تحلیل مروری پایداری سیستمهای غیرخطی بر اساس روش مستقیم لیاپانوف
DOI:
https://doi.org/10.67753/frsj.v3i2.212Keywords:
پایداری، پایداری مجانبی، تابع لیاپانوف، روش مستقیم لیاپانوف، سیستمهای غیرخطیAbstract
A Review Analysis of the Stability of Nonlinear Systems Based on Lyapunov’s Direct Method
Mohammad Yazanparast, Farhad hakimi, Ahmad Wali Noori
Abstract
Nonlinear systems appear in many mathematical, engineering, physical, and population-related phenomena. Their stability analysis is important for understanding system behavior near equilibrium, especially when explicit solutions of the differential equations are difficult or unavailable. This article presents an analytical review of nonlinear system stability based on Lyapunov’s direct method. The study focuses on the Lyapunov function, its positive definiteness, the behavior of its derivative along system trajectories, the difficulty of constructing a suitable function, recent computational approaches, and several simple illustrative examples. The study relies on reliable sources published after 2015, covering stability theory, nonlinear control, fractional-order systems, neural networks, optimization and methods, and also population and epidemiological models. The review shows that Lyapunov’s direct method can assess stability, asymptotic stability, and, under stronger assumptions, exponential stability without explicitly solving the system equations. Nevertheless, constructing a suitable Lyapunov function remains the most difficult part of the method, which has motivated the development of computational methods, machine-learning approaches, and control Lyapunov functions. The article concludes that Lyapunov’s direct method, despite its practical limitations, remains important for both the analysis and teaching of nonlinear system stability.
Keywords: Asymptotic stability, Lyapunov direct method, Lyapunov function, Nonlinear systems, Stability
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