Q$2.$ Determine the closed$-$loop optimal control using HJB equation when the performance index is$,?$

$J={\displaystyle \underset{0}{\overset{}{}}}(x^T\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]x+M^i\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]M)dt$

where, $M$ is the input and the system equation is$,$

$x^{}=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]x+\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]M$

Also explain the advantage of the matrix$-$Riccati equation developed from the Hamilton$-$Jacobi$-$Bellman approach.