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Observability of the linear Zakharov--Kuznetsov equation

Roberto de A. Capistrano Filho, Vilmos Komornik, Ademir F. Pazoto

Abstract

We study the linear Zakharov--Kuznetsov equation with periodic boundary conditions. Employing some tools from the nonharmonic Fourier series we obtain several internal observability theorems. Then we prove various exact controllability and rapid uniform stabilization results by applying a duality principle and a general feedback construction. The method presented here introduces a new insight into the control of dispersive equations in two-dimensional cases and may be adapted to more general equations.

Observability of the linear Zakharov--Kuznetsov equation

Abstract

We study the linear Zakharov--Kuznetsov equation with periodic boundary conditions. Employing some tools from the nonharmonic Fourier series we obtain several internal observability theorems. Then we prove various exact controllability and rapid uniform stabilization results by applying a duality principle and a general feedback construction. The method presented here introduces a new insight into the control of dispersive equations in two-dimensional cases and may be adapted to more general equations.
Paper Structure (12 sections, 19 theorems, 139 equations, 3 figures)

This paper contains 12 sections, 19 theorems, 139 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Problem setting
  4. Notation and main results
  5. Structure of the article
  6. Well-posedness and eigenfunction expansions
  7. A short review of Ingham type theorems
  8. Observability on segments
  9. Observability on small balls
  10. A review on observation, control and stabilization
  11. Controllability and stabilization results
  12. Further comments

Key Result

Theorem 1.1

For any fixed $x_0\in\mathbb T$ and bounded interval $I$ of length $|I|>0$, there exist two constants $C_1, C_2>0$ such that all solutions of 13 with $u_0\in L^2(\mathbb T\times\mathbb T)$ satisfy the following estimates:

Figures (3)

  • Figure 1: (iv) The region containing $B_{2,4,7}$
  • Figure 2: The region $-9<y^2-x^2<9$
  • Figure : Theorems \ref{['t11']}, \ref{['t12']} and \ref{['t14']}

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Remark 2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1
  • ...and 26 more