This document discusses Lyapunov theory for analyzing the stability of nonlinear systems. It introduces concepts like Lyapunov stability, positive definite functions, and Lyapunov stability theorems. The document notes that Lyapunov stability theorems relate the existence of a Lyapunov function to the stability of solutions or trajectories of a system. It states that finding Lyapunov functions typically involves choosing a functional form and determining parameter values such that the required properties are satisfied.

1. Lyapunov Theory
Sistem Non Linear
Aditya A. Purnama / 1410501023
Dosen Pembimbing : R. Suryoto Edy Raharjo, S.T., M.Eng

2. Table of content
• Stability
• Positive definite functions
• Lyapunov stability theorems
• Converse Lyapunov theorems
• Finding Lyapunov functions

3. Stability
• Stability of solutions of differential equations and of trajectories of
dynamical systems under small perturbations of initial conditions.
• In partial differential equations one may measure the distances
between functions using Lp norms or the sup norm, while in differential
geometry one may measure the distance between spaces using the
Gromov–Hausdorff distance.
• In dynamical systems, an orbit is called Lyapunov stable if the forward
orbit of any point is in a small enough neighborhood or it stays in a
small (but perhaps, larger) neighborhood.

4. Positive definite functions

5. Lyapunov stability theorems

6. Lyapunov stability theorems

7. Converse Lyapunov theorems
• A typical converse Lyapunov theorem has the form

8. Finding Lyapunov functions
• There are many different types of Lyapunov theorems the key in all cases is to find a
Lyapunov function and verify that it has the required properties
• There are several approaches to finding Lyapunov functions and verifying the properties
• Decide form of Lyapunov function (e.g., quadratic), parametrized by some parameters
(called a Lyapunov function candidate)
• Try to find values of parameters so that the required hypotheses hold

9. TERIMA KASIH 
Daftar Pustaka :
https://en.wikipedia.org/wiki/Stability_theory
https://en.wikipedia.org/wiki/Lyapunov_function
https://en.wikipedia.org/wiki/Lyapunov_equation
http://stanford.edu/class/ee363/lectures/lyap.pdf
http://stanford.edu/class/ee363/lectures/lq-lyap.pdf