Controllability of NLS in the vicinity of solitary wave solutions

2006/08/05 by Horst Lange, Lange, Horst, Holger Teismann +1
Engineering · Mathematics · #35Q55 #93B05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #math.AP #math.OC #msc:35Q55 #msc:93B05

paper · pdf · doi:10.48550/arxiv.math/0608135

27 pages

arxiv created 2006/08/05 · openalex publication_date 2006/08/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Local exact controllability of the 1D NLS (subject to zero boundary conditions) with distributed control is shown to hold in a H¹--neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to J.-L. Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail.

Citations

• Asymptotic stability of N-soliton states of NLS

Related

• Local boundary controllability in classes of differentiable functions for the wave equation
• Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions
• Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control
• Experimental Study of the HUM Control Operator for Linear Waves
• Rigorous numerics for NLS: bound states, spectra, and controllability