turkmath.org

In this talk, we propose a solution to a well-known open problem in control theory related with the backstepping boundary stabilization of the Korteweg-de Vries (KdV) equation on a finite domain. A backstepping controller is obtained from an integral transformation involving a specially constructed kernel. However, this method fails as the order of the PDE gets higher or the location of controller is changed, and there are serious mathematical challenges in finding the right kernel. The purpose of this talk is to introduce ''pseudo-backstepping,'' a technique using only an imperfect kernel, which is obtained by relaxing some of the conditions enforced in the standard backstepping method. We prove that the boundary controllers constructed via these imferfect kernels still exponentially stabilize the system with the cost of a slower rate of decay. Secondly, we design an observer for the KdV equation when there is no access to the medium. We only assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. Using only this partial information available, we construct Dirichlet-Neumann boundary controllers for the original system acting at the right endpoint so that the system becomes exponentially stable. Finally, we give numerical simulations illustrating the efficacy of our controllers. This is a joint work with A. Batal (IZTECH) and this research is funded by IZTECH BAP Grant# 2017IYTE14.