<p>This paper investigates the stability and control of a nonlinear dispersive partial differential equation, specifically the generalized Kawahara (GK) equation, under periodic boundary conditions. Unlike earlier studies that primarily focused on linear stabilization or low-order models, we develop Lyapunov-based output feedback controllers for both single-input single-output (SISO) and multiple-input multiple-output (MIMO) configurations. These controllers are designed to drive the system toward zero equilibrium and enhance its convergence rate, even in the presence of weak or localized damping. A further contribution is the systematic application of the Fourier Galerkin method to derive an implementable finite-dimensional framework that enables rigorous analysis and control design for the GK equation. The proposed control strategies are applied to the GK equation under two scenarios: with and without a localized damping term. The Fourier Galerkin method is used to approximate solutions, providing a practical framework for implementing the control design. Numerical simulations validating the effectiveness of the controllers, highlighting their performance across different damping conditions and offering insights into the stabilization of higher-order nonlinear dispersive systems are presented. Overall, this work presents one of the first comprehensive and systematic control frameworks for the GK equation. It extends methods previously applied to the KdV equation to the generalized Kawahara equation, a fifth-order nonlinear dispersive PDE with sign-varying localized damping, and demonstrates that a finite-dimensional output feedback controller can stabilize both the linearized and nonlinear systems without spillover effects. In doing so, the paper advances the understanding and control of higher-order nonlinear dispersive systems beyond existing literature.</p>