The equivalence problem for control systems under non-linear 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. Unfortunately, the invariants of this geometry were known only in low dimensions and the fundamental

more »

... eorem of such submanifolds needed to be established. Applying the solution to the fiber geometry induced by nstates and (n − 1)-controls leads in a surprisingly simply 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 in earlier work [3], [7] is clearly explained with a precise geometric meaning. Similar analysis of the general scalar control problem has also been worked out and required a solution of the fundamental theorem of curves in centro-affine n-space, and again gives a solution to the equivalence problem on the total space, which will not be described in this note. The original solution of the equivalence problem for n-states and (n − 1)controls, due to Robert Bryant and the first author, was sufficently complicated that a complete proof was never published, although an outline exists in [1] . This approach had the disadvan-