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We prove rapid stabilizability to the ground state solution for a class ofabstract parabolic equations of the form \begin{equation*}u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\geq0 \end{equation*} where the operator $-A$is a self-adjoint accretive operator on a Hilbert space and $p(\cdot)$ is thecontrol function. The proof is based on a linearization argument. We prove thatthe linearized system is exacly controllable and we apply the moment method tobuild a control $p(\cdot)$ that steers the solution to the ground state infinite time. Finally, we use such a control to bring the solution of thenonlinear equation arbitrarily close to the ground state solution with doublyexponential rate of convergence.  We give several applications of our result to different kinds of parabolicequations.

Superexponential stabilizability of evolution equations of parabolic type via bilinear control