In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE.

Lyapunov theory is used to make conclusions about trajectories of a system ̇x = f (x) (e.g., G.A.S.) without finding the trajectories (i.e., solving the differential equation)

Lyapunov function actually allows to gain some information about the global behavior of orbits. First, let me define the basin of attraction of an asymptotically stable equilibrium at the set of all initial conditions leading to the long term behavior that approach this equilibrium.

Lyapunov functions, titled after Aleksandr Lyapunov, are scalar functions that can be used to verify the stability of equilibrium of an ordinary differential equation in the concept of ordinary differential equations (ODEs).

4 days ago · A Lyapunov function is a scalar function V (y) defined on a region D that is continuous, positive definite, V (y)>0 for all y!=0), and has continuous first-order partial derivatives at every point of D.

For conservative systems in physics, the total energy is always a Lyapunov function. Even for non-conservative systems, it is frequently important to look for energy-like ex-pressions as Lyapunov function candidates. One can say that Lyapunov functions have an explicit upper bound (zero) imposed on their increments along system trajectories:

Lyapunov analysis is used to study either the passive dynamics of a system or the dynamics of a closed-loop system (e.g. the plant and controller connected in a feedback loop). We will see generalizations of the Lyapunov function ideas to input-output systems later …

Sep 26, 2003 · Lecture 7: Finding Lyapunov Functions1 This lecture gives an introduction into basic methods for finding Lyapunov functions and storage functions for given dynamical systems.

Sep 19, 2003 · Lecture 5: Lyapunov Functions and Storage Functions 1 This lecture gives an introduction into system analysis using Lyapunov functions and their generalizations.

A Lyapunov function is a scalar function defined on the phase space, which can be used to prove the stability of an equilibrium point. The Lyapunov function method is applied to study the stability of various differential equations and systems.