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

TL;DR

This paper investigates the spectral properties of functions related to the KdV equation on a star-graph, linking spectral analysis to controllability and providing new insights into PDE control theory.

Contribution

It introduces a detailed analysis of entire functions associated with the spectral problem of the KdV equation on star graphs, a novel context for such spectral studies.

Findings

Spectral functions associated with the KdV on star graphs are characterized as entire functions.

Controllability of the KdV on star graphs is connected to the spectral analysis of these functions.

The study provides conditions under which certain spectral functions are entire, impacting controllability results.

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.