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

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\begin{center}
{\Huge{\bf THE PONTRYAGIN\\
$\,$\\
\vspace{-5mm}
MAXIMUM PRINCIPLE}}\\
$\,\,$\\
{\Large{\it H\'ector J. Sussmann} --- November 29, 2000}
\end{center}

\medbreak
\begin{center}
{\Large\bf CONTENTS}
\end{center}

\medbreak

\thispagestyle{empty}
\noindent 1.\ CARATH\'EODORY'S MISSED OPPORTUNITY\hfill 1 

\medbreak
\noindent 2.\ THE MAXIMUM PRINCIPLE\hfill 2

\noindent \ 2.1\ A classical version for fixed time 
interval problems \hfill 3

\noindent \ 2.2\ A classical version for variable 
time interval problems \hfill 5

\noindent \ 2.3\ An example \hfill 7

\medbreak


{\large 

\noindent The purpose of this note is to present a precise statement of a
technically simple version of the Pontryagin maximum principle. But
before we get there we will add one more example to our list of ``missed
opportunities,'' by showing how Carath\'eodory failed to discover the
Hamiltonian maximization condition because of his insistence on using the
``wrong'' Hamiltonian.


\ssection{Carath\'eodory's missed opportunity}\label{subsec-carat} 

\noindent {\bem Constantin Carath\'eodory} (1873-1950) published his
work on the calculus of variations in a 1935 book,
cf. \cite{carath}. His discussion of the excess function, and of how
Weierstrass' condition can be expressed in Hamiltonian form, is worth
quoting in detail\Ft{\quocar}, because it shows that the author
understood the importance of expressing the Weierstrass condition in
Hamiltonian form, but was unable to find its simple Hamiltonian
meaning because he was using an inappropriate Hamiltonian formalism.

Carath\'eodory uses $x_i$ for the position variables (that is, our $q^i$) 
and $y_i$ for the momenta (that is, our $p_i$). He works in the neighborhood
of a ``regular line \mbox{element}'' $e=(t,x_i,\dot x_i)$,   where the
``regularity'' assumption is the requirement that the Hessian matrix 
$\{L_{\dot x_i,\dot x_j}\}_{1\le i,j\le n}$ be positive definite
at $e$.   This condition guarantees that, near $e$, one can express
everything in ``canonical coordinates'' $(t,x_i,y_i)$, i.e., that
the map $(t,x,\dot x)\to (t,x,y)$ is invertible, if  we let
$y_i=\chi_i(t,x,\dot x)$, where 
$\chi_i(t,x,\dot x)=\frac{\partial L}{\partial \dot x_i}(t,x,\dot x)$.

He then expresses the Weierstrass condition, which involves 
the ``line element $(t,x_i,\dot x_i)$'' together with 
a nearby ``line element'' $(t,x_i,x'_i)$,  in terms of the corresponding 
canonical coordinates $(t,x_i,y_i)$ and 
$(t,x_i,y'_i)$.  He uses $H$ to represent what he calls
``the Hamiltonian'' (that is, our classical Hamiltonian), given by 
the formula $H=\sum_{i=1}^ny_i\dot x_i-L(t,x,\dot x)$, where 
$\dot x_i=\varphi_i(t,x,y)$, and $\varphi$ is the inverse 
function of $\chi$ (that is, $\varphi(t,x,\chi(t,x,u))=u$ 
and $\chi(t,x,\varphi(t,x,y))=y$). In addition, he uses 
$H'$ to represent the Hamiltonian evaluated at the other 
point $(t,x,y')$  in phase space, that is,
$H'=\sum_{i=1}^ny'_ix'_i-L(t,x,x')$, where 
$x'_i=\varphi_i(t,x,y')$, $y'_i=\chi_i(t,x,x')$.

With these notations, the Weierstrass excess function becomes 
\begin{eqnarray*}
\mathcal{E}(t,x,\dot x,x')&=&
L(t,x,x')-L(t,x,\dot x)-\frac{\partial L}{\partial\dot x}
(t,x,\dot x)\cdot(x'-\dot x)\\
&=&L(t,x,x')-L(t,x,\dot x)-y\cdot(x'-\dot x)\\
&=&y\cdot\dot x-L(t,x,\dot x)-\Big(y\cdot x'-L(t,x,x')\Big)\\
&=&y\cdot\dot x-L(t,x,\dot x)-\Big(y'\cdot x'-L(t,x,x')\Big)-(y-y')\cdot x'\\
&=&H-H'-(y-y')\cdot x'\,.
\end{eqnarray*} 
Since
\begin{eqnarray*}
\frac{\partial H'}{\partial y'_j}&=&
\frac{\partial\quad}{\partial y'_j}\Big(\sum_{i=1}^ny'_ix'_i-L(t,x,x')\Big)\\
&=&x'_j+\sum_{i=1}^ny'_i\frac{\partial\varphi_i}{\partial y'_j}(t,x,y')
-\frac{\partial\quad}{\partial y'_j}\ Big(L(t,x,\varphi(t,x,y'))\Big)\\
&=&x'_j+\sum_{i=1}^ny'_i\frac{\partial\varphi_i}{\partial y'_j}(t,x,y')
-\sum_{i=1}^n\frac{\partial L}{\partial x'_i}(t,x,x')
\frac{\partial\varphi_i}{\partial y'_j}(t,x,x')\\
&=&x'_j+\sum_{i=1}^ny'_i\frac{\partial\varphi_i}{\partial y'_j}(t,x,y')
-\sum_{i=1}^ny'_i\frac{\varphi_i}{\partial y'_j}(t,x,x')\\
&=&x'_j\,,
\end{eqnarray*} 
Carath\'eodory concluded that 
``the equation
\begin{eqnarray*}
\mathcal{E}(t,x,\dot x,x')&=&L'-L-L_{\dot x_j}(x'_j-\dot x_j)\\
&=&H-H'-H'_{y'_j}(y_j-y'_j)
\end{eqnarray*} 
therefore holds, by which the $\mathcal{E}$ function is represented
in the desired form.''


\ssection{The maximum principle}

\noindent  The first rigorous statement and proof of the maximum principle
appears in the book \cite{pontrybook}. This ``classical'' version was
then improved by other authors, e.g. L. Cesari \cite{cesari},
M. Hestenes \cite{hestenesbook}, E. B. Lee and L. Markus
\cite{leemarkus}, and L. D. Berkovitz \cite{berkovitz}.  

Here we shall present a couple of simple versions of the result, 
under technical assumptions that will be considerably weakened later.

\ssubsection{A classical version for fixed time 
interval problems}

\noindent We begin by quoting a relatively simple version.

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.] $\bar x$, $\hat x$ are given points 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 $\xi_*(b)=\hat x$,

\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 $\xi(b)=\hat x$.
\end{itemize}
\end{itemize}

\begin{theorem}\label{theo-timenovar}
 Assume that the data $n$, $m$, $Q$, $U$, $a$, $b$, $f$, 
$L$, $\bar x$, $\hat x$ satisfy conditions C1-C7, $TCP_{[a,b]}(Q,U,f)$ 
and $J$ are defined by C8-C9, and $\gamma_*=(\xi_*,\eta_*)$ satisfies C10.
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 $t\in[a,b]$.
\end{itemize} 
\end{theorem}

\begin{remark} {\rm
It is clear that
$$
\frac{\partial H}{\partial p}(x,u,p,p_0,t)=f(x,u,t)\,.
$$
Therefore the adjoint equation, together with the equation of C8-c, say that
the pair $(\xi_*,\pi)$ is a solution of  {\bem Hamilton's equations}}

\noindent\begin{minipage}{9.5cm}
\begin{eqnarray}
\dot\xi_*(t)&=&\phantom{-}\frac{\partial H}{\partial p}
(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)\,,\\
\dot\pi(t)&=&-\frac{\partial H}{\partial x}(\xi_*(t),\eta_*(t),\pi(t),
\pi_0,t)\,.
\end{eqnarray}\end{minipage}\begin{minipage}{2.8cm}
\null$\quad$\\
\null$\quad$\\
\null$\qquad\quad\,\,\qquad\quad\diamondsuit$\end{minipage}
\end{remark}
\begin{remark}{\rm 
The map $\pi$ is often referred to in the control theory literature as 
the ``adjoint vector.'' Its proper status from the differential-geometric 
point of view is that of a {\bem field of covectors} along $\xi_*$.} \proofend
\end{remark}
\begin{remark}{\rm 
The number $\pi_0$ is the ``abnormal multiplier.''}\proofend
\end{remark}
\begin{remark}{\rm 
A trajectory-control pair for which there exists a pair $(\pi,\pi_0)$ 
that satisfies E1-E5 is 
called an {\bem extremal}. Therefore the Maximum Principle 
says that a necessary condition for a TCP to be a solution of the problem of 
minimizing the functional $J(\xi,\eta)$ subject to the conditions 
$(\xi,\eta)\in TCP_{[a,b]}(Q,U,f)$ and the endpoint conditions
C10-b is that $(\xi,\eta)$ be an extremal.}\proofend
\end{remark}
\begin{remark}{\rm 
An extremal is {\bem normal} if the pair $(\pi,\pi_0)$ can be chosen so that
$\pi_0\neq 0$, and {\bem abnormal} if the pair $(\pi,\pi_0)$ can be chosen 
so that $\pi_0=0$, The pair $(\pi,\pi_0)$ need not be unique, so in particular
it can happen that an extremal is both normal and abnormal.}\proofend
\end{remark}
\begin{remark}{\rm 
Property E3 is referred to as the ``nontriviality condition.''}\proofend
\end{remark}
\begin{remark}{\rm 
The adjoint equation says that
$$
\dot\pi(t)=-\pi(t)\cdot
\frac{\partial f}{\partial x}(\xi_*(t),\eta_*(t),t)+\pi_0
\frac{\partial L}{\partial x}(\xi_*(t),\eta_*(t),t)\,.
$$
In particular, if $(\pi(t),\pi_0)=(0,0)$ for one value $\tau$ of $t$, 
then $\pi_0=0$, so the equation reduces to 
$$
\dot\pi(t)=-\pi(t)\cdot
\frac{\partial f}{\partial x}(\xi_*(t),\eta_*(t),t)\,,
$$
which is a linear homogeneous time-varying ordinary differential equation for 
$\pi$. Therefore the fact that $\pi(\tau) =0$ implies that $\pi(t)=0$ for 
all $t$, so $(\pi(t),\pi_0)=(0,0)$ for all $t$. Hence the nontriviality 
condition could equally well have been stated by saying ``for some $t$'' 
rather than ``for all $t$.''}\proofend
\end{remark}
\begin{remark}\label{pointed}{\rm 
If we let 
$$
h(t)=H(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)\,,
$$
then, formally, the derivative of $h$ is given by
\begin{eqnarray*}
\dot h(t)&=&\frac{\partial H}{\partial x}(\Gamma(t))\cdot\dot\xi_*(t)\!+\!
\frac{\partial H}{\partial u}(\Gamma(t))\cdot
\dot\eta_*(t)\!+\!\frac{\partial H}{\partial p}(\Gamma(t))\cdot\dot\pi(t)
\!+\!\frac{\partial H}{\partial t}(\Gamma(t))\\
&=&\frac{\partial H}{\partial u}(\Gamma(t))\cdot\dot\eta_*(t)
+\frac{\partial H}{\partial t}(\Gamma(t))\,,
\end{eqnarray*}
where we have written $\Gamma(t)=(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)$
and used the fact that $\dot\xi_*=\frac{\partial H}{\partial p}$ 
and $\dot\pi=-\frac{\partial H}{\partial x}$ by Hamilton's equations. 
If we disregard the fact that $\dot\eta_*(t)$ need not exist
(because $\eta_*$ is not assumed to be diferentiable or even continuous),
and in addition we pretend that 
``$\frac{\partial H}{\partial u}(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)$
must vanish because the function $u\mapsto H(\xi_*(t),u,\pi(t),\pi_0,t)$
has a maximum at $u=\eta_*(t)$,'' then we end up with 
$$
\dot h(t)=\frac{\partial H}{\partial t}(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)\,.
$$
This leads to the conjecture---that we are very far from having proved---that
{\bem if the dynamical law $f$ and the Lagrangian $L$ do not depend on 
$t$ then the function
$[a,b]\ni t\mapsto H(\xi_*(t),\eta_*(t),\pi(t),\pi_0,t)$ is constant.}
We will see later that this can be made precise and proved rigorously.}
\end{remark}

\ssubsection{A classical version 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.] $\bar x$, $\hat x$ are given points 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'') are
$a_*$, $b_*$ 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 $\xi_*(b_*)=\hat x$;

\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 $\xi(b)=\hat x$.
\end{itemize}
\end{itemize}

\begin{theorem}\label{theo-timevar}
Assume that the data $n$, $m$, $Q$, $U$, $f$, 
$L$, $\bar x$, $\hat x$ satisfy conditions C1-C6, $TCP(Q,U,f)$ 
and $J$ are defined by C7-C8, and $a_*$, $b_*$,  
$\gamma_*=(\xi_*,\eta_*)$ 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 $t\in[a,b]$.
\end{itemize} 
\end{theorem}

\begin{remark}{\rm
If we fix $a_*$ and $b_*$, and consider only TCPs $(\xi,\eta)$ belonging
to $TCP_{[a_*,b_*]}(Q,U,f)$, then $\gamma_*$ is obviously a solution
of the minimization problem considered in Theorem 
\ref{theo-timenovar}. So all the conditions given by that theorem
should be satisfied here. And, indeed, this is what happens, because
{\bem the only difference between the conclusion 
of Theorem \ref{theo-timenovar} and that 
of Theorem \ref{theo-timevar} is that in 
Theorem \ref{theo-timevar} we obtain the additional condition that the
value of the Hamiltonian must vanish.}}\proofend
\end{remark}
\begin{remark}{\rm
We already pointed out in Remark \ref{pointed} that, at least formally, 
the conclusions of Theorem \ref{theo-timenovar} already imply 
that the Hamiltonian function
$t\mapsto H(\xi_*(t),\eta_*(t),\pi(t),\pi_0)$ is constant, since in
our problem $f$ and $L$ do not depend on $t$. So the only novelty 
of Theorem \ref{theo-timevar} is that the constant now turns out to be equal 
to zero.}\proofend
\end{remark}

\ssubsection{An example}

We consider the ``one-dimensional soft 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 $(0,0)$ in minimum time.

This is, of course, a problem of the kind discussed in Theorem 
\ref{theo-timevar}, so we shall solve it by applying the theorem.

In this case, the configuration space $Q$ is $\rr^2$, the control set
$U$ is the compact interval $[-1,1]$, the dynamical law $f$ is given
by 
$$
f(x,y,u)=(y,u)\,,
$$
and the Lagrangian is identically equal to $1$. 

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. (Notice that we are using the fact 
that $\pi_x$ is absolutely continuous!) 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.

Now that we know that $A$ and $B$ cannot both vanish, there are two
possibilities. Either $A=0$ or $A\neq 0$. 

Suppose first that $A=0$. Then $B\neq 0$, and
Hamiltonian maximization tells us that $[-1,1]\ni u\mapsto Bu$ is
maximized by $u=\eta_*(t)$. If $B>0$, this implies that $\eta_*(t)=1$
for all $t$. If $B<0$ then it follows that $\eta_*(t)=-1$ for all $t$.

Now assume that $A\neq 0$. Then $t\mapsto B-At\equaldef\varphi(t)$ is
a nonconstant linear function of $t$. Therefore $\varphi$ vanishes at
most once on the interval $[a_*,b_*]$. If $\varphi$ never vanishes on
$[a_*,b_*]$, or vanishes at one of the endpoints, then $\eta_*(t)$ is
either always equal to $1$ (if $\varphi(t)>0$) or always equal to $-1$
(if $\varphi(t)<0$).  If $\varphi(\tau)=0$ for some
$\tau\in\,]a_*,b_*[$, then $\eta_*(t)$ will be equal to $1$ for
$t<\tau$ and to $-1$ for $t>\tau$ (if $\varphi$ changes sign at $\tau$
from positive to negative) or 
$\eta_*(t)$ will be equal to $-1$ for
$t<\tau$ and to $1$ for $t>\tau$ (if $\varphi$ changes sign at $\tau$
from negative to positive).  

So we have proved that $\eta_*$ is of one of the following 
four types:

\medbreak
\begin{itemize}
\item[a.] constantly equal to $1$,

\item[b.] constantly equal to $-1$,

\item[c.] constantly equal to $-1$ for $t<\tau$ and to $1$ for 
$t>\tau$ for some $\tau\in\,]a_*,b_*[$, 

\item[d.] constantly equal to $1$ for $t<\tau$ and to $-1$ for 
$t>\tau$ for some $\tau\in\,]a_*,b_*[$. 
\end{itemize}

\medbreak\noindent For a problem where the control set $U$ is a
compact convex subset of $\rr^m$, a control $\eta_*$ that takes values
in the set of extreme points of $U$ is said to be a {\bem bang-bang
control}.  In our case, we have proved that {\bem all optimal
controls are bang-bang and either constant of piecewise constant 
with one switching}. 

Notice that the switchings of the optimal control $\eta_*$ are determined
by the function $\varphi$, which in this example happens to be 
$\pi_y$. That is why this function is called the {\bem switching function}
for this problem.

Notice also that {\bem we have made essential use of all the conditions
given by Theorem \ref{theo-timevar}}. In particular, 
{\bem the nontriviality condition was crucial, and the fact 
that the value of the Hamiltonian is equal to zero was also decisive.}


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\end{document}

