Without being aware of this, many technological applications of everyday use involve quantum systems. An example is magnetic resonance (MRI) used in medical imaging. An MRI examination is the repetition of a few hundred to a few thousand observations. Improved transfer time would make it possible to obtain better shots in less time, thus better use of equipment to treat more patients, who would be exposed for less time to intense magnetic fields. The CONSTAT project aims to lay the groundwork for the technological applications used in the medical and other fields. It is at the interface of several disciplines: basic mathematical research with an application purpose in health (imaging, treatments), and multiple applications in photonic research, nanosciences, quantum models etc... The study of the modeling of quantum, algebraic and geometric systems will make it possible to obtain analytical and numerical frames of the minimum control time for invariant bilinear systems. This work aims to identify and circumvent the difficulties of these large-scale numerical estimates. The terminals obtained will provide guidance on the minimum time required for the transfer of populations to a finite-dimensional bilinear quantum system, and will improve the usual applications resulting from them.