  The aim of this paper is to prove the superexponential stabilizability to the
ground state solution of a degenerate parabolic equation of the form
\begin{equation*}
u_t(t,x)+(x^{\alpha}u_x(t,x))_x+p(t)x^{2-\alpha}u(t,x)=0,\qquad
t\geq0,x\in(0,1) \end{equation*} via bilinear control $p\in
L_{loc}^2(0,+\infty)$. More precisely, we provide a control function $p$ that
steers the solution of the equation, $u$, to the ground state solution in small
time with doubly-exponential rate of convergence.\\ The parameter $\alpha$
describes the degeneracy magnitude. In particular, for $\alpha\in[0,1)$ the
problem is called weakly degenerate, while for $\alpha\in[1,2)$ strong
degeneracy occurs. We are able to prove the aforementioned stabilization
property for $\alpha\in [0,3/2)$. The proof relies on the application of an
abstract result on rapid stabilizability of parabolic evolution equations by
the action of bilinear control. A crucial role is also played by Bessel's
functions.
