Consider the LTI system

x$=$

$01$

$01$

x $+$

$1$

$0$

u x$(0)=$ x$0$

and infinite horizon cost

R

$0$x$2+$ ru$2$dt$,$ where r $>0.$

$($a$)$ Determine what Theorem $10\mathrm{.}6$ implies about the LQR for this problem.

$($b$)$ Compute by hand the LQR matrix $$ P$,$ the optimal feedback control law, the closed loop

state matrix Acl, and the poles of the closed loop system.

$($c$)$ Comment on how the feedback law and poles vary as r gets very large $($expensive control$)$

or very small $($cheap control$).$

$($d$)$ Find the eigenvalues of the Hamiltonian matrix H by hand. Is your answer consistent

with part $($b$)?$