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.
Contents
- Definition of a Lyapunov function
- Further discussion of the terms arising in the definition
- Locally asymptotically stable equilibrium
- Stable equilibrium
- Globally asymptotically stable equilibrium
- Example
- References
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 function
A Lyapunov function for an autonomous dynamical system
with an equilibrium point at
Further discussion of the terms arising in the definition
Lyapunov functions arise in the study of equilibrium points of dynamical systems. In
for some smooth
An equilibrium point is a point
Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at
By the chain rule, for any function,
A function
Let
be an equilibrium of the autonomous system
and use the notation
which is the time derivative of the Lyapunov-candidate-function
Locally asymptotically stable equilibrium
If
The converse is also true, and was proved by J. L. Massera.
Stable equilibrium
If the Lyapunov-candidate-function
for some neighborhood
Globally asymptotically stable equilibrium
If the Lyapunov-candidate-function
then the equilibrium is proven to be globally asymptotically stable.
The Lyapunov-candidate function
(This is also referred to as norm-coercivity.)
Example
Consider the following differential equation with solution
Considering that
This correctly shows that the above differential equation,