Control and stabilization problem for a class of fourth-order nonlinear Schr\"odinger equation on boundaryless compact manifold

In this paper, we study the stabilization property and large time controllability for a class of fourth-order Schrödinger equations on a compact manifold without boundary in dimensions $1\leq d\leq5$: \begin{align} i\partial_tu+(\Delta_g^2-\beta\Delta_g)u=-|u|^{2k}u, \,\,x\in M\tag{4NLS}\label{4NLS1} \end{align} where $k\in\mathbb{N}$ and $\beta\in\mathbb{R}_{+}$ when $1\leq d\leq 4$ but $\beta\in\mathbb{Q}_{+}$ for $d=5$.

We adapt the strategy in Macia [Vietnam J.

Math.

(2021)] to establish observability and the propagation of singularities.

Moreover, we use these propagation estimates to deduce the unique continuation property for $\eqref{4NLS1}$.

By the classical Hilbert Uniqueness Method (HUM) and the Picard iteration, the stabilization and large time controllability hold under the Geometric Control Condition (GCC) and the Unique Continuation Property (UCP) for the linearized equation.

To obtain the controllability and stabilization at the $H^2$ level with $d=5$, we will focus on $M=\Bbb S^5$ with $k=1$.

Our results extend those of Laurent [SIAM J.

Math.

Anal.

(2009)] and Capistrano Filho-Pampu [Math.

Z., 2022].