$9.$The first order discrete system

$x(k+1)=0\mathrm{.}5x(k)+u(k)$

is to be transferred from initial state $x(0)=-2$ to final state $x(2)=0$

in two states while the performance index

$J={\displaystyle \underset{k=0}{\overset{1}{}}}|x(k)|+5|u(k)|$

is minimized.

Assume that the admissible control values are only

$-1,0\mathrm{.}5,0,0\mathrm{.}5,1$

Find the optimal control sequence

$u^{**}(0),u^{**}(1)$