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Controllability for a non-local formulation of surface gravity waves

M. A. Fontelos, R. Lecaros, J. López-Ríos, A. Pérez

Abstract

In this paper, we study the approximate controllability of a system governed by an evolution problem known as the sloshing problem. This problem involves a spatial, nonlocal differential operator inherent in the dynamics of a two-dimensional, incompressible, non-viscous fluid within a confined domain. Our work establishes unique continuation results that enable the application of source control localized in an interior domain, allowing the aforementioned controllability.

Controllability for a non-local formulation of surface gravity waves

Abstract

In this paper, we study the approximate controllability of a system governed by an evolution problem known as the sloshing problem. This problem involves a spatial, nonlocal differential operator inherent in the dynamics of a two-dimensional, incompressible, non-viscous fluid within a confined domain. Our work establishes unique continuation results that enable the application of source control localized in an interior domain, allowing the aforementioned controllability.
Paper Structure (9 sections, 9 theorems, 108 equations, 1 figure)

This paper contains 9 sections, 9 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. A non-local, integro-differential formulation of the Laplace equation
  4. Equation analysis through an orthogonal basis
  5. Well-posedness of the Cauchy problem
  6. Properties of the operator $\mathcal{A}$
  7. Existence of weak solutions
  8. Proof of Proposition \ref{['I_cp']} and the approximate controllability
  9. Comments

Key Result

Proposition 1

Let $T>0$, $[\phi_0,\phi_1]\in H_{w^{-1}}^{1/2}\times L^2$ and $f:\mathbb{R}\longrightarrow \mathbb{R}$ Lipschitz, i.e. there exists $c>0$ a positive constant such that $|f(x)|\le c|x|$. If $\phi=0$ in an open subset $M\subset (0,T)\times (-1,1)$, where $\phi\in C((0,T);H^{1/2}_{w^{-1}})$ is the sol Then

Figures (1)

  • Figure 1: The reference domain of the fluid enclosed by side walls.

Theorems & Definitions (21)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 11 more