Lyapunov Functions
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In the theory of
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
(ODEs), Lyapunov functions, named after
Aleksandr Lyapunov Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в, – 3 November 1918) was a Russian mathematician, mechanician and physicist. He was the son of the astronomer Mikhail Lyapunov and the brother of t ...
, are scalar functions that may be used to prove the stability of an
equilibrium Equilibrium may refer to: Film and television * ''Equilibrium'' (film), a 2002 science fiction film * '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film * "Equilibrium" (''seaQuest 2032'') * ''Equilibr ...
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 differ ...
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, such as in a parametric curve. Examples include the mathematical models ...
s and
control theory Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
. A similar concept appears in the theory of general state-space
Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
s 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 mass-energy, conservation of linear momen ...
s can often be used to construct Lyapunov functions for
physical system A physical system is a collection of physical objects under study. The collection differs from a set: all the objects must coexist and have some physical relationship. In other words, it is a portion of the physical universe chosen for analys ...
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, such as in a parametric curve. Examples include the mathematical models ...
:\beging:\R^n \to \R^n & \\ \dot = g(y) \end with an equilibrium point at y=0 is a
scalar function In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical q ...
V:\R^n\to\R that is continuous, has continuous first derivatives, is strictly positive for y\neq 0, and for which the time derivative \dot = \nabla\cdot g is non positive (these conditions are required on some region containing the origin). The (stronger) condition that -\nabla\cdot g is strictly positive for y\neq 0 is sometimes stated as -\nabla\cdot g is ''locally positive definite'', or \nabla\cdot g 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 \R^n, 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, such as in a parametric curve. Examples include the mathematical models ...
can be written as :\dot = g(y) for some smooth g:\R^n \to \R^n. An equilibrium point is a point y^* such that g\left(y^*\right) = 0. Given an equilibrium point, y^*, there always exists a coordinate transformation x = y - y^*, such that: :\begin \dot = \dot = g(y) = g\left(x + y^*\right) = f(x) \\ f(0) = 0 \end Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at 0. By the chain rule, for any function, H:\R^n \to \R, the time derivative of the function evaluated along a solution of the dynamical system is : \dot = \frac H(x(t)) = \frac\cdot \frac = \nabla H \cdot \dot = \nabla H\cdot f(x). A function H 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. Definition 1 Let \mathbb be the set of real numbers and \mathbb be the set of complex numbers. A function f: \mathbb \to \mathbb is ...
(in the sense of dynamical systems) if both H(0) = 0 and there is a neighborhood of the origin, \mathcal, such that: :H(x) > 0 \quad \forall x \in \mathcal \setminus\ .


Basic Lyapunov theorems for autonomous systems

Let x^* = 0 be an equilibrium point of the autonomous system :\dot = f(x). and use the notation \dot(x) to denote the time derivative of the Lyapunov-candidate-function V: :\dot(x) = \frac V(x(t)) = \frac\cdot \frac = \nabla V \cdot \dot = \nabla V\cdot f(x).


Locally asymptotically stable equilibrium

If the equilibrium point is isolated, the Lyapunov-candidate-function V is locally positive definite, and the time derivative of the Lyapunov-candidate-function is locally negative definite: :\dot(x) < 0 \quad \forall x \in \mathcal(0)\setminus\, for some neighborhood \mathcal(0) of origin, then the equilibrium is proven to be locally asymptotically stable.


Stable equilibrium

If V 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. ...
.


Globally asymptotically stable equilibrium

If the Lyapunov-candidate-function V is globally positive definite, radially unbounded, the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite: :\dot(x) < 0 \quad \forall x \in \R ^n\setminus\, 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.) The converse is also true, and was proved by
José Luis Massera José Luis Massera (Genoa, Italy, June 8, 1915 – Montevideo, September 9, 2002) was a Uruguayan dissident and mathematician who researched the stability of differential equations. Massera's lemma is named after him. He published over 40 pap ...
(see also
Massera's lemma In stability theory and nonlinear control, Massera's lemma, named after José Luis Massera, deals with the construction of the Lyapunov function to prove the stability of a dynamical system In mathematics, a dynamical system is a system in w ...
).


Example

Consider the following differential equation on \R: :\dot x = -x. Considering that x^2 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^2 on \R . Then, :\dot V(x) = V'(x) \dot x = 2x\cdot (-x) = -2x^2< 0. This correctly shows that the above differential equation, x, 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 (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
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 t ...
*
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 ...
* Lyapunov optimization


References

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External links


Example
of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function {{Authority control Stability theory