Spectral properties of the Second Variation of an optimal control problem

In this thesis we study the spectral properties of the Second Variation of an optimal control problem. In particular we focus on three aspects: the asymptotic distribution of the spectrum on the real line, the change in Morse index of an extremal subject to di erent boundary conditions and the determinant. We provide, under some regularity assumptions, an exhaustive Weyl-type law for the eigenvalue of the Second Variation. We prove a formula for the change of Morse index of an extremal satisfying di erent sets of boundary conditions. We apply it to get iteration formulas for periodic extremals and discretization formulas that reduce the problem of computing the index to a nite dimensional one. Moreover we present some ideas on how to apply the theory to variational problems on graphs. Finally we provide a way to compute the determinant of the Second Variation in terms of the fundamental solution of a system of ODEs, proving a generalized Hill-type formula. Application to stability are discussed.