Mathematical Sciences: Control Systems Involving C-O-R Generalized Inputs with Applications to Splines and Mechanics

在技术中，经常需要构造控制系统/自动机/计算机程序来精确地或近似地跟踪轨迹，该轨迹具有一些缺乏平滑性，例如，加速度导数中的不连续性，并且其在非线性流形上取值。 在这样的背景下，通过平滑控制系统生成这样的轨迹需要接受Colombeau、Oberguggenberger和Rosinger意义上的广义函数的输入的控制系统。可替代地，可以使用非线性样条算法来构造这样的轨迹。 本研究将沿着三个方向进行：（i）在非线性控制系统的背景下发展基本的Colombeau广义函数理论;（ii）将在平坦仿射几何背景下发展的Bezier和deCasteljau样条算法扩展到更一般的Lie理论和路径几何理论情况;及（iii）将（i）及（ii）所发展的理论应用于空中交通管制技术。 空中交通控制算法的发展涉及精确或近似处理分段光滑的非线性机械系统的轨迹。 本研究将追求两个方向的数学分析，这样的轨迹，一个通过反馈控制理论，其他的非线性推广样条算法开发的计算机图形学方面的Bezier和deCasteljau。 通过控制方法产生这样的轨迹需要处理的力量和反馈法的非线性控制系统，这是广义函数的意义上的Colombeau，Oberguggenberger和Rosinger。 在本研究中，将初步探讨其数学基础理论。 还将研究和发展将Bezier-deCasteljau样条方法扩展到飞机控制所需的数学情况。 该项目由应用数学计划和几何分析计划共同支持。 ***

9424353 Hermann Often in technology it is necessary to construct a control system/automata/computer program to track exactly or approximately a trajectory which has some lack of smoothness, for example, discontinuities in the derivative of acceleration, and which takes values on a nonlinear manifold. In such a context, generating such a trajectory via a smooth control system requires a control system which accepts inputs which are generalized functions in the sense of Colombeau, Oberguggenberger and Rosinger. Alternately, such trajectories may be constructed using a nonlinear spline algorithm. This research will pursue three directions: (i) the development of the underlying Colombeau generalized function theory in the context of nonlinear control systems; (ii) extension of the spline algorithms of Bezier and deCasteljau developed in the context of flat affine geometry to more general Lie-theoretic and path geometry-theoretic situations; and (iii) application the theories developed in (i) and (ii) to air traffic control technology. Development of air traffic control algorithms involves dealing exactly or approximately with piecewise smooth trajectories of nonlinear mechanical systems. This research will pursue two directions of mathematical analysis of such trajectories, one via feedback control theory, the other a nonlinear generalization of the spline algorithms developed in a computer graphics context by Bezier and deCasteljau. Generating such trajectories via control methodology requires dealing with forces and feedback laws for nonlinear control systems which are generalized functions in the sense of Colombeau, Oberguggenberger and Rosinger. The underlying mathematical theory will be initiated in this research. Extending the Bezier-deCasteljau spline approach to the mathematical situations required for aircraft control will also be studied and developed. This project is jointly supported by the Applied Mathematics Program and the Geometric Analysis Program. ***