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

TL;DR

This paper investigates control and stabilization of a fourth-order nonlinear Schrödinger equation on compact manifolds, extending previous results by removing assumptions and applying advanced analysis techniques across various dimensions.

Contribution

It proves stabilization results under geometric control conditions for dimensions 1 to 4 and extends these results to the 5-dimensional sphere, removing previous assumptions.

Findings

Stabilization achieved under geometric control condition (GCC).

Removal of unique continuation assumption in stabilization proofs.

Extension of control results to 5-dimensional sphere.

Abstract

In this paper, we study the control and stabilization problem for a class of fourth-order Schr\"odinger equation on compact manifold without boundary with dimensions $d\in[1,5]$: \begin{align*} i\partial_tu+(\Delta_g^2-\beta\Delta_g)u=|u|^{2k}u, \end{align*} where $k\in\Bbb N$. For $1\leq d\leq4$ and $k\geq1$, we combine the method proposed by Loyola and semiclassical analysis to prove the stabilization result only under the geometric control condition (GCC), which removes the unique continuation assumption in Capistrano-Filho-Pampu [Math. Z. (2022)]. For $d=5$, we focus on a special case, i.e. $\Bbb S^5$. Establishing the propagation of singularity in Bourgain space, we prove the similar control and stabilization result in energy space as lower dimensions, which generalizes the result of Laurent [SIAM J. Math. Anal. (2009)].