Pseudo-backstepping and its application to the control of Korteweg-de Vries equation from the right endpoint on a finite domain

TL;DR

This paper introduces the pseudo-backstepping method for boundary control of the Korteweg-de Vries equation at the right endpoint, overcoming overdetermined kernel issues and achieving exponential stabilization with controllable decay rates.

Contribution

The paper proposes the pseudo-backstepping approach for right-endpoint boundary control of the KdV equation, addressing kernel overdetermination and enabling effective stabilization.

Findings

Pseudo-backstepping kernel satisfies all but one boundary condition.

Exponential stabilization achieved with a single Dirichlet controller.

Numerical simulations confirm theoretical results.

Abstract

In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries (KdV) equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping controllers that act at the right endpoint of the domain is more challenging than its left endpoint counterpart. The standard application of the backstepping method fails, because corresponding kernel models become overdetermined. In order to deal with this difficulty, we introduce the pseudo-backstepping method, which uses a pseudo-kernel that satisfies all but one desirable boundary condition. Moreover, various norms of the pseudo-kernel can be controlled through a parameter in one of its boundary conditions. We prove that the boundary controllers constructed via this pseudo-kernel still exponentially stabilize the system with the cost of a low exponential rate of decay. We show that a single Dirichlet controller is sufficient for exponential stabilization with a slower rate of decay. We also consider a second order feedback law acting at the right Dirichlet boundary condition. We show that this approach works if the main equation includes only the third order term, while the same problem remains open if the main equation involves the first order and/or the nonlinear term(s). At the end of the paper, we give numerical simulations to illustrate the main result.