Jacobi Fields in Optimal Control

In this thesis we study how the information about the Hessian of optimal control problems can be encoded in some special curves in the Lagrangian Grassmanian associated to the problem. This allows to generalize the famous Morse theorem in the classical calculus of variations to a variety of very singular problems, which relates the number of conjugate points on an extremal curve to the index of the Hessian. The thesis consists of four chapters. In the first one the notion of L-deritvatives is explained that is used later as a generalization of Jacobi fields. In the second chapter we apply those techniques to characterize the situations when the curve in the Lagrangian Grassmanian can be defined using the flow of some Hamiltonian system, generalizing directly the classical case. In the third chapter a simple degenerate case is studied. Finally in the last chapter as an application of the previous discussion minimality of abnormal extremals in Engel sub-Riemannian structures in considered.