Bilinear systems -- A new link to mathcal H₂-norms, relations to\n stochastic systems and further properties

2019/10/31 by Martin Redmann, Redmann, Martin
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1910.14427

openalex publication_date 2019/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove several new results that give new insights into\nbilinear systems. We discuss conditions for asymptotic stability using\nprobabilistic arguments. Moreover, we provide a global characterization of\nreachability in bilinear systems based on a certain Gramian. Reachability\nenergy estimates using the same Gramian have only been local so far. The main\nresult of this paper, however, is a new link between the output error and the\n mathcal H₂-error of two bilinear systems. This result has several\nconsequences in the field of model order reduction. It explains why mathcal\nH₂-optimal model order reduction leads to good approximations in terms of the\noutput error. Moreover, output errors based on the mathcal H₂-norm can now\nbe proved for balancing related model order reduction schemes in this paper.\nAll these new results are based on a Gronwall lemma for matrix differential\nequations that is established here.\n

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