Mathematical Sciences: The Geometry of Highly Nonlinear Control Systems

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This research program is aimed at developing a comprehensive theory of the geometrical foundations that underlie the dynamics of highly nonlinear control system, that can only very poorly be described by linear approximations. It is intended to focus first on the practical purposes of feedback stabilization of highly-nonlinear systems. Of particular immediate interest is a generalization of the classical zero- dynamics manifold only by some high-order condition (immediately related to tracking through singularities). This is envisioned to parallel the development of condition for controllability, from linear, over first order nonlinear to eventually high-order nonlinear conditions. A second main direction involves the development of tools in terms of elements of a free Lie algebra (and the relations they satisfy) that allow one to concisely describe conditions for stabilizability, and also characterize controllability and optimality. Most important are the characterization of filtrations and of bases of a free nilpotent Lie algebra (on a finite number of generators) that are suitable for the problems considered. This work is a continuation of a research program that has expanded from controllability, to the visualization of geometrical and topological properties of the control system.