x,u)\in\rr$ are continuous maps
such that 
\begin{itemize}
\item[(\tA 4.a)] the maps $\Omega\ni x\mapsto f(x,u)\in\rr^n$ and 
$\Omega\ni x\mapsto L(x,u)\in\rr$ are of class $C^1$ for each fixed
$u$,
\item[(\tA 4.b)] the maps $\Omega\times U\ni(x,u)\mapsto 
\frac{\partial f}{\partial x}(x,u)\in\rr^{n\times n}$ and 
$\Omega\times U\ni(x,u,t)\mapsto\frac{\partial L}{\partial x}(x,u)
\in\in\rr_n$ are continuous.
\end{itemize}
\item[(\tA 5)] $\alpha>0$ and 
$\mathcal{U}=\bigcup_{T_*-\alpha\le T\le T_*+\alpha}
C^{0,pc}([0,T],U)$.
\item[(\tA 6)] $S$ is a subset of $\Omega$.	
\item[(\tA 7)] $(\xi_*,\eta_*)\in TCP(\Omega,U,\mathcal{U},f)$,
$\eta_*\in C^{0,pc}([0,T_*],U)$, and $\xi_*(T_*)\in S$. 
\item[(\tA 8)] $C$ is a tangent cone to $S$ at $\xi_*(T_*)$. 
\item[(\tA 9)] $J(\xi_*,\eta_*)\le J(\xi,\eta)$ for all 
$(\xi,\eta)\in TCP(\Omega,U,\mathcal{U},f)$ such that $\xi(b)\in S$,
where
$$
J(\xi,\eta)\equaldef\int_0^TL(\xi(t),\eta(t),t)\,dt\,\,
{\rm if}\,\,\eta\in C^{0,pc}([0,T],U)\,..
$$
\end{itemize}

\medbreak
\noindent{\bf THEOREM OC-VT:}  {\bf\em (Optimal control theorem, variable
time.)} {\em Assume that (\tA 1,2,3,4,5,6,7,8,9) hold. 
Define the {\bf\em Hamiltonian} 
$\Omega\times\rr_n\times U\times\rr\ni(x,p,u,p_0,t)\mapsto 
H(x,p,u,p_0)\in\rr$ by letting 
$$
H(x,p,u,p_0)\equaldef p\cdot f(x,u)-p_0(x,u)\,.
$$
Then there exists  a pair $(\pi,\pi_0)$ 
such that 
\begin{itemize}
\item[(\#)] $\pi:[a,b]\mapsto\rr_n$ is a continuous map 
and $\pi_0\in\rr$;
\item[(AE)] (the "adjoint equation") 
\begin{equation}\label{adjeq-ocvt}
\dot\pi(t)=-
\pi(t)\cdot\frac{\partial f}{\partial x}(\xi_*(t),\eta_*(t))
+\pi_0\frac{\partial L}{\partial x}(\xi_*(t),\eta_*(t))
\end{equation}
at all points of continuity $t$ of $\eta_*$;
\item[(HM)] (the "Hamiltonian maximization condition")
$$
H(\xi_*(t),\pi(t),\eta_*(t),\pi_0)=
\max\Big\{H(\xi_*(t),\pi(t),u,\pi_0):u\in U\}
$$ 
at all points of continuity $t$ of $\eta_*$; 
\item[(ZH)] (the "zero Hamiltonian condition")
$$
H(\xi_*(t),\pi(t),\eta_*(t),\pi_0)=0
$$ 
at all points of continuity $t$ of $\eta_*$; 
\item[(TC)] (the "transversality condition") 
$$
\pi(b)\in -C^\dagger\,;
$$ 
\item[(NT)] (the "nontriviality condition") 
$(\pi(t),\pi_0)\neq(0,0)$ for some $t$ (which implies
that $(\pi(t),\pi_0)\neq(0,0)$ for all $t$);
\item[(NN)] (the "nonnegativtiy condition") $\pi_0\ge 0$.
\end{itemize}}

\vspace{1cm}
\noindent{\bf ASSUMPTIONS FOR THEOREM SEP-FT:} 
\begin{itemize}
\item[(S1)] $\Omega$ is an open subset of $\rr^n$.
\item[(S2)] $U$ is a metric space.
\item[(S3)] $a,b\in\rr$ and $a<b$.
\item[(S4)] $\Omega\times U\times[a,b]\ni(x,u,t)\mapsto f(x,u,t)\in\rr^n$ 
is a continuous map such that 
\begin{itemize}
\item[(S4.a)] the map $\Omega\ni x\mapsto f(x,u,t)\in\rr^n$ 
is of class $C^1$ for each fixed $(u,t)$,
\item[(S4.b)] the map $\Omega\times U\times[a,b]\ni(x,u,t)\mapsto 
\frac{\partial f}{\partial x}(x,u,t)\in\rr^{n\times n}$ is continuous.
\end{itemize}
\item[(S5)] $\mathcal{U}=C^{0,pc}([a,b],U)$.
\item[(S6)] $S$ is a subset of $\Omega$.	
\item[(S7)] $(\xi_*,\eta_*)\in TCP(\Omega,U,\mathcal{U},f)$
and $\xi_*(b)\in S$. 
\item[(S8)] $C$ is a tangent cone of $S$ at $\xi_*(b)$. 
\item[(S9)] $\mathcal{R}_{\Omega,U,\mathcal{U},f}(\xi_*(a))
\cap S=\{\xi_*(b)$, where
$$
\mathcal{R}_{\Omega,U,\mathcal{U},f}(\xi_*(a))\equaldef
\Big\{\xi(b):(\xi,\eta)\in TCP(\Omega,U,\mathcal{U},f)\Big\}\,.
$$
\end{itemize}

\medbreak
\noindent{\bf THEOREM SEP-FT:}  {\bf\em (Separation theorem, fixed time.)}
{\em Assume that conditions (S1,2,3,4,5,6,7,8,9) hold. 
Define the {\bf\em Hamiltonian} 
$$
\Omega\times\rr_n\times U\times\rr\times[a,b]\ni(x,p,u,p_0,t)\mapsto 
H(x,p,u,t)\in\rr
$$ by letting 
$$
H(x,p,u,t)\equaldef p\cdot f(x,u,t)\,.
$$
Then there exists a $\pi$ such that 
\begin{itemize}
\item[(\#)] $\pi:[a,b]\mapsto\rr_n$ is a continuous map;
\item[(AE)] (the "adjoint equation") 
\begin{equation}\label{adjeq-sepft}
\dot\pi(t)=-
\pi(t)\cdot\frac{\partial f}{\partial x}(\xi_*(t),\eta_*(t),t)
\end{equation}
at all points of continuity $t$ of $\eta_*$;
\item[(HM)] (the "Hamiltonian maximization condition")
$$
H(\xi_*(t),\pi(t),\eta_*(t),t)=
\max\Big\{H(\xi_*(t),\pi(t),u,t):u\in U\}
$$ 
at all points of continuity $t$ of $\eta_*$; 
\item[(TC)] (the "transversality condition") 
$$
\pi(b)\in -C^\dagger\,;
$$ 
\item[(NT)] (the "nontriviality condition") 
$\pi(t)\neq 0$ for some $t$ (which implies
that $\pi(t)\neq 0$ for all $t$);
\end{itemize}}

\medbreak
\noindent{\bf REMARK:}  The adjoint equation (\ref{adjeq-sepft})
can also be written as
\begin{equation}\label{ae-ham-noab}
\dot\pi(t)=-\frac{\partial H}{\partial x}(\xi_*(t),\pi(t),\eta_*(t),t)\,.
\end{equation}
The dynamical equation (\ref{dyneq}) is equivalent to
\begin{equation}\label{dyneq-ham-noab}
\dot\xi_*(t)=\frac{\partial H}{\partial p}(\xi_*(t),\pi(t),\eta_*(t),
t)\,.
\end{equation}
The pair of equations (\ref{dyneq-ham-noab}), (\ref{ae-ham-noab}) is equivalent
to the single equation
\begin{equation}\label{full-ham-noab}
\dot\Theta(t)=\vec{H}_{\eta_*(t),t}(\Theta(t))\,,
\end{equation}
evolving in $\Omega\times\rr_n$, i.e., $(x,p)$ space
(the cotangent bundle of $\Omega$, where $\Theta(t)=(\xi_*(t),\pi(t))$,
$H_{u,t}(x,p)\equaldef H(x,p,u,t)$, 

\bigbreak

\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
$\qquad$
\begin{center}
{\bf THE INTRINSIC FORM OF THE ADJOINT EQUATION:}
\end{center}
The adjoint equation (\ref{adjeq-sepft}) 
is equivalent to the following statement:
\begin{center}\begin{minipage}{10.5cm}
{\large\em For every vector field $X$ of class $C^1$ on 
$\Omega$, the equality
$$
\frac{d\,\,}{dt}\Big(\pi(t)\cdot X(\xi_*(t))\Big)
=\pi(t)\cdot[f_{\eta_*(t),t},X](\xi_*(t))
$$
holds, where
$$
f_{u,t}(x)\equaldef f(x,u,t)\,.
$$}
\end{minipage}\end{center}
\end{minipage}}}\end{center}

\vspace{1cm}
\noindent{\bf ASSUMPTIONS FOR THEOREM SEP-VT:} 
\begin{itemize}
\item[(\tS 1)] $\Omega$ is an open subset of $\rr^n$.
\item[(\tS 2)] $U$ is a metric space.
\item[(\tS 3)] $T_*\in\rr$ and $T_*>0$.
\item[(\tS 4)] $\Omega\times U\ni(x,u)\mapsto f(x,u)\in\rr^n$ 
is a continuous map such that 
\begin{itemize}
\item[(\tS 4.a)] the map $\Omega\ni x\mapsto f(x,u)\in\rr^n$ 
is of class $C^1$ for each fixed $u$,
\item[(\tS 4.b)] the map $\Omega\times U\ni(x,u)\mapsto 
\frac{\partial f}{\partial x}(x,u)\in\rr^{n\times n}$ is continuous.
\end{itemize}
\item[(\tS 5)] $\alpha>0$ and
$\mathcal{U}=\bigcup_{T_*-\alpha\le T\le T_*+\alpha}
C^{0,pc}([0,T],U)$.
\item[(\tS 6)] $S$ is a subset of $\Omega$.	
\item[(\tS 7)] $\eta_*\in C^{0,pc}([0,T_*],U)$, 
$(\xi_*,\eta_*)\in TCP(\Omega,U,\mathcal{U},f)$,
and $\xi_*(T_*)\in S$. 
\item[(S8)] $C$ is a tangent cone of $S$ at $\xi_*(b)$. 
\item[(S9)] $\mathcal{R}_{\Omega,U,\mathcal{U},f}(\xi_*(0))
\cap S=\{\xi_*(T)$, where
$$
\mathcal{R}_{\Omega,U,\mathcal{U},f}(\xi_*(0))\equaldef
\Big\{\xi(T):\eta\in C^{0,pc}([0,T],U), 
(\xi,\eta)\in TCP(\Omega,U,\mathcal{U},f)\Big\}\,.
$$
\end{itemize}

\medbreak
\noindent{\bf THEOREM SEP-VT:}  {\bf\em (Separation theorem, variable time.)}
{\em Assume that (\tS 1,2,3,4,5,6,7,8,9) hold. 
Define the {\bf\em Hamiltonian} 
$$
\Omega\times\rr_n\times U\times\rr\times[a,b]\ni(x,p,u,p_0,t)\mapsto 
H(x,p,u)\in\rr
$$ 
by letting 
$$
H(x,p,u)\equaldef p\cdot f(x,u)\,.
$$
Then there exists a $\pi$ such that 
\begin{itemize}
\item[(\#)] $\pi:[a,b]\mapsto\rr_n$ is a continuous map;
\item[(AE)] (the "adjoint equation") 
\begin{equation}\label{adjeq-sepvt}
\dot\pi(t)=-
\pi(t)\cdot\frac{\partial f}{\partial x}(\xi_*(t),\eta_*(t))
\end{equation}
at all points of continuity $t$ of $\eta_*$;
\item[(HM)] (the "Hamiltonian maximization condition")
$$
H(\xi_*(t),\pi(t),\eta_*(t))=
\max\Big\{H(\xi_*(t),\pi(t),u):u\in U\}
$$ 
at all points of continuity $t$ of $\eta_*$; 
\item[(ZH)] (the "zero Hamiltonian condition")
$$
H(\xi_*(t),\pi(t),\eta_*(t))=0
$$ 
at all points of continuity $t$ of $\eta_*$; 
\item[(TC)] (the "transversality condition") 
$$
\pi(b)\in -C^\dagger\,;
$$ 
\item[(NT)] (the "nontriviality condition") 
$\pi(t)\neq 0$ for some $t$ (which implies
that $\pi(t)\neq 0$ for all $t$)\,.
\end{itemize}}

\vspace{2cm}
\begin{center}
\fbox{\fbox{
\begin{minipage}{10cm}
$$
\left.\begin{array}{ccc}
{\rm Theorem\,SEP-FT}&\Longrightarrow&{\rm Theorem\,SEP-VT}\\
\Downarrow&\,\,&\Downarrow\\
{\rm Theorem\,OC-FT}&\Longrightarrow&{\rm Theorem\,OC-VT}\end{array}\right.
$$
\end{minipage}}}
\end{center}

\end{document}

