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

9424353赫尔曼通常在技术上，有必要构建一个控制系统/自动机/计算机程序来准确或近似地跟踪具有一些不光滑性的轨迹，例如，加速度导数中的不连续，并在非线性流形上取值。在这样的背景下，通过平滑的控制系统产生这样的轨迹需要一个控制系统接受输入，这些输入是科伦博、Oberguggenberger和Rosinger意义上的广义函数。或者，可以使用非线性样条法来构建这样的轨迹。这项研究将追求三个方向：(I)在非线性控制系统的背景下发展基本的Columbeau广义函数理论；(Ii)将在平坦仿射几何背景下发展的Bezier和deCasteljau的样条算法推广到更一般的李论和路径几何理论的情况；以及(Iii)将(I)和(Ii)中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地处理非线性机械系统的分段光滑轨迹。这项研究将追求这类轨迹的两个数学分析方向，一个是通过反馈控制理论，另一个是Bezier和DeCasteljau在计算机图形学背景下开发的样条算法的非线性推广。通过控制方法产生这样的轨迹需要处理非线性控制系统的力和反馈律，这些力和反馈律是Columbeau，Oberguggenberger和Rosinger意义下的广义函数。基本的数学理论将在这项研究中启动。还将研究和开发将Bezier-deCasteljau样条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联合资助。***

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. ***