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

N. Smaoui; E. Almufadhal; R. Al Jamal; N. Smaoui; E. Almufadhal; R. Al Jamal

doi:10.3934/math.2026743

AIMS Mathematics

2026, Volume 11, Issue 6: 18280-18303. doi: 10.3934/math.2026743

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Research article

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

1.
Department of Mathematics, Faculty of Science, Kuwait University, Sabah Al Salem University City, P.O. Box 5969, Safat 13060, Shadadiya, Kuwait
2.
College of Integrative Studies, Abdullah Al Salem University, Khaldiya, Kuwait

Received: 25 April 2026 Revised: 08 June 2026 Accepted: 11 June 2026 Published: 22 June 2026
MSC : 35K55, 35B35, 93D15, 93C20, 35Q93

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.
- generalized Kuramoto–Sivashinsky equation,
- boundary control,
- adaptive control,
- nonlinear partial differential equations
Citation: N. Smaoui, E. Almufadhal, R. Al Jamal. Adaptive and non-adaptive boundary stabilization of the generalized Kuramoto–Sivashinsky equation[J]. AIMS Mathematics, 2026, 11(6): 18280-18303. doi: 10.3934/math.2026743

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Abstract

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.

References

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