SIAM Journal on Control and Optimization, Volume 59, Issue 4, Page 2477-2497, January 2021.
In this paper, we prove several new results that give new insights into bilinear systems. We show under which conditions a bilinear system is asymptotically stable. 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.

中文翻译：

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双线性系统---与 $\mathcal H_2$-范数、与随机系统的关系以及更多属性的新链接

SIAM Journal on Control and Optimization，第 59 卷，第 4 期，第 2477-2497 页，2021 年 1 月。
在本文中，我们证明了几个新结果，这些结果为双线性系统提供了新的见解。我们展示了双线性系统在哪些条件下渐近稳定。此外，我们提供了基于某个 Gramian 的双线性系统可达性的全局特征。到目前为止，使用相同 Gramian 的可达性能量估计只是局部的。然而，本文的主要结果是在两个双线性系统的输出误差和 $\mathcal H_2$ 误差之间建立了新的联系。这个结果在模型降阶领域有几个后果。它解释了为什么 $\mathcal H_2$-最优模型阶数减少导致输出误差方面的良好近似。此外，现在可以证明基于 $\mathcal H_2$-norm 的输出误差来平衡本文中的相关模型降阶方案。