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, who introduced them in his doctoral thesis General Problem of the Stability of Motion, the method of Lyapunov functions (also called the Lyapunov’s second method for stability) is important to stability theory of dynamical systems and control theory. Actually, it is the only universal method for the investigation of the stability of nonlinear dynamical systems of general configuration.
For many classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability.
Informally, a Lyapunov function is a function that takes positive values everywhere except at any stasis in question, and decreases (or is non-increasing) along every trajectory of the ODE. The principal merit of Lyapunov function-based stability analysis of ODEs is that the actual solution (whether analytical or numerical) of the ODE is not required.
- 1 Definition of a Lyapunov candidate function
- 2 Definition of the equilibrium point of a system
- 3 Basic Lyapunov theorems for autonomous systems
- 4 Example
- 5 Canonical generalization of Lyapunov's second method or general procedure of utilization of Lyapunov functions
- 6 See also
- 7 References
- 8 External links
Definition of a Lyapunov candidate function
with being a neighborhood region around
Definition of the equilibrium point of a system
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
Main article: Lyapunov stability
be an equilibrium of the autonomous system
be the time derivative of the Lyapunov-candidate-function .
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
If the Lyapunov-candidate-function is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:
Globally asymptotically stable equilibrium
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.)
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.
Canonical generalization of Lyapunov's second method or general procedure of utilization of Lyapunov functions
Although until 2014 there had been no a general procedure developed for constructing Lyapunov functions for ODEs but in a number of specific cases, the construction of Lyapunov functions had been 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. In 2014 the paper  was posted on ArXiv proposing a solution to this old problem considered insurmountable by many researchers. Two years later the monograph  presenting it as a generalization of the Lyapunov’s second method for stability came out. Actually, its concept is based on three key ideas, namely
1. The representation of the extended phase space of an original (n+1)-dimensional non-autonomous dynamical system
with some integral curve under investigation for stability in the terms of foliations, where is phase vector, is vector field, is time, is a particular solution to the original system with a given initial point of the phase trajectory .
2. Extension of the Lyapunov’s transformation or change of phase variables rectifying only the integral curve via
to a new transformation flattening the n entire n-dimensional invariant manifolds or hypersurfaces forming the one by intersecting each other, where . The new transformation is called the cascade of sequential flattening diffeomorphisms resulting in the canonizing diffeomorphism denoted as , where . The last name means that the original dynamic system under its action assumes a special form called canonical where all n invariant hypersurfaces turn in the n corresponding invariant hyperplanes.
3. The topological classification of the covering maps of the coverings associated with the ODEs describing the canonical form of the original system.
Figures A, B, C, D illustrate the ideas for a 3D original system and its canonical form governed by the following systems of equations ,
where , , if and if .
The first idea is graphically explained by Figure A. The second one is illustrated with Figures B and C. The third idea is expressed by Figure D, where .
- Lyapunov, A. M. (1992-08-28). General Problem of the Stability Of Motion. CRC Press. ISBN 9780748400621.
- Sparavalo, Myroslav (2014-03-23). "The Lyapunov Concept of Stability from the Standpoint of Poincare Approach: General Procedure of Utilization of Lyapunov Functions for Non-Linear Non-Autonomous Parametric Differential Inclusions". arXiv:1403.5761 [cs.SY].
- Sparavalo, Myroslav K. (2016-04-19). Lyapunov Functions in Nonlinear Unsteady Dynamics and Control: Poincaré's Approach from Metaphysical Theory to Down-to-Earth Practice (1 ed.). Myroslav K. Sparavalo. ISBN 9780692694244.
- Weisstein, Eric W., "Lyapunov Function", MathWorld.
- Khalil, H.K. (1996). Nonlinear systems. Prentice Hall Upper Saddle River, NJ.
- This article incorporates material from Lyapunov function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.