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Approximation theorem for the Kawahara operator and its application in control theory

Roberto de A. Capistrano Filho, Luan S. de Sousa, Fernando A. Gallego

Abstract

Control properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in $(0,+\infty)$. Following Rosier SIAM Simon (2000), the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.

Approximation theorem for the Kawahara operator and its application in control theory

Abstract

Control properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in . Following Rosier SIAM Simon (2000), the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.
Paper Structure (13 sections, 8 theorems, 141 equations)

This paper contains 13 sections, 8 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Problem set
  3. Historical background
  4. Main results
  5. Final comments and paper's outline
  6. Preliminaries
  7. Auxiliary lemma
  8. Observability inequality via Ingham inequality
  9. Global Carleman estimate
  10. Proof of Theorem \ref{['main1']}
  11. Approximation Theorem
  12. Approximation Theorem applied in control problem
  13. Proof of Theorem \ref{['main3']}

Key Result

Theorem 1.1

There exist constants $s_0=s_{0}(L,T)>0$ and $\tilde{C}= \tilde{C}(L,T)>$ such that for any $q\in\mathcal{D}(P)$ and all $s \geq s_{0}$, one has

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 4.1
  • ...and 4 more