Index theorems for graph-parametrized optimal control problems

Abstract

In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Some applications such as the iteration formulas and lower bounds for the index are proved.

Morse index of block Jacobi matrices via optimal control

“…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 «…”