Stabilize system about origin using Backstepping Design I would like to use backstepping design to formulate a control $u(x_1,x_2)$ to globally stabilize the following system about the origin $(0,0)$. $$ \dot x_1 = -x_1+\bigg(\dfrac{1}{12}x_1-1\bigg)x_2$$ $$ \dot x_2 = u(x_1,x_2) $$
I am familiar with the process of backstepping design, but for this specific problem I can't seem to come up with a desired value for the virtual control $x_2$. At first I tried using a desired value $x_{2 \rm d}\triangleq 0$, but after the usual process (define error, change coords, evaluate lyapunov function derivative) I get the control $ u = 1 - \dfrac{1}{12}x_1 -x_2 $ which does stabilize the system, but not about the origin, since the control does not reduce to zero at the origin.
 A: For $V_1=\frac{1}{2}x_1^2$ we have
$$\dot{V}_1=x_1\dot{x}_1=-x_1^2+\left(\frac{x_1^2}{12}-x_1\right)x_2$$
If we now define the new error variable $z_2=x_2-\alpha_1$ with virtual control $\alpha_1$ we can write equivalently
$$\dot{V}_1=-x_1^2+\left(\frac{x_1^2}{12}-x_1\right)\alpha_1+\left(\frac{x_1^2}{12}-x_1\right)z_2$$
The above equation guides us to select
$$\alpha_1(x_1)=-\left(\frac{x_1^2}{12}-x_1\right)$$
so that
$$\dot{V}_1=-x_1^2-\left(\frac{x_1^2}{12}-x_1\right)^2+\left(\frac{x_1^2}{12}-x_1\right)z_2$$
For the dynamics of $z_2$ we have
$$\dot{z}_2=\dot{x}_2+\left(\frac{x_1}{6}-1\right)\dot{x}_1=u+\left(\frac{x_1}{6}-1\right)\left(-x_1-x_2+\frac{x_1x_2}{12}\right)$$
For $V_2:=V_1+\frac{1}{2}z_2^2$ we then have
$$\dot{V}_2= -x_1^2-\left(\frac{x_1^2}{12}-x_1\right)^2+\left[\frac{x_1^2}{12}-x_1+u+\left(\frac{x_1}{6}-1\right)\left(-x_1-x_2+\frac{x_1x_2}{12}\right)\right]z_2$$
Choosing therefore
$$u=x_1-\frac{x_1^2}{12}+\left(\frac{x_1}{6}-1\right)\left(x_1+x_2-\frac{x_1x_2}{12}\right)-z_2\\
=2x_1-x_2-\frac{x_1^2}{6}+\left(\frac{x_1}{6}-1\right)\left(x_1+x_2-\frac{x_1x_2}{12}\right)$$
we have that
$$\dot{V}_2\leq -2V_2$$
and $V_2$ converges exponentially fast to zero. From this one easily deduces the exponential convergence of $x_1,x_2$ to zero.

Edit: As pointed by Thomas $\alpha_1=0$ can also be used as a virtual control law. Then, 
$$\dot{V}_1=-x_1^2+\left(\frac{x_1^2}{12}-x_1\right)z_2.$$
Also, the dynamics of $z_2$ are
$$\dot{z}_2=\dot{x}_2=u$$
and
$$\dot{V}_2= -x_1^2+\left(\frac{x_1^2}{12}-x_1+u\right)z_2$$
Therefore choosing

$$u=x_1-\frac{x_1^2}{12}-z_2=x_1-\frac{x_1^2}{12}-x_2$$

we have that
$$\dot{V}_2= -2V_2$$
and $V_2,x_1,x_2,u$ converge exponentially fast to zero.
