标题： 控制理论与线性偏微分算子的参数化 显示英文标题

标题： Control Theory and Parametrizations of Linear Partial Differential Operators

摘要： 当 ${\cal{D}}:\xi \rightarrow \eta$ 是一个线性 OD 或 PD 算子时，"直接问题"是找到相容性条件（CC）作为算子 ${\cal{D}}_1:\eta \rightarrow \zeta$，使得 ${\cal{D}}\xi=\eta$ 推出 ${\cal{D}}_1\eta=0$。 当 ${\cal{D}}$ 是奇异的，该过程在维度 $n$中提供连续的一阶奇异算子 ${\cal{D}}_1, ... , {\cal{D}}_n$。 相反地，当给出${\cal{D}}_1$时，一个更为困难的“逆问题”是寻找一个具有生成 CC${\cal{D}}_1\eta=0$的算子${\cal{D}}: \xi \rightarrow \eta$。 当由${\cal{D}}_1$定义的微分模是“{\it 无挠性}”时，就可以做到这一点，可以说${\cal{D}}_1$是由${\cal{D}}$参数化的。 对微分算子的伴随算子的系统使用提供了一个构造性测试。 一个控制系统是可控的{\it 当且仅当}它可以被参数化。 因此，任何 OD 或 PD 控制系统的可控性是一个“{\it 内置}”性质，不依赖于系统变量中输入和输出变量的选择。 在 OD 情况下当${\cal{D}}_1$为形式满射时，可控性仅相当于 $ad({\cal{D}}_1)$的单射性。 在应用中，柯西应力算子的参数化吸引了许多著名科学家，从 G.B. Airy 在 1863 年对于$n=2$到 A. Einstein 在 1915 年对于$n=4$。 我们证明了所有这些工作都已经明确使用了自伴爱因斯坦算子{\it 无法参数化的}，并且是基于从Bianchi算子${\cal{D}}_2$引入的算子$div$与柯西算子之间的混淆，柯西算子是任意$n$的Killing算子${\cal{D}}$的伴随算子。这一纯粹的数学结果深刻地质疑了引力波的起源和存在。 显示英文摘要

摘要： When ${\cal{D}}:\xi \rightarrow \eta$ is a linear OD or PD operator, a "direct problem" is to find compatibility conditions (CC) as an operator ${\cal{D}}_1:\eta \rightarrow \zeta$ such that ${\cal{D}}\xi=\eta$ implies ${\cal{D}}_1\eta=0$. When ${\cal{D}}$ is involutive, the procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ in dimension $n$. Conversely, when ${\cal{D}}_1$ is given, a much more difficult " inverse problem " is to look for an operator ${\cal{D}}: \xi \rightarrow \eta$ having the generating CC ${\cal{D}}_1\eta=0$. This is possible when the differential module defined by ${\cal{D}}_1$ is " {\it torsion-free} ", one shall say that ${\cal{D}}_1$ is parametrized by ${\cal{D}}$. The systematic use of the adjoint of a differential operator provides a constructive test. A control system is controllable {\it if and only if} it can be parametrized. Accordingly, the controllability of any OD or PD control system is a " {\it built in} " property not depending on the choice of the input and output variables among the system variables. In the OD case when ${\cal{D}}_1$ is formally surjective, controllability just amounts to the injectivity of $ad({\cal{D}}_1)$. Among applications, the parametrization of the Cauchy stress operator has attracted many famous scientists from G.B. Airy in 1863 for $n=2$ to A. Einstein in 1915 for $n=4$. We prove that all these works are already explicitly using the self-adjoint Einstein operator {\it which cannot be parametrized} and are based on a confusion between the $div$ operator induced from the Bianchi operator ${\cal{D}}_2$ and the Cauchy operator, adjoint of the Killing operator ${\cal{D}}$ for an arbitrary $n$. This purely mathematical result deeply questions the origin and existence of gravitational waves.