本論文為探討PDE控制系統的等價方法。利用Lie-Backlund對稱方法推導出微分方程之間的等價關係，描述非線性PDE控制系統的等價變換過程。
等價的線性PDE系統的線性形式是由u的函數所構成。對於非線性PDE系統用延拓方法求出其單參數李群的極小生成元，透過極小生成元可以建構可變映射u的函數。
最後探討在非線性PDE等價變換為線性PDE過程中，得到可線性化的必要條件，並且應用Lie-Backlund變換與可變映射u建構線性非時變的PDE控制系統。

This paper presents the Lie-Backlund symmetry method to give the equivalence between differential equations and describe the equivalent transformation procedure of nonlinear control systems of partial differential equations.
The equivalent linear systems found by solving the infinitesimal generator of one-parameter Lie groups with prolongations and the infinitesimal generator are used to construct the parameters of invertible mapping u. And the equivalence linear form of the nonlinear system is constructed via u.
Some necessary conditions for mapping a nonlinear control system of PDE’s to a linear control system of PDE’s are discussed, and application of Lie-Backlund symmetries and invertible mapping u constructed linear time-invariant control system of partial differential equations.

目錄 Ⅰ
符號表 Ⅱ
論文摘要（中文） Ⅳ
論文摘要（英文） Ⅴ
第一章 序論 1
第一節 前言 1
第二節 文獻回顧 3
第三節 研究動機 6
第二章 李變換群 7
第一節 李群的變換 7
第二節 微分方程的不變群 17
第三節 延拓向量場 23
第四節 應用Mathematica程式 42
第三章 對稱與可變映射 45
第一節 Noether’s Theorem 46
第二節 對稱 47
第三節 建構微分方程的映射關係 58
第四節 可變映射 62
第四章 等價方法的應用 66
第一節 非線性微分方程的可線性化必要條件 68
第二節 非線性PDE系統可線性化與可控制性質 69
第三節 可變映射 的應用 70
第四節 Lie-Bäcklund對稱於 映射的應用 77
第五節 討論 83
第五章 總結 89
第一節 結論 89
第二節 未來研究方向 91
附錄A 微分流形的基礎理論 92
第一節 n維歐氏空間與流形 93
第二節 切空間 98
第三節 切映射 106
附錄B 李群 116
第一節 拓樸群 116
第二節 李群與李代數 121
第三節 結構方程 129
第四節 李群的同態與李子群 139
參考文獻 140

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