Boundary Controllability of the Korteweg-de Vries Equation on a Bounded Domain

TL;DR

This paper investigates boundary controllability of the Korteweg-de Vries equation on a finite interval, analyzing how different boundary controls affect the ability to steer the system, including linear and nonlinear cases.

Contribution

It provides new results on boundary controllability for the Korteweg-de Vries equation with various boundary control configurations, including nonlinear local controllability.

Findings

Linear system is exactly controllable with two or three boundary controls.

Single boundary control on the left end achieves null controllability.

Single boundary control on the right end is exactly controllable except for critical domain lengths.

Abstract

This paper is devoted to study boundary controllability of the Korteweg-de Vries equation posed on a finite interval, in which, because of the third-order character of the equation, three boundary conditions are required to secure the well-posedness of the system. We consider the cases where one, two, or all three of those boundary data are employed as boundary control inputs. The system is first linearized around the origin and the corresponding linear system is shown to be exactly boundary controllable if using two or three boundary control inputs. In the case where only one control input is allowed to be used, the linearized system is known to be only \emph{null} controllable if the single control input acts on the left end of the spatial domain. By contrast, if the single control input acts on the right end of the spatial domain, the linearized system is exactly controllable if and only if the length of the spatial domain does not belong to a set of critical values. Moreover, the nonlinear system is shown to be locally exactly boundary controllable via contraction mapping principle if the associated linearized system is exactly controllable.