The aim of this paper is to provide a short introduction to modern issues in
the control of infinite dimensional closed quantum systems, driven by the
bilinear Schr\"odinger equation. The first part is a quick presentation of some
of the numerous recent developments in the fields. This short summary is
intended to demonstrate the variety of tools and approaches used by various
teams in the last decade. In a second part, we present four examples of
bilinear closed quantum systems. These examples were extensively studied and
may be used as a convenient and efficient test bench for new conjectures.
Finally, we list some open questions, both of theoretical and practical
interest.Comment: 6 page