Local controllability in finite time and the controllable time of the Korteweg-De Vries equation using the right Neumann controls

TL;DR

This paper studies the boundary controllability of the Korteweg-de Vries equation with right Neumann controls, revealing limitations at critical lengths and establishing new controllability times for most cases.

Contribution

It extends previous work by analyzing local controllability at critical lengths and determining new controllability times for the KdV equation with Neumann boundary controls.

Findings

The KdV system is not locally null-controllable in small time at certain critical lengths.

New controllability times are established for all but two critical lengths.

Previous results on controllability time are extended and refined.

Abstract

We investigate the local boundary controllability of the Korteweg-de Vries (KdV) equation with right Neumann boundary controls at critical lengths. We show that the KdV system is not locally null-controllable in small time for all critical lengths for which the unreachable subspace of the linearized system has dimension at least two. This result extends the work of Coron, Koenig, and Nguyen, who established it for a subclass of these lengths. We also obtain a new controllability time for such systems for all but two critical lengths. It is worth noting that the latest results on the controllability time prior to this work date back to the work of Cerpa (2007) and Cerpa and Crepeau (2009).