$\dot{x_{1}}=x_{2}^2-3\sin(x_{1})x_{2}$
$\dot{x_{2}}=x_{1}^3-3x_{2}\cos(x_{1})+u^{1/2}$
Question: Using the backstepping method and Lyapunov function, design the controller $u$ that will make the following system asymptotically stable.
For backstepping control, $x_{1}$ must be in this form;
$\dot{x_{1}}=f(x_{1})+g(x_{1})x_{2}$
then,
$e_{1}=x_{1}-x_{1}^d,\space \dot{e_{1}}=\dot{x_{1}}-\dot{x_{1}}^d$, where $e_{1}: error, \space x_{1}^d: desired\space x_{1}$
and Lyapunov function is $L_{1}=\frac{1}{2}e_{1}^2, \space \dot{L_{1}}=e_{1}\dot{e_{1}}$. For asymptotically stability, $\dot{L_{1}}=-k_{1}e_{1}^2, k_{1}>0$. Now we have a $x_{2}$ that extracted from $\dot{L_{1}}$ and this $x_{2}=x_{2}^d$ for $e_{2}=x_{2}-x_{2}^d$. Where, $\dot{L_{2}}=e_{2}\dot{e_{2}}^2$ and $\dot{L_{2}}=-k_{2}e_{2}^2, k_{2}>0$. Finally we can extract $u$ from $\dot{L_{2}}$.
But in this question $\dot{x_{1}}$ is not in form $\dot{x_{1}}=f(x_{1})+g(x_{1})x_{2}$. So how can I solve this question with backstepping control?