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