Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles

2010/02/05 by Lionel Rosier, Rosier, Lionel, Bing‐Yu Zhang +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1002.1126

openalex publication_date 2010/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of n intervals (n≥ 1). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as far as the localization of the control region and the smoothness of the state space are concerned. It is also proved that for the linear Schrödinger equation with Dirichlet control, the exact controllability holds in H^-1(Ω) whenever the control region contains a neighborhood of a vertex.

Citations

• Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3
• Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

Related

• Global controllability and stabilization for the nonlinear Schrodinger equation on an interval
• Internal control of the Schrödinger equation
• Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions
• Controllability of the Schrödinger equation on unbounded domains without geometric control condition
• Local controllability of 1D linear and nonlinear Schr "odinger equations\n with bilinear control