Stability and Lyapunov Theory - Karayiannidis, Yiannis ; 0000-0001-5129-342X (Encyclopedia of Systems and Control Engineering (First Edition), 2025)
In this chapter, we explore mathematical tools for assessing the stability, convergence, and boundedness of trajectories in generally nonlinear dynamical systems. We delve into the seminal theorems introduced by A. Lyapunov which are primarily concerned with the stability of equilibrium points. The chapter progresses to discuss the extension of Lyapunov's Theory, due to LaSalle, aimed at assessing stability attributes of invariant sets, and the utilization of the Lyapunov equation, for investigating the stability of linear systems and systems amenable to linearization. Additionally, we extend stability analysis to systems with time-varying dynamics and how external disturbances influence the system's trajectory behavior through the concept of Input-to-State Stability.
Keywords: Asymptotic stability; Autonomous systems; Continuous-time differential equations; Control systems; Exponential stability; Input-to-state stability; Invariant and limits sets; Lyapunov theory; Nonlinear dynamical systems; Stability analysis; Uniform ultimate boundedness