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

Türker Özsarı, Ahmet Batal

Published 2018-01-22Version 1

In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries (KdV) equation which act at the right endpoint of the domain. Controlling the KdV equation from the right endpoint of the domain is a mathematically more challenging problem than its left endpoint counterpart from the point of constructing backstepping controllers. 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 are able to 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. At the end of the paper, we give numerical simulations to illustrate our main result.