Adaptive and non-adaptive boundary stabilization of the generalized Kuramoto–Sivashinsky equation

This paper investigates the boundary stabilization of the generalized Kuramoto–Sivashinsky (GKS) equation on a bounded domain. A zero-mean formulation is first introduced to eliminate the spatial drift and to provide a suitable framework for control design. Several nonlinear non-adaptive boundary feedback laws are then proposed and shown, via Lyapunov-based analysis, to guarantee global exponential stability of the closed-loop system in $L^2(0,1)$ when the system parameters are known. To address the case of uncertain coefficients, adaptive boundary feedback laws are subsequently developed, allowing the feedback gains to be adjusted dynamically while preserving exponential stabilization. Numerical simulations are presented to validate the theoretical results and to demonstrate the effectiveness of the proposed controllers. A comparative study between the best-performing adaptive and non-adaptive boundary control laws is also provided, highlighting the trade-off between convergence speed and robustness with respect to parameter uncertainty. Finally, for completeness, a derivation of the GKS equation corresponding to the case $n=4$ is also presented.