A Pseudo-group Isomorphism between Control Systems and Certain Generalized Finsler Structures

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. Similarly the equivalence problem for Finsler structures is shown to lead to a fiber geometry over the base inducing a centro-affine geometry. The appearance of a centro-affine fiber geometry in both control systems and Finsler structures will be explained after establishing a canonical pseudo-group isomorphism between arbitrary control systems and certain generalized Finsler geometries, that is variational problems with non-holonomic constraints. The generalized Finsler structure turns out to be the geometry of the constrained variational problem arising from the variational problem of time optimal control along control trajectories. Further analysis will show that a classical Finsler structure will correspond to regular control systems with m-states and (m − 1)-controls. The term regular will be made precise in section 3, but some centro-affine geometry is needed for the definition. The original solution of the feedback equivalence problem for the regular system in m-states and (m − 1)-controls, due to Robert Bryant and the first author, was sufficiently complicated that a complete proof was never published, although an outline exists in [Ga89]. This approach had the disadvantage that the meanings of even the simplest invariants were not visible. 1991 Mathematics Subject Classification. Primary 53B40, 53C60, 93B29, 93B52.