Operators arising as Second Variation of optimal control problems and their spectral asymptotics

TL;DR

This paper analyzes the spectral asymptotics of operators derived from the second variation in optimal control problems, providing a finite-dimensional characterization and applying it to singular extremals.

Contribution

It introduces a novel spectral analysis of second variation operators in optimal control, linking their eigenvalues to finite-dimensional data and applying results to singular extremals.

Findings

Eigenvalues characterized by finite-dimensional data

Spectral asymptotics computed for specific operators

Application to singular extremal cases

Abstract

We compute the asymptotic for the eigenvalues of a particular class of compact operators deeply linked with the second variation of optimal control problems. We characterize this family in terms of a set of finite dimensional data and we apply this results to a particular class of singular extremal to get a nice description of the spectrum of the second variation.