Mathematical Sciences: Global Decompositions of Nonlinear Control Systems

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The problem of decomposing a large-scale nonlinear control system into an interconnected family of smaller-scale subsystems is a natural one and has received the attention of many researchers. The vast majority of the work done on this problem thus far has concentrated on local decompositions. By decomposition one usually means a change of variables that produces an equivalent system that has at least two independent but interconnected parts. Decomposition is very important from a practical point of view, because it simplifies the analysis of complex systems. The investigator proposes a notion of global system decomposition which is reasonable from a conceptual point of view and has potential of producing results that can be applied to specific examples. He will attempt to extend known results on local decompositions to a global setting. Both necessary and sufficient conditions of existence of global decompositions will be sought. A study of symmetry groups associated with a control system will constitute a portion of this research. Applications of global decompositons to the study of global controllability of nonlinear control systems will also be pursued. This research attempts to develop further the applications of differential geometry to control systems. If succesful, it could produce results of considerable importance for such applications of control theory as robot manipulators and helicopter autopilots.