We will state the Pontryagin Maximum Principle (PMP) and show some of its consequences. As a first order necessary condition for optimality it allows finding extremals in optimal control problems. It is also a basic tool for analyzing their properties.

In the second part of the seminar, we intend to describe some fundamental notions and facts in sub-Riemannian geometry. Applying PMP we will briefly describe the structure of geodesics, especially of abnormal ones which do not appear in Riemannian geometry.