标题： 算符值微分黎卡提方程的李分裂和斯特兰分裂收敛性分析 显示英文标题

标题： Convergence analysis of Lie and Strang splitting for operator-valued differential Riccati equations

摘要： 微分Riccati方程（DREs）是一类具有二次非线性项的半线性矩阵值或算子值微分方程。它们出现在许多不同的领域，在线性二次调节器的最优控制中尤为重要，因为它们提供了最优反馈控制律。在偏微分方程控制的背景下，这些Riccati方程是算子值的。为了近似其解，需要同时进行空间和时间离散化。虽然前者已经在文献中得到了很好的分析，但针对应用于DRE的时间步进方法的严格收敛分析非常少，尤其是在无限维、算子值的情况下。鉴于此，我们在这种情况下分析了两种数值时间步进方案——Lie分裂法和Strang分裂法。该分析依赖于未控系统通过生成解析半群的算子演化的假设，并且初始条件足够光滑或者DRE中的非线性项足够光滑化。这些假设较为温和，因为它们不足以保证DRE精确解在算子范数下的连续性。然而，它们暗示了某种点态意义上的正则性，这可以用来证明以经典阶数在算子范数下的收敛性。结果通过四个数值实验进行了说明，其中收敛性与预期阶数的相关性与满足相关假设一致。实验还表明，作为算子值DRE空间离散化的矩阵值DREs的行为类似，除非离散化较粗。 显示英文摘要

摘要： Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic non-linearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic regulators, where they provide the optimal feedback control laws. In the context of control of partial differential equations, these Riccati equations are operator-valued. To approximate their solutions, both spatial and temporal discretizations are needed. While the former have been well analyzed in the literature, there are very few rigorous convergence analyses of time stepping methods applied to DREs, particularly in the infinite-dimensional, operator-valued setting. In view of this, we analyze two numerical time-stepping schemes, the Lie and Strang splitting methods, in such a setting. The analysis relies on the assumption that the uncontrolled system evolves via an operator that generates an analytic semigroup, and that either the initial condition is sufficiently smooth, or the nonlinearity in the DRE is sufficiently smoothing. These assumptions are mild, in the sense that they are not enough to even guarantee continuity in operator-norm of the exact solution to the DRE. However, they imply certain regularity in a pointwise sense, which can be leveraged to prove convergence in operator-norm with the classical orders. The results are illustrated by four numerical experiments, where convergence with the expected order is correlated with the relevant assumptions being fulfilled. The experiments also demonstrate that matrix-valued DREs which arise as spatial discretizations of operator-valued DREs behave similarly, unless the discretization is coarse.