turkmath.org

We first describe the algorithm for designing a backstepping boundary controller for stabilizing partial differential equations (PDEs). A backstepping controller is obtained from an integral transformation involving a Hilbert-Schmidt kernel. We show that this stabilization method can easily fail for higher order PDEs such as Korteweg-de Vries (KdV) equation due to the overdetermined nature of the related kernel models. In order to deal with the associated mathematical challenges, we 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 may still exponentially stabilize the system with the cost of a slower rate of decay. In the second part of the talk, we consider the case in which the full state of the system cannot be measured. We show that the pseudo-backstepping kernel can also be used to design an efficient observer for the reconstruction and stabilization of the original system. Numerical simulations illustrating the efficacy of the pseudo-backstepping method are provided.