Lyapunov function

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, ^[1] 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.

A similar concept appears in the theory of general state space Markov Chains, usually under the name Foster-Lyapunov functions.

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.

Definition of a Lyapunov candidate function

Let

$V:\mathbb{R}^n \to \mathbb{R}$

be a continuous scalar function.
$V$ is a Lyapunov-candidate-function if it is a locally positive-definite function, i.e.

$V(0) = 0 \,$

$V(x) > 0 \quad \forall x \in U\setminus\{0\}$

with $U$ being a neighborhood region around $x = 0.$

Definition of the equilibrium point of a system

Let

$g : \mathbb{R}^n \to \mathbb{R}^n$

$\dot{y} = g(y) \,$

be an arbitrary autonomous dynamical system with equilibrium point $y^* \,$:

$0 = g(y^*). \,$

There always exists a coordinate transformation $x = y - y^* \,$, such that:

$\dot{x} = \dot{y} = g(y) = g(x + y^*) = f(x) \,$

$f(0) = 0. \,$

So the new system $f(x)$ has an equilibrium point at the origin.

Basic Lyapunov theorems for autonomous systems

Let

$x^* = 0 \,$

be an equilibrium of the autonomous system

$\dot{x} = f(x). \,$

And let

$\dot{V}(x) = \frac{d}{dt} V(x(t)) = \frac{\partial V}{\partial x}\cdot \frac{dx}{dt} = \nabla V \cdot \dot{x} = \nabla V\cdot f(x)$

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

Stable equilibrium

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

$\dot{V}(x) \leq 0 \quad \forall x \in \mathcal{B}\setminus\{0\}$

for some neighborhood $\mathcal{B}$ of $0$, then the equilibrium is proven to be stable.

Locally asymptotically stable equilibrium

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

$\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}\setminus\{0\}$

for some neighborhood $\mathcal{B}$ of $0$, 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

If the Lyapunov-candidate-function $V$ is globally positive definite, radially unbounded and the time derivative of the Lyapunov-candidate-function is globally negative definite:

$\dot{V}(x) < 0 \quad \forall x \in \mathbb{R}^n\setminus\{0\},$

then the equilibrium is proven to be globally asymptotically stable.

The Lyapunov-candidate function $V(x)$ is radially unbounded if

$\| x \| \to \infty \Rightarrow V(x) \to \infty.$

(This is also referred to as norm-coercivity.)

Example

Consider the following differential equation with solution $x$ on $\mathbb{R}$:

$\dot x = -x.$

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 $V(x)=|x|$ on $\mathbb{R}\setminus\{0\}$. Then,

$\dot V(x) = V'(x) f(x) = \mathrm{sgn}(x)\cdot (-x) = -|x|<0.$

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.

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 ^[2] was posted on ArXiv proposing a solution to this old problem considered insurmountable by many researchers. Two years later the monograph ^[3] 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

${dx \over dt} = f(t, x)$

with some integral curve $x_t = (t, x(t, x_0))$ under investigation for stability in the terms of foliations, where $x = (x_1, ...,x_n )$ is phase vector, $f(t, x) = (f_1(t, x), ...,f_n(t, x))$ is vector field, $t\in[t_0; +\infty[$ is time, $x(t, x_0)$ is a particular solution to the original system with a given initial point of the phase trajectory $x_0$.

2. Extension of the Lyapunov’s transformation $\vartheta$ or change of phase variables rectifying only the integral curve $x_t$ via

$\vartheta:z\longrightarrow{x} = (z + x(t, x_0))$

to a new transformation flattening the n entire n-dimensional invariant manifolds or hypersurfaces forming the one by intersecting each other, where $z = (z_1, ..., z_n)$. The new transformation is called the cascade of sequential flattening diffeomorphisms resulting in the canonizing diffeomorphism denoted as $x = \psi(y)$, where $y = (y_1, ..., y_n)$. 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 $\{p_1, ..., p_n\}$ 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 $\psi:\{{dx_1 \over dt} = f(t, x), {dx_2 \over dt} = f(t, x)\}\longleftrightarrow\{{dy_1 \over dt} = f_1^2(t, y), {dy_2 \over dt} = f_2^2(t, y)\}$,

where $x = (x_1, x_2)$, $y = (y_1, y_2)$, $f_1^2(t, y)\equiv0$ if $y_1 = 0$ and $f_2^2(t, y)\equiv0$ if $y_2 = 0$.

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 $p_1:S_1^+{\cup}S_1^0{\cup}S_1^-{\longrightarrow}Y_1^+{\cup}Y_1^0{\cup}Y_1^+$ .

See also

• Ordinary differential equations
• Control-Lyapunov function
• Foster's theorem
• Lyapunov optimization

References

1. Lyapunov, A. M. (1992-08-28). General Problem of the Stability Of Motion. CRC Press. ISBN 9780748400621.
2. 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].
3. 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 edition 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.

External links

• Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
• Some Lyapunov diagrams