LaSalle's invariance principle (also known as the invariance principle,
Barbashin-Krasovskii-LaSalle principle,
or Krasovskii-LaSalle principle) is a criterion for the
asymptotic 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. T ...
of an autonomous (possibly nonlinear)
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 ...
.
Global version
Suppose a system is represented as
:
where
is the vector of variables, with
:
If a
(see
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
)
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
can be found such that
:
for all
(negative semidefinite),
then the set of
accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s of any trajectory is contained in
where
is the union of complete trajectories contained entirely in the set
.
If we additionally have that the function
is positive definite, i.e.
:
, for all
:
and if
contains no trajectory of the system except the trivial trajectory
for
, then the origin is
asymptotically 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 ...
.
Furthermore, if
is radially unbounded, i.e.
:
, as
then the origin is globally
asymptotically 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 ...
.
Local version
If
:
, when
:
hold only for
in some neighborhood
of the origin, and the set
:
does not contain any trajectories of the system besides the trajectory
, then the local version of the invariance principle states that the origin is locally
asymptotically 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 ...
.
Relation to Lyapunov theory
If
is negative definite, then the global asymptotic stability of the origin is a consequence of
Lyapunov's second theorem. The invariance principle gives a criterion for asymptotic stability in the case when
is only negative semidefinite.
Examples
Simple example
Example taken from.
Consider the vector field
in the plane. The function
satisfies
, and is radially unbounded, showing that the origin is globally asymptotically stable.
Pendulum with friction
This section will apply the invariance principle to establish the local
asymptotic 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. T ...
of a simple system, the pendulum with friction. This system can be modeled with the differential equation
[Lecture notes on nonlinear control](_blank)
University of Notre Dame, Instructor: Michael Lemmon, lecture 4.
:
where
is the angle the pendulum makes with the vertical normal,
is the mass of the pendulum,
is the length of the pendulum,
is the
friction coefficient
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
, and
''g'' is acceleration due to gravity.
This, in turn, can be written as the system of equations
:
:
Using the invariance principle, it can be shown that all trajectories that begin in a ball of certain size around the origin
asymptotically converge to the origin. We define
as
:
This
is simply the scaled energy of the system.
Clearly,
is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
in an open ball of radius
around the origin. Computing the derivative,
:
Observe that
. If it were true that
, we could conclude that every trajectory approaches the origin by
Lyapunov's second theorem. Unfortunately,
and
is only
negative semidefinite since
can be non-zero when
. However, the set
:
which is simply the set
:
does not contain any trajectory of the system, except the trivial trajectory x = 0. Indeed, if at some time
,
, then because
must be less than
away from the origin,
and
. As a result, the trajectory will not stay in the set
.
All the conditions of the local version of the invariance principle are satisfied, and we can conclude that every trajectory that begins in some neighborhood of the origin will converge to the origin as
.
History
The general result was independently discovered by
J.P. LaSalle (then at
RIAS) and
N.N. Krasovskii, who published in 1960 and 1959 respectively. While
LaSalle was the first author in the West to publish the general theorem in 1960, a special case of the theorem was communicated in 1952 by Barbashin and
Krasovskii, followed by a publication of the general result in 1959 by
Krasovskii.
[Vidyasagar, M. ''Nonlinear Systems Analysis,'' SIAM Classics in Applied Mathematics, SIAM Press, 2002.]
See also
*
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 ...
*
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. ...
Original papers
*
LaSalle, J.P. ''Some extensions of Liapunov's second method,'' IRE Transactions on Circuit Theory, CT-7, pp. 520–527, 1960.
PDF)
*
* Krasovskii, N. N. ''Problems of the Theory of Stability of Motion,'' (Russian), 1959. English translation: Stanford University Press, Stanford, CA, 1963.
Text books
*
*
*
*
Lectures
*
Texas A&M University
Texas A&M University (Texas A&M, A&M, or TAMU) is a public, land-grant, research university in College Station, Texas. It was founded in 1876 and became the flagship institution of the Texas A&M University System in 1948. As of late 2021, T ...
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*
NC State University
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notes on LaSalle's invariance principle
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*
Caltech
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notes on LaSalle's invariance principle
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*
MIT
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OpenCourseware notes on Lyapunov stability analysis and the invariance principle
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*
Purdue University
Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money ...
notes on stability theory and LaSalle's invariance principle
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References
{{DEFAULTSORT:Krasovskii-LaSalle principle
Stability theory
Dynamical systems
Principles