Marching Over Poles: Innovative Ways to Solve Matrix Differential Riccati Equations

矩阵微分黎卡提方程（MDRE）在应用数学、科学和工程中经常出现。 特别是，它们在最优控制，滤波和估计，基本粒子物理和微分方程的两点边值问题中发挥着重要作用。 在过去的25年中，已经提出了许多算法来数值求解MDRE。 这些包括通用的数值方法重新调整，以利用今天的计算技术，以获得更好的性能。 然而，这些方法，由于其通用性，固有地没有并且不能充分利用MDRE的许多有价值的结构性质，其中一些仍有待发现。 这个项目的目标是建立一个更好的数学理解MDRE的长期解决方案的行为，特别是那些有奇异性的解决方案，并得出新的数值方法，能够优雅地处理奇异性，不同于常用的中止和重新启动程序。 一个新的MDRE求解程序包作为一个更好的数学理解，新的算法，并在今天的微处理器的IEEE浮点异常处理能力的开发的结果，预计将被公开发布后，该项目的成功完成，供科学界使用。 研究生与新兴的专业知识在数值微分方程将参与。

Matrix Differential Riccati Equations (MDREs) arise frequently throughout applied mathematics, science, and engineering. In particular, they play major roles in optimal control, filtering and estimation, elementary particle physics, and two-point boundary value problems of differential equations. A number of algorithms have been proposed over the past 25 years for solving MDREs numerically. These include general-purpose numerical methods retooled to take advantage of today's computing technology for better performance. Nevertheless, these methods, because of their generality, inherently do not and cannot fully exploit many valuable structural properties of MDREs, some of which remain to be discovered. The objectives of this project are to build a better mathematical understanding of the long-term solution behavior of MDREs, especially those that have singularities in the solutions, and to derive new numerical methods that are able to gracefully handle singularities, unlike the commonly used abort-and-restart procedures. A new MDRE solver package as the result of a better mathematical understanding, new algorithms, and the exploitation of IEEE floating-point exception handling capabilities in today's microprocessors is expected to be released publicly for the use of scientific communities upon the successful completion of the project. Graduate students with emerging expertise in numerical differential equations will be involved.