Geometric aspects of nonlinear filtering

Future military systems must meet new and higher demands on precision, safety, economic and environmental concerns. Vehicle guidance and control clearly illustrate this by having a natural intrinsic geometry that calls for general geometric methods in modeling, filtering and control. This report addresses one particular line of research, differential geometric filtering, giving a picture of the present state of the research efforts in this area. It will be followed by a more complete report. The present work includes a generalization of the Zakai equation, a coordinate free construction of jets of mappings and a completely intrinsic construction of the geodesic spray of an affine connection. Some technical questions concerning the necessity and sufficiency of certain conditions for projections filters are also addressed.