From Control Theory to Gravitational Waves

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 Dξ = η implies D₁η = 0. When D is involutive, the procedure provides successive first-order involutive operators D₁,...,D_n 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:ξ→η having the generating CC D₁η = 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 η 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