Jacobi Fields in Optimal Control I: Morse and Maslov Indices
Abstract
In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations, but which also works for non-smooth extremals. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.
Keywords
Cite
@article{arxiv.1810.02960,
title = {Jacobi Fields in Optimal Control I: Morse and Maslov Indices},
author = {Andrei Agrachev and Ivan Beschastnyi},
journal= {arXiv preprint arXiv:1810.02960},
year = {2021}
}
Comments
Section 2 about the gluing formula completely removed, additional sources added to the introduction, examples added, various typos corrected