Consider the second order affine control system

${\displaystyle \underset{a}{\overset{?}{}}}$:$\{\begin{array}{cc}{x}_1^{},& =x_2^3\\ x_2^{},& =-x_1^3+x_{2}u.\end{array}$

$($a$)$ Use the KYP Lemma to find a virtual output $y=h(x)$ that renders the input$-$output system

passive.

$($b$)$ Use a passivity$-$based method to design a globally stabilizing state feedback control law $u=$

$u(x_1,x_2),$ with $u(0,0)=0,$ for ${\displaystyle \underset{a}{\overset{?}{}}}.$

$($c$)$ Use the result obtained in $($b$)$ to find a smooth controller $v(x_1,x_2,x_3),$ with $v(0,0,0)=0,$ which

globally stabilizes the nonlinear system

${\displaystyle \underset{?}{\overset{?}{}}}$:$\{\begin{array}{c}{x}_1^{}=x_2^3\\ x_2^{}=-x_1^3+x_2{x}_3\\ x_3^{}=v+x_{2}sin{x}_3\end{array}$