Section: New Results

Necessary conditions for dynamic equivalence

Participant : Jean-Baptiste Pomet.

If two control systems on manifolds of the same dimension are dynamic equivalent (see section  3.2.3 ), we prove in [24] that either they are static equivalent –i.e. equivalent via a classical diffeomorphism– or they are both ruled; for systems of different dimensions, the one of higher dimension must be ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is “trivial” (i.e., linear controllable).

This is an important contribution because it is difficult in general to prove that two systems are not dynamic equivalent. No general general necessary condition (or obstruction) was known; that condition is also the only general obstruction known for flatness.