In this talk I will discuss some results concerning spectral properties of the Second Variation of an optimal control problem.

The first topic I will discuss is a formula to compute how the Morse Index changes under different boundary conditions. For instance, this result can be used to produce a certain type of discretization formulae to reduce the Morse Index computation to a nite dimensional problem. It can be specialized to the case of periodic extremals to get iteration formulae. Moreover, it is useful when dealing with Variational problems on graphs since it can be employed to reduce the complexity of the domain.

The second topic I will discuss is a possible definition of the determinant of the Second Variation for an optimal control problem with general smooth boundary conditions.

One of the technical points is a precise understanding of the asymptotic behaviour of the spectrum. It turns out that the second variation is not in general a trace class operator and the standard approach using nite rank approximations does not immediately apply. Instead of working with regularized determinants, we provide a formalism to compute the determinant using the symplectic structure of the problem.

This talk is based on joint works with A. Agrachev and I. Beschastnyi.