Several different frameworks for the mathematical theory of linear control systems have been developed over the last three decades. This project combines two important and well-developed approaches: the function-theoretic methodology and the techniques of commutative algebra and module theory. The specific technical research issues which will be adressed include the geometric study of poles and transmission zeros of singular systems; construction and application of a local algebraic version of the fundamental pole-zero exact sequence of a singular linear system; characterization and construction of globally minimal generalized inverses of linear systems; algebraic and analytic studies of Wiener-Hopf factorization; studies of singular values and spectral factorization; and interpolation of matrix functions. The theory of feedback control systems, originally motivated by problems in engineering, occupies a central place in applied mathematics and is fundamental to control engineering applications. The present project examines mathematical questions for the generalized or "singular" linear systems which arise naturally when complicated networks are constructed from smaller components. The poles of these systems govern stability and long- term behavior, while the transmission zeros govern adaptability and transient responses. Techniques both from functional analysis and the modern theory of abstract algebraic structures will be combined in these studies to provide the conceptual foundations and new technical results for these systems.
