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Critical lengths for the linear Kadomtsev-Petviashvili II equation

Roberto de A. Capistrano-Filho, Fernando Gallego, Ricardo Muñoz

TL;DR

The paper addresses boundary controllability and stabilization of the linear KP-II equation on a rectangle, uncovering a precise critical-length phenomenon connected to Paley–Wiener theory. By proving boundary observability inequalities and employing the Hilbert uniqueness method, it establishes exact boundary controllability and boundary exponential stabilization for lengths $L$ outside an explicitly characterized set $\mathcal{R}$. A Lyapunov-based analysis yields an explicit exponential decay rate, enriching the understanding of dissipative control for dispersive 2D models. The results extend the one-dimensional KdV critical-length theory to KP-II, offering concrete guidelines for control design on rectangular domains and signaling directions for nonlinear KP-II research.

Abstract

The critical length phenomenon of the Korteweg-de Vries equation is well known; however, in higher dimensions, it is unknown. This work explores this property in the context of the Kadomtsev-Petviashvili equation, a two-dimensional generalization of the Korteweg-de Vries equation. Specifically, we demonstrate observability inequalities for this equation, which allow us to deduce the exact boundary controllability and boundary exponential stabilization of the linear system, provided that the spatial domain length avoids certain specific values, a direct consequence of the Paley-Wiener theorem. To the best of our knowledge, our work introduces new results by identifying a set of critical lengths for the two-dimensional Kadomtsev-Petviashvili equation.

Critical lengths for the linear Kadomtsev-Petviashvili II equation

TL;DR

The paper addresses boundary controllability and stabilization of the linear KP-II equation on a rectangle, uncovering a precise critical-length phenomenon connected to Paley–Wiener theory. By proving boundary observability inequalities and employing the Hilbert uniqueness method, it establishes exact boundary controllability and boundary exponential stabilization for lengths outside an explicitly characterized set . A Lyapunov-based analysis yields an explicit exponential decay rate, enriching the understanding of dissipative control for dispersive 2D models. The results extend the one-dimensional KdV critical-length theory to KP-II, offering concrete guidelines for control design on rectangular domains and signaling directions for nonlinear KP-II research.

Abstract

The critical length phenomenon of the Korteweg-de Vries equation is well known; however, in higher dimensions, it is unknown. This work explores this property in the context of the Kadomtsev-Petviashvili equation, a two-dimensional generalization of the Korteweg-de Vries equation. Specifically, we demonstrate observability inequalities for this equation, which allow us to deduce the exact boundary controllability and boundary exponential stabilization of the linear system, provided that the spatial domain length avoids certain specific values, a direct consequence of the Paley-Wiener theorem. To the best of our knowledge, our work introduces new results by identifying a set of critical lengths for the two-dimensional Kadomtsev-Petviashvili equation.
Paper Structure (17 sections, 21 theorems, 161 equations)

This paper contains 17 sections, 21 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Notations and main results
  4. Paper's outline
  5. Well-posedness for the KP-II
  6. Homogeneous system
  7. Adjoint system
  8. KP-II equation with feedback
  9. Nonhomegenous system
  10. Boundary observations
  11. Observation for controllability
  12. Observation for stabilization
  13. Control results of KP-II equation
  14. Boundary controllability
  15. Boundary stabilization
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\beta$ and $\alpha^{(j)}$, for $j = 1,\dots, N$, denote $n$-dimensional multi-index with non-negative-integer-valued components. Suppose that $1 < p^{(j)} < \infty$, $1 < q < \infty$, $0 < \mu_j < 1$ with Then, for $f(x)\in C_0^\infty(\mathbb{R}^n)$, $\left\lVert D^\beta f \right\rVert_{q} \leq C \prod_{j=1}^N \left\lVert D^{\alpha^{(j)}}f \right\rVert_{p^{(j)}}^{\mu_j},$ where for non-neg

Theorems & Definitions (39)

  • Theorem 1.1: besov79
  • Lemma 1.2: Rosier
  • Theorem 1.3
  • Theorem 1.4: Uniform exponential stabilization
  • Theorem 1.5: Explicit decay rate
  • Remark 1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 29 more