# Boundary Effects on the Controllability of Coupled KdV Systems

## TL;DR

This paper investigates the boundary controllability of a coupled nonlinear KdV system modeling gravity waves, using spectral analysis and the contraction mapping theorem to establish local controllability results.

## Contribution

It introduces a novel approach combining spectral analysis and entire function theory to analyze controllability of coupled KdV systems with boundary controls.

## Key findings

- Controllability is achieved for the linearized system using duality and hidden regularity.
- Spectral problem solved via Paley-Wiener method and entire function analysis.
- Local controllability of the nonlinear system is established.

## Abstract

We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley-Wiener method introduced by Rosier. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system.