Control and stabilization of kdv-kdv and kp type systems

Abstract

This thesis presents a study on the boundary stabilization and control of several nonlinear dispersive systems, including the Boussinesq KdV-KdV type system, the Hirota-Satsuma system, the Kadomtsev-Petviashvili (KP) equation and its higher-order variant, the Kawahara-KP (K- KP) equation. For the Boussinesq KdV-KdV type system and Hirota-Satsuma system, we design feedback laws at the boundary that combine damping mechanisms and delay terms, demonstrating the exponential decay of the energy associated with the system, given small initial data. For this purpose, we use the Lyapunov method and fixed-point arguments. In the context of the KP equation, we explore the critical length phenomenon, deriving observability inequalities that lead to boundary controllability and exponential stabilization. These results depend on the spatial length and are demonstrated using the Paley-Wiener Theorem. Finally, for the K-KP equation, we establish local and global exponential stability results through two different approaches, providing optimal constants and the minimum time to ensure exponential decay of the energy.