Mathematical Sciences: Transformations of Linear Systems: Classification and Orbit Closure

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9404641 O'Halloran Feedback and equivalence are transformations of linear systems of differential or difference equations which may be viewed as algebraic group actions on varieties. For a given algebraic group action, two basic questions are classification (a method for determining whether or not two elements are in the same orbit) and orbit closure (a description of the elements in the topological closure of an orbit). The main purpose of this research is to solve the classification problem for the output feedback action on the space of state space systems, using methods from algebraic geometry, algebraic geometry software, and linear algebra. For the action of the full feedback group on the space of generalized state space systems, a classification has been established; this project will solve the orbit closure problem in this setting. The principle behind many automated systems, from aircraft control systems to thermostat regulated heating systems, is that of control by feedback, in which current information about the device and its environment is used as input in order to effect necessary adjustments. "State feedback" refers to the situation in which full information is available; when only partial information is available, adjustments are made based on "output feedback." "High gain feedback" occurs when reiteration of feedback causes an evolution of the system in which feedback input is playing a progressively larger role. This research focuses on mathematical models of feedback control systems, in which two major projects will be pursued. The first is to determine which mathematical models of control systems can be obtained from a given one by applying output feedback, thus reflecting the range of situations that could arise in a physical system when output feedback is utilized. The second project addresses the identification of those mathematical models of control systems which can be obtained from a given one when high gain state feedb ack is applied. ***