Consider the linear system

$x^{}(t)=Ax(t)+Bu(t),$

with $x(t)in{R}^5,u(t)inR,$ and where

$A=\left[\begin{array}{ccccc}-7& 3& 3& 1& 4\\ 2& 2& 1& 0& -2\\ -9& 1& 1& 5& 4\\ -8& 2& 3& 1& 5\\ -6& 3& 4& -1& 4\end{array}\right],B=\left[\begin{array}{c}1\\ 0\\ 1\\ 1\\ 1\end{array}\right].$

Note. 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 $u(t)=Fx(t)$ such that the closed$-$loop system

$x^{}(t)=(A+BF)x(t)$

is asymptotically stable.