李雅普诺夫函数

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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, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.

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, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.

在常微分方程 ordinary differential equations理论中，李雅普诺夫函数 Lyapunov functions是可用来证明常微分方程平衡点 equilibrium point稳定性的标量函数。以俄罗斯数学家亚历山大·李亚普诺夫的名字命名的李雅普诺夫函数(也称为稳定性的李雅普诺夫第二种方法)对于动态系统 dynamical system的稳定性理论 stability theory和控制论 control theory是很重要的。在一般状态空间马尔可夫链 Markov chains理论中也出现了类似的概念，通常称为 Foster-Lyapunov 函数。

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 provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.

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 provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.

对于某些类型的常微分方程，李雅普诺夫函数的存在性是其稳定性的充要条件。尽管对于常微分方程没有构造李雅普诺夫函数的一般方法，但在许多具体情况下，李雅普诺夫函数的构造是已知的。例如，二次函数 quadratic function满足一个状态的系统的李雅普诺夫函数；一个特定线性矩阵不等式的解为线性系统提供了李雅普诺夫函数；守恒律 Conservation law通常可以用来构造物理系统的李雅普诺夫函数。

Definition of a Lyapunov function 李雅普诺夫函数的定义

A Lyapunov function for an autonomous dynamical system

A Lyapunov function for an autonomous dynamical system

对于一个自治动力系统

[math]\displaystyle{ \begin{cases} g : \R ^n \to \R ^n \\ \dot{y} = g(y) \end{cases} }[/math]

[math]\displaystyle{ \begin{cases} g : \R ^n \to \R ^n \\ \dot{y} = g(y) \end{cases} }[/math]

开始{ cases } g: r ^ n to r ^ n dot { y } g (y) end { cases } / math

with an equilibrium point at [math]\displaystyle{ y=0 }[/math] is a scalar function [math]\displaystyle{ V:\R^n\to\R }[/math] that is continuous, has continuous first derivatives, is locally positive-definite, and for which [math]\displaystyle{ -\nabla{V}\cdot g }[/math] is also locally positive definite. The condition that [math]\displaystyle{ -\nabla{V}\cdot g }[/math] is locally positive definite is sometimes stated as [math]\displaystyle{ \nabla{V}\cdot g }[/math] is locally negative definite.

with an equilibrium point at [math]\displaystyle{ y=0 }[/math] is a scalar function [math]\displaystyle{ V:\R^n\to\R }[/math] that is continuous, has continuous first derivatives, is locally positive-definite, and for which [math]\displaystyle{ -\nabla{V}\cdot g }[/math] is also locally positive definite. The condition that [math]\displaystyle{ -\nabla{V}\cdot g }[/math] is locally positive definite is sometimes stated as [math]\displaystyle{ \nabla{V}\cdot g }[/math] is locally negative definite.

[math]\displaystyle{ y=0 }[/math] 是它的一个平衡点，其李雅普诺夫函数是一个标量函数 scalar function：[math]\displaystyle{ V:\R^n\to\R }[/math]，它是连续的并且有连续的一阶导数，且是局部正定的，以及[math]\displaystyle{ -\nabla{V}\cdot g }[/math]也是局部正定的。有时把 [math]\displaystyle{ -\nabla{V}\cdot g }[/math] 局部正定的条件表述为 [math]\displaystyle{ \nabla{V}\cdot g }[/math] 是局部负定的。

--Jxzhou（讨论）上述英文和维基百科不同，暂且按本文的英文翻译。

Further discussion of the terms arising in the definition 定义中出现的术语的进一步讨论

Lyapunov functions arise in the study of equilibrium points of dynamical systems. In [math]\displaystyle{ \R^n, }[/math] an arbitrary autonomous dynamical system can be written as

Lyapunov functions arise in the study of equilibrium points of dynamical systems. In [math]\displaystyle{ \R^n, }[/math] an arbitrary autonomous dynamical system can be written as

李雅普诺夫函数出现在动力系统平衡点的研究中。在 [math]\displaystyle{ \R^n, }[/math] 空间中，一个任意的自治动力系统可以被写成

[math]\displaystyle{ \dot{y} = g(y) }[/math]

[math]\displaystyle{ \dot{y} = g(y) }[/math]

数学(y) / 数学

for some smooth [math]\displaystyle{ g:\R^n \to \R^n. }[/math]

for some smooth [math]\displaystyle{ g:\R^n \to \R^n. }[/math]

对于一些平滑的函数 [math]\displaystyle{ g:\R^n \to \R^n. }[/math]

An equilibrium point is a point [math]\displaystyle{ y^* }[/math] such that [math]\displaystyle{ g(y^*)=0. }[/math] Given an equilibrium point, [math]\displaystyle{ y^*, }[/math] there always exists a coordinate transformation [math]\displaystyle{ x = y - y^*, }[/math] such that:

An equilibrium point is a point [math]\displaystyle{ y^* }[/math] such that [math]\displaystyle{ g(y^*)=0. }[/math] Given an equilibrium point, [math]\displaystyle{ y^*, }[/math] there always exists a coordinate transformation [math]\displaystyle{ x = y - y^*, }[/math] such that:

平衡点是一个满足[math]\displaystyle{ g(y^*)=0. }[/math]的点[math]\displaystyle{ y^* }[/math]。给定一个平衡点[math]\displaystyle{ y^*, }[/math]，总是存在一个坐标变换[math]\displaystyle{ x = y - y^*, }[/math]，使得：

[math]\displaystyle{ \begin{cases} \dot{x} = \dot{y} = g(y) = g(x + y^*) = f(x) \\ f(0) = 0 \end{cases} }[/math]

[math]\displaystyle{ \begin{cases} \dot{x} = \dot{y} = g(y) = g(x + y^*) = f(x) \\ f(0) = 0 \end{cases} }[/math]

数学开始{情况} dot { y } g (x + y ^ *) f (x) f (0)0 end { cases } / math

Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at [math]\displaystyle{ 0 }[/math].

Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at [math]\displaystyle{ 0 }[/math].

因此在研究平衡点时，只要假设平衡点出现在[math]\displaystyle{ 0 }[/math]处就足够了。

By the chain rule, for any function, [math]\displaystyle{ H:\R^n \to \R, }[/math] the time derivative of the function evaluated along a solution of the dynamical system is

By the chain rule, for any function, [math]\displaystyle{ H:\R^n \to \R, }[/math] the time derivative of the function evaluated along a solution of the dynamical system is

根据链式法则，对于任意函数[math]\displaystyle{ H:\R^n \to \R, }[/math]，函数沿动力学系统解取值的时间导数为

[math]\displaystyle{ \dot{H} = \frac{d}{dt} H(x(t)) = \frac{\partial H}{\partial x}\cdot \frac{dx}{dt} = \nabla H \cdot \dot{x} = \nabla H\cdot f(x). }[/math]

[math]\displaystyle{ \dot{H} = \frac{d}{dt} H(x(t)) = \frac{\partial H}{\partial x}\cdot \frac{dx}{dt} = \nabla H \cdot \dot{x} = \nabla H\cdot f(x). }[/math]

Math dot { h frac { dt } h (x (t)) frac { partial x } cdot frac { dt nabla h nabla h dot f (x) . / math

A function [math]\displaystyle{ H }[/math] is defined to be locally positive-definite function (in the sense of dynamical systems) if both [math]\displaystyle{ H(0) = 0 }[/math] and there is a neighborhood of the origin, [math]\displaystyle{ \mathcal{B} }[/math], such that:

A function [math]\displaystyle{ H }[/math] is defined to be locally positive-definite function (in the sense of dynamical systems) if both [math]\displaystyle{ H(0) = 0 }[/math] and there is a neighborhood of the origin, [math]\displaystyle{ \mathcal{B} }[/math], such that:

函数数学 [math]\displaystyle{ H }[/math] 被定义为局部正定函数 positive-definite function(在动力系统的意义上)，如果数学 [math]\displaystyle{ H(0) = 0 }[/math]并且有一个邻域[math]\displaystyle{ \mathcal{B} }[/math]使得

[math]\displaystyle{ H(x) \gt 0 \quad \forall x \in \mathcal{B} \setminus\{0\} . }[/math]

[math]\displaystyle{ H(x) \gt 0 \quad \forall x \in \mathcal{B} \setminus\{0\} . }[/math]

数学 h (x)0对于数学中的所有 x-0. /

Basic Lyapunov theorems for autonomous systems 自治系统的基本李雅普诺夫定理

Let [math]\displaystyle{ x^* = 0 }[/math] be an equilibrium of the autonomous system

Let [math]\displaystyle{ x^* = 0 }[/math] be an equilibrium of the autonomous system

令[math]\displaystyle{ x^* = 0 }[/math]是如下自治系统的平衡点

[math]\displaystyle{ \dot{x} = f(x). }[/math]

[math]\displaystyle{ \dot{x} = f(x). }[/math]

Math x } f (x) . / math

and use the notation [math]\displaystyle{ \dot{V}(x) }[/math] to denote the time derivative of the Lyapunov-candidate-function [math]\displaystyle{ V }[/math]:

and use the notation [math]\displaystyle{ \dot{V}(x) }[/math] to denote the time derivative of the Lyapunov-candidate-function [math]\displaystyle{ V }[/math]:

并使用记号[math]\displaystyle{ \dot{V}(x) }[/math]表示李雅普诺夫候选函数[math]\displaystyle{ V }[/math]的时间导数:

[math]\displaystyle{ \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). }[/math]

[math]\displaystyle{ \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). }[/math]

(x) frac { dt } v (x (t)) frac { partial x } dot frac { dt } nabla v nabla v nabla f (x) / math

Locally asymptotically stable equilibrium 局部渐进稳定平衡点

If the equilibrium is isolated, the Lyapunov-candidate-function [math]\displaystyle{ V }[/math] is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:

If the equilibrium is isolated, the Lyapunov-candidate-function [math]\displaystyle{ V }[/math] is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative definite:

如果平衡点是孤立的，李雅普诺夫候选函数[math]\displaystyle{ V }[/math]是局部正定的，并且李雅普诺夫候选函数的时间导数是局部负定的：

[math]\displaystyle{ \dot{V}(x) \lt 0 \quad \forall x \in \mathcal{B}\setminus\{0\} }[/math]

[math]\displaystyle{ \dot{V}(x) \lt 0 \quad \forall x \in \mathcal{B}\setminus\{0\} }[/math]

对于数学中的所有 x-0-0 / math

for some neighborhood [math]\displaystyle{ \mathcal{B} }[/math] of origin then the equilibrium is proven to be locally asymptotically stable.

for some neighborhood [math]\displaystyle{ \mathcal{B} }[/math] of origin then the equilibrium is proven to be locally asymptotically stable.

对于原点的某些邻域[math]\displaystyle{ \mathcal{B} }[/math]，那么可以证明平衡点是局部渐近稳定的。

Stable equilibrium 稳定平衡点

If [math]\displaystyle{ V }[/math] is a Lyapunov function, then the equilibrium is Lyapunov stable.

If [math]\displaystyle{ V }[/math] is a Lyapunov function, then the equilibrium is Lyapunov stable.

如果[math]\displaystyle{ V }[/math]是李雅普诺夫函数，那么平衡点是李雅普诺夫稳定的。

The converse is also true, and was proved by J. L. Massera.

The converse is also true, and was proved by J. L. Massera.

反之亦然，J. L. Massera证明了这一点。

Globally asymptotically stable equilibrium 全局渐进稳定平衡点

If the Lyapunov-candidate-function [math]\displaystyle{ V }[/math] is globally positive definite, radially unbounded, the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:

If the Lyapunov-candidate-function [math]\displaystyle{ V }[/math] is globally positive definite, radially unbounded, the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:

如果李雅普诺夫候选函数数学[math]\displaystyle{ V }[/math] 是全局正定、径向无界的，平衡点是孤立的并且李雅普诺夫候选函数的时间导数是全局负定的：

[math]\displaystyle{ \dot{V}(x) \lt 0 \quad \forall x \in \R ^n\setminus\{0\}, }[/math]

[math]\displaystyle{ \dot{V}(x) \lt 0 \quad \forall x \in \R ^n\setminus\{0\}, }[/math]

对于所有的 x 在 r ^ n setminus 0，/ math

then the equilibrium is proven to be globally asymptotically stable.

then the equilibrium is proven to be globally asymptotically stable.

那么可以证明平衡点是全局渐近稳定的。

The Lyapunov-candidate function [math]\displaystyle{ V(x) }[/math] is radially unbounded if

The Lyapunov-candidate function [math]\displaystyle{ V(x) }[/math] is radially unbounded if

李雅普诺夫候选函数[math]\displaystyle{ V(x) }[/math]是径向无界的如果满足

[math]\displaystyle{ \| x \| \to \infty \Rightarrow V(x) \to \infty. }[/math]

[math]\displaystyle{ \| x \| \to \infty \Rightarrow V(x) \to \infty. }[/math]

数学下划线 v (x)下划线 v (x)。数学

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

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

(这也被称为范数强制。)

Example 例子

Consider the following differential equation with solution [math]\displaystyle{ x }[/math] on [math]\displaystyle{ \R }[/math]:

Consider the following differential equation with solution [math]\displaystyle{ x }[/math] on [math]\displaystyle{ \R }[/math]:

考虑下面具有[math]\displaystyle{ \R }[/math]上的解[math]\displaystyle{ x }[/math]的微分方程：

[math]\displaystyle{ \dot x = -x. }[/math]

[math]\displaystyle{ \dot x = -x. }[/math]

Math dot x-x / math

Considering that [math]\displaystyle{ x^2 }[/math] is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study [math]\displaystyle{ x }[/math].

Considering that [math]\displaystyle{ x^2 }[/math] is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study [math]\displaystyle{ x }[/math].

考虑到[math]\displaystyle{ x^2 }[/math]在原点附近总是正的，这是一个帮助我们研究[math]\displaystyle{ x }[/math]性质的自然的李雅普诺夫候选函数。

So let [math]\displaystyle{ V(x)=x^2 }[/math] on [math]\displaystyle{ \R }[/math]. Then,

So let [math]\displaystyle{ V(x)=x^2 }[/math] on [math]\displaystyle{ \R }[/math]. Then,

令[math]\displaystyle{ V(x)=x^2 }[/math]在[math]\displaystyle{ \R }[/math]上。然后有

[math]\displaystyle{ \dot V(x) = V'(x) \dot x = 2x\cdot (-x) = -2x^2\lt 0. }[/math]

[math]\displaystyle{ \dot V(x) = V'(x) \dot x = 2x\cdot (-x) = -2x^2\lt 0. }[/math]

math dot v (x) v’(x) dot x2x cdot (- x-rrb--2 x ^ 20. / math

This correctly shows that the above differential equation, [math]\displaystyle{ x, }[/math] is asymptotically stable about the origin. Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable.

This correctly shows that the above differential equation, [math]\displaystyle{ x, }[/math] is asymptotically stable about the origin. Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable.

这正表明上面的微分方程 [math]\displaystyle{ x }[/math]关于原点是渐近稳定的。注意，使用相同的李雅普诺夫候选函数可以证明该平衡点也是全局渐近稳定的。

See also 参见

• Lyapunov stability 李雅普诺夫稳定性

• Ordinary differential equations 常微分方程

• Control-Lyapunov function 控制李雅普诺夫函数

• Chetaev function 切塔耶夫函数

• Foster's theorem 福斯特定律

• Lyapunov optimization 李雅普诺夫优化

References

• 模板:Mathworld

• Khalil, H.K. (1996). Nonlinear systems. Prentice Hall Upper Saddle River, NJ. 

• La Salle, Joseph; Lefschetz, Solomon (1961). Stability by Liapunov's Direct Method: With Applications. New York: Academic Press. 

• 模板:PlanetMath attribution

External links

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

Category:Stability theory

范畴: 稳定性理论

This page was moved from wikipedia:en:Lyapunov function. Its edit history can be viewed at 李雅普诺夫函数/edithistory