Numerical Linear Algebra and Control

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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.