Index theorems for graph-parametrized optimal control problems

TL;DR

This paper establishes Morse index theorems for constrained variational problems on graphs, providing formulas to compare indices across different graphs or boundary conditions, with applications in physics and geometry.

Contribution

It introduces new Morse index formulas for graph-based variational problems, extending classical theorems to a broader class of constrained graph problems.

Findings

Formulas for the difference of Morse indices between two Hessians

Applications include iteration formulas and index lower bounds

Theorems applicable to various physical and geometric 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. Several applications such as the iteration formulas or lower bounds for the index are proved.