Consider an n-dimensional Riemannian manifold (you may think of a region in Rⁿ) together with a collection of vector fields V_i, i=1,2, ... m, m< n. We will be concerned with the following questions:

a. Given two points in M, does there exist a (smooth) path connecting the points and such that its tangent vector at every point is a linear combination of V_i's?

b. If such paths exist, what is the shortest one?

c. What are the properties of the metric spaces whose distance function is defined as the length of shortest path tangent to the span of V_i's at every point?

d. How can we analyze particular examples?

One of the motivations for this set-up comes from applied problems with the number of controls smaller than the dimension of the configuration space of objects to control. A classical example is parallel parking: the driver has only steering wheel and acceleration pedal in his/her disposal, while the space of positions of the car is three-dimensional. It is even more striking for a truck with several trailers: the configuration space of a trailer train with k trailers is (3+k)-dimensional. We will discuss many other examples of such systems (planographers, bicycles, rolling a ball, falling cats, particles in magnetic field etc.) My main reason to choose Control Theory for my mini-course is that it belongs to the intersection of many mathematical topics, giving an excellent opportunity to give geometric introduction into these disciplines and then show how they can work if one puts them together. These mathematical topics include: geometry of Length Spaces; theory of Connections; Variational Methods; Nilpotent Groups. At the same time, there are many nice real-life examples and applications.

This is an approximate plan of the course:

1. Our main model example: the three-dimensional Euclidean coordinate space viewed as the Heisinger group, and a distribution of two-planes invariant under the group action. In this example we will see how it is possible that, having only two-dimensional space of available directions at every point, one still can find a path connecting any two given points (and such that its velocity vector at every point belongs to the two-plane of our distribution at that point.)
2. Lie brackets of vector fields and integrability/nonintegrability conditions of Frobenius and Chow.
3. Important examples (and related notions and theories): connections in vector bundles (holonomies and curvature); contact structures.
4. Length spaces. Carnot-Caratheodory spaces (length spaces arising from control theory). Local structure of Carnot-Caratheodory spaces (box-ball theorem, metric tangent cones).
5. Examples.

I do not want to suggest reading any books on this topic before the conference. It will be helpful, however, if students refresh their knowledge of differential equations (existence and uniqueness of solutions, smooth dependence of initial data and parameters), and multi-dimensional calculus (inverse and implicit function theorems, basics in differential forms and smooth manifolds). However, our exposition is planned to be almost self-contained, we will begin from the scratch and review almost everything that we need to use.

Tentative List of Topics

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