Lyapunov function

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 candidate function[edit]

Let

be a continuous scalar function.

is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.

with being some neighbourhood around .

Definition of the equilibrium point of a system[edit]

Let

be an arbitrary autonomous dynamical system with equilibrium point :

There always exists a coordinate transformation , such that:

So the new system has an equilibrium point at the origin.

Basic Lyapunov theorems for autonomous systems[edit]

Main article: Lyapunov stability

Let

be an equilibrium of the autonomous system

And let

be the time derivative of the Lyapunov-candidate-function .

Stable equilibrium[edit]

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

for some neighborhood of , then the equilibrium is proven to be stable.

Locally asymptotically stable equilibrium[edit]

If 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 , then the equilibrium is proven to be locally asymptotically stable. The converse is also true, and was proved by J. L. Massera.

Globally asymptotically stable equilibrium[edit]

If the Lyapunov-candidate-function is globally positive definite, radially unbounded 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.)

Example[edit]

Consider the following differential equation with solution on :

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

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

See also[edit]

References[edit]

External links[edit]