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.
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 strictly positive, and for which is also strictly positive. The condition that is strictly positive is sometimes stated as is "locally positive definite", or is "locally negative definite".
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:
Let be an equilibrium of the autonomous system
and use the notation to denote the time derivative of the Lyapunov-candidate-function :
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.
If is a Lyapunov function, then the equilibrium is Lyapunov stable.
The converse is also true, and was proved by J. L. Massera.
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.