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. 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：非结构化线性时不变系统$\mathcal{H}_2$最优降阶建模的插值必要最优性条件已广为人知。基于静态参数问题$\mathcal{L}_2$最优降阶建模的前期工作，本文研究并发展了结构化线性时不变系统（特别是一阶系统、端口哈密顿系统和时滞系统）的$\mathcal{H}_2$最优降阶建模最优性条件。我们证明，在所有上述不同结构设定下，双侧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统一的最优性框架。