Functional determinants for the second variation

Morse index of block Jacobi matrices via optimal control

“…3 we can conclude that elements in W ? k  W kC1 are in one to one correspondence with trajectories of ˆkC1 having Let us compute now h«x; xi for x 2 W ?…”

“…The techniques used in our proofs are deeply connected with symplectic geometry. One can extend them to much more general settings [1][2][3]. More precisely, exploiting this connection, we prove a formula linking the negative inertia index ind .« / (i.e., the number of negative eigenvalues of «…”

Index theorems for graph-parametrized optimal control problems

“…This formula requires to know the index and a suitable generalization of the determinant of the second variation at a critical point [29,32]. The index is computed in the current paper, while the determinant is investigated in [16].…”

“…The idea was to introduce extra vertices inside the single edge and apply an iterative procedure consisting in fixing each of the new vertices one by one. Note that, if we had fixed all of the vertices at the same time, a direct application of equation (16) would have resulted in a computation of the Maslov index in a very big symplectic space. Instead, the recursive nature of the proof allows us to reduce greatly the dimensionality of the problem.…”

“…We can now apply directly corollary 1 to the boundary conditions ∆ × Ñ and ∆ × N. In order to see that everything indeed reduces to formula (16), we show how to rewrite each term of equation (45), without writing explicitly the lengthy formula here. Let us go term by term starting from the ones involving dimensions.…”