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Global Approximate Controllability of the Camassa-Holm Equation by a Finite Dimensional Force

Shirshendu Chowdhury, Rajib Dutta, Debanjit Mondal

Abstract

In this paper, we consider the Camassa-Holm equation posed on the periodic domain $\mathbb{T}$. We show that Camassa-Holm equation is globally approximately controllable by three dimensional external force in $H^s(\mathbb{T})$ for $ s > \frac{3}{2}$ . The proof is based on Agrachev-Sarychev approach in geometric control theory.

Global Approximate Controllability of the Camassa-Holm Equation by a Finite Dimensional Force

Abstract

In this paper, we consider the Camassa-Holm equation posed on the periodic domain . We show that Camassa-Holm equation is globally approximately controllable by three dimensional external force in for . The proof is based on Agrachev-Sarychev approach in geometric control theory.
Paper Structure (8 sections, 8 theorems, 129 equations)

This paper contains 8 sections, 8 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Existence and properties.
  3. Proof of Theorem \ref{['main_thm']}
  4. Proof of the Propositions
  5. Proof of Proposition \ref{['prp_existance']}
  6. Proof of Proposition \ref{['prp_cntity']}
  7. Proof of Proposition \ref{['prp_limit']}
  8. Appendix

Key Result

Theorem 1.1

For $s > \frac{3}{2},$ equation ctrl_equn is approximately controllable in $H^s(\mathbb{T})$ by a piece wise constant controls with values in $\mathcal{H} ,$ where

Theorems & Definitions (17)

  • Remark 1.1
  • Definition 1.1
  • Theorem 1.1
  • Proposition 2.1: Well-posedness
  • Proposition 2.2: Stability
  • Remark 2.1
  • Definition 2.1
  • Proposition 2.3: Density
  • proof
  • Proposition 2.4: Asymptotic property
  • ...and 7 more