Boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations with critical size restrictions on the spatial domain

TL;DR

This paper investigates the boundary controllability of a nonlinear coupled Korteweg-de Vries system on a bounded interval, revealing critical length restrictions and controllability conditions with different control configurations.

Contribution

It extends previous results by demonstrating the existence of a critical length phenomenon and controllability with fewer controls under specific domain sizes.

Findings

Existence of critical length phenomenon for nonlinear system.

Controllability with a single boundary control for certain domain sizes.

Sharp control spaces due to hidden regularity in the adjoint system.

Abstract

This article is dedicated to improve the controllability results obtained by Cerpa et al. in Commun. Contemp. Math 13 (2011) and by Micu et al. in Commun. Contemp. Math 11 (5) (2009) for a nonlinear coupled system of two Korteweg-de Vries (KdV) equations posed on a bounded interval. Initially, in Micu et al., the authors proved that the nonlinear system is exactly controllable by using four boundary controls without any restriction on the length L of the interval. Later on, in Cerpa et al., two boundary controls were considered to prove that the same system is exactly controllable for small values of the length L and large time of control T. Here, we use the ideas contained in Capistrano-Filho et al. (arXiv 1508.07525) to prove that, with another configuration of four controls, it is possible to prove the existence of the so-called critical length phenomenon for the nonlinear system, i. e., whether the system is controllable depends on the length of the spatial domain. In addition, when we consider only one control input, the boundary controllability still holds for suitable values of the length L and time of control T. In both cases, the control spaces are sharp due a technical lemma which reveals a hidden regularity for the solution of the adjoint system.