Lyapunov function

In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.

For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.

Definition of a Lyapunov functionEdit

A Lyapunov function for an autonomous dynamical system


with an equilibrium point at   is a scalar function   that is continuous, has continuous first derivatives, is locally positive-definite, and for which   is also locally positive definite. The condition that   is locally positive definite is sometimes stated as   is locally negative definite.

Further discussion of the terms arising in the definitionEdit

Lyapunov functions arise in the study of equilibrium points of dynamical systems. In   an arbitrary autonomous dynamical system can be written as


for some smooth  

An equilibrium point is a point   such that   Given an equilibrium point,   there always exists a coordinate transformation   such that:


Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at  .

By the chain rule, for any function,   the time derivative of the function evaluated along a solution of the dynamical system is


A function   is defined to be locally positive-definite function (in the sense of dynamical systems) if both   and there is a neighborhood of the origin,  , such that:


Basic Lyapunov theorems for autonomous systemsEdit

Let   be an equilibrium of the autonomous system


and use the notation   to denote the time derivative of the Lyapunov-candidate-function  :


Locally asymptotically stable equilibriumEdit

If the equilibrium is isolated, the Lyapunov-candidate-function   is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:


for some neighborhood   of origin then the equilibrium is proven to be locally asymptotically stable.

Stable equilibriumEdit

If   is a Lyapunov function, then the equilibrium is Lyapunov stable.

The converse is also true, and was proved by J. L. Massera.

Globally asymptotically stable equilibriumEdit

If the Lyapunov-candidate-function   is globally positive definite, radially unbounded, the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:


then the equilibrium is proven to be globally asymptotically stable.

The Lyapunov-candidate function   is radially unbounded if


(This is also referred to as norm-coercivity.)


Consider the following differential equation with solution   on  :


Considering that   is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study  . So let   on  . Then,


This correctly shows that the above differential equation,   is asymptotically stable about the origin. Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable.

See alsoEdit


  • Weisstein, Eric W. "Lyapunov Function". MathWorld.
  • Khalil, H.K. (1996). Nonlinear systems. Prentice Hall Upper Saddle River, NJ.
  • La Salle, Joseph; Lefschetz, Solomon (1961). Stability by Liapunov's Direct Method: With Applications. New York: Academic Press.
  • This article incorporates material from Lyapunov function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External linksEdit

  • Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function