Problems in Muldimensional and Nonlinear Systems Theory

自20世纪60年代以来，从许多角度研究的一个基本对象是“Schur-class函数”及其算子值推广，即定义在单位盘上的全纯函数，其值是一个Hilbert空间到另一个Hilbert空间的压缩算子映射。在Livsic-Brodskii， Sz的算子模型理论中，schur类函数作为收缩算子的特征函数出现。-Nagy-Foias和de Branges-Rovnyak，作为单一综合（或节能的离散时间，输入-状态-输出线性系统）的传递函数，以及作为Lax-Phillips和Adamjan-Arov意义上的离散时间散射系统的散射函数。该项目的一个主要主题是为各种多变量系统开发相同的思想融合。在第一个多变量集合中，广义Schur-class函数在多维复欧几里德空间的单位多盘上是解析的和压缩的，是Roesser所考虑的类型的节能多维系统的传递函数。散射系统实际上是由保守的Roesser输入-状态-输出系统产生的几何结构，有助于深入了解多盘上的收缩函数满足冯·诺伊曼不等式（即，推导出将交换收缩算子元组映射到另一个收缩算子的函数演算）。在第二种形式中，广义Schur-class函数是有限多个非交换变量的幂级数，是具有有限多个非交换生成子的自由半群中的时间参数系统的传递函数。这一分析将给（在系统和散射理论方面）对最近在多维复欧几里德空间中球的行收缩和函数理论的模型理论的工作提供额外的见解。在第三种形式中，广义Schur-class函数实际上是嵌入在复欧几里德空间的代数曲线上定义的两个parahermite束之间的压缩束映射。这种分析将导致离散时间版本的Livsic多变量系统理论和非酉算子交换元组的模型理论，以及另一种多变量散射理论。最后，该项目包括将这些思想中的一些应用于非线性设置，特别是针对具有停止成本，切换成本和/或边界状态空间约束的非线性系统的非线性h∞控制理论的发展。长期以来，能量守恒（或更一般地说，耗散）系统的概念一直是许多学科（例如，经典力学和量子力学以及电路理论）的基本概念。这一概念的最新推广允许系统通过输入和输出信号与外部环境相互作用，并扩大能量记录以考虑与外部环境的能量交换。这个概念，以及与之相关的函数和算子理论，在鲁棒控制理论的最新进展中已经成为基础，在鲁棒控制理论中，工程师寻求设计一个反馈控制来保证系统的令人满意的性能，即使在外部环境中存在未建模的干扰和/或系统数学描述中的建模错误。该项目的目标是将这些思想推向基本的新方向，即：(1)新型多维系统和散射，以及(2)由(a)控制设置的瞬时切换或(b)由边界反射引起的状态动力学的瞬时跳跃引起的不连续的非线性系统。第一个方向直接应用于在时间和空间维度上演化的系统，以及具有建模参数变化不确定性的不确定系统和某些类型的非线性系统。第二个方向与物理系统有关，其中瞬时跳跃发生在两个或多个连续模型描述（混合系统）之间（例如在高速公路十字路口的交通信号设置中），以及某些物理量的可容许值集具有导致系统描述中边界反射的边界的系统（例如排队网络中队列长度非负的约束）。

9987636BallA 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).