李雅普诺夫函数

在常微分方程理论中，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 scalar function。李雅普诺夫函数是以俄罗斯数学家亚历山大·李亚普诺夫 Aleksandr Lyapunov的名字命名，也称为稳定性的李雅普诺夫第二方法。对于动态系统 dynamical system的稳定性理论 stability theory和控制论 control theory十分重要。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，但在许多特定情况下，李雅普诺夫函数的构造是已知的。例如，二次函数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。

李雅普诺夫函数的定义

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]

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 A 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 B is also locally positive definite. The condition that C is locally positive definite is sometimes stated as D 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：[math]\displaystyle{ V:\R^n\to\R }[/math]，该函数连续、具有连续的一阶导数且局部正定，[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。

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

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，任意一个自治动力系统都可以被写成：

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

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

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]

因此在研究平衡点时，只要需要假设平衡点出现在[math]\displaystyle{ 0 }[/math]处。

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]

函数 [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 [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]

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

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

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

并使用[math]\displaystyle{ \dot{V}(x) }[/math]表示李雅普诺夫候选函数 Lyapunov-candidate-function[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]

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

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，其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]

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]，可以证明平衡点是局部渐近稳定的。

稳定平衡点 Stable equilibrium

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.

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