Lyapunov function

Lyapunov function
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. There is no general technique for constructing Lyapunov functions for ODEs, however, depending on formulation type, a systematic method to construct Lyapunov functions for ordinary differential equations using their most general form in autonomous cases was given by Prof. Cem Civelek. Though, in many specific cases the construction of Lyapunov functions is known. For instance, according to a lot of applied mathematicians, for a dissipative gyroscopic system a Lyapunov function could not be constructed. However, using the method expressed in the publication above, even for such a system a Lyapunov function could be constructed as per related article by C. Civelek and Ö. Cihanbegendi. In addition, 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
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 for , and for which the time derivative is non positive (these conditions are required on some region containing the origin). The (stronger) condition that is strictly positive for is sometimes stated as is locally positive definite, or is locally negative definite.
Further discussion of the terms arising in the definition
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 systems
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 equilibrium
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 equilibrium
If is a Lyapunov function, then the equilibrium is Lyapunov stable. The converse is also true, and was proved by José Luis Massera.
Globally asymptotically stable equilibrium
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.)
Example
Consider the following differential equation 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 also
Lyapunov stability
Ordinary differential equations
Control-Lyapunov function
Chetaev function
Foster's theorem
Lyapunov optimization
References
Civelek, C. (2018). Archives of Control Sciences, volume 28 (LXIV), No. 2, pages 201–222 Doi:10.24425/123456
Civelek, C.; Cihanbeğendi, Ö. (2020). Frontiers of Information Technology & Electronic Engineering, volume 21, pages 629–634 Doi: 10.1631/FITEE.1900014
External links
Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
Stability theory