Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on $\mathcal{L}_2$-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. Under certain diagonalizability assumptions, we show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.

翻译：对于无结构线性时不变（LTI）系统的$\mathcal{H}_2$最优降阶建模，其插值型必要最优性条件已广为人知。基于先前关于稳态参数问题$\mathcal{L}_2$最优降阶建模的研究，本文针对结构LTI系统（特别是二阶系统、端口哈密顿系统和时滞系统）的$\mathcal{H}_2$最优降阶建模，发展并研究其最优性条件。在一定的可对角化假设下，我们证明在所有不同的结构设置中，双切向埃尔米特插值是最优性的共同形式，从而为结构化降阶建模建立了一个统一的最优性框架。