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。与大多数现有工作不同，所考虑的系统无需在离散时间或连续时间下保持稳定。因此，允许有理传递函数在复平面的一般域中存在极点。特别地，这涵盖了特定保守偏微分方程的情况，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$\mathcal{H}_2$进行适当修正，引入了一个新的类$\mathcal{H}_2$最优模型降阶问题，并推导了一阶最优性条件。与经典$\mathcal{H}_2$情况类似，这些条件呈现出有理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。