Numerical Linear Algebra and Control

本项目研究光滑矩阵值函数的光滑变奇异值分解的存在性、唯一性及其计算方法。光滑奇异值分解在微分代数方程中有应用，包括广义系统的时变最优控制问题。初步工作与真实的解析函数被扩展到一个更大的类的光滑函数和更有效的数值方法跟踪所有或部分的光滑变化的奇异值分解正在调查。该项目还开发了计算可控性半径、规则性半径和计算控制中其他感兴趣的距离的数值方法。它正在开发有效的，可证明可靠的，数值稳定的方法。这里的成功将推广到一种手段，估计距离从一个通用的矩阵铅笔代数品种的非通用铅笔和一般条件估计。

This project studies existence, uniqueness and methods for calculating a smoothly varying singular value decomposition of a smooth matrix valued function. The smooth singular value decomposition has applications in differential algebraic equations including time dependent optimal control problems for descriptor systems. Preliminary work with real analytic functions is to be extended to a larger class of smooth functions and more efficient numerical methods for tracking all of or part of a smoothly varying singular value decomposition are being investigated. This project is also developing numerical methods for calculating the controllability radius, regularity radius, and other distances of interest in computational control. It is developing efficient, provably reliable, numerically stable methods. Success here will generalize to a means of estimating the distance from a generic matrix pencil to an algebraic variety of nongeneric pencils and to general condition estimators.