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Coupled linear Schrödinger equations: Control and stabilization results

K. Bhandari, R. de A. Capistrano-Filho, S. Majumdar, T. Y. Tanaka

Abstract

This article presents some controllability and stabilization results for a system of two coupled linear Schrödinger equations in the one-dimensional case where the state components are interacting through the Kirchhoff boundary conditions. Considering the system in a bounded domain, the null boundary controllability result is shown. The result is achieved thanks to a new Carleman estimate, which ensures a boundary observation. Additionally, this boundary observation together with some trace estimates, helps us to use the Gramian approach, with a suitable choice of feedback law, to prove that the system under consideration decays exponentially to zero at least as fast as the function $e^{-2ωt}$ for some $ω>0$.

Coupled linear Schrödinger equations: Control and stabilization results

Abstract

This article presents some controllability and stabilization results for a system of two coupled linear Schrödinger equations in the one-dimensional case where the state components are interacting through the Kirchhoff boundary conditions. Considering the system in a bounded domain, the null boundary controllability result is shown. The result is achieved thanks to a new Carleman estimate, which ensures a boundary observation. Additionally, this boundary observation together with some trace estimates, helps us to use the Gramian approach, with a suitable choice of feedback law, to prove that the system under consideration decays exponentially to zero at least as fast as the function for some .
Paper Structure (20 sections, 10 theorems, 118 equations)

This paper contains 20 sections, 10 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Literature review
  4. Control results for Schrödinger equation
  5. Control results for coupled system
  6. Rapid stabilization of PDEs
  7. Main results and further comments
  8. Structure of the article
  9. Boundary controllability
  10. Global Carleman estimate
  11. Observability inequality and its application
  12. Rapid exponential stabilization
  13. Gramian method
  14. Verification of the hypotheses
  15. Proof of Theorem \ref{['th-main-1']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let the set where $\gamma_1, \gamma_2>0$ are as appearing in linear. For any $T>0$, initial data $\left(u_0, v_0\right) \in \mathcal{H}'$ and parameters $\gamma_1, \gamma_2, \alpha_1, \alpha_2, \alpha$, and for any $\sigma \in \mathfrak{S}$, there exists a control $h \in L^2(0,T)$ such that the solution $(u, v)

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5: Carleman estimate
  • proof : Proof of Theorem \ref{['Thm.1']}
  • ...and 13 more