Ivan Beschastnyi (CMAP)
Title: Jacobi fields in optimal control
Abstract. In the classical calculus of variations and Riemannian geometry, Jacobi fields play a special a role in the study of optimality of geodesics and derivation of invariants. This technique can be extended to constrained variational problems if one considers not just a single Jacobi field, but the space of all solutions to the Jacobi equations with certain boundary conditions. Interpreting this space as a Lagrangian plane and considering its evolution in time allows to find the negative inertia index of the second variation. In order to do so, one has to deal with the non-smoothness of the Hamiltonian along an extremal curve and to develop an intersection theory for discontinuous curves in the Lagrangian Grassmanian. In a recent work with A. Agrachev we extend some of his earlier ideas to construct a theory that covers all the possible cases encountered in optimal control.