Model reduction of controlled fokker-planck and liouville-von neumann equations

We study and compare two different model reduction techniques for bilinear systems, specifically generalized balancing and H₂-based model reduction, and apply it to semi-discretized controlled Fokker-Planck and Liouville-von Neumann equations. For this class of transport equations, the control enters the dynamics as an advection term that leads to the bilinear form. A specific feature of the systems is that they are stable, but not asymptotically stable, and we discuss aspects regarding structure and stability preservation in some depth as these aspects are particularly relevant for the equations of interest. Another focus of this article is on the numerical implementation and a thorough comparison of the aforementioned model reduction methods.