On the blowing-up of stably free behaviours

It is well-known that a time-varying controllable ordinary differential linear system is flat outside some singularities. In this paper, we prove that every time-varying controllable linear system is a projection of a flat system. We give an explicit description of a flat system which projects onto a given controllable one. This phenomenon is similar to a classical one largely studied in algebraic geometry and called the blowing-up of a singularity. These results simplify the ones obtained in [6] and generalize them to MIMO multidimensional systems. Finally, we prove that every controllable multi-input ordinary differential linear system with polynomial coefficients is flat.