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 Schr\"odinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space $\mathcal{H}_2$, a new $\mathcal{H}_2$ like optimal model reduction problem is introduced and first order optimality conditions are derived. As in the classical $\mathcal{H}_2$ 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.

翻译：研究了大规模线性动力系统的最优模型降阶问题。与现有大多数工作不同，本文所考虑的系统不需要在离散时间或连续时间下稳定。因此，相应的有理传递函数允许在复平面的一般区域存在极点。特别地，这涵盖了特定保守偏微分方程的情形，例如谱位于虚轴上的线性薛定谔方程和无阻尼线性波动方程。通过对经典连续时间 Hardy 空间 $\mathcal{H}_2$ 进行适当修改，引入了一个新的类似 $\mathcal{H}_2$ 的最优模型降阶问题，并导出了一阶最优性条件。与经典 $\mathcal{H}_2$ 情形类似，这些条件具有有理 Hermite 插值结构，并据此提出了一种迭代模型降阶算法。数值算例验证了新方法的有效性。