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An analysis of the entire functions associated with the operator of the KdV equation

Roberto de A. Capistrano Filho, Hugo Parada, Jandeilson Santos da Silva

TL;DR

The paper addresses exact controllability of the linear KdV equation on a star-shaped network by reducing controllability to an observability inequality for the adjoint, which becomes a spectral problem involving entire-function quotients. It develops a detailed framework to analyze when these quotients are entire, yielding a precise characterization of critical lengths through sets \mathcal{N}, \mathcal{N}^*, and the newly introduced \mathcal{N}^\dagger under six boundary-control configurations. The main contribution is a complete delineation of when exact controllability holds for mixed Neumann/Dirichlet, full Neumann, and full Dirichlet boundary conditions, in terms of length values, together with the first rigorous appearance of the set \mathcal{N}^\dagger in this context. The results advance spectral/entire-function methods in control on networks and provide a concrete bridge between network geometry, boundary control placement, and length-induced controllability obstructions, with potential implications for waveguides and quantum graphs.

Abstract

It is well known that the controllability property of partial differential equations (PDEs) is closely linked to the proof of an observability inequality for the adjoint system, which, sometimes, involves analyzing a spectral problem associated with the PDE under consideration. In this work, we study a series of spectral issues that ensure the controllability of the renowned Korteweg-de Vries equation on a star-graph. This investigation reduces to determining when certain functions, associated with this spectral problem, are entire. The novelty here lies in presenting this detailed analysis in the context of a star graph structure.

An analysis of the entire functions associated with the operator of the KdV equation

TL;DR

The paper addresses exact controllability of the linear KdV equation on a star-shaped network by reducing controllability to an observability inequality for the adjoint, which becomes a spectral problem involving entire-function quotients. It develops a detailed framework to analyze when these quotients are entire, yielding a precise characterization of critical lengths through sets \mathcal{N}, \mathcal{N}^*, and the newly introduced \mathcal{N}^\dagger under six boundary-control configurations. The main contribution is a complete delineation of when exact controllability holds for mixed Neumann/Dirichlet, full Neumann, and full Dirichlet boundary conditions, in terms of length values, together with the first rigorous appearance of the set \mathcal{N}^\dagger in this context. The results advance spectral/entire-function methods in control on networks and provide a concrete bridge between network geometry, boundary control placement, and length-induced controllability obstructions, with potential implications for waveguides and quantum graphs.

Abstract

It is well known that the controllability property of partial differential equations (PDEs) is closely linked to the proof of an observability inequality for the adjoint system, which, sometimes, involves analyzing a spectral problem associated with the PDE under consideration. In this work, we study a series of spectral issues that ensure the controllability of the renowned Korteweg-de Vries equation on a star-graph. This investigation reduces to determining when certain functions, associated with this spectral problem, are entire. The novelty here lies in presenting this detailed analysis in the context of a star graph structure.
Paper Structure (24 sections, 24 theorems, 256 equations, 6 figures, 1 table)

This paper contains 24 sections, 24 theorems, 256 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dispersive systems on graph structure
  3. Background of control theory
  4. Setting of problem and functional framework
  5. Main result and heuristic
  6. Outline
  7. Well-posedness analysis
  8. Weak solutions
  9. The adjoint system
  10. Trace estimates
  11. Regularity
  12. A study of entire functions
  13. Observability inequality
  14. Entire function related to each condition
  15. Case N=2 and $m=1$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $T>0$ and $u^0,u^T\in \mathbb{L}^2(\mathcal{T})$. Consider $\alpha=N$, $m=0,\dots,N$ and $l_j=L$ for $j=1,\dots,N$.

Figures (6)

  • Figure 1: Network with $2$ edges and mixed controls.
  • Figure 2: Network with $N$ edges for $m=1$.
  • Figure 3: Network with $N$ edges for $m=N-1$.
  • Figure 4: Network with $N$ edges for $m=2$.
  • Figure 5: Network with $N$ edges with control on Neumann condition.
  • ...and 1 more figures

Theorems & Definitions (56)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 2.1
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 46 more