用图论方法识别多分散随机多组模型的局部拓扑结构

摘要

本文研究了具有多重离散的随机多群模型的局部拓扑识别问题。基于自适应钉扎控制和图论方法，得到了具有多重离散的随机多群模型局部拓扑识别的一些充分判据。也就是说，可以成功地识别出未知的局部拓扑结构。最后，通过数值算例验证了理论结果的有效性。

1.简介

近年来，多群模型因其在生物学、流行病学等许多不同领域的广泛应用而受到越来越多的关注[1,2]. 多组模型的不同动力学行为已被广泛研究，参见[三,4]为了全球稳定[5]用于同步和[6]用于平稳分布。

在许多多种群模型中，特别是在多径环境中，多重扩散具有很大的影响，并且物种之间总是存在不同的扩散。此外，自然界中的系统必须受到随机扰动的影响[7,8,9,10]. 具有多重分散性的随机多群模型是有效的数学模型，已引起越来越多的关注，参见[11,12,13]. 应该注意的是，许多研究中的拓扑结构[三,4,5,6,8,9,10,11,12,13]已知。事实上，在许多实际应用中，拓扑结构通常是未知的或不确定的。因此，识别具有多重扩散的随机多群模型的未知拓扑结构非常重要。

在本文中，我们考虑以下具有多重分散的随机多群模型

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& \left[{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \\ \hfill & \mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=",">,</trans>\hfill \end{array}$$

(1)

哪里${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是的状态向量我-中的第个组件k个-第个组。我们定义$\mathrm{<trans\; data-src="m">米</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="s">秒</trans>}}{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$对于${\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}$.${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}}$表示的状态向量k个-第th组。${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}}<trans\; data-src="\times ">\times </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src="\to ">\to </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$代表的性能我-的第个组件k个-第个组。${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="\times ">\times </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src="\to ">\to </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$显示了上的扰动强度我-的第个组件k个-第个组。${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$是指我-第个组件小时-第个组到k个-第个组。在这里，${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$如果没有分散我-第个组件小时-第个组到k个-第个组。${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=":">:</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="\times ">\times </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="\times ">\times </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src="\to ">\to </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是顶点的影响小时在顶点上k个,$\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans><trans\; data-src="·">·</trans><trans\; data-src=")">)</trans>$是一维布朗运动。为了更好地理解，我们画了一个由四个群和两个分散体组成的图，参见图1在这里，${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="11">11</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="13">13</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="14">14</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="22">22</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="31">31</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="33">33</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="34">34</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="44">44</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="11">11</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="14">14</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="21">21</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="22">22</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="33">33</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="41">41</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="42">42</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="44">44</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$

在本文中，数学模型有若干组。如果我们为每个组添加一个控制器，将导致高控制成本。众所周知，固定控制策略是减少受控组数量的有效技术[14,15,16]. 另一方面，如果大量群具有未知的部分拓扑结构，或者我们只对部分拓扑结构感兴趣，那么钉扎控制是一种有效的技术。因此，我们将尝试使用固定控制机制来识别未知的部分拓扑结构(1).

值得一提的是，Lyapunov方法是研究局部拓扑识别的有效工具[17,18,19]. 然而，很难为(1)由于多组和分散。令人鼓舞的是，Li和他的合作者将图论和Lyapunov方法相结合，间接地为耦合系统构建了全局Lyapunow函数。同时，他们使用这种方法来研究耦合系统的全局稳定性[20]. 这种方法称为图形理论方法。据我们所知，利用图论方法对随机多群模型进行局部拓扑识别的工作很少。

基于上述讨论，本文旨在利用图理论方法识别具有多重离散的随机多群模型的未知局部拓扑结构。主要贡献如下。

• 该数学模型是通用的，包括多重分散和随机扰动。
• 钉扎控制器具有成本效益并且可以减少受控组。
• 局部拓扑识别的图形理论方法是一种新颖的方法。
• 成功地识别了具有多重扩散的随机多群模型的未知局部拓扑结构。

本文的其余部分组织如下。在第2节，显示一些初步信息。主要结果介绍于第3节.英寸第4节，用两个数值算例验证了理论结果的有效性。结论见第5节.

2.前期工作

图论和随机微分方程的一些必要基础知识

有向图$\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=")">)</trans>$包含一个集合$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\}">\}</trans>$顶点和集合P（P）弧的$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>$，其中$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>$是从初始顶点算起的弧我到端子顶点j个.给定一个有向图$\mathrm{<trans\; data-src="M">M（M）</trans>}$具有N个顶点，我们定义加权矩阵$\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}$谁的条目${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}$等于弧的权重$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>$如果从顶点有弧我到顶点j个，否则为0。带权矩阵的有向图单位描述为$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src=")">)</trans>$拉普拉斯矩阵$\mathbf{<trans\; data-src="L">我</trans>}$属于$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src=")">)</trans>$定义为$\mathbf{<trans\; data-src="L">我</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="l">我</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}$，其中${\mathrm{<trans\; data-src="l">我</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="=">=</trans>{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="l">我</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}$.

假设系数${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$和${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$第页，共页(1)满足局部Lipschitz条件和线性增长条件[21,22]. 然后，对于初始值${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$, (1)具有唯一的连续解，表示为$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=";">;</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans>$此外，如果${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$和${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后(1)承认一个微不足道的解决方案$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=";">;</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

我们定义微分算子$\mathrm{<trans\; data-src="L">我</trans>}$执行${\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src="\times ">\times </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src=";">;</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src=")">)</trans>$以及(1)作为

$$\begin{array}{c}\mathrm{<trans\; data-src="L">我</trans>}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="+">+</trans>\frac{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}\left[{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]\\ <trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="Trace">跟踪</trans>}\left[{{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\left(\frac{{<trans\; data-src="\partial ">\partial </trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}}\right){\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]<trans\; data-src=",">,</trans>\end{array}$$

(2)

哪里

$\frac{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="=">=</trans>\left(\frac{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="i">我</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans>}}<trans\; data-src=",">,</trans>\frac{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="i">我</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\frac{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="i">我</trans>}}_{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=")">)</trans>}}\right)$,$\frac{{<trans\; data-src="\partial ">\partial </trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}}<trans\; data-src="=">=</trans>{\left(\frac{{<trans\; data-src="\partial ">\partial </trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="i">我</trans>}}_{\mathrm{<trans\; data-src="p">第页</trans>}}<trans\; data-src=")">)</trans>}<trans\; data-src="\partial ">\partial </trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="i">我</trans>}}_{\mathrm{<trans\; data-src="q">q个</trans>}}<trans\; data-src=")">)</trans>}}\right)}_{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\times ">\times </trans>{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$.

柠檬 1

([21]).假设有一个函数$\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src="\times ">\times </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src=";">;</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src=")">)</trans>$，一个函数$\mathrm{<trans\; data-src="\vartheta ">\vartheta </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src=";">;</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src=")">)</trans>$和一个连续函数$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src=":">:</trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src="\to ">\to </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}$这样的话

$$\underset{<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="|">|</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim">林</trans>}\underset{\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="<"><</trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="inf">inf公司</trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=",">,</trans>$$

和微分算子$\mathrm{<trans\; data-src="L">我</trans>}$作用于V的轨迹(1)满足

$$\mathrm{<trans\; data-src="L">我</trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="\vartheta ">\vartheta </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src="\times ">\times </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}_{<trans\; data-src="+">+</trans>}<trans\; data-src=".">.</trans>$$

此外，${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}$被限定为$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}$然后，对于每个初始值${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="N">N个</trans>}}$,${<trans\; data-src="lim">林</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=";">;</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$存在并且几乎肯定是有限的。此外，

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim">林</trans>}\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=";">;</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=".">.</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$$

柠檬 2

（中的定理2.2[20]).假设$\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="2">2</trans>}$.${\mathrm{<trans\; data-src="\nu ">\nu </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}$是第k个对角线元素的余因子$\mathbf{<trans\; data-src="L">我</trans>}$。然后，以下身份保持不变：

$$\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\nu ">\nu </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}{\mathbf{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src="\in ">\in </trans>\mathbb{<trans\; data-src="Q">问</trans>}}{<trans\; data-src="\sum ">\sum </trans>}\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src=")">)</trans>\underset{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">电子</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="C">C类</trans>}}_{\mathrm{<trans\; data-src="Q">问</trans>}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\sum ">\sum </trans>}{\mathbf{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

在这里，${\mathbf{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}$是任意函数，$\mathbb{<trans\; data-src="Q">问</trans>}$是的所有生成单圈图的集合$<trans\; data-src="(">(</trans>\mathcal{<trans\; data-src="G">G公司</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src=")">)</trans>$是的重量$\mathrm{<trans\; data-src="Q">问</trans>}$、和${\mathrm{<trans\; data-src="C">C类</trans>}}_{\mathrm{<trans\; data-src="Q">问</trans>}}$表示的定向循环$\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src=".">.</trans>$

3.主要成果

在本节中，我们将研究基于自适应钉扎同步和图形理论方法的具有多重分散的随机多组模型的局部拓扑识别问题。

为了识别具有多重分散的随机多群模型的局部拓扑结构，取(1)作为驱动系统，带有自适应固定控制器的响应系统描述为

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& \left[{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\hfill \\ \hfill & <trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(3)

${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$和${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$扰动强度和函数控制我-的第个组件k个-分别为响应网络中的第th组。${\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="2">2</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是的状态向量我-中的第个组件k个-第个组。${\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}}$表示的状态向量k个-第个组。${\mathrm{<trans\; data-src="B">B类</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}$表示未知部分耦合矩阵的估计${\mathrm{<trans\; data-src="A">A类</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\left({\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$.${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$是通用的自适应固定控制器。在不损失一般性的情况下，识别由前沿组成的多分散随机多群模型的局部拓扑结构就足够了我群体及其扩散，即，${<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}$.让$\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="y">年</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="l">我</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}$是同步错误，其中${\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}$$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=")">)</trans>$.我们表示${\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=")">)</trans>$.系统间同步误差的动力系统(1)和(三)可以写为

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& <trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill & <trans\; data-src="-">-</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill <trans\; data-src="=">=</trans>& <trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill & <trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \\ \hfill & \mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\hfill \end{array}$$

(4)

哪里${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$.

假设以下条件适用于每个$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}$和$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}$.

• （A1）有常数${\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}$如下不等式适用于任何${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="m">米</trans>}}$:

  $${\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}\left({\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)<trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="j">j个</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=".">.</trans>$$
• （A2）假设${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$被限定为${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$存在一个常数${\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$这样的话

  $$\left|{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left|{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right|<trans\; data-src=".">.</trans>$$
• （A3）存在正常数${\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$和${\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$这样的话

  $$<trans\; data-src="|">|</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="|">|</trans><trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="|">|</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="|">|</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\in ">\in </trans>{\mathbf{<trans\; data-src="R">R（右）</trans>}}^{{\mathrm{<trans\; data-src="m">米</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=".">.</trans>$$
• （A4）假设每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=")">)</trans>$,${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$在轨道上线性无关${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$外部同步歧管的${<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>}_{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src=".">.</trans>$

接下来，我们给出了一些自适应钉扎控制器和更新律$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}$,$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}$.

$${\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(5)

$${\stackrel{<trans\; data-src="\dot{}">\dot{}</trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

$$\begin{array}{c}\hfill {\stackrel{<trans\; data-src="\dot{}">\dot{}</trans>}{\mathrm{<trans\; data-src="b">b条</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left\{\begin{array}{cc}\hfill & <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill & <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=",">,</trans>\hfill \end{array}\right.\end{array}$$

(6)

哪里${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$和${\mathrm{<trans\; data-src="\eta ">\eta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$是任意正常数。

定理 1

如果（A1）–（A4）保持不变$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="l">我</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$每个都紧密相连$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$，然后是未知的局部拓扑结构$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$耦合网络的(1)可以通过以下方式识别$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$控制器下方(5)和更新法律(6)概率为1。也就是说，它适用于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$那个

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim">林</trans>}\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=".">.</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$$

证明。 

我们定义

$$\begin{array}{cc}\hfill & {\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\frac{{\left({\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}}{{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\frac{{\left({\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}}{{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}{\left({\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\hfill \end{array}$$

在哪儿${\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}$是一个足够大的正数。然后，它从微分算子的定义出发$\mathrm{<trans\; data-src="L">我</trans>}$那个

$$\begin{array}{cc}\hfill & \mathrm{<trans\; data-src="L">我</trans>}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill <trans\; data-src="=">=</trans>& {\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill & <trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="Trace">跟踪</trans>}\left[{\left({\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="\times ">\times </trans>\left({\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)\right]\hfill \\ \hfill & <trans\; data-src="-">-</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\left({\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}\right){\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill <trans\; data-src="\le ">\le </trans>& \underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="j">j个</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{\left({\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}\hfill \\ \hfill <trans\; data-src="\le ">\le </trans>& \underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="j">j个</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{\left({\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|\left({\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|\right)\hfill \\ \hfill <trans\; data-src="=">=</trans>& \underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="j">j个</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left({\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}\right)<trans\; data-src="+">+</trans>\left(\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left({\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{\left({\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}\right){\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=".">.</trans>\hfill \end{array}$$

我们定义

$$\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

哪里${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$是的辅因子k个-的拉普拉斯矩阵的第th个对角元素$\left(\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{\left({\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="l">我</trans>}}\right)<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$那么，不难推导出

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="L">我</trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& \underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="L">我</trans>}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill <trans\; data-src="\le ">\le </trans>& \underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="j">j个</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left({\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}\right)\hfill \\ \hfill & <trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left(\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left({\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{\left({\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}\right){\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}\hfill \\ \hfill <trans\; data-src="\triangleq ">\triangleq </trans>& \mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="I">我</trans>}\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="I">我</trans>}\mathrm{<trans\; data-src="I">我</trans>}\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

现在，我们计算两个部分，我和$\mathrm{<trans\; data-src="I">我</trans>}\mathrm{<trans\; data-src="I">我</trans>}$.

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="=">=</trans>& \underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="i">我</trans>}}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left[\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="i">我</trans>}}\right]{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(7)

根据引理2，它产生

$$\begin{array}{cc}\hfill & \underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\left[\underset{{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}{<trans\; data-src="\sum ">\sum </trans>}\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>\underset{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">电子</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="C">C类</trans>}}_{{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\sum ">\sum </trans>}{\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="u">u个</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="v">v（v）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\right]<trans\; data-src=",">,</trans>\hfill \end{array}$$

哪里${\mathbb{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是的所有生成单圈图的集合$\left(\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{\left({\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="l">我</trans>}}\right)$,$\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$是的重量${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$和${\mathrm{<trans\; data-src="C">C类</trans>}}_{{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}$是指${\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$.采取

$${\mathrm{<trans\; data-src="F">F类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\left({\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}\right)<trans\; data-src=",">,</trans>$$

它认为

$$\mathrm{<trans\; data-src="I">我</trans>}\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\left[\underset{{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="\in ">\in </trans>{\mathbb{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}{<trans\; data-src="\sum ">\sum </trans>}\mathrm{<trans\; data-src="\omega ">\omega </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>\underset{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="E">电子</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="C">C类</trans>}}_{{\mathrm{<trans\; data-src="Q">问</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}}<trans\; data-src=")">)</trans>}{<trans\; data-src="\sum ">\sum </trans>}\left|{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="u">u个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="v">v（v）</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}\right)\right]<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

(8)

然后，可以获得

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="L">我</trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>& \underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left[\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="i">我</trans>}}\right]{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left(\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left({\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{\left({\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}\right){\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}\hfill \\ \hfill <trans\; data-src="=">=</trans>& \underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left[\frac{\mathrm{<trans\; data-src="1">1</trans>}}{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}}\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left({\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)<trans\; data-src="+">+</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{\left({\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}\right]{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=".">.</trans>\hfill \end{array}$$

作为${\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}$足够大，存在${\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\left({<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="s">秒</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="i">我</trans>}}\right)<trans\; data-src="/">/</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>$${<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="l">我</trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\left({\mathrm{<trans\; data-src="B">B类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)<trans\; data-src="+">+</trans>{\left({\mathrm{<trans\; data-src="\beta ">\beta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$因此，

$$\begin{array}{c}\hfill \mathrm{<trans\; data-src="L">我</trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="s">秒</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left|{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right|}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\le ">\le </trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\triangleq ">\triangleq </trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\end{array}$$

(9)

哪里$\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$是一个常数，由${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$和${\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}$此外，上述分析表明

$$\underset{<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="|">|</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim">林</trans>}\underset{\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="<"><</trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="inf">inf公司</trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="\infty ">\infty </trans><trans\; data-src=".">.</trans>$$

因此，从引理1，${<trans\; data-src="lim">林</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=";">;</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$存在并且几乎肯定是有限的。它还认为${<trans\; data-src="lim">林</trans>}_{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=".">.</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$结合拉萨尔不变性原理、假设（A4）和误差系统(4)，可以得到集合$\mathbb{<trans\; data-src="M">M（M）</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\}">\}</trans>$是最大的不变集${\mathbb{<trans\; data-src="M">M（M）</trans>}}^{<trans\; data-src="\prime ">\prime </trans>}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans>$因此，对于误差系统的任何初始值(4)，轨迹渐近收敛到$\mathbb{<trans\; data-src="M">M（M）</trans>}$概率为1[21]. 证明了具有多重扩散的随机多群模型(1)和(三)自适应控制器下渐近实现完全外同步(5)和更新法律(6). 此外，未知的多重拓扑结构$\left(\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{\left({\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}\right)<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$已由成功识别$\left(\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{\left({\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}\right)<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$概率为1，这就完成了这个证明。□

备注 1

众所周知，李亚普诺夫方法在局部拓扑识别的研究中起着重要的作用[17,18,19]. 然而，本文考虑了多重扩散和随机扰动，这使得数学模型更加复杂。因此，对于多群模型，很难直接构造全局Lyapunov函数。动机[20,23,24]，我们通过顶点Lyapunov函数的加权求和来构造全局Lyapunow函数${\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}$以…的形式$\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="l">我</trans>}}{<trans\; data-src="\sum ">\sum </trans>}_{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="s">秒</trans>}}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="V">V（V）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}$在这里，${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$是拉普拉斯矩阵第k对角元素的余因子$\left(\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{\left({\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\right)}_{\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="l">我</trans>}}\right)<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$显然，这种方法与网络的拓扑结构密切相关。由于这种方法使用了图论的一些结果，所以通常称为图论方法。该方法可用于研究许多其他网络的动态行为。例如，在[20]利用图论方法研究了具有扩散的单种群生态模型的稳定性。他们获得的全球稳定结果比[25,26]. 此外，利用图理论方法还可以得到耦合振子和多匹配捕食-被捕食模型的全局渐近稳定性。

备注 2

虽然自适应控制方法可以实现多组模型的拓扑识别，但它需要为每个组添加一个控制器[27,28]. 然而，在本文中，该模型包含了大量组。将控制器添加到所有组有时很难实施，并且控制成本可能更高。因此，本文提出的钉扎控制是有用的。一方面，它可以降低控制成本，因为只需要控制一小部分群体。另一方面，更可行的方法是只向感兴趣的组添加控制器来识别相应的未知拓扑结构。

什么时候？$\mathrm{<trans\; data-src="l">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="N">N个</trans>}$，我们可以获得整个拓扑识别。具体来说，驱动系统的相应响应系统(1)可以通过以下特征

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& \left[{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\hfill \\ \hfill & <trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(10)

然后，驱动系统之间的错误系统(1)和响应系统(10)可以表示为

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& <trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill & <trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>\left[{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(11)

自适应控制器及其更新律$\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}$如下所示：

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="\dot{}">\dot{}</trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\end{array}$$

(12)

$$\begin{array}{c}\hfill {\stackrel{<trans\; data-src="\dot{}">\dot{}</trans>}{\mathrm{<trans\; data-src="b">b条</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=",">,</trans>\end{array}$$

(13)

哪里${\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$是任意正常数。然后，我们得到以下推论。

推论 1

如果（A1）–（A4）保持不变$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="D">D类</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$每个都紧密相连$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$，然后是未知的整体拓扑结构$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$耦合网络的(1)可以通过以下方式识别$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=",">,</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$控制器下方(12)和更新法律(13). 也就是说，它适用于每个$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>$那个

$$\underset{\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim">林</trans>}\underset{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="|">|</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=".">.</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=".">.</trans>$$

这个证明类似于定理1，我们在这里省略了它。

如果${\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$，则可以在中找到相应的整个拓扑识别结果[27]. 因此，本文的数学模型更具一般性。此外，本文的理论结果是通用的，可以得到局部拓扑识别和整体拓扑识别。

4.仿真结果

在这一部分中，给出了两个仿真实例来验证理论结果的有效性。我们使用Lorenz系统来描述每个组的动态。显然，洛伦兹系统满足（A1）[29].

例子 1

我们考虑一个具有四个群和三种扩散的一般耦合系统。驱动系统可以采用以下形式：

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="=">=</trans>\left[{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="4">4</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=",">,</trans>\hfill \end{array}$$

(14)

哪里${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="T">T型</trans>}}<trans\; data-src=",">,</trans>$ ${\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="g">克</trans>}\left\{\mathrm{<trans\; data-src="0.1">0.1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0.1">0.1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0.1">0.1</trans>}\right\}$.

• ${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="10">10</trans>}\left({\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)$,${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.5">0.5</trans>}<trans\; data-src="sin">罪</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$;
• ${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="28">28</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$,${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.3">0.3</trans>}<trans\; data-src="cos">余弦</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$;
• ${\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="8">8</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="3">三</trans>}$,${\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.1">0.1</trans>}<trans\; data-src="cos">余弦</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>$.

一些砝码${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}$配置矩阵的${\mathrm{<trans\; data-src="A">A类</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans>$可以任意选择为

• ${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="12">12</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="34">34</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="43">43</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="31">31</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=";">;</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="13">13</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="21">21</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="34">34</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="12">12</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="34">34</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>$${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="43">43</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=";">;</trans>$
• ${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="32">32</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="41">41</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="12">12</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="43">43</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="31">31</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=";">;</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="42">42</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="21">21</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=";">;</trans>$
  其他值设置为0。在不损失通用性的情况下，我们将控制器添加到前两个组中。然后，我们只需要识别部分权重配置矩阵${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="4">4</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>$$\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$因此，具有自适应钉扎控制器的响应系统可以表示为

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& \left[{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}}{\overset{\mathrm{<trans\; data-src="4">4</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}\hfill \\ \hfill & <trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(15)

我们定义${\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$然后，误差系统(14)和(15)可以显示为

$$\begin{array}{cc}\hfill \mathrm{<trans\; data-src="d">d日</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>& <trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\hfill \\ \hfill & <trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}}{\overset{\mathrm{<trans\; data-src="4">4</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\mathrm{<trans\; data-src="H">H（H）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="+">+</trans><trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathbb{<trans\; data-src="W">W公司</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(16)

可从定理1获得三个权重配置矩阵${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="4">4</trans>}}$可以通过以下方式进行估算${\mathrm{<trans\; data-src="B">B类</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="4">4</trans>}}$根据以下控制器和更新规则$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}$.

$$\begin{array}{c}\hfill {\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="\dot{}">\dot{}</trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\end{array}$$

(17)

$${\stackrel{<trans\; data-src="\dot{}">\dot{}</trans>}{\mathrm{<trans\; data-src="b">b条</trans>}}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left\{\begin{array}{cc}\hfill & <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill & <trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}{\left({\mathrm{<trans\; data-src="e">e（电子）</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans>\right)}^{\mathrm{<trans\; data-src="T">T型</trans>}}{\mathrm{<trans\; data-src="\Gamma ">\Gamma </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}\right.$$

(18)

拿${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="p">第页</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\delta ">\delta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\eta ">\eta </trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.1">0.1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$我们可以得到一些模拟结果。驱动系统同步错误(14)和响应系统(15)如所示图2.图2a显示第一组的同步错误图2b表示第二组的同步错误。显然，随着时间的推移，所有误差曲线都接近0，这意味着驱动响应系统可以达到同步。图3给出了时间演变${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans>$在系统中(15).图3a显示了的识别结果${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans>$观察到曲线稳定在三个常数：2${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}$和${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="21">21</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}$，1用于${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="12">12</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}$的和0${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="11">11</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="14">14</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="22">22</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="23">23</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}$.图3b显示了${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans>$我们可以看到${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="21">21</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}$倾向于4，${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="12">12</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}$倾向于3，${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}$趋于1且${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="11">11</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="13">13</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="14">14</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="22">22</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="23">23</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=")">)</trans>}$趋于0。图3c显示了${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans>$如我们所见，曲线${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="12">12</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}$收敛到2，曲线${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}$收敛到1和曲线${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="11">11</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="13">13</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="14">14</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="21">21</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="22">22</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="23">23</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}$收敛到0。因此${\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src=")">)</trans>$可以收敛到一个实值，这意味着局部拓扑识别是成功的。

例子 2

在这个例子中，我们假设网络结构遵循Newman和Watts提出的小世界算法[30]. 考虑系统(14)由八个组组成，固定组的数量与上面的示例相同。为了更好地查看，拓扑结构如所示图4，其中图4a遵守带参数的小世界算法$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="8">8</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.1">0.1</trans>}<trans\; data-src=")">)</trans>$,图4b遵守带参数的小世界算法$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="8">8</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}<trans\; data-src=")">)</trans>$,图4c遵守带参数的小世界算法$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="8">8</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.3">0.3</trans>}<trans\; data-src=")">)</trans>$其中，

$$\begin{array}{cc}\hfill & {\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="12">12</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="13">13</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="17">17</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="18">18</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="21">21</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="23">23</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="24">24</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="28">28</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="31">31</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="32">32</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="34">34</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="35">35</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="36">36</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="42">42</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\hfill \\ \hfill & {\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="43">43</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="45">45</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="46">46</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="53">53</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="54">54</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="56">56</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="57">57</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="63">63</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="64">64</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="65">65</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="67">67</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="67">67</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="68">68</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; 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data-src="81">81</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="82">82</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="85">85</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="86">86</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="87">87</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

其他的${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$此外，一些参数可以任意设置如下：${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}<trans\; data-src=",">,</trans>$${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.3">0.3</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="3">三</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.1">0.1</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="4">4</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="0.6">0.6</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="5">5</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="6">6</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.2">0.2</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="7">7</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.3">0.3</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="8">8</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.4">0.4</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.8">0.8</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="y">年</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0.7">0.7</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1.5">1.5</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="8">8</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$其他参数与上述示例相同。

在与上例相同的条件下，可以得出定理1的所有假设都满足。这意味着三个重量配置矩阵${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="8">8</trans>}}$可以通过以下方式进行估算${\mathrm{<trans\; data-src="B">B类</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="8">8</trans>}}$在控制器和更新法则下(17)和(18)的$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="8">8</trans>}$.

图5显示了的仿真结果${\mathrm{<trans\; data-src="B">B类</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="b">b条</trans>}}_{\mathrm{<trans\; data-src="k">k个</trans>}\mathrm{<trans\; data-src="h">小时</trans>}}^{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans>}_{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="8">8</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="3">三</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="h">小时</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\dots ">\dots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="8">8</trans>}<trans\; data-src=")">)</trans>$很明显，所有曲线都稳定在实值，这意味着局部拓扑识别是成功的。

5.结论

总之，本文的内容有四个方面。

(1)

该模型考虑了多群模型、多重扩散和随机扰动。

(2)

利用图论方法，可以间接地构造具有多重离散的随机多群模型的全局李亚普诺夫函数。特别是，该方法可以用于研究大规模复杂网络的许多动力学行为，如耦合振子的稳定性和多匹配捕食者-食饵模型。

(3)

利用钉扎控制可以成功地识别随机多群模型的未知局部拓扑结构。

(4)

通过数值算例，可以看出本文得到的理论结果是有效的。

在本文中，噪声是白噪声。然而，在实际应用中也存在颜色噪声。因此，带彩色噪声的多组模型局部拓扑识别将是未来的工作之一。