Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space \(\varvec{\mathcal {H}}_{\varvec{2}}\), a new \(\varvec{\mathcal {H}}_{\varvec{2}}\)-like optimal model reduction problem is introduced and first-order optimality conditions are derived. As in the classical \(\varvec{\mathcal {H}}_{\varvec{2}}\) case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method.

研究了大规模线性动力系统的最优模型简化。与大多数现有的工作相比，所考虑的系统不需要稳定，无论是离散时间还是连续时间。因此，底层有理传递函数允许在复平面的一般域中具有极点。特别是，这涵盖了特定保守偏微分方程的情况，例如线性薛定谔和虚轴上谱的无阻尼线性波动方程。通过对经典连续时间 Hardy 空间\(\varvec{\mathcal {H}}_{\varvec{2}}\)的适当修改，一个新的\(\varvec{\mathcal {H}}_{\varvec引入了类似{2}}\)的最优模型约简问题并导出了一阶最优性条件。与经典的\(\varvec{\mathcal {H}}_{\varvec{2}}\)情况一样，这些条件表现出合理的 Hermite 插值结构，为此提出了迭代模型简化算法。数值例子证明了新方法的有效性。