Jacobi fields in optimal control: Morse and Maslov indices. (January 2022)

• Record Type: Journal Article
• Authors: Agrachev, Andrei; Beschastnyi, Ivan
• Is Part Of: Nonlinear analysis. Volume 214(2022)
• Publication Date: 2022-01
• Subjects: 58J30 -- 49K15 -- 53D12; Morse theorems -- Spectral flow -- Jacobi fields -- Optimal control -- Maslov index -- Lagrangian Grassmanian; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112608
• Languages: English
• ISSNs: 0362-546X

Abstract

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 generalisation 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.