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第5行： − 在常微分方程理论中，李雅普诺夫函数是可用来证明常微分方程平衡点稳定性的标量函数。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(也称为李雅普诺夫稳定性的第二种方法)对于动态系统的稳定性理论和控制论是很重要的。在一般状态空间马尔可夫链理论中也出现了类似的概念，通常称为 Foster-Lyapunov 函数。+ 第13行：
第13行： − 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；守恒律通常可以用来构造物理系统的李雅普诺夫函数。+ 第39行：
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无编辑摘要
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<font color="#ff8000">常微分方程 ordinary differential equations</font>理论中，<font color="#ff8000">李雅普诺夫函数 Lyapunov functions</font>是可用来证明常微分方程<font color="#ff8000">平衡点 equilibrium point</font>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(也称为李雅普诺夫稳定性的第二种方法)对于<font color="#ff8000">动态系统 dynamical system</font>的<font color="#ff8000">稳定性理论 stability theory</font>和<font color="#ff8000">控制论 control theory</font>是很重要的。在一般状态空间<font color="#ff8000">马尔可夫链 Markov chains</font>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，<font color="#ff8000">二次函数 quadratic function</font>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；<font color="#ff8000">守恒律 Conservation law</font>通常可以用来构造物理系统的李雅普诺夫函数。
with an equilibrium point at <math>y=0</math> is a scalar function <math>V:\R^n\to\R</math> that is continuous, has continuous first derivatives, is locally positive-definite, and for which <math>-\nabla{V}\cdot g</math> is also locally positive definite. The condition that <math>-\nabla{V}\cdot g</math> is locally positive definite is sometimes stated as <math>\nabla{V}\cdot g</math> is locally negative definite.
with an equilibrium point at <math>y=0</math> is a scalar function <math>V:\R^n\to\R</math> that is continuous, has continuous first derivatives, is locally positive-definite, and for which <math>-\nabla{V}\cdot g</math> is also locally positive definite. The condition that <math>-\nabla{V}\cdot g</math> is locally positive definite is sometimes stated as <math>\nabla{V}\cdot g</math> is locally negative definite.
<math>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>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>-\nabla{V}\cdot g</math>也是局部正定的。有时把 <math>-\nabla{V}\cdot g</math> 局部正定的条件表述为 <math>\nabla{V}\cdot g</math> 是局部负定的。
<math>y=0</math> 是它的一个平衡点，其李雅普诺夫函数是一个<font color="#ff8000">标量函数 scalar function</font>：<math>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>-\nabla{V}\cdot g</math>也是局部正定的。有时把 <math>-\nabla{V}\cdot g</math> 局部正定的条件表述为 <math>\nabla{V}\cdot g</math> 是局部负定的。
--[[用户:Jxzhou|Jxzhou]]（[[用户讨论:Jxzhou|讨论]]）上述英文和维基百科不同，暂且按本文的英文翻译。
--[[用户:Jxzhou|Jxzhou]]（[[用户讨论:Jxzhou|讨论]]）上述英文和维基百科不同，暂且按本文的英文翻译。
A function <math>H</math> is defined to be locally positive-definite function (in the sense of dynamical systems) if both <math>H(0) = 0</math> and there is a neighborhood of the origin, <math>\mathcal{B}</math>, such that:
A function <math>H</math> is defined to be locally positive-definite function (in the sense of dynamical systems) if both <math>H(0) = 0</math> and there is a neighborhood of the origin, <math>\mathcal{B}</math>, such that:
函数数学 <math>H</math> 被定义为局部正定函数(在动力系统的意义上)，如果数学 <math>H(0) = 0</math>并且有一个邻域<math>\mathcal{B}</math>使得
函数数学 <math>H</math> 被定义为局部<font color="#ff8000">正定函数 positive-definite function</font>(在动力系统的意义上)，如果数学 <math>H(0) = 0</math>并且有一个邻域<math>\mathcal{B}</math>使得