Pointwise control of the linearized Gear-Grimshaw system

TL;DR

This paper establishes the conditions under which a coupled Korteweg-de Vries system on the circle can be controlled pointwise using Dirac measures, providing explicit controllability times and extending to cases with a single control.

Contribution

It introduces a spectral analysis approach to prove pointwise controllability of the linearized Gear-Grimshaw system with one or two controls, including sharp controllability times.

Findings

Pointwise controllability is achieved under general physical parameter assumptions.

Controllability with a single internal control is possible due to a proven uniqueness property.

Sharp controllability times are identified under certain coefficient conditions.

Abstract

In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.