Superexponential stabilizability of degenerate parabolic equations via bilinear control

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 tgeq0,xin(0,1) end{equation*} via bilinear control $pin L_{loc}^2(0,+infty)$. 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 $alphain[0,1)$ the problem is called weakly degenerate, while for $alphain[1,2)$ strong degeneracy occurs. We are able to prove the aforementioned stabilization property for $alphain [0,3/2)$. 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 Bessels functions.