Consider the linear system

x$($t$)=$ Ax$($t$)+$ Bu$($t$),$

with x$($t$)$ belonging to R$^5,$ and u$($t$)$ belonging to R$,$ and where

A $=$

$[[73314]$

$[22102]$

$[91154]$

$[82315]$

$[63414]]$ so it $5*5$ matrix and B being $5*1$ matrix

$[1$

$0$

$1$

$1$

$1]$

In the questions below, you may use any software $($e$.$g$.,$ Python, Matlab$)$ to perform

standard matrix computations such as the computation of eigenvalues, matrix$-$vector multiplications, and the computation of matrix rank. However, clearly explain your reasoning. Also, do not

use any build$-$in functions for the automatic computation of controllers, if available.

$($a$)$ Is the system asymptotically stable?

$($b$)$ Is the system controllable?

$($c$)$ Is the system stabilizable?

$($d$)$ Following a systematic procedure that exploits the appropriate canonical forms, design a

static state feedback controller u$($t$)=$ F x$($t$)$ such that the closed$-$loop system

x$($t$)=($A $+$ BF$)$x$($t$)$

is asymptotically stable.

Hint. See the proof that is in the photo. $.$