$($e$)$ Using state$-$feedback, we want to stabilise the system described by

the matrices $(3),$ while minimising the cost functional

${\displaystyle \underset{0}{\overset{+}{}}}x(t{)}^{TT}Qx(t)+u(t{)}^{TT}Ru(t)dt,$

where $Q=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],R=1.$

Write down the Algebraic Riccati Equation $($ARE$)$ for the optimal con$-$

trol problem associated with $(3)$ and $(4).$

You do not need to solve the ARE; just state it$.$

$[5$ marks$]$

$($f$)$ Denote by $P$ the solution of the ARE of Question $2.($e$).$ Give an ex$-$

pression for the feedback vector $K$ corresponding to the optimal in$-$

put.

$[4$ marks$]$

$($g$)$ Write a Matlab script that:

Defines the system associated with the matrices $(3)$;

Solves the ARE for the cost functional $(4)$;

Computes the optimal gain;

Plots the impulse$-$response of the closed$-$loop system.

Your script must, as much as possible, use the functions available in

the Matlab Control Toolbox. Using the correct Matlab syntax is

not essential; however, it is important that the program shows clearly

that you know which functions to use, and how to use them.