In the usual framework of control-affine systems, one makes a choice of drift
vector field and control vector fields, then studies system properties in
terms of this choice. However, it is the case that different choices of
drift vector field and control vector fields can generate the same class of
trajectories; two such choices are called *feedback equivalent*. To
understand the geometry of a control-affine system, one should study the
equivalence class and not a choice of representative given by a particular
set of drift and control vector fields. Associated to an equivalence class
under feedback equivalence is an affine subbundle of the tangent bundle of
the state manifold.
In this work, a setting for studying the local structure of affine subbundles
is presented. Such a framework should be useful for understanding geometric
properties of control-affine systems; for example obstructions to local
controllability and stabilisability. To give some idea of the tools one
might use in such investigations, we initiate the study of local
controllability using jet bundles of affine subbundles. The use of jet
bundles leads to algebro-geometric conditions for, and obstructions to, local
controllability.

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Last Updated: Tue Jun 18 09:14:27 2024