Exercise $2.$ Let $(x)in{R}^{+}$and consider the discrete time control system over infinite horizon.

$k(s+1)=Ak(s)-c(s),st,$

$k(t)=xin{R}_{+},$

under the control and state constraints $c(*)0$ and $k(*)0.$ The objective functional to

maximize is

${\displaystyle \underset{s=0}{\overset{+}{}}}{}^s\frac{c(s{)}^{1-}}{1-},$

for $,in(0,1).$

$($i$)$ Define $g(s)$:$=\frac{k(s+1)-k(s)}{k(s)}$ the discrete growth rate of $k$ at time s$.$ What can we say about

the maximal $($varying the control$)g$ at time $t?$

$($ii$)$ Write the Bellman Equation

$($iii$)$ Find a solution of the Bellman equation of the form $v(k)=a{k}^{1-}$ for some $a>0.$ What

is th value of $a?$

$($iv$)$ Express the optimal control in feedback form

$($v$)$ What is the growth rate of $k$ along the optimal trajectory