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→Basic Lyapunov theorems for autonomous systems
数学 h (x)0对于数学中的所有 x-0. /
数学 h (x)0对于数学中的所有 x-0. /
== Basic Lyapunov theorems for autonomous systems==
== Basic Lyapunov theorems for autonomous systems 自治系统的李雅普诺夫基本定理==
{{main article|Lyapunov stability}}
{{main article|Lyapunov stability}}
Let <math>x^* = 0</math> be an equilibrium of the autonomous system
Let <math>x^* = 0</math> be an equilibrium of the autonomous system
令<math>x^* = 0</math>是如下自治系统的平衡点
:<math>\dot{x} = f(x).</math>
:<math>\dot{x} = f(x).</math>
and use the notation <math>\dot{V}(x)</math> to denote the time derivative of the Lyapunov-candidate-function <math>V</math>:
and use the notation <math>\dot{V}(x)</math> to denote the time derivative of the Lyapunov-candidate-function <math>V</math>:
并使用记号<math>\dot{V}(x)</math>表示李雅普诺夫候选方程<math>V</math>的时间导数:
:<math>\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>\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>
===Locally asymptotically stable equilibrium===
===Locally asymptotically stable equilibrium 局部渐进稳定平衡点===
If the equilibrium is isolated, the Lyapunov-candidate-function <math>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>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>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>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>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>\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}\setminus\{0\}</math>
:<math>\dot{V}(x) < 0 \quad \forall x \in \mathcal{B}\setminus\{0\}</math>
for some neighborhood <math>\mathcal{B}</math> of origin then the equilibrium is proven to be locally asymptotically stable.
for some neighborhood <math>\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>\mathcal{B}</math>，那么可以证明平衡点是局部渐近稳定的。
===Stable equilibrium===
===Stable equilibrium 稳定平衡点===
If <math>V</math> is a Lyapunov function, then the equilibrium is [[Stability theory|Lyapunov stable]].
If <math>V</math> is a Lyapunov function, then the equilibrium is [[Stability theory|Lyapunov stable]].
If <math>V</math> is a Lyapunov function, then the equilibrium is Lyapunov stable.
If <math>V</math> is a Lyapunov function, then the equilibrium is Lyapunov stable.
如果<math>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===
===Globally asymptotically stable equilibrium 全局渐进稳定平衡点===
If the Lyapunov-candidate-function <math>V</math> is globally positive definite, [[Radially unbounded function|radially unbounded]], the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:
If the Lyapunov-candidate-function <math>V</math> is globally positive definite, [[Radially unbounded function|radially unbounded]], the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:
If the Lyapunov-candidate-function <math>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>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>V</math> 是全局正定、径向无界的，并且李雅普诺夫候选函数的平衡点是孤立的以及时间导数是全局负定的：
:<math>\dot{V}(x) < 0 \quad \forall x \in \R ^n\setminus\{0\},</math>
:<math>\dot{V}(x) < 0 \quad \forall x \in \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>V(x)</math> is radially unbounded if
The Lyapunov-candidate function <math>V(x)</math> is radially unbounded if
李雅普诺夫候选函数<math>V(x)</math>是径向无界的如果满足
:<math>\| x \| \to \infty \Rightarrow V(x) \to \infty. </math>
:<math>\| x \| \to \infty \Rightarrow V(x) \to \infty. </math>
(这也被称为范数强制。)
(这也被称为范数强制。)
==Example==
==Example==