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Control and stabilization problem for a class of fourth-order nonlinear Schrödinger equation on boundaryless compact manifold

Yilin Song, Jiqiang Zheng, Ruihan Zhou

TL;DR

The paper addresses controllability and stabilization of the fourth-order nonlinear Schrödinger equation on compact manifolds without boundary, governed by $i\partial_tu+(\Delta_g^2-\beta\Delta_g)u=|u|^{2k}u$ with $d\in[1,5]$. It combines Strichartz estimates, semiclassical observability, and the Hilbert Uniqueness Method (HUM) to derive GCC-based observability and global controllability for $d\le4$, and extends a Bourgain-space approach to the $d=5$ case on $\mathbb{S}^5$, with analogous stabilization results under damping. The main contributions are (i) a removal of unique continuation assumptions under GCC for low dimensions via Loyola-type semiclassical analysis, (ii) a Bourgain-space propagation/observability framework enabling stabilization and controllability in the $d=5$ setting on the sphere, and (iii) a unified strategy combining propagation of compactness/regularity with nonlinear observability to obtain local controllability and exponential stabilization. The results highlight the intricate interplay between high-order dispersion, nonlinear effects, and geometry, expanding the scope of control theory for dispersive PDEs on manifolds and providing tools for stabilization in higher-order nonlinear models.

Abstract

In this paper, we study the control and stabilization problem for a class of fourth-order Schrödinger equation on compact manifold without boundary with dimensions $d\in[1,5]$: \begin{align*} i\partial_tu+(Δ_g^2-βΔ_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)].

Control and stabilization problem for a class of fourth-order nonlinear Schrödinger equation on boundaryless compact manifold

TL;DR

The paper addresses controllability and stabilization of the fourth-order nonlinear Schrödinger equation on compact manifolds without boundary, governed by with . It combines Strichartz estimates, semiclassical observability, and the Hilbert Uniqueness Method (HUM) to derive GCC-based observability and global controllability for , and extends a Bourgain-space approach to the case on , with analogous stabilization results under damping. The main contributions are (i) a removal of unique continuation assumptions under GCC for low dimensions via Loyola-type semiclassical analysis, (ii) a Bourgain-space propagation/observability framework enabling stabilization and controllability in the setting on the sphere, and (iii) a unified strategy combining propagation of compactness/regularity with nonlinear observability to obtain local controllability and exponential stabilization. The results highlight the intricate interplay between high-order dispersion, nonlinear effects, and geometry, expanding the scope of control theory for dispersive PDEs on manifolds and providing tools for stabilization in higher-order nonlinear models.

Abstract

In this paper, we study the control and stabilization problem for a class of fourth-order Schrödinger equation on compact manifold without boundary with dimensions : \begin{align*} i\partial_tu+(Δ_g^2-βΔ_g)u=|u|^{2k}u, \end{align*} where . For and , 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 , we focus on a special case, i.e. . 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)].

Paper Structure

This paper contains 20 sections, 37 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof
  4. Structure of the article
  5. Preliminaries
  6. Propagation of compactness and regularity
  7. Propgation of compactness
  8. Propagation of compactness in energy space
  9. Propagation of compactness in Bourgain space
  10. Propagation of regularity
  11. Abstract framework
  12. Propagation of regularity in energy space
  13. Propogation of regularity in Bourgain space
  14. Observability
  15. Local Controllability
  16. ...and 5 more sections

Key Result

Theorem 1.1

When $d\leq4$, for any open subset $\omega\subset M$ satisfying Geometric Control Condition (GCC) and for any $R_{0}>0$, there exist $T(R_{0})>0$ and $C>0$ such that for every $u_{0}$ and $u_{1}$ in $H^{2}(M)$ with there exists a control $h\in L^{2}([0,T]; H^{2}(M))$ with $\|h\|_{L^{2}([0,T]; H^{2}(M))}\leq C$ supported in $[0,T]\times\bar{\omega}$ such that the unique solution $u$ in $E_{T}$ of

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 51 more