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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亚普诺夫函数的一般方法，但在许多具体情况下，李亚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; 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。+
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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理论是很重要的。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。
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。
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。