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.
