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Consider the following differential equation with solution <math>x</math> on <math>\R </math>:

Consider the following differential equation with solution <math>x</math> on <math>\R </math>:

考虑一下下面的关于数学 x / math 和数学 r / math 的微分方程:

考虑下面具有<math>\R </math>上的解<math>x</math>的微分方程：

Considering that <math>x^2</math> is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study <math>x</math>.

Considering that <math>x^2</math> is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study <math>x</math>.

考虑到数学 x ^ 2 / math 在原点周围总是正的，这是一个帮助我们学习数学 x / math 的自然候选李亚普诺夫函数。

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

So let <math>V(x)=x^2</math> on <math>\R </math>. Then,

So let <math>V(x)=x^2</math> on <math>\R </math>. Then,

So let <math>V(x)=x^2</math> on <math>\R </math>. Then,

So let <math>V(x)=x^2</math> on <math>\R </math>. Then,

让数学 v (x) x ^ 2 / math on math r / math。然后,

令<math>V(x)=x^2</math>在<math>\R </math>上。然后有

This correctly shows that the above differential equation, <math>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>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.

这正确地表明，上面的微分方程，math x，/ math 关于原点是渐近稳定的。注意，使用相同的李亚普诺夫候选者可以证明平衡点也是全局渐近稳定的。

这正表明上面的微分方程 <math>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==

==See also==