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Exact controllability of two-dimensional hydroelastic waves

Lizhe Wan, Jiaqi Yang

Abstract

We prove the exact controllability of two-dimensional hydroelastic waves in the periodic setting. We show that if the initial data and the final data are small, for exterior pressure whose support is any non-empty open set $ω$, the two-dimensional hydroelastic wave system is exactly controllable in arbitrary short time.

Exact controllability of two-dimensional hydroelastic waves

Abstract

We prove the exact controllability of two-dimensional hydroelastic waves in the periodic setting. We show that if the initial data and the final data are small, for exterior pressure whose support is any non-empty open set , the two-dimensional hydroelastic wave system is exactly controllable in arbitrary short time.
Paper Structure (20 sections, 29 theorems, 290 equations)

This paper contains 20 sections, 29 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Ideas of the proof
  4. Linearized hydroelastic waves
  5. Reduction of the linearized operator at leading and sub-leading order
  6. Change of space variable
  7. Time reparametrization and reduction at top order
  8. Symmetrization of top order
  9. Symmetrization of lower orders
  10. Reduction of the linearized operator at lower orders
  11. Translation of space variable
  12. Reduction at lower orders
  13. Well-posedness of the linear system
  14. Observability of the linearized system
  15. Controllability of the linearized system
  16. ...and 5 more sections

Key Result

Lemma 1.1

For every $T>0$, there exists a positive constant $C_1(T)$ such that, for all $(w_n)_{n\in \mathbb{Z}} \in \ell^2(\mathbb{Z}, \mathbb{C})$, $m\geq c$ for some constant $c>0$, one has the following Ingham inequality: Let $\mathfrak{t}(\xi)$ be a symbol that is defined in SymbolTau, then

Theorems & Definitions (33)

  • Lemma 1.1: zbMATH05046349
  • Theorem 1.2
  • Lemma 2.1: MR2138139MR3060183
  • Lemma 2.2: MR3776276
  • Lemma 2.3: MR2138139
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 5.4
  • ...and 23 more