A fundamental object of study since the 1960s from a number of points of view has been that of a ``Schur-class function'' and its operator-valued generalizations, i.e., a holomorphic function defined on the unit disk whose values are contractive operators mapping one Hilbert space to another. Schur-class functions arise as the characteristic function for a contraction operator in the operator-model theory of Livsic-Brodskii, Sz.- Nagy-Foias and de Branges-Rovnyak, as the transfer function of a unitary colligation (or energy-conserving discrete-time, input-state-output linear system), and as the scattering function of a discrete-time scattering system in the sense of Lax-Phillips and Adamjan-Arov. A major theme of the project is to develop this same confluence of ideas for various multivariable systems. In the first such multivariable setting, the generalized Schur-class function is analytic and contractive on the unit polydisk in multidimensional complex Euclidean space, and is the transfer function of an energy-conserving multidimensional system of the type considered by Roesser. The geometry of which scattering systems actually arise from a conservative Roesser input-state-output system lends insight into which contractive functions on the polydisk satisfy a von Neumann inequality (i.e., induce a functional calculus which maps tuples of commuting contraction operators to another contraction operator). In the second incarnation, the generalized Schur-class function is a power series in finitely many noncommuting variables, and is the transfer function for a system with time parameter taken to lie in a free semigroup with finitely many noncommuting generators. This analysis will give additional insight (on the system and scattering theory side) to recent work on model theory for row contractions and function theory on the ball in multidimensional complex Euclidean space. In the third incarnation, the generalized Schur-class function is actually a contractive bundle map between two parahermitian bundles defined over an algebraic curve embedded in complex Euclidean space. This analysis will lead to a discrete-time version of the Livsic multivariable system theory and a model theory for commuting tuples of nonunitary operators, as well as another multivariable scattering theory. Finally, the project includes an application of some of these ideas to nonlinear settings, specifically, the development of a nonlinear H-infinity control theory for nonlinear systems with stopping cost, switching cost, and/or boundary state-space constraints.
The notion of an energy-conserving (or more generally, dissipative) system has long been a fundamental notion in a number of disciplines (e.g., classical and quantum mechanics and circuit theory). A more recent generalization of this notion allows interaction of the system with an outside environment through input and output signals, and the enlargement of the energy bookkeeping to take into account energy exchange with the outside environment. This notion in turn, together with the function and operator theory associated with it, has been fundamental in recent advances in the theory of robust control, where the engineer seeks to design a feedback control to guarantee satisfactory performance of a system even in the presence of unmodeled disturbances in the outside environment and/or modeling errors in the mathematical description of the system. The goal of this project is to push these ideas in fundamental, new directions, namely: (1) new types of multidimensional systems and scattering, and (2) nonlinear systems with discontinuities arising from (a) instantaneous switching of the control setting, or (b) instantaneous jumps in the state dynamics caused by boundary reflections. The first direction has direct application to systems that evolve in both time and in a spatial dimension, as well as to uncertain systems with a modeled parameter- variation uncertainty and certain types of nonlinear systems. The second direction is relevant to physical systems where instantaneous jumps occur between two or more continuous-model descriptions (hybrid systems) (such as in the setting of a traffic signal at a highway intersection), as well as to systems where the set of admissible values for some physical quantity has a boundary which leads to a boundary reflection in the system description (such as the constraint that the queue lengths be nonnegative in a queueing network).
