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 functions (also called 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 lawss can often be used to construct Lyapunov functions for physical systems.

Definition

A Lyapunov function for an autonomous dynamical system

is a scalar function is locally positive definite is locally negative definite.

Further discussion of the terms arising in the definition

an arbitrary autonomous dynamical system can be written as

such that:

.

the time derivative of the function evaluated along a solution of the dynamical system is

is defined to be locally positive-definite function, such that:

Basic Lyapunov theorems for autonomous systems

be an equilibrium point of the autonomous system

:

Locally asymptotically stable equilibrium

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

of origin, then the equilibrium is proven to be locally asymptotically stable.

Stable equilibrium

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

Globally asymptotically stable equilibrium

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.

is radially unbounded if

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

The converse is also true, and was proved by José Luis Massera (see also Massera's lemma).

Example

:

. Then,

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 also

• Lyapunov stability

• Ordinary differential equations

• Control-Lyapunov function

• Chetaev function

• Foster's theorem

• Lyapunov optimization

References

External links

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