Application of fuzzy logic controls on hyperbolic differential equations

Abstract

The selection of suitable fuzzy logic control (FLC) systems for stabilizing hyperbolic conservation laws (HCLs) remains an open issue in computational fluid dynamics (CFD), particularly for shock-capturing schemes. This work addresses this gap by adopting a dual methodological strategy: (i) a systematic review of over 50 studies (2000–2025) on flux-limiting FLC approaches, and (ii) a comparative benchmark of Mamdani- and Sugeno-type FLCs applied to discontinuous solutions of HCLs. Our results demonstrate that, using weighted average defuzzification, Sugeno-type systems achieve approximately a 20% reduction in mean square error compared to the Mamdani centroid-based method in shock-dominated regimes. This performance gain aligns with adaptive CFD practices that prioritize rule-based, computationally inexpensive smoothing. By integrating theoretical analysis with experimental validation, this work strengthens the mathematical foundations of fuzzy control in PDE-driven modelling.