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      \newtheorem{xca}{\protect\exercisename}
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\usepackage{babel}
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\begin{document}
\title{Exercise for Optimal control -- Week 2\date{}}
\author{Choose \textbf{one }problem to solve.}
\maketitle
\begin{xca}[Insect control]
Let $w(t)$ and $r(t)$ denote, respectively, the worker and reproductive
population levels in a colony of insects, e.g. wasps. At any time
$t$, $0\le t\le T$ in the season the colony can devote a fraction
$u(t)$ of its effort to enlarging the worker force and the remaining
fraction $u(t)$ to producing reproductives. The per capita mortality
rate of workers is $\mu$ and the per capita natality rate is $b$
when full effort is put on the worker population. Assume $\mu<b$.
The two populations are governed by the equations
\begin{align*}
\dot{w} & =(bu-\mu)w\\
\dot{r} & =c(1-u)w
\end{align*}
with $(w(0),r(0)=(1,0)$, where $u$ satisfies the constraint $0\le u(t)\le1$.
The objective is to maximize $r(T)$ or minimize
\[
J=-r(T).
\]
\end{xca}
%
\begin{xca}[Time optimal control of a lunar lander]
Study the time optimal control of the moon lander problem (see lecture
notes for the model). In addition, argue that there is an optimal
feedback controller.
\end{xca}
%
\begin{xca}[Minimum fuel and time control]
 Consider the planar system
\begin{align*}
\dot{x}_{1} & =x_{2}\\
\dot{x}_{2} & =u
\end{align*}
where $u$ satisfies the constraint $|u(t)|\le1$ for all $t\in[0,t_{f}]$.
For any given initial state $(\xi_{1},\xi_{2})$, find an optimal
control $u_{*}$ which drives the state to $(0,0)$ while minimizing
\[
J=t_{f}+\int_{0}^{t_{f}}|u(t)|{\rm d}t=\int_{0}^{t_{f}}(1+|u(t)|{\rm d}t.
\]

(Be careful, this exercise is difficult! In this example, we penalize
both the terminal time and fuel consumption.)
\end{xca}

\end{document}
