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.