Lyapunov Stability Theory: Energy Functions and Stability Analysis

This lecture introduces Lyapunov stability theory, focusing on energy-like functions and their role in determining the stability of linear time-invariant systems. The instructor explains how to define energy functions and their relationship to the distance of the system state from the origin. The lecture covers the properties of positive-definite matrices, the concept of Lyapunov functions, and the theorems related to system stability. Through examples and proofs, the lecture demonstrates the application of Lyapunov theory in analyzing stability and provides insights into the optimization problems involved. Additionally, the lecture explores stability concepts for linear time-varying systems and the implications of Lyapunov theory in nonlinear and switched systems.