In the theory of
ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
(ODEs), Lyapunov functions, named after
Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Lia ...
, 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
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial diffe ...
of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s and
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
. 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
In convex optimization, a linear matrix inequality (LMI) is an expression of the form
: \operatorname(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\succeq 0\,
where
* y= _i\,,~i\!=\!1,\dots, m/math> is a real vector,
* A_0, A_1, A_2,\dots,A_m are n\times n ...
provides Lyapunov functions for linear systems; and
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
s can often be used to construct Lyapunov functions for
physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
s.
Definition
A Lyapunov function for an autonomous
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
:
with an equilibrium point at
is a
scalar function
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ( ...
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
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
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 mathematics, a positive-definite function is, depending on the context, either of two types of function.
Most common usage
A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such ...
(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
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. T ...
. The converse is also true, and was proved by
J. L. 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
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...
*
Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s
*
Control-Lyapunov function In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is ''(Lyapunov) stable'' or (more ...
*
Chetaev function The Chetaev instability theorem for dynamical systems states that if there exists, for the system \dot = X(\textbf) with an equilibrium point at the origin, a continuously differentiable function V(x) such that
# the origin is a boundary point of th ...
*
Foster's theorem
In probability theory, Foster's theorem, named after Gordon Foster, is used to draw conclusions about the positive recurrence of Markov chains with countable state spaces. It uses the fact that positive recurrent Markov chains exhibit a notion o ...
*
Lyapunov optimization This article describes Lyapunov optimization for dynamical systems. It gives an example application to optimal control in queueing networks.
Introduction
Lyapunov optimization refers to the use of a Lyapunov function to optimally control a dynam ...
References
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*
*
*
External links
Exampleof determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
{{Authority control
Stability theory