Lyapunov stability theory

Abstract

Stability of nonlinear systems are discussed in this chapter. Lyapunov stability, asymptotic stability, and exponential stability of an equilibrium point of a nonlinear system are defined. The Lyapunov’s direct method is introduced as an indispensable tool for analyzing stability of nonlinear systems. The Barbashin–Krasovskii theorem provides a method for global stability analysis. The LaSalle’s invariant set theorem provides a method for analyzing autonomous systems with invariant sets. Stability of non-autonomous systems involves the concepts of uniform stability, uniform boundedness, and uniform ultimate boundedness. The Barbalat’s lemma is an important mathematical tool for analyzing asymptotic stability of adaptive control systems in connection with the concept of uniform continuity of a real-valued function.