Problems in Muldimensional and Nonlinear Systems Theory

自20世纪60年代以来，从多个角度来看，一个基本的研究对象是“Schur类函数”及其算子值推广，即定义在单位圆盘上的全纯函数，其值是将一个希尔伯特空间映射到另一个希尔伯特空间的压缩算子。在Livsic-Brodskii，Sz.-Nagy-FOIAs和de Brange-Rovnyak的算符模型理论中，Schur类函数作为压缩算子的特征函数，作为么正综合(或能量守恒的离散时间、输入-状态-输出线性系统)的传递函数，以及在Lax-Phillips和Adamjan-Arov意义下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同样的思想汇流。在第一类多变量情形下，广义Schur类函数在高维复欧氏空间中的单位多圆盘上是解析压缩的，是Roesser所考虑的这类能量守恒多维系统的传递函数。散射系统实际上产生于保守的Roesser输入-状态-输出系统的几何给出了多面体上哪些压缩函数满足von Neumann不等式的见解(即，诱导一个将交换压缩算子的元组映射到另一个压缩算子的函数演算)。在第二个实例中，广义Schur类函数是有限多个非对易变量的幂函数，是时间参数取在具有有限多个非对易生成元的自由半群中的系统的传递函数。这一分析将(在系统和散射理论方面)为最近关于行压缩的模型理论和多维复欧氏空间中球上的函数理论的工作提供更多的见解。在第三个实例中，广义Schur类函数实际上是定义在嵌入在复欧氏空间中的代数曲线上的两个拟厄尔米丛之间的压缩丛映射。这一分析将导致Livsic多变量系统理论的离散版本和用于交换非么正算子元组的模型理论，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到非线性环境中，具体地说，发展了具有停止成本、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_无穷控制理论。能量守恒(或者更广泛地说，耗散)系统的概念长期以来一直是许多学科(例如经典和量子力学和电路理论)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，并扩大能源簿记以考虑与外部环境的能量交换。这一概念，连同与之相关的函数和算子理论，一直是鲁棒控制理论的最新进展的基础，其中工程师试图设计反馈控制，以确保即使在外部环境中存在未建模的干扰和/或系统的数学描述中存在建模错误的情况下，系统也具有令人满意的性能。这个项目的目标是将这些想法推向基本的、新的方向，即：(1)新类型的多维系统和散射，以及(2)由于(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而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，以及具有模型化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。第二个方向涉及在两个或多个连续模型描述(混合系统)之间发生瞬时跳跃的物理系统(例如在高速公路交叉口的交通信号设置中)，以及其中某些物理量的允许值的集合具有导致系统描述中的边界反射的边界的系统(例如排队网络中排队长度为非负的约束)。

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).