## 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.