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\parindent=0pt

\centerline{{\mymagtwobf FINAL EXAM}}

\bigbreak
\centerline{{\mymagtwoit H\'ector J. Sussmann---The Weizmann Institute---January 2001}}

\bigbreak
This is a take-home exam, consisting of FOUR problems. You can send
your solutions to me by e-mail, to {\it sussmann@math.rutgers.edu}$\,$.
Send either a Plain TeX or LateX file {\bf without using any special
Weizmann-Institute macros}, or a Postscript file. When you e-mail your file,
{\bf do not use any ``MIME-encoding'' or Microsoft-dependent stuff.}

\bigbreak

PROBLEM 1: Do Problem 3 of the list in the first handout, dated 
November 1, 2000.  (Recall: a {\it norm} on $\rr^n$ is a function
$$
\rr^n\ni x\mapsto \|x\|\in\rr
$$
such that (i)~$\|x\|>0$ for all  $x\in\rr^n$ such that $x\neq 0$,
(ii)~$\|x+y\|\le\|x\|+\|y\|$ for all $x$, $y$, (iii)~$\|r\cdot x\|=
|r|\cdot\|x\|$ whenever $r\in\rr$ and $x\in\rr^n$.)

\medbreak
PROBLEM 2: Do Problem 11 of the list in the first handout, dated 
November 1, 2000.

\medbreak
PROBLEM 3: For Problem 16 of the list in the first handout, dated 
November 1, 2000, prove the following:


{\parindent=18pt

\medbreak
\item{(a)} For every optimal trajectory-control pair $(\xi,\eta)$ with a 
continuous control, the control $\eta$ is is fact constant and equal 
to $+1$, $-1$ or $0$; moreover, the zero control case can only occur 
where $\omega=0$. 

\item{(b)} For an optimal trajectory-control pair $(\xi,\eta)$, it is 
{\bf not} possible for the control $\eta$ to have a jump discontinuity from 
$0$ to $1$ or $-1$ or from $1$ or $-1$ to zero. (In other words, there are 
no optimal
``jump junctions'' between bang and singular arcs.)

}

\medbreak{\it NOTE: Result (b) is fairly counterintuitive. One would 
imagine in view of (a) that, if you want to control the vehicle from a
certain initial position $(x_-,y_-,\theta_-,\omega_-)$ to a terminal
position $(x_+,y_+,\theta_+,\omega_+)$ which is very far away, then
the right thing to do is to follow a straight line,
if possible, and if this cannot be done then you follow a straight line
most of the time, except that at the beginning and at the end you use some 
bang-bang manoeuvers to get into the line. Now, motion along a 
straight line corresponds  to $\theta={\rm constant}$, so 
$\omega=0$ and $u=0$. Result (b) says that the ``bang-bang manoeuvers''
cannot be finite concatenations of $+1$ and $-1$ controls.}

\medbreak OPTIONAL PROBLEM: Give a complete description of the optimal 
trajectories for Problem 3. NOTE: this is a {\bf hard open problem}. If
you solve it correctly then this will be a major mathematical achievement 
that will make you famous.

\medbreak
PROBLEM 4: Fill in the details (for the fixed time interval case) 
for the result on ``more general
endpoint conditions'' outlined in the remark in Page 7 of the January 3
handout. In particular: (a)~give a complete statement of the theorem, 
(b)~give a complete proof that the statement follows from Theorem 1.3.1
of the handout.

\end