Bilinear systems -- A new link to $\mathcal H_2$-norms, relations to stochastic systems and further properties

Abstract: 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.