Control Theory and Parametrizations of Linear Partial Differential Operators

Abstract

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.