In this thesis we study the spectral properties of the Second Variation of an optimal control problem. In particular we focus on three aspects: the asymptotic distribution of the spectrum on the real line, the change in Morse index of an extremal subject to di erent boundary conditions and the determinant. We provide, under some regularity assumptions, an exhaustive Weyl-type law for the eigenvalue of the Second Variation. We prove a formula for the change of Morse index of an extremal satisfying di erent sets of boundary conditions. We apply it to get iteration formulas for periodic extremals and discretization formulas that reduce the problem of computing the index to a nite dimensional one. Moreover we present some ideas on how to apply the theory to variational problems on graphs. Finally we provide a way to compute the determinant of the Second Variation in terms of the fundamental solution of a system of ODEs, proving a generalized Hill-type formula. Application to stability are discussed.

Spectral properties of the Second Variation of an optimal control problem / Baranzini, Stefano. - (2022 Nov 07).

Spectral properties of the Second Variation of an optimal control problem

Baranzini, Stefano
2022

Abstract

In this thesis we study the spectral properties of the Second Variation of an optimal control problem. In particular we focus on three aspects: the asymptotic distribution of the spectrum on the real line, the change in Morse index of an extremal subject to di erent boundary conditions and the determinant. We provide, under some regularity assumptions, an exhaustive Weyl-type law for the eigenvalue of the Second Variation. We prove a formula for the change of Morse index of an extremal satisfying di erent sets of boundary conditions. We apply it to get iteration formulas for periodic extremals and discretization formulas that reduce the problem of computing the index to a nite dimensional one. Moreover we present some ideas on how to apply the theory to variational problems on graphs. Finally we provide a way to compute the determinant of the Second Variation in terms of the fundamental solution of a system of ODEs, proving a generalized Hill-type formula. Application to stability are discussed.
Agrachev, Andrey
Supervisor: Ivan Beschastnyi
Baranzini, Stefano
File in questo prodotto:
File Dimensione Formato  
main_thesis.pdf

accesso aperto

Descrizione: tesi di Ph.D.
Tipologia: Tesi
Licenza: Non specificato
Dimensione 1.15 MB
Formato Adobe PDF
1.15 MB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/129990
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact