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

2025/11/25 by Yilin Song, Jiqiang Zheng, Song, Yilin +3
#math.AP

paper · pdf · doi:10.48550/arxiv.2511.19890

Abstract

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≤ d≤5: i∂_tu+(Δ_g²-βΔ_g)u=-|u|^2ku, x∈ M where k∈ℕ and β∈ℝ₊ when 1≤ d≤ 4 but β∈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 \eqref4NLS1. 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 H² level with d=5, we will focus on M=\Bbb S⁵ with k=1. Our results extend those of Laurent [SIAM J. Math. Anal. (2009)] and Capistrano Filho-Pampu [Math. Z., 2022].

Citations

• Bilinear eigenfunction estimates and the nonlinear Schr�dinger equation on surfaces
• Exact Controllability, Stabilization and Perturbations for Distributed Systems

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