Classical geometries arising in feedback equivalence

The equivalence problem for control systems under nonlinear feedback is recast as a problem involving the determination of the invariants of submanifolds in the tangent bundle of state space under fiber preserving transformations. This leads to a fiber geometry described by the invariants of submanifolds under the general linear group, which is the classical subject of centro-affine geometry. Applying the solution to the fiber geometry induced by n-states and (n-1)-controls leads in a surprisingly simple way to the solution of the equivalence problem on the whole total space. In particular, mysterious results on the existence of feedback invariant pseudo-Riemannian geometries uncovered by Gardner, Shadwick and Wilkens (1989) and Wilkens (1990) are clearly explained with a precise geometric meaning. The original solution of the equivalence problem for n-states and (n-1)-controls was sufficiently complicated that a complete proof was never published, although an outline by Gardner exists (1989). This approach had the disadvantage that the meanings of the various invariants uncovered were not visible. The new ideas make these results accessible and in this case lead to the Finsler geometry of a generalized variational problem arising from the variational problem of time optimal control along control trajectories.< >