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

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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. ***