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Control of three dimensional water waves

Hui Zhu

Abstract

We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously obtained by Alazard, Baldi, and Han-Kwan for 2-D water waves. Our proof combines an iterative scheme, that reduces the controllability of the original quasi-linear equation to that of a sequence of linear equations, with a semiclassical approach for the linear control problems.

Control of three dimensional water waves

Abstract

We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously obtained by Alazard, Baldi, and Han-Kwan for 2-D water waves. Our proof combines an iterative scheme, that reduces the controllability of the original quasi-linear equation to that of a sequence of linear equations, with a semiclassical approach for the linear control problems.

Paper Structure

This paper contains 38 sections, 78 theorems, 475 equations.

Table of Contents

  1. Introduction
  2. From Euler to Zakharov / Craig--Sulem
  3. Outline of Paper
  4. Strategy of Proof and Some Notations
  5. Paralinearization and Reduction to Null Controllability
  6. Iterative Scheme.
  7. Hilbert Uniqueness Method
  8. Semiclassical Observability
  9. Weak $L^2$-Observability
  10. Unique Continuation and Strong $L^2$-Observability
  11. $H^{s}$-Controllability
  12. Reformulation of Problem
  13. Paralinearization of Dirichlet-Neumann Operator
  14. Paralinearization of Surface Tension
  15. Equation for the Good Unknown
  16. ...and 23 more sections

Key Result

Theorem 1.2

Suppose that $d \ge 1$, $T > 0$, $s$ is sufficiently large, and $\omega \subset \mathbb{T}^d$ satisfies the geometric control condition, then for some $\varepsilon_0 > 0$ sufficiently small and for all $(\eta_i,\psi_i) \in H^{s+1/2}(\mathbb{T}^d) \times H^{s}(\mathbb{T}^d)$ satisfying and $\int_{\mathbb{T}^d} \eta_0 \,\mathrm{d} x = \int_{\mathbb{T}^d} \eta_1 \,\mathrm{d} x = 0$, there exists $P_

Theorems & Definitions (163)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 153 more