Abstract The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.

中文翻译：

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无限维 LQR 问题的数值解和相关的 Riccati 微分方程

摘要 抛物线偏微分方程线性二次调节器设计问题的数值分析需要求解Riccati方程。在有限时间范围的情况下，出现 Riccati 微分方程 (RDE)。所得 RDE 的系数矩阵通常具有给定的结构，例如稀疏或低秩。关联的 RDE 通常非常僵硬，因此在这种情况下应该使用隐式方案。在本文中，我们基于 BDF 和 Rosenbrock 方法的矩阵值实现，推导出了用于求解能够利用这种结构的 RDE 的有效数值方法。我们通过仅处理解决方案的近似低秩因子来表明这些方法适用于大规模问题。我们还在求解 RDE 的数值算法中加入了步长和阶次控制。此外，我们表明在 Galerkin 投影框架内，有限维 RDE 的解在强算子拓扑中收敛到无限维 RDE 的解。数值实验表明了所提出方法的性能。