%0 Journal Article
%T From Control Theory to Gravitational Waves
%A Jean-Francois Pommaret
%J Advances in Pure Mathematics
%P 49-100
%@ 2160-0384
%D 2024
%I Scientific Research Publishing
%R 10.4236/apm.2024.142004
%X When D:¦Î¡ú¦Ç 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 D1:¦Ç¡ú¦Î such that D¦Î = ¦Ç implies D1¦Ç = 0. When D is involutive, the procedure provides successive first-order involutive operators D1,...,Dn when the ground manifold has dimension n. Conversely, when D1 is given, a much more difficult ¡°inverse problem¡± is to look for an operator D:¦Î¡ú¦Ç having the generating CC D1¦Ç = 0. If this is possible, that is when the differential module defined by D1 is ¡°torsion-free¡±, that is when there does not exist any observable quantity which is a sum of derivatives of ¦Ç that could be a solution of an autonomous OD or PD equation for itself, one shall say that the operator D1 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 D1 is formally surjective, controllability just amounts to the formal injectivity of ad(D1), even in the variable coefficients case, a result still
%K Differential Operator
%K Differential Sequence
%K Killing Operator
%K Riemann Operator
%K Bianchi Operator
%K Cauchy Operator
%K Control Theory
%K Controllability
%K Elasticity
%K General Relativity
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=131578