Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of non-parametric linear time-invariant (LTI) systems are known and well-investigated. In this work, using the general framework of $\mathcal{L}_2$-optimal reduced-order modeling of parametric stationary problems, we derive interpolatory $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimality conditions for parametric LTI systems with a general pole-residue form. We then specialize this result to recover known conditions for systems with parameter-independent poles and develop new conditions for a certain class of systems with parameter-dependent poles.

翻译：非参数化线性时不变（LTI）系统的$\mathcal{H}_2$最优降阶建模的插值型必要最优性条件已被广泛认知并深入研究。本文利用参数化平稳问题的$\mathcal{L}_2$最优降阶建模的一般框架，推导了具有一般极点-留数形式的参数化LTI系统的插值型$\mathcal{H}_2 \otimes \mathcal{L}_2$最优性条件。随后，我们对该结果进行特化，重构了参数无关极点系统的已知条件，并针对特定类别的参数相关极点系统发展了新的条件。