A Robust Discuss on Lyapunov Stability and LaSalle's Invariance Principle

Abstract: This paper provides a rigorous and systematic exposition of Lyapunov stability theory and LaSalle’s invariance principle for nonlinear autonomous dynamical systems. The analysis begins with the foundational formulation of equilibrium points and the construction of Lyapunov functions, emphasizing their roles as generalized energy measures for dynamical systems. Key structural properties, including positive definiteness and radial unboundedness, are introduced and used to establish global boundedness and stability properties of system trajectories. The classical Lyapunov direct method is developed to establish global asymptotic stability under strict decay conditions on the Lyapunov function derivative. This framework is subsequently extended through LaSalle’s invariance principle, which relaxes the requirement of strict negativity and allows for semidefinite dissipation. In this setting, convergence is characterized in terms of the largest invariant subset of theV set where the Lyapunov derivative vanishes. Supporting results are presented concerning monotonicity of Lyapunov functions along trajectories, compactness of sublevel sets, and invariance properties of limit sets. Particular emphasis is placed on stability conditions of the form ≤ – k and ≤ 0 and their implications for asymptotic behaviour. A geometric interpretation is provided in which Lyapunov functions define a generalized energy landscape, and system trajectories evolve toward invariant sets where dissipation ceases. The results demonstrate that Lyapunov’s direct method and LaSalle’s invariance principle together form a unified and powerful analytical framework for establishing global asymptotic stability in nonlinear control systems, optimization dynamics, and dissipative mechanical systems. Keywords: Lyapunov stability; LaSalle invariance principle; nonlinear dynamical systems; global asymptotic stability; Lyapunov function; radial unboundedness; equilibrium points; invariant sets; nonlinear control theory; stability analysis. Title: A Robust Discuss on Lyapunov Stability and LaSalle’s Invariance Principle Author: Ogbonna Nnamuchi International Journal of Novel Research in Physics Chemistry & Mathematics ISSN 2394-9651 Vol. 13, Issue 2, May 2026 - August 2026 Page No: 35-41 Novelty Journals Website: www.noveltyjournals.com Published Date: 29-May-2026 DOI: https://doi.org/10.5281/zenodo.20448002 Paper Download Link (Source) https://www.noveltyjournals.com/upload/paper/A%20Robust%20Discuss%20on%20Lyapunov%20Stability-29052026-5.pdf