Abstract:My goal here is to talk about a tight relationship between some models in control theory, and some other classification problems in non-holonomic geometry.
Parallel parking geometry, or non-holonomic geometry, is the type of geometric thinking you and me have to perform in order to parallel park our vehicles in a narrow spot. In geometric terms, in the 3D configuration space of the car the velocities are restricted to lie within a 2D plane field which is non-integrable.
Now if one considers a car and a trailer model (e.g. an airport luggage cart) the geometry gets even more interesting. Frederic Jean showed in the mid 90's that certain configurations of these car plus trailer system were "singular" in a control theoretic sense. One such configuration is the jackknifed one where the trailer axis is perpendicular to the car's symmetry axis.
Jean's discovery had resurfaced a class of 2D non-integrable distributions predating E. Cartan, and are commonly associated to the name of Edouard Goursat. Nowadays we refer to them as "Goursat Distributions." It was a common (mis-)belief till around the 70's that there was a single normal form for Goursat distributions, namely the Cartan distribution in the jet spaces $J^k(\R,\R)$. However Giaro, Kumpera and Ruiz gave an example of a Goursat plane field non-equivalent to the Cartan distribution.
After a short introduction, I'll briefly mention how R. Montgomery and M. Zhitomirskii offered very recently a complete solution to the classification problem of enumerating all possible Goursat structures. Their solution appeals to the local analysis of finite germs of Legendrian curves in contact \R^3. I plan describe this key duality between Legendrian curves and distributions which is the main step in their proof.
I will conclude my talk by mentioning my recent work with Richard Montgomery on the extension of his work with Zhitomirskii to the class of so-called special Goursat k-flags. These first steps involved an analogy with V. Arnold's list of simple curve singularities in \R^n, \C^n etc. and I hope time will also permit me to briefly mention some open problems left in the general classification problem.