Superexponential stabilizability of degenerate parabolic equations via bilinear control

Abstract

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.