Interpolatory Necessary Optimality Conditions for Reduced-order Modeling of Parametric Linear Time-invariant Systems

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.