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\begin{document}
\title{Exercise for Optimal control -- Week 5}
\author{Choose \textbf{2 }problems to solve.}
\maketitle
\begin{xca}
A public company has in year $k$ profits amounting to $x_{k}$ SEK.
The management then distributes $u_{k}$ to the shareholders and invests
$x_{k}-u_{k}$ in the company itself. Each SEK invested in such way
will increase the company profit by $\theta>0$ the following year
so that
\[
x_{k+1}=x_{k}+\theta(x_{k}-u_{k}).
\]
Suppose $x_{0}\ge0$ and $0\le u_{k}\le x_{k}$ so that $x_{k}\ge0$
for each $k$. The objective of the management is to maximize the
total amount distributed to the shareholders over a period of $N$
years, i.e.,
\[
\max_{u_{k}}\sum_{k=0}^{N-1}u_{k}
\]
subject to $u_{k}\in[0,x_{k}]$. 
\end{xca}

\begin{xca}
Derive the complete value iteration procedure -- with $J_{0}=0$
-- for the optimal control problem
\[
\min\sum_{i=1}^{\infty}(x_{k}^{\top}Qx_{k}+u_{k}^{\top}Ru_{k})
\]
under the constraint:
\[
x_{k+1}=Ax_{k}+Bu_{k}.
\]
(there is no constraint on $u$).

\emph{Hint: 1) Write $J_{k}$, $k\ge1$ as $J_{k}(x)=x^{\top}P_{k}x$,
and $u_{k}=-K_{k}x$ where 
\[
u_{k}\in\arg\min_{u}\{x^{\top}Qx+u^{\top}Ru+J_{k}(Ax+Bu)\}.
\]
2) find the iteration formula for $K_{k}$ and $P_{k}$. Don't forget
the boundary conditions.}
\end{xca}

\begin{xca}
Show that free terminal time optimal control problem can be turned
into a fixed terminal time problem. Why is this useful in numerical
computation? \emph{Hint: consider a rescaling of time $\tau=\frac{t}{t_{f}}$. }
\end{xca}

\begin{xca}
Derive the maximum principle for the Bolza form cost by utilizing
the maximum principle for the Mayer form.
\end{xca}

\begin{xca}
Prove the maximum principle for the case that $t_{f}$ is free. You
may consider the Mayer type problem. \emph{Hint: all the necessary
conditions for $t_{f}$ fixed are still necessary. One only needs
to derive the additional condition that $H\equiv0$ along the optimal
solution. You can either use the trick in Ex2 or consider a new variation
in $t_{f}$: $x_{\epsilon}(t_{f}+\epsilon\mu)\in\Omega_{1}$ where
$x_{\epsilon}(\cdot)$ is some needle variation and $\mu$ some real
number. }
\end{xca}


\end{document}
