From Control Theory to Gravitational Waves

Abstract

When D : ξ → η is a linear ordinary differential (OD) or partial differential (PD) operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D 1: η → ξ such that D ξ = η implies D 1 η = 0. When D is involutive, the procedure provides successive first-order involutive operators D 1 ,..., D n when the ground manifold has dimension n . Conversely, when D 1 is given, a much more difficult “inverse problem” is to look for an operator D : ξ → η having the generating CC D 1 η = 0. If this is possible, that is when the differential module defined by D 1 is “ torsion-free ”, that is when there does not exist any observable quantity which is a sum of derivatives of η that could be a solution of an autonomous OD or PD equation for itself , one shall say that the operator D 1 is parametrized by D . The parametrization is said to be “ minimum ” if the differential module defined by D does not contain a free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test with five steps using double differential duality . We prove and illustrate through many explicit examples the fact that a control system is controllable if and only if it can be parametrized. Accordingly, the controllability of any OD or PD control system is a “ built in ” property not depending on the choice of the input and output variables among the system variables. In the OD case and when D 1 is formally surjective, controllability just amounts to the formal injectivity of ad ( D 1 ), even in the variable coefficients case, a result still not acknowledged by the control community. Among other applications, the parametrization of the Cauchy stress operator in arbitrary dimension n has attracted many famous scientists (G. B. Airy in 1863 for n = 2, J. C. Maxwell in 1870, E. Beltrami in 1892 for n = 3, and A. Einstein in 1915 for n = 4 ). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized and the comparison needs no comment. As a byproduct, they are all based on a confusion between the so-called div operator D 2 induced from the Bianchi operator and the Cauchy operator, adjoint of the Killing operator D which is parametrizing the Riemann operator D 1 for an arbitrary n . This purely mathematical result deeply questions the origin and existence of gravitational waves, both with the mathematical foundations of general relativity. As a matter of fact, this new framework provides a totally open domain of applications for computer algebra as the quoted test can be studied by means of Pommaret bases and related recent packages.