From Kalman to Einstein and Maxwell: the Structural Controllability Revisited

Abstract

In the Special Relativity paper of Einstein (1905), only a footnote provides a reference to the conformal group of space-time for the Minkowski metric $\omega$. We prove that General Relativity (1915) will depend on the following {\it cornerstone} result of differential homological algebra (1990). Let $K$ be a differential field and $D=K[d_1,...,d_n]$ be the ring of differential operators with coefficients in $K$. If $M$ is the differential module over $D$ defined by the Killing operator ${\cal{D}} :T \rightarrow S_2T^*: \xi \rightarrow \Omega = {\cal{L}}(\xi) \omega$ and $N$ is the differential module over $D$ defined by the $Cauchy = ad(Killing)$ adjoint operator with torsion submodule $t(N)$, then $t(N) \simeq {ext}^1_D(M) = 0$ and the Cauchy operator can be thus parametrized by stress functions having strictly nothing to do with $\Omega$. This result is largely superseding the Kalman controllability test in classical OD control theory and is showing that controllability is a structural "{\it built-in}" property of an OD/PD control system not depending on the choice of inputs and outputs, contrary to the engineering tradition. It also points out the {\it terrible confusion} done by Einstein (1915) while following Beltrami (1892), both of them using the Einstein operator but ignoring that it was self-adjoint in the framework of differential double duality (1995). We finally prove that the structure of electromagnetism and gravitation only depends on the nonlinear {\it elations} of the conformal group of space-time, showing thus that {\it nothing is left from the mathematical foundations of both general relativity and gauge theory}.