  In this paper, we prove several new results that give new insights into
bilinear systems. We discuss conditions for asymptotic stability using
probabilistic arguments. Moreover, we provide a global characterization of
reachability in bilinear systems based on a certain Gramian. Reachability
energy estimates using the same Gramian have only been local so far. The main
result of this paper, however, is a new link between the output error and the
$\mathcal H_2$-error of two bilinear systems. This result has several
consequences in the field of model order reduction. It explains why $\mathcal
H_2$-optimal model order reduction leads to good approximations in terms of the
output error. Moreover, output errors based on the $\mathcal H_2$-norm can now
be proved for balancing related model order reduction schemes in this paper.
All these new results are based on a Gronwall lemma for matrix differential
equations that is established here.
