Ciprian Foias Livres

Classical H~ interpolation theory was conceived at the beginning of the century by C. Caratheodory, L. Fejer and I. Schur. The basic method, due to Schur, in solving these problems consists in applying the Mobius transform to peel off the data. In 1967, D. Sarason encompassed these classical interpolation problems in a representation theorem of operators commuting with special contractions. Shortly after that, in 1968, B. Sz.- Nagy and C. Foias obtained a purely geometrical extension of Sarason's results. Actually, their result states that operators intertwining restrictions of co-isometries can be extended, by preserving their norm, to operators intertwining these co-isometries; starring with R. G. Douglas, P. S. Muhly and C. Pearcy, this is referred to as the commutant lifting theorem. In 1957, Z. Nehari considered an L ~ interpolation problern which in turn encompassed the same classical interpolation problems, as well as the computation of the distance of a function f in L ~ to H~. At about the sametime as Sarason's work, V. M.

This book offers a unified approach to both stationary and nonstationary interpolation problems in finite or infinite dimensions, utilizing the commutant lifting theorem from operator theory and the state space method from mathematical system theory. The authors initially aimed to address nonstationary interpolation issues of Nevanlinna-Pick and Nehari types by transforming these problems into stationary ones for operator-valued functions with operator arguments, employing classical commutant lifting techniques. This reduction necessitated a review and further development of classical results for stationary problems within a broader framework. System theory proved essential for structuring the problems and providing natural state space formulas for the solutions. Consequently, the scope of the research expanded beyond the original intent. The final results of this extensive effort are compiled here. The research, initiated in 1994 with financial support from the "NWO-stimulansprogramma" for the Thomas Stieltjes Institute for Mathematics in the Netherlands, was also bolstered by contributions from Indiana University, Purdue University, Tel-Aviv University, and the Vrije Universiteit at Amsterdam. Special thanks are extended to Dr. A. L. Sakhnovich for his thorough review of the manuscript and to Sharon Wise for her meticulous preparation of the troff file.

Since its inception in the early 1980s, H( optimization theory has become the control methodology of choice in robust feedback analysis and design. The purpose of this monograph is to present, in a tutorial fashion, a self contained operator theoretic approach to the H( control for disturbed parameter systems, that is, systems which admit infinite dimensional state spaces. Such systems arise for problems modelled by partial differential equations or which have time delays. Besides elucidating the mathematics of H( control, extensive treatment is given to its physical and engineering underpinnings. The techniques given in the book are carefully illustrated by two benchmark problems: an unstable system with a time delay which comes from the control of the X-29, and the control of a Euler-Bernoulli flexible beam with Kelvin-Voigt damping.