Geometry of Optimal Control Problems and Hamiltonian Systems

These notes are based on the mini-course given in June 2004 in Cetraro, Italy, in the frame of a C.I.M.E. school. Of course, they contain much more material that I could present in the 6 h course. The idea was to explain a general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian Geometry. This concerns Jacobi fields, Morse's index formula, Levi-Civita connection, Riemannian curvature and related topics.

I tried to make the presentation as light as possible: gave more details in smooth regular situations and referred to the literature in more complicated cases. There is an evidence that the results described in the notes and treated in technical papers we refer to are just parts of a united beautiful subject to be discovered on the crossroads of Differential Geometry, Dynamical Systems, and Optimal Control Theory. I will be happy if the course and the notes encourage some young ambitious researchers to take part in the discovery and exploration of this subject.

Acknowledgments.I would like to express my gratitude to Professor Gamkrelidze for his permanent interest to this topic and many inspiring discussions and to thank participants of the school for their surprising and encouraging will to work in the relaxing atmosphere of the Mediterranean resort.