A Survey on Asymptotic and Exponential Stability of Nonlinear Differential Equations

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Published ID: IJNRDTH00250

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Bhushan Devidas Kumawat

Asymptotic Stability; Exponential Stability; Lyapunov Methods; Nonlinear Differential Equations; LaSalle's Invariance Principle; Lyapunov Function Construction; Stability of Nonlinear Systems; Control Systems; Perturbation Theory; Stochastic Stability

The investigation of asymptotic and exponential stability, one of the most basic problems existing in a qualitative theory for nonlinear differential equations, The proliferation of dynamical systems of increasing sophistication in engineering, biology, physics, and control science has created an enormous need for stability frameworks that are not only rigorous and systematic but also computationally tractable. This survey gives a detailed, critical review of theoretical results, recent developments and applications for asymptotic and exponential stability of nonlinear ordinary and partial differential equations. Starting with a rigorous approach to the mathematical definition of equilibrium definitions, Lyapunov stability and classical hierarchy of stability concepts, the survey methodically covers asymptotic stability via Lyapunov's direct method, LaSalle's invariance principle and perturbation in Section 5, before addressing the topic of exponential stability through decay rate bounds (uniform versus non uniform), as well as Lyapunov based sufficient conditions. The analytical methods are assessed, along with some of their strengths and well established limitations, by regarding approaches that include linearization, Lyapunov function construction algorithms, fixed point theorems and comparison principles. In terms of high dimensional and non smooth systems, numerical and computational approaches, such as simulation based verification (SBV) and software assisted analysis, are reviewed. Domain specific examples are handled spanning application domains including control engineering, power systems, robotics, biological modeling and mechanical systems. They close the work with an organized synthesis of recent developments, highlighting open challenges and proposing frontier research areas including (but not limited to) stability of stochastic systems, high dimensional dynamic systems, and novel structured Lyapunov function synthesis methods based on large amounts of data. The work is to be useful as a reference for researchers working in this space (and also to be analytically rigorous enough to serve as an entry point into the field).

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Bhushan Devidas Kumawat (May-2026). A Survey on Asymptotic and Exponential Stability of Nonlinear Differential Equations. INTERNATIONAL JOURNAL OF NOVEL RESEARCH AND DEVELOPMENT, 11(5), 395-447. https://ijnrd.org/papers/IJNRDTH00250.pdf

Volume 11 Issue 5, May-2026

Pages : 395-447

Paper Reg. ID: IJNRD_325423

Published Paper Id: IJNRDTH00250

Research Area: Applied Mathematics

Author Type: Indian Author

Country: Tal Chopda Dist Jalg, Maharashtra, India

Published Paper PDF: https://ijnrd.org/papers/IJNRDTH00250.pdf

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