Control results for Korteweg-de Vries type systems

Abstract

This thesis investigates boundary controllability for dispersive systems governed by the Korteweg- de Vries (KdV) type equation. The main goal is to steer the system’s state using boundary

controls. We first focus on the well-known KdV equation in a bounded domain with purely Neumann boundary conditions and a single control input. A central difficulty arises when the spatial domain length is critical, rendering the associated linear system uncontrollable. To address this, we employ the return method to establish controllability of the nonlinear system. The second problem considers the KdV equation on a star-shaped graph, modeled as a system of N KdV-type equations defined on intervals (0, lj ), coupled through a condition at the central node. We demonstrate controllability using N boundary controls, which may be Neumann, Dirichlet, or a combination of both. We identify the corresponding sets of critical lengths through detailed spectral analysis for each boundary configuration. Lastly, we explore the controllability of the fifth-order KdV equation, also known as the Kawahara equation, using two boundary controls. Here, we adopt the flatness method—a

nonstandard approach that bypasses the need for an observability inequality. This method ex- presses the state and control variables in terms of so-called “flat outputs” in Gevrey spaces.

Within this framework, we address two key problems: achieving null controllability and char- acterizing the set of states reachable from zero, thereby identifying a functional space where

exact controllability holds.